Branching Time and the Semantics of Future Contingents

Jacek Wawer

PhD dissertation prepared under the supervision of Prof. dr hab. Tomasz Placek

Institute of Philosophy Jagiellonian University Kraków, Poland Acknowledgments

First and foremost, I would like to express my gratitude to Professor Tomasz Placek, the supervisor of my dissertation. Over a decade ago, he introduced me to the world of branching and has helped me to travel it ever since. The assistance he provided during my work cannot be overestimated. The philosophical community was impoverished due to his dedication, but I benefited enormously. Also, thanks to his continuous support, I had the best possible environment to develop my thought. I have never been an obedient student and often challenged, rather than developed his philosophical views. Nonetheless, he only doubled his support and helped me to express my views better than I could have done it myself. I count myself to be immensely lucky to be his student. I would like to thank Professors Fabrice Correia, John MacFarlane, and Thomas Müller who mentored me during my research visits in Geneva, Berkeley, and Konstanz respectively. Many of their helpful suggestions influenced my views. I am indebted to Alex Malpass, Leszek Wronski,´ Juliusz Doboszewski, Antje Rumberg, and Michał Marczyk for their feedback on various fragments of this work and many inspiring discussions which helped me clarify my ideas and recognize new paths. I would have not accomplished this work, if not for the continuous support and encouragement of my wife, Karolina. My research was possible due to the financial, administrative, and scientific support of many institutions. Most importantly, the Jagiellonian University, and also (in alphabetic order) of the Foundation for Polish Science, Ministry of Science and Higher Education of the Republic of Poland, the Polish National Science Centre, the Polish- U.S. Fulbright Commission, and the Rector’s Conference of the Swiss Universities supplemented my research. Last, but not least, I would like to thank Professor Nuel Belnap and Professor Peter Øhrstrøm. I had a chance to personally discuss philosophy with them only a few times. Nonetheless, their influence and inspiration will be visible at virtually every page of this work. I disagree with some of their views (always with considerable discomfort), but the problems I undertake are their problems, expressed in their vocabulary, and approached using their methods. If I was able to see anything at all, it is only because I stood on the shoulders of the giants. Contents

1 Introduction1

2 Branching Realism5 2.1 Naïve Branching Realism...... 7 2.1.1 Futures in abundance...... 8 2.1.2 Problem with the trousers universe...... 11 2.2 Genuine Branching Realism...... 13 2.2.1 Two perspectives...... 16 2.2.2 Two languages...... 18 2.2.3 Which perspective is basic?...... 20 2.2.4 Are possible histories possible?...... 23 2.2.5 Indexical actuality...... 25 2.2.6 Towards Branching Actualism...... 27

3 Ockhamist semantics 31 3.1 Branching structure...... 31 3.2 Ockhamist truth...... 32 3.3 A few remarks on the logic of Ockhamism...... 37 3.4 Future tense operator...... 39 3.5 Modal operator...... 41 3.6 Sentences and propositions...... 44 3.7 From semantics to postsemantics...... 45

4 Semantics of Branching Realism 47 4.1 Metaphysical constraint of semantics...... 47 4.2 Extremism...... 49 4.3 Modalism...... 59 4.4 Many-valued semantics...... 62 4.5 Supervaluations...... 76 4.6 Assessment relativism...... 83 4.7 History relativism...... 93 4.8 Local relativism...... 99 4.8.1 Recognized possibilities...... 99 4.8.2 Counterfactual branches...... 105

iii CONTENTS

4.8.3 Continuations...... 110 4.8.4 Sets of transitions...... 119 4.8.5 A problem with local relativism...... 126

5 Thin Red Line 129 5.1 Metaphysics of the Thin Red Line...... 129 5.2 Semantic impact...... 133 5.3 Objections to the Thin Red Line...... 137 5.3.1 Metaphysics...... 137 5.3.2 Epistemology...... 137 5.3.3 Actuality...... 138 5.3.4 Semantics...... 139 5.3.5 Postsemantics...... 146

6 Branching Actualism 166 6.1 Metaphysical background...... 166 6.2 Semantic impact...... 173 6.3 Response to objections...... 179 6.3.1 Metaphysics...... 179 6.3.2 Epistemology...... 180 6.3.3 Actuality...... 181 6.3.4 Semantics...... 186 6.3.5 Postsemantics...... 188 6.3.6 Possible predictions...... 192 6.4 Branching possibilities...... 198 6.4.1 Are actualist possibilities suﬃciently real?...... 198 6.4.2 Are genuine possibilities suﬃciently real?...... 203 6.4.3 The nature of branching possibilities...... 206 6.4.4 Divergence or branching?...... 213 6.5 Localism and trans-localism...... 217

Summary 230

APPENDIX 232

iv Chapter 1

Introduction

Aristotle has stated the famous problem of the so-called future contingents in section 9 of De Interpretatione. Future contingents are statements concerning possible future eventualities. Aristotle describes them as “affirmations regarding particulars that are going to be” with respect to which “both possibilities are open, both being and not being, and consequently, both coming to be and not coming to be” (18a28–18a32 and 19a10-19a12). When Aristotle reflected on such affirmations, he noticed that we are torn by two conflicting ideas. On the one hand, if something might happen and it might not happen, then we cannot truly say that it will happen, nor can we truly say that it will not happen. If the affirmation were true, then the affirmed state would necessarily have to happen. If the negation were true, then the state would necessarily have to not to happen. Therefore, neither the affirmation, nor the negation of a contingent event is true (De Interpreta- tione, 18a34–18b16). On the other hand, the contingent occurrence either will happen, or will not happen. If it will happen, then the affirmation of the future occurrence is true and if it will not happen, then the negation of the future occurrence is true. It is thus absurd to say that neither the affirmation, nor the negation is true because, “Take a sea-battle: it would have neither to happen nor not to happen” (De Interpretatione, 18b17–18b25).1 Aristotle identified the tension between our notions of truth, possibility, and time. Every solution generates some substantial conceptual costs (otherwise, the issue would not have been so fervently disputed for over almost two and a half millennia in both European and Arab philosophy). In fact, it is not even entirely clear what was Aris- totle’s own reaction to the dialectical situation he outlined, since his comments are vague enough to allow mutually exclusive interpretations. This work addresses Aristo- tle’s problem yet again, this time in the form it assumed in the modern temporal logic. Specifically, in the context of the model of branching time. To get a better taste of Aristotle’s puzzle, let us use a simple example. The Greeks triremes were berthed at the harbor of Salamis. The captains were awaiting the orders, ready to fight the approaching Persian fleet. As Herodotus re-

1Throughout my dissertation, I use the translations of Aristotle’s work contained in (Aristotle, 1991).

1 CHAPTER 1. INTRODUCTION counts, Queen Artemisia of Caria advised emperor Xerxes against attacking the Greeks at Salamis. As she had provided very good reasons, she could have swayed him in favor of another tactics. At that point, the faith of the naval campaign was unsettled. Had Xerxes listened to Artemisia, there would not have been a sea battle, had he rejected her advice, there would have been one. At the same time, the Greek leaders—Themistocles and Eurybiades—had observed the maneuvers of the Persian ﬂeet from a hill nearby Salamis and Themistocles said to Eurybiadeds, “There will be a sea battle tomorrow.” Let me depict the story with a very simple, branching model:

no sea battle m3 m2 sea battle

There will be a sea m0 battle tomorrow m1 Xerxes and Artemisia Confer

Point m0 represents the moment at which the future sea battle is being decided. The right “branch” of the treelike model represents the continuation of this moment which holds the sea battle, the left branch represents the opposite continuation. At moment m1, Themistocles says, “There will be a sea battle tomorrow.” The Aristotelian question is: is the sentence true? Even a quick glance on the model suffices to recognize that the issue is highly problematic. All that we have at our disposal is two alternative continuations of moment m1, which seem to be of little help if we want to answer the Aristotelian question. Based on the picture alone, one can easily argue that the sentence might be true and it might be false, but there is no clear way of deciding whether it is true. The formally most attractive semantic theory of branching (which I discuss in chapter3) o ffers a somewhat evasive answer to the Aristotelian question—it depends! With respect to the “right” branch, the sentence is true; with respect to the “left” branch, it is false. In a sense, such an answer is definitely correct. Indeed, in the battle-possibility, it is true that there will be a sea battle, and in the peace-possibility, it is true that there will be none. But is it really the answer we were looking for? Many philosophers were unconvinced. There seems to be a gap between the common-sense notion of truth and the notion of truth embodied in the most accurate semantic theory. A real question is: how (and if) to bridge this gap? Any answer to this question requires taking a side in the debate which originated with Aristotle’s puzzle. A majority of branching theorists side, in one way or another, with the first of Aristotle’s intuitions. They argue that none of the branches extending Themistocles’ utterance can be somehow privileged. Therefore, they all concede that no sentence regarding the contingent future event is true. Their theories can be subsumed under the umbrella term—antifuturism (I discuss them in chepter4). There is also a weaker, but persistent trend in the other direction. Philosophers and logicians in this community argue that there exists a very natural way to distinguish one of the branches. After all, only one of the alternative future continuations will actually

2 CHAPTER 1. INTRODUCTION take place and it is the continuation that is relevant for semantic purposes. The sentence about the future is true if and only if what it says will actually happen. This brand of theorists side with the second of Aristotle’s intuitions. They might be characterized as futurists. The branching environment is generally believed to be hostile to the futuristic approach. It seems unlikely to distinguish the actual continuation of the utterance given that both continuations have the utterance as their parts. I partially agree with this assessment. Under one interpretations of the branching model—which I call branching realism—there indeed is no place for the distinguished actuality. In this account, the temporal and modal reality is best represented as a branching object. Or, as Nuel Belnap likes to put it, Our World is branching. There is a lot of misconceptions regarding branching realism. One of them is that according to branching realism, actual world is branching. In this view, the branching tree represent the ordering of actual events. This leads to unacceptable consequences and the branching realists rightly rejected this interpretation. There is another misunderstanding of branching realism that is far less recognized, however. According to the second misrepresentation of branching realism, it holds that the branching tree represents the ordering of possible events. I might be sticking my neck out too far, given that many branching realists themselves describe the elements of the structure as possible events, but I think that they must mean it at most metaphorically. I base my opinion on an Aristotelian precept that one can talk of possibility only if one is ready to complement it with actuality. The second misinterpretation of branching realism might have led some futurists astray. On the one hand, they held that our world is branching, and on the other, since they believed that the branching represents possible events, they insisted that it needs to be supplemented with actual events. They ended up with a controversial view that our world is branching, but the actual part of our world is not. These vague comments need to suﬃce at this point. I describe in chapter5 how this particular brand of futurism has evolved, and, more importantly, how it was criticized by branching realists. I mostly agree with their criticism. Indeed, if you believe in the branching world, you cannot distinguish the actual branch. The realists conclude that the idea of a distinguished actual branch needs to be abandoned. As they say, one philosopher’s modus ponens is another’s modus tollens. I accept the implication mentioned above, but I derive the opposite conclusion. I do want to think about the elements of the branching structure as the possible events. Then, in accordance with the Aristotelian idea, I complement the possible with the actual. Then, I distinguish one of the branches as the possible branch that gets actualized. Therefore, I abandon the idea that the world is branching. Thus, I accept branching among the possible events and reject branching among the actual events (where the diﬀerence between the two is absolute). I call this position branching actualism. The branching pictures might easily suggest a realist picture. Remember that when I presented you with the Aristotelian question in the branching setting, I suggested that Themistocles utters a sentence at moment m1. It suggests, in turn, that a concrete person—Themistocles—occupies a realm of partially overlapping worlds. In the actualist setting, however, Themistocles does not live on the branching tree, he lives in a non-branching actual world and when he says what will happen, he says what will

3 CHAPTER 1. INTRODUCTION happen in the actual world that he lives in. Therefore, I shall argue that branching actualism offers a much more hospitable environment for futurism than branching realism does. The most general aim of this work is to establish futurism as a viable semantic theory in the context of branching possibilities. Let me end with a short road map of the remaining material. In the next chapter, I will recount the history of the idea of the branching model. As I present the proces- sions of the idea, I will explain how the realist understanding has gradually become the prevalent account of branching. Then, I will explicate branching realism in more detail and defend it against possible misconceptions. Since the remaining part of the world is largely technical in character, I will devote chapter3 to the exposition of the formal apparatus I am going to use. In particular, I will discuss the formal properties of the branching structure and the basic properties of the Ockhamist semantics. In chapter 4, I will describe how branching realism naturally leads to antifuturism and discuss a number of specific proposals which rationalize the position. The discussion is both expository and critical in character. Chapter5 is devoted to the discussion of the so- called Thin Red Line theory. It is a theory which tries to combine branching realism with futurism. I give the historical background of the theory and its technical details. Then, I recount the criticism that was targeted at the Thin Red Line. Finally, chapter 6 advocates a futuristic theory based on the actualist account of branching. I introduce branching actualism in more detail and examine how it can be used to support futurism. I also explain what is the actualist approach to possibility and how the actualist-based futurism defends itself against the attacks mounted against the Thin Red Line. Finally, I argue that while branching actualism—in contrast to branching realism—does not excludes futurism, it also does not enforce it. Branching actualism admits both futuristic and antifuturistic theories. I end with a tentative suggestion that the discrepancy among the actualists results from a disagreement regarding the temporal character of truth. A fraction of this work has already appeared in print. In particular, fragments of (Wawer, 2014) are incorporated, in a modified form, as sections 5.3.1– 5.3.4, 6.3.1– 6.3.4, while sections 7.8– 7.11 of the appendix are a portion of (Malpass and Wawer, 2012).

4 Chapter 2

Branching Realism

The earliest mention of the branching structure as a semantic device is due to Saul Kripke. As a high-school student in Omaha, Nebraska, Kripke read Prior’s Time and Modality (1957), and impressed by the content, wrote a letter to the author. In the letter, he suggested to Prior a “tensed” interpretation of one of Kripke’s own semantics of modal logic: Now in an indetermined system, we perhaps should not regard time as a linear series, as you have done. Given the present moment, there are several possibilities for what the next moment may be like—and for each possible next moment, there are several possibilities for the next moment after that. Thus the situation takes the form, not of a linear sequence, but of a “tree” (. . . ) The whole tree then represents the entire set of possibilities for present and future; and every point determines a subtree consisting of its own present and future. (Kripke, 1958) There is a characteristic hesitation inherent in this short note that was preserved for a few decades to come. On the one hand, the first sentence suggests that the elements of the structure are times and explicitly says that time is non-linear. On the other hand, the second sentence recommends a different interpretation—the elements of the structure represent possibilities (what moments may be like). This reading is further supported by the last sentence, which explicitly says that the tree represents “the entire set of possibilities for present and future.” These two readings: (a) branching structure as a set of times, and (b) branching structure as possible developments in time, were often confused throughout the history of application of branching. The textual work reveals an element of similar conceptual ambiguity implicit in many early works on the subject. It might be partly explained by the novelty of the relational semantics for modal and temporal logic in general. The early writers seem to have been so thrilled by the new semantic tool that they often rushed to apply it in their logical investigation, setting aside the conceptual interpretation. It is not uncommon to see but a few lines devoted to the description of the branching structure before an author proposes a semantic definition of this or that modal connective and investigates its logical properties. The interpretation of the structure is left to the reader. Since

5 CHAPTER 2. BRANCHING REALISM the structure is so innovative, however, the interpretation is far from self-evident and diﬀerent accounts might render diﬀerent intuitions. One of the best examples of a philosopher who had a rather instrumental approach to branching structures was the addressee of the letter, Arthur Prior. He put them to extensive use in his logical investigations, but introduced them with a few laconic sentences:

[W]e may deﬁne an Ockhamist model as a line without beginning or end which may break up into branches as it moves from left to right (i.e., from past to future), though not the other way; so that from any point on it there is only one route to the left (into the past) but possibly a number of alternative routes to the right (into the future). (Prior, 1967, p. 126)

The quote is open to a number of alternative interpretation concerning both the metaphysical status of the structure and its detailed formal properties. Prior describes a line which branches as it moves from past to future, but it is not entirely clear what the line is meant to represent: times, possible times, possible developments, the world, or something still different. Elsewhere (Prior, 1967, p. 53), he is slightly more informative and claims that the “tree” structure represents “branching futures” or “alternative routes into the future.” It might suggest branching of time view, but in a following fragment, Prior consistently refers to those futures as “possible futures,” which suggests a modal reading.1 The branching model was precisely formally defined in an influential work of Rich- mond Thomason(1970). The paper explicitly defines (and extends) the Ockhamist semantics, but As far as the interpretation of the structure is concerned, he is also ambiguous.

[F]or many philosophical (and perhaps even some scientiﬁc) purposes it is more interesting to consider the case in which time may be nonlinear. Such an account of time will permit instances in which a time α has alternative possible futures. (. . . ) [N]onlinear time puts these alternatives into the ontological structure of time. (Thomason, 1970, p. 265)

On the one hand, Thomason explicitly espouses a non-linear account of time (he argues that such an account is enforced by indeterminism). On the other hand, even this short fragment contains a hint for an alternative interpretation. After all, the author writes that “α has alternative possible futures,” rather than “α has many futures,” which would be a more accurate description of a branching of time. The constant ambiguity as to what the structure represents remained unresolved at this early stage. John Burgess, for example, describes it as follows:

1It is not entirely surprising that Prior was so scarce in his interpretation of the structure, given that he treated the relational models of tense logic with considerable reserve. As Goldblatt(2006, p. 27) notices, already in 1958 Prior firmly claimed that the relational structure of times has little, if any, metaphysical significance, (Prior, 1958, pp. 115–116). He held this view ever since and reaffirmed it as late as (Prior and Fine, 1977, p. 37). Interestingly, regardless of his explicit reservation about the relational structures and their philosophical importance, Prior was considered an advocate of the branching of time. This view was attributed to Prior by allies of the view (Thomason, 1970) and by its adversaries (Rescher and Urquhart, 1971).

6 CHAPTER 2. BRANCHING REALISM

If the determinist sees Time as a line, the indeterminist sees it as a system of forking paths (. . . ) paths from left to right represent possible courses of history. (. . . ) [T]he picture only included courses of events that at some point or other were possibilities. (Burgess, 1978, pp. 159–160)

The beginning of the ﬁrst sentence clearly states that it is the time that branches (and, presumably, the events in time as well). He also conjectures that indeterminism requires such a decision. The remaining part, nevertheless, supports another view: that a branching diagram depicts possible courses of events. A similar confusion can be found in Gabbay et al.’s (2000) monograph on temporal logic.

Much of the motivation for considering time to be branching comes from the idea of using diﬀerent branches to represent diﬀerent possible histories of some part of the world. (Gabbay et al., 2000, p. 63)

Let me ﬁrst explore the literalist understanding, i.e., that (space)time is literally branching in an indeterministic world.

2.1 Naïve Branching Realism

Throughout the years, it has become more popular to refer to the theory of branching structures as Branching Time theory. In an important monograph on history of temporal logic, Øhrstrøm and Hasle(1995) o ﬀer the following summary of the history of branching:

[D]uring the last decades a number of intellectuals have suggested a new kind of time models. According to these models time is viewed as a branching system—a tree-structure. (Øhrstrøm and Hasle, 1995, p. 180)2

I think that the continuous tendency among the branching theorists to reify the structures and to treat them as a representation of a temporal reality is largely due to a historical coincidence. The branching model was often considered a generalization of the linear temporal model. Since the line means to represents the succession of times, the tree was also taken to represent the succession of times. The analogy is misleading, however, and a very simple argument is suﬃcient to show it. First of all, as long as the focus is on purely temporal relations, it is natural to assume that the following four statements are analytically equivalent:

1. Time t1 is earlier than time t2.

2. Time t2 is later than time t1.

3. Time t2 is in the future of time t1.

2Notably, Peter Øhrstrøm admitted in personal communication in September 2014 that “there is obviously a need for clarity” regarding the accurate interpretation of the branching structures. I hope that my work will at least partly answer the need.

7 CHAPTER 2. BRANCHING REALISM

4. Time t1 is in the past of time t2.

In my view, the interrelations encoded above partly constitute the meaning of temporal notions. In temporal logic, in fact, these interrelations are built into the very notion of a model. We assume in such models (a) that the earlier-later relation is the inverse of the later-earlier relation, (b) that the earlier-later relation is the accessibility relation of future-operator F, and (c) that the later-earlier relation is the accessibility relation of past-operator P. Keeping these deﬁnitions in mind, let us now consider a typical application of the branching model and represent an indeterministic fragment of the Greek-Persian war.

t4 t5

Greeks Persians win win

no sea battle t3 t2 sea battle

t1 Xerxes decides whether to ﬁght

Let us now assume that the branching is indeed about time. That is, let us assume that the ordering relation in the branching structure represents the relation of temporal precedence among times, earlier-later-than. Based on this assumption, we can conclude that there are times in the branching structure which have more than one future! If the diagram above represents the branching time, we infer that as of time t1, there is time t2, later than t1, at which there is a sea battle, and there is a time t3, later than t1, at which there is no sea battle (times t2 and t3 are not comparable by temporal relations). It means, by the above-presented equivalence, that as of t1, there will be a sea battle in the future and there will be no sea battle in the future. Furthermore, the sea battle will be followed by the Greek victory and it will be followed by the Greek defeat. It is the bizarre conclusions which result from the idea that the branching structure represents succession of times. I call such an interpretation of the structure Naïve Branching Realism.

2.1.1 Futures in abundance Naïve Branching Realism was criticizes along similar lines by David Lewis(1986). He deﬁned branching as follows:

In branching, worlds are like Siamese twins. There is one initial spatiotemporal segment; it is continued by two diﬀerent futures—diﬀerent both numerically and qualitatively—and so there are two overlapping worlds. One world consist of the initial segment plus one of its futures; the other world

8 CHAPTER 2. BRANCHING REALISM

consists of the identical initial segment plus the other future. (Lewis, 1986, p. 206).

Thereby, Lewis considers a temporal series of events S 0 followed by two, mutually incomparable, temporal series of events, S 1 and S 2. Both S 1 and S 2 are temporal continuations of S 0. Lewis calls S 0 + S 1 a “world,” S 0 + S 2 is also called a “world.” Lewis implicitly assumes a form of equivalence described above. If an event e1 is in a spatiotemporal continuation of event e2, then e2 is in the future of e1. He then concludes that the events in S 0 have two distinct futures. The consequence of this view is rather unwelcome: The trouble with branching exactly is that it conflicts with our ordinary presupposition that we have a single future. If two futures are equally mine, one with a sea fight tomorrow and one without, it is nonsense to wonder which way it will be — it will be both ways — and yet I do wonder. The theory of branching suits those who think this wondering is nonsense. (Lewis, 1986, pp. 207–208). Thus, from so construed branching model, Lewis derives the conclusion that any indeterministic moment has many futures, all of which actually happen. He raises two concerns about this view (i) it is in conflict with common sense, (ii) some attitudes, such as wonder (but presumably also hope, desire, guess, or expectation) are pointless, since they presuppose a single future. When Peter Øhrstrøm(1981) considers a similar theory, he agrees with Lewis’ diagnosis and states that Many will probably agree with me that with respect to the future operator, [it] is not a theory that pays due regard to everyday language and generally accepted ideas. (Øhrstrøm, 1981, p. 87) When confronted with Lewis’ objection, the theorists of branching often turn linguistic. They proceed to demonstrate, in contrast to Lewis’ suggestion, that no sentence of the form “Tomorrow:φ and Tomorrow: non-φ” is ever true in branching models. A characteristic line of thought is presented by Belnap et al.(2001): Lewis misdescribes the theory of branching time in saying of such a situation that “it will be both ways.” Branching time is entirely clear that “Tomorrow there will be a sea fight and tomorrow there will not be a sea fight” is a contradiction. (Belnap et al., 2001, p. 206) The “theory of branching time” that Belnap et al.(2001) have in mind is Ock- hamism. I define it in detail in section 3.2. For now it is sufficient to note that in Ockhamism the truth of a sentence is relative to a “moment” on a tree and a “branch” of a tree. The branch is a maximal line through a tree, typically called a history or a chronicle. Histories resemble what Lewis calls “worlds” in the above-cited fragment. Since the truth value of a sentence is relative to a history, then even if a moment is followed by a sea battle in one of the histories and is not followed by a sea battle in another history, the contradictory sentence is not true. Belnap et al.(2001) argue that Lewis failed to recognize that the notion of truth-at-moment does not make sense.

9 CHAPTER 2. BRANCHING REALISM

One must relativize truth to the history parameter as well. The reason is that only thus can we make sense, in branching time, out of plain (linear) future-tense sentences such as “There will be a sea battle tomorrow.” (Belnap et al., 2001, p. 225)

In a nutshell, they explain that if a moment is part of many histories, it makes no sense to talk about the future. Consequently, sentences in future tense are nonsensical, unless relativized to a history. While so relativized, they are perfectly meaningful and they behave just as the common sense commands. Essentially, the same semantic line of defense against Lewis’ objection is embraced by Placek(2012)

To state things bluntly, in constructing a semantic model for this Humphrey story, we will take care to preclude that at some valuation point e/h the two sentences “Humphrey has five fingers on his left hand” and “Humphrey has six fingers on his left hand” were true. The Lewis objection does not, therefore, demonstrate any contradiction or some other logical problem resulting from the concept of branching individuals. (Placek, 2012, p. 36)

Most recently, a similar argument was made by John MacFarlane:

All we conclude from the datum that it won’t be both ways is that our semantic theory must avoid making (4) Tomorrow it will be sunny here and won’t be sunny here. Tomorrow (Here is sunny ∧ ¬Here is sunny) true at any context. (MacFarlane, 2014, p. 211)

The Ockhamist semantics guarantees that such a sentence is indeed self-contradictory (as do numerous other semantic theories discussed in chapter4). Nonetheless, I do not ﬁnd the purely semantic line of defense entirely convincing. In my view, it does not address the underlying metaphysical concern. To make my meaning clear, let me elaborate an analogy. If you stand at the foot of the obelisk in the center of Piazza del Popolo and face south, you will clearly see two churches in front of you (the “twin churches” of Santa Maria dei Miracoli and Santa Maria in Montesanto). You will be inclined to say that the sentence “There are two churches in front of me” is true and the sentence “There is one church in front of me” is false. Let us consider, however, a semantic theory—a spatial analog of Ockhamism— according to which a sentence of the form “in front of me, φ” is not simply true or false. According to this theory, the truth of a sentence needs to be relativized to a person and to a particular spatial angle. Let us say that angle zero is determined by a person’s sagittal axis and that all the angles ranging from −45◦ to +45◦ can be used to determine the truth value of the sentence “In front of me, φ.” Let us call such theory Euclidianism. Let us return to the foot of the obelisk. In Euclidianism, the sentence “There is a church in front of me” is true relative to some angles and false relative to other angles. Importantly, there is no angle relative to which the sentence is both true and false.

10 CHAPTER 2. BRANCHING REALISM

Hence, Euclidianism guarantees that it is not both ways in front of me. Importantly though, the sentence, “There are two churches in front of me,” is true relative to no angle. Therefore, we can say that it is definitely false that there are two churches in front of me (it is false at all angles). Clearly, however, none of the two churches in front of you is going to disappear due to my technical trick. No matter what the semantic theory tells you, you are still facing the two of them. The analogy with branching is evident. One could argue that Ockhamism relies on the same trickery as Euclidianism. We have introduced an artificial semantic parameter—“history,” analogous to the “angle”—that bring about the result we desire. Making a future tense sentence relative to histories does guarantee that the sentence, “Tomorrow it will be sunny here and won’t be sunny here,” is false, but one can easily argue that it does not make one of the futures disappear. There still are two different future ahead of us, it will be this way and it will be the other way, we have just gerrymandered the semantic apparatus to conceal this inconvenient reality.

2.1.2 Problem with the trousers universe Not only the common sense, but also advanced science speaks against branching of time (or, as is more appropriate in the context of relativity theory, spacetime). John Earman(2008) provides an argument, grounded in philosophy of physics, against the idea of “branching in individual spacetime models” (p. 189, he also calls it “individual branching”). He identiﬁes two worries with such a position. The ﬁrst is a conceptual one:

[T]here is no necessary connection, in either direction, between determinism and individual branching. (. . . ) [B]ranching in individual spacetime by itself need not entail indeterminism. (Earman, 2008, p. 192)

Thus, Earman argues, against Thomason or Burgess, that branching (space)time is not an appropriate representation of physical indeterminism. He claims that an individual branching structure might be deterministic and that a non-branching structure might be indeterministic. The quote reveals that Earman construes the ordering as a non-modal, spatiotemporal relation within a single (space)time. Under this reading, it is hard to disagree with the author’s concern: a branched spatiotemporal structure does not indicate indeterminism. The second argument against individual branching is motivated by more speciﬁc physical considerations.

One might contemplate a literal branching of a relativistic spacetime as pictured in Figure 10.1, which shows an upside down “trousers universe” for which the “trunk” bifurcates into two “legs.” However, such a contemplation involves a change in the spatial topology and, thus, it runs up against no-go results for topology change. (Earman, 2008, pp. 193)

Subsequently, Earman mounts a number of topological results relevant for the theory of general relativity which show that acceptance of branching within a spacetime incurs very high costs—we need to give up a number of physically relevant properties

11 CHAPTER 2. BRANCHING REALISM which we would like any spacetime to posses. Similar argument along was earlier proposed by Gordon McCabe(2005). Both authors refer to Penrose(1979), who pursued the same general strategy. I trust the scientiﬁc expertise of the authors and conclude that acceptance of branching within the spacetime structure is a very costly decision and that we should prefer philosophical theories which do not make such a commitment, i.e., reject Naïve Branching Realism. It is an open question, deserving a more thorough exegesis, if anyone has ever held such a position. It is sometimes attributed to the so-called “many-worlds” interpretation of quantum mechanics. Some of the their statements do encourage such a reading, e.g.,

Of the three main proposals for solving this dilemma [i.e., the measurement problem—JW], I shall focus on one that pictures the universe as con- tinually splitting into a multiplicity of mutually unobservable but equally real worlds. (DeWitt, 1973, p. 155)

“[R]elative state formulation of quantum mechanics” was and is still more radical: it claims that the formalism of quantum mechanics, taken completely literally, describes a reality where every macroscopic superposition of quantum states is really a splitting of the universe into parallel copies. (Barrett et al., 2010, p. vii)

These suggestions are nonetheless sufficiently general to allow multiple interpretations and I am not competent enough to render a definite judgment (Belnap and Müller, 2010, offer a reading of the many-worlds interpretation which does not require branching within a world, but a branching of worlds). Earman(2008) attributes individual branching to Storrs McCall(1994), which is at least disputable, given that McCall writes “[T]he model to be presented consists of a branching set of many space-time manifolds” (McCall, 1994, p. 2).3 Earman hesitates whether his arguments apply to the the so-called Branching Space-Time theory of Nuel Belnap. For an extensive argument that it does not, see (Placek and Belnap, 2012). We might thus suspect that Naïve Branching Realism is a paradigmatic straw man. A theory that has been often discredited, but never endorsed. A theory resembling Naïve Branching Realism was taken most seriously, perhaps, by Borghini and Torrengo (2013). The authors write that

Granted: the two events ex and ey are both in the same world [i.e., the branching world]; if we take such world to be the actual world, then ex and ey are both actual. (Borghini and Torrengo, 2013, p. 109)

3It is a view much closer to what Earman calls “ensemble branching,” i.e., a branching among models rendered by an appropriate isomorphism between appropriate fragments of the model. Some of McCall’s claims suggest individual branching, e.g.,: “In general there will be many such futures. If for example a draw for a lottery takes place on 31 December 1999, and a million different people have purchased tickets for a prize of a million dollars, then, assuming that the procedure of drawing the winning ticket is a truly random one, there will be a million different physically possible outcomes, in each of which a different person wins. Every one of these futures branches off from a single space-time manifold” (p. 3). However, this fragment also admits many interpretations and, as far as I can see, McCall’s theory is compatible with a version of Earman’s ensemble branching based on identity isomorphism—a version that Earman does not disqualify.

12 CHAPTER 2. BRANCHING REALISM

Thus, the authors conclude that the actual world is indeed branching. It resembles the version of branching criticized by Lewis and Earman.4 To summarize, a number of arguments, both common sense and scientiﬁc, speak strongly against the idea of branching (space)time. Therefore, one should not endorse such a version of branching realism. As I have already noted, not many people do. When confronted with the type of problems described above, branching theorists typically turn to the other insight present in the citation from Kripke, and insist that the relation encoded by the branching structure is not a purely temporal relation, but it includes a modal component. It means, however, that it is (at least) slightly misleading to call the theory of branching structures Branching Time.5 This criticism extends to my work. When I decided on the title, I chose to follow, somewhat recklessly, perhaps, the established terminology and risk misunderstanding. Nonetheless, in the course of this work, I will avoid such façon the parler.

2.2 Genuine Branching Realism

When confronted with diﬃculties above, the proponents of branching often answer that the odd conclusions ensue from misconception. They turn to the second understanding of branching inherent in Kripke’s letter and argue that the tree has a modal character. The modal component of branching has been strongly stressed by the very person who propagated the branching-of-time view, i.e., Richmond Thomason. In 1984, he proposes the following interpretation of the model:

These treelike frames represent ways in which thing can evolve indeterministically. (Thomason, 1984, p. 213)

Thus, he explicitly promotes the idea which I ﬁnd particularly appealing: That the branching structure represents possible scenarios. He further explicates his notion in terms of overlapping possible worlds:

I like to think of possible worlds as overlapping, so that the same moment may have alternative futures. (Thomason, 1984, p. 207, n. 5)

The notion of a “possible world” admits many diﬀerent interpretations and Thoma- son chooses the Lewisian concrete worlds sharing initial segments. He does not develop his metaphysical picture in much detail, but is generally sympathetic towards Lewisian account of modality. He writes, for example, that “the prose of philosophical

4The authors do not stop there. They want to defend a version of the Thin Red Line (TRL): “TRL and branching time theory share the same treelike topological structure of time, but the former adds a special entity: the thin red line (R), representing that special future which will be the case.” (Borghini and Torrengo, 2013, p. 110). So, the authors want to combine the view that all possible futures are actual with the claim that only one of the futures will be the case. I ﬁnd such a combination hardly defensible. Regardless, it should be noted that Borghini and Torrengo’s account of branching is not a pure Naïve Branching Realism. 5This might be part of the reason why Belnap changed the name of his theory from Branching Space Time to Branching Space Times (see e.g., Belnap, 2003a).

13 CHAPTER 2. BRANCHING REALISM modal realists, such as D. Lewis, is much more judicious than that in which the physi- cists sometimes indulge” (Thomason, 1984, p. 211). It means that Thomason rejects the Naïve idea of the branching actual world and replaces it with a branching among possible worlds. Nonetheless, he retains the “objective” flavor of the theory by adopting the most realist account of possible worlds available at the philosophical market, where the worlds are understood as maximal concrete objects. The realist attitude towards branching is taken over by Nuel Belnap. It culminates in (Belnap et al., 2001), where we can find the most elaborate discussion of the structure. The authors introduce an important terminological shift, however. What is called a possible world in (Thomason, 1984) is termed a history in (Belnap et al., 2001), while the Thomason’s collection of overlapping possible worlds is called Our World by Bel- nap et al.(2001). It is important to keep the terminological di fference in mind to avoid confusion since Belnap et al.’s (2001) Our World is crucially different from Thoma- son’s possible world. In Facing the Future, we can find a thorough discussion of the foundations of indeterminism, involving, among other things, a description of what the authors refer to as the “ontology,” “metaphysics,” or “extra-linguistic” portion of the theory of branching. An extensive quote is worthwhile:

There are three fundamental ideas, already employed in earlier chapters: moments, the causal ordering relation, and Our World. First there is the idea of a moment (we use “m”); a moment is an instantaneous concrete event with unlimited (presumably inﬁnite) spatial extent. (. . . ) The second idea is the causal ordering relation, also called the earlier- later-than relation, m1 ≤ m2. This is a B-order relation, which we postulate to be branching rather than linear because of indeterminism. (. . . ) The ﬁnal idea is Our World. Start with this very moment (yours or ours; at this level of idealization it does not matter). Now form the set of all moments that are connected to this very moment by means of any zigzag combination of the causal ordering or its converse. That is, include all moments that you can reach by means of a “causal path,” no matter how complicated. That is what we mean by “Our World” construed as a set of moments. (Belnap et al., 2001, pp. 139–140)

Thus, according to this metaphysical picture, our worlds consists of a huge bunch of concrete events connected by the causal relation “earlier-later-than.” Some of the events are “compatible” (i.e., those which are connected by a linear path), some are not. My writing this words, just as your reading them, are both parts of our world (these two events are compatible). Importantly, the concrete situation in which I studied economics, instead of philosophy, is also a part of our world (a part incompatible with your reading these words). Adopting John Divers’s (2002) terminology, I call such conception Genuine Branching Realism.6

6Belnap(2003b) also o ﬀers a version of Genuine Branching Realism compatible with special theory of relativity (so called BST). The interpretation of the generalized structure is very similar: “A single, individual model of BST theory represents many pairwise-incompatible branching courses of events (each course of events imagined as a spacetime with content)” (Placek and Belnap, 2012, p. 445).

14 CHAPTER 2. BRANCHING REALISM

A cursory reading might leave the reader with an impression that Belnap et al. (2001) endorse a version of Naïve Branching Realism. Nonetheless, the authors indicate on numerous occasions that the branching structure should be understood in modal terms. When they comment on the ordering relation, for example, they explain that “[G]iven m1 < m2, (. . . ) one should say that m2 is in the ‘future of possibilities’ of 7 m1—not simply in its ‘future.’ ” (pp. 139–140). Elsewhere, they write that:

What branching time says is that the captain “has it both ways” in the entirely innocuous sense that he lives through a sea battle on history h1 and lives through no-sea battle on history h2. That just says that there are at m0 two possibilities for him, a fact about our world that we must keep. (Belnap et al., 2001, p. 207)

It turns out, then, that the idea of branching is completely non-controversial. It is meant to capture a simple-minded idea that, in some cases, more than one option is possible. How could anyone argue with that? In fact, not many people do. The opponents of branching are rarely the raging determinists who argue that possibilities are illusory. Many of them are ready to admit, for example, that the third world war could rally have happened in the last century. They object to the further claim of Belnap et al.(2001) that the possible third world war is a concrete event in our world. This means that they object to statements like:

It is good to think of a moment as a possible event, a possible momentary event. Momentary events automatically have their locus in the causal structure of our world, so that it makes sense to think of them as concrete. (Belnap et al., 2001, p. 190)

Many people, myself included, would disagree with this statement. I think that WWII sadly does have its “locus in the causal structure of the world,” while WWIII fortunately does not. It does not seem as if both wars are equally real parts of Our World. Admittedly, the view according to which our world contains all the mutually incompatible events is rather unusual. Even the authors agree that it is an extraordinary conception of the world, distinct from “that of Kripke 1959 (etc.), of Lewis 1986 (etc.), or of the standard four-dimensional concept derived (we suppose) from Newton by way of Einstein and Minkowski” (Belnap et al., 2001, p. 179). In light of the comment, it is easier to sympathize with John Earman’s remark: “I have been unable to get a ﬁx on what Belnap branching involves” (Earman, 2008, p. 192). On the one hand, Our World is described as a representation of a completely

7Incidentally, it means that Belnap et al.(2001) severe the link between “earlier,” “later,” “past,” and “future,” and replace it with an alternative equivalence:

1. Moment m1 is earlier than moment m2.

2. Moment m2 is later than moment m1.

3. Moment m2 is in the future of possibilities of m1.

4. Moment m1 is in the settled past of moment m2.

15 CHAPTER 2. BRANCHING REALISM innocuous idea that many scenarios are possible. On the other hand, it is (less in- nocuously) depicted as a branching structure of concrete events connected by spatiotemporal and causal relations. It is just not easy to get one’s head around this idea. Let me now spend a few more pages trying to ﬁgure out what Genuine Branching Realism comes down to.

2.2.1 Two perspectives It is easy to misrepresent Belnap et al.’s (2001) idea of the branching world, unless we distinguish two distinct perspectives, “internal” and “external.” Belnap(2011) himself describe these perspectives as “standpoints.” He metaphorically describes the external standpoint as “godlike,” “scientific,” or “outside of Our World”(Belnap, 2011, p. 86), while the internal standpoint is always “located at some point event (paradigmatically the one to which “here-now” refers) in Our World”(Belnap, 2011, p. 87). This kind of distinction of two “standpoints” is familiar from the philosophy of time. It is inherent in McTaggart’s (1908) famous distinction of two ways to order times, in shape of the so-called A-series and B-series. The basic concepts defining A- series of times are those of past, present, and future. While the basic concept used to define the B-series of times is the earlier-later relation. The A-concepts might be called “internal,” since they can be intelligibly interpreted only from a particular temporal location, while the B-concepts might be call “external” since they abstract from a particular location in the temporal series. In fact, Belnap et al.(2001) themselves admit that “in considering indeterminism we concentrate on a generalization of McTaggart’s ‘B-series’ ” (Belnap et al., 2001, p. 134). Seemingly, they just replace a linear order with a partial order, but the generalization they advocate is much more substantial. The crucial addition they make to McTaggart is that when they describe reality from the “external” standpoint, they abstracts not only from the particular temporal location, but also from the particular modal location. This makes their account more unusual. They argue that from the external standpoint we can neither see what happens now, nor can we see what happens actually. All we can see is a number of overlapping temporal paths of evolution.8 This line of thought is well-phrased by Tomasz Placek:

From the external perspective, a branching model contains a plethora of possibilities, all on par, with no distinction between possible and actual. It is somewhat similar to the physicist’s study of the possible evolutions of a given system; the study does not ask which of these possible evolutions the system actually travels. This is a “scientiﬁc view” or “a view from nowhere.” (Placek, 2012, 36–7)

8It is not the usual “godlike” perspective on reality. In the typical account, it is clear from the “godlike” perspective, whether I live my life righteously and deserve salvation, or whether my life is a disgrace and I should be condemned. The God of Branching Realism cannot answer if I should be condemned or redeemed. All he can do is to point to diﬀerent regions of the world and say “I shall condemn him over here and I redeem him over there.” Nor is it the usual account of the world in the philosophy of science. The collection of all the scenarios is sometimes considered a representation of a scientiﬁc theory (as in the so-called semantic view of theories, see, van Fraassen, 1980), but it is rarely seen as the representation of the world itself.

16 CHAPTER 2. BRANCHING REALISM

Not only is it the view from no-where (i.e., neither here, nor there), but it also a view from no-when (i.e., neither now, nor then), and from no-how (i.e., neither actually, nor possibly). In this respect, Genuine Branching Realism closely resembles the realist project of David Lewis(1970a).

If we take a timeless point of view and ignore our own location in time, the big diﬀerence between the present time and other times vanishes. That is not because we regard all times as equally present, but rather because if we ignore our own location in time we cannot use temporally indexical terms like “present” at all. And similarly, I claim, if we take an a priori point of view and ignore our own location among the worlds, the big diﬀerence between the actual world and other worlds should vanish. (Lewis, 1983, pp. 19–20)9

Our World looks differently if we look at it from the “internal” standpoint, i.e., from the perspective of a particular moment on the tree. Then, the difference between the present, past, and future becomes very vivid, just as the difference between the actual and the possible. The events on the tree no longer seem on a par. The events taking place at the moment are much more “lively.” The world looks different from perspectives of various moments. For example: from the perspective of the Treaty of Versailles, the second world war is a distressing possibility; from the perspective of the Anschluss of Austria, the war is an imminent danger; and from the perspective of the Berlin Blockade, the war is a horrible past. Similarly, from our present perspective, the third world war in the 20th century is a horrifying, but unrealized, possibility, while from perspective of an alternative possibility, the third world war has wreaked havoc on the human race. The crucial claim of Genuine Branching Realism is that none of these perspectives is privileged. None of them has a more accurate vista on Our World. In particular, the fact that there was no WWIII in the last century is relative. It holds from our viewpoint, but it does not from different points of view. The attitude towards branching was clearly endorsed by Richmond Thomason

Consider two diﬀerent branches b1 and b2, through t, with t < t1 ∈ b1 and t < t2 ∈ b2. From the standpoint of t1, b1 is actual (at least up to t1). From the standpoint of t2, b2 is actual (at least up to t2). And neither standpoint is correct in any absolute sense. (Thomason, 1984, p. 215)

The fragment encapsulates the essence of Genuine Branching Realism. First of all, we cannot evaluate what is actual, unless we specify the “standpoint” and, secondly, no particular standpoint is “correct in any absolute sense.” It means, in particular, that we should not “absolutize” our particular modal perspective on reality and should not

9Warning, I quote a reprint of the article. The original contains a potentially misleading typo: (. . . ) if we ignore our own location among the worlds we cannot use temporally indexical terms like “present” at all. (. . . ) [I]f we take an a priori point of view and ignore our own location in time, the big diﬀerence between the actual world and other worlds should vanish. (Lewis, 1970a, pp. 186–187) It is clear that the reprint gets it right.

17 CHAPTER 2. BRANCHING REALISM claim that the facts we consider to be actual are absolutely diﬀerent from the facts we consider to be merely possible. Due to this commitment of Genuine Branching Realism, Our World resembles a tree, rather than a line or a point. A version of the view has been defended by Nuel Belnap and many of his collaborators (which is particularly resonant in their arguments against the Thin Red Line which I shall discuss in section 5.3). A version of this view was recently rephrased by John MacFarlane:

From today’s point of view, we (on the sunny branch) can rightly assess yesterday’s prediction of sunny weather as accurate. But equally, the “branched” versions of ourselves (on the rainy branch) can rightly assess it as inaccurate. (MacFarlane, 2014, p. 202)

In a way, such a metaphysical vision is very natural for a modal logician to have. After all, when we construct a Kripke model, we simply take a set of “points” or “cases” (times, worlds, epistemic states, states of a computer program, etc.) and deﬁne relations between those points. We do not distinguish any of these points as “real.” For example, in a temporal model, none of the times is distinguished as The Present, in a model of possibility, none of the worlds is distinguished as The Actual etc.10 From the “outside” all the cases are on a par. Only when we select a particular case, we can speak about what presently or actually happens. Genuine Branching Realists raise the semantic model of modal logic to the metaphysical level. Just as no particular moment or history is semantically privileged, no particular moment or history is privileged in reality.

2.2.2 Two languages To explicate the idea of the two perspectives, Belnap(2011) appeals to a linguistic criterion. Suppose we “step outside of branching time.” To do this is to confine ourselves to language that has no trace of indexicality, a perfectly proper thing to do. (Belnap et al., 2001, p. 207) Hence, the litmus paper distinguishing the external perspective is that “we are not entitled to use either tense expressions or differentially applicable modal expressions” (Belnap, 2011, p. 86). Just as it makes no sense to distinguish past, present, and future moments in the ordinary B-series, ”it makes no sense to distinguish two kinds of point events, the ‘possible’ and the ‘actual’ ” (Belnap, 2011, p. 86) in the generalized B- series. The notions like “past,” “present,” “future,” “possible,” “settled,” “actual” make sense only when used at a particular location “within” a branching model. None of the internal, non-relative, tense and modal concepts makes any sense from the outside. Some languages distinguish not only tenses that differentiate between past, present, and future events but also an indicative and hypothetical mood. The former is used to talk about what actually is the case, while the latter to talk about what could have been the case. When we “step outside of branching time,” we need to speak in a “tenseless”

10Interestingly, Kripke himself, when he ﬁrst introduced the relational semantics for modal logic, explicitly distinguished the actual world as a special element of the model (cf. Kripke, 1959).

18 CHAPTER 2. BRANCHING REALISM and “moodless” language, a language whose sentences are in neither past, present, nor future tense and in neither indicative, nor hypothetical mood. I will distinguish between “mooded” sentences that refer to what is actual and possible and “moodless” sentences that, even if grammatically in an indicative mood, are not intended to describe the actual situation. If we read tenseless sentences as if they were in the present tense, we could misconstrue the B-series. For example, we could understand a sentence like, “The night is the shortest on Midsummer,” as a bizarre tensed sentence “The night is presently the shortest on Midsummer.” It might even suggest a peculiar view that, in B-theory, all times are present. Similarly, if we read moodless claims as if they were claims in the indicative mood, we could easily misconstrue the branching model. We could understand a claim like, “Moment m has two incompatible continuation,” as, “Moment m actually has two incompatible continuations,” which would reduce Genuine Branching Realism to Naïve Branching Realism. It seems to me that David Lewis committed this very mistake, when he criticized the idea of overlapping worlds. Interestingly, John Divers observed that Lewis’ own modal realism could be criticized in a similar vein:

Genuine Realism treats our non-modal claims about ordinary individuals such as donkeys, swans, planets, etc., as implicitly world-restricted claims. Thus (5) There are donkeys is to be interpreted—by default, in ordinary contexts of use—as the (true) claim that the actual world has donkeys as parts (. . . ) However, there are non-modal existential claims about properties, numbers, propositions which GR takes to be true—e.g., (6) There is a plurality of worlds. (7) Natural properties exist. Genuine Realism cannot construe the quantiﬁers in these cases as (invari- ably) world-restricted, for if read that way, they express—what are from the Genuine Realist standpoint—falsehoods. (Divers, 2002, p. 48)

It means that the claim that there is a plurality of worlds cannot be understood as an indicative claim that there actually is a plurality of worlds (actually, there is only one world). Similarly, the branching claim that many events have many incompatible continuations cannot be read as the claim that many events actually have many incompatible continuations. Therefore, if Lewis’ criticism were to be effective against Branching Realism, Divers’s criticism would be equally effective against Lewis’ realism. Things are considerably more intricate in the modal case than in the temporal. Firstly, we are more familiar with tenses then with moods, so understanding a tenseless expression is somewhat easier than understanding a moodless expression. Secondly, we have names of times in form of specific dates and hours, while we have no analogous linguistic devices to refer to elements of modal space. Thirdly, in the temporal case, we clearly distinguish the internal notions like “past,” “present,” and “future” from the external notions like “earlier,” “later,” “before,” “after.” In the modal case, we do not make an analogous distinction. We have the internal modal notions “necessary”

19 CHAPTER 2. BRANCHING REALISM

“possible” and “actual,” but we do not have the external modal analog of “earlier-later” relation.11 A particularly popular strategy is to externally characterize the elements of the modal space as “possible.” David Lewis characterized his worlds as “possible worlds,” while Nuel Belnap characterizes his moments as “possible moments” and histories as “possible histories.”12 We need to keep in mind, however, that it might be a misleading terminology, since it introduces a dangerous equivocation of the term “possible.” On the one hand, “possible” is the internal concept that can be interpreted only from a particular location on the tree, and on the other hand it is the “external” concept applicable to all the elements of the tree (it would be comparably misleading to describing all the events on the tree as “past events”). In fact, I will argue that some philosophers have indeed been misled by such a characterization of the tree. I would claim that a Branching Realist should simply call the elements of the model events, moments, or histories, without further modal qualiﬁcation, and I will do so, when I present their views. I imagine that Belnap et al.(2001) chose to call them “possible” to avoid the impression that the all moments on the tree are actual moments. However, even if Belnap et al.(2001) call their moments, “possible moments,” it is important to keep in mind that they do so in a special sense of “possible” and that they are in neither possible, nor actual in the ordinary sense.

2.2.3 Which perspective is basic? We have introduced two standpoints: external and internal. We have also introduced two types of languages to describe the world: tenseless, moodless language and tensed, mooded language. We might wonder which of the standpoints is more accurate and which language more fit to describe reality. Is Our World fundamentally constituted of tenseless and moodless facts like, “The Persians are winning the Battle of Salamis at moment m in history h,” or is it fundamentally constituted of tensed and mooded facts like, “The Persians could have won the battle of Salamis”? Belnap et al.(2001) do not give any definite answer to this question. They use tenseless and moodless conceptual apparatus in their theoretical investigations, but with regards to the metaphysical question about which kind of facts is more fundamental, they say, “We aim to avoid this question altogether” (Belnap et al., 2001, p. 134). We should note, however, that modal neutrality for which Belnap et al.(2001) argue seems to privilege moodless facts. Observe that from our everyday perspective, we clearly distinguish indicative from hypothetical facts. For example, the fact that the war now takes place in Syria is clearly distinct from the fact that the war now could have taken place in Iran. The difference is lost in Belnap et al.’s (2001) model of Our World. In this model, none of the facts are distinguished, as the facts corresponding to the indicative modal facts—as the facts that actually take place. If the mooded facts were primitive, then we would loose a substantial amount of information when switching to the external perspective. The description from the external perspective would thus

11These linguistic observations might suggest a certain metaphysical dis-analogy between modal and temporal case. 12See e.g., (Belnap and Green, 1994; Belnap et al., 2001; Belnap, 2002b, 2011, 2003b; Placek and Belnap, 2012).

20 CHAPTER 2. BRANCHING REALISM be highly incomplete. It would have to be completed with the data indicating which events on the tree are actual and which are merely possible. As we shall see, Belnap et al.(2001) ﬁercely argue against such completion of their model. It suggests that their privilege the moodless perspective on reality. The vocabulary chosen by Nuel Belnap is also evocative. He describes the branching model (as represented from the external, moodless perspective) as Our World with the capital “O” and the capital “W”. Not much can be based on this observation, but it gives you a general idea of what gets stressed. If Branching Realists chose moodless over the mooded facts, they could share David Lewis’ greatest achievement in the philosophy of possibility and declare that there is no primitive modality. Phillip Bricker explicates this thesis as follows:

[The thesis that there is no primitive modality] demands that our total theory, our best account of the whole of reality, can be stated without recourse to modal notions, that the (primitive) ideology of our total theory be non- modal. (Bricker, 2008, p. 115) Belnap et al.(2001) tend to theorize along these moodless lines. I have already mentioned that in the B-theoretic model of Our World which they develop, “it makes no sense to distinguish two kinds of point events, the ‘possible’ and the ‘actual’ ” (Belnap, 2011, p. 86). Also, in the generalized model of Our World, the so-called Branching Space Times, it is also presumed that the best model of the world should be expressed in non-modal terms:

In what follows I will try to avoid indexical language. In particular, I will not draw a distinction (inevitably indexical when not relational) between the actual and the possible—except in motivating or giving examples. “Possible point events” are just “point events.” These point events are to be taken not as mere spatiotemporal positions open to alternative concrete ﬁllings, but as themselves concrete particulars. (Belnap, 2003b, pp. 4–5) The addition that the events are “concrete particulars” is as symptomatic. Bricker (2008, p. 114) explains that Lewis’ major incentive to think of possibilities as concrete entities was to get rid of primitive modality.13 Modal neutrality thus suggests the moodless account of reality. Nonetheless, it does not enforce it. It is theoretically possible to combine neutrality with the view that there are fundamentally mooded facts. To do so, one could adopt what Kit Fine calls non-standard realism. I will explore the possibility, since it elucidates a great number of discussions that has taken place in the branching community. Fine developed non- standard realism in the domain of time, rather than possibility. Non-standard tense- realism holds that 13I hesitate to declare in full conﬁdence that the B-theory of Belnap works entirely without primitive modality, since it is not clear to me, if the causal relation < between events can be explained in entirely non-modal terms. Also, Belnap says that he will “try to avoid indexical language,” but does not declare if he thinks that it is possible to eradicate indexicality altogether.

21 CHAPTER 2. BRANCHING REALISM

(i) Among basic constituents of reality, there are tensed facts (each oriented towards a time), and (ii) “No single time is privileged, the tensed facts that constitute reality are not oriented towards one time as opposed to another” (Fine, 2005, p. 271).

Thesis (i) accounts for “realism,” while thesis (ii) accounts for “non-standard.” The standard form of tense-realism—presentism—has it that all the tensed facts that constitute reality are oriented towards one time—the present time. Tense antirealism holds that the facts constituting reality are tenseless. By analogy, non-standard form of realism in the philosophy of modality holds that

(i) Among basic constituents of reality, there are mooded facts (each oriented towards a possibility), and (ii) No single possibility is privileged, the mooded facts that constitute reality are not oriented towards one possibility as opposed to another.

The standard form of mode-realism—actualism—insists that all the mooded facts that constitute reality are oriented towards one possibility—the actuality. According to mood antirealism, the basic constituents of reality are moodless.14 Certain claims of Branching Realists suggest that they might be closer to non- standard mood realism than to a straightforward mood antirealism. For example, Tomasz Placek writes that

Lewis’ objection that branching individuals appear absurd has some sub- stance if considered from the external standpoint. This objection, however, is overturned if interpreted from the internal standpoint. For branching theory the concept of indexically-given modalities and tenses is essential. (Placek, 2012, pp. 37, emphasis mine)

If we accept non-standard mood realism, we need to somehow deal with the problem that among basic constituents of reality, there is a fact that there is WWIII in the 20th century and also the fact that there is no WWIII in the 20th century. We either need to conclude that reality is inconsistent or accept that mooded facts are not absolute. Belnap et al.(2001) would deﬁnitely prefer the second option. After all, they write that:

[B]ranching time then seems to say that the captain has it both ways, both living through a sea battle and living through no-sea-battle. The reductio is, however, an illusion. Omitting the relativization to histories is intolerable. (Belnap et al., 2001, 207).

14Admittedly, it is diﬃcult to clearly diﬀerentiate between non-standard mood realism and standard mood antirealism. According to antirealism, reality consists of moodless fact like that-the-captain-is-living- through-a-sea-battle-in-history-h. According to non-standard mood realism, reality contains mooded facts like that-the-captain-is-living-through-a-sea-battle, but these facts hold relative to a history. Schemati- cally, antirealism depicts basic constituents of reality as FACT(p-at-h), while non-standard realism as At-h(FACT(p)). In contrast, actualism holds that basic constituents of reality have the form FACT(p).

22 CHAPTER 2. BRANCHING REALISM

When considered in this context, relativization to a history amounts to much more than just a technical, semantic trick. It amounts to a metaphysical thesis that a mooded fact like that the captain is living through a sea battle is not absolute, but relative to a modal standpoint, i.e., to a history. One can then combine mood realism with modal neutrality and argue that branching theory of Belnap et al.(2001) is in fact a non-standard tense and mood realism in the style of Fine. This interpretation could be supported by Fine’s observation that non- standard realism naturally generates the distinction between the internal and external account of reality which Belnap(2011) and Placek(2012) advocated:

Each of the non-standard positions is committed, in its own way, to a distinction between a single comprehensive über-reality and a plurality of more particular realities. (Fine, 2005, p. 282)

In the context of Branching Realism, the über-reality reality is the reality depicted by a branching structure of concrete events, while the particular realities are the facts perceived as actual and present through the perspective of the moments in the structure. If like me, you struggle with the idea of two realities, you might ﬁnd consolation in Fine’s remark that “it is very hard to say what this distinction comes to” (Fine, 2005, p. 282). In the end, Fine proposes a vague and metaphorical account of the diﬀerence between the external and internal perspective generated by non-standard realism:

One might say that über-reality “manifests itself” in the form of the particular realities, that it becomes “alive” or “vivid” through the particular realities obtaining. Each particular reality presents itself as the whole of reality. It creates the illusion, if you like, that there are no further facts, even though there are many such realities and each is equally real. But it should be acknowledged that these remarks merely gesture in the direction of a certain idea and that, if we have here a viable conception of pluralistic universe, then none of the usual models for making sense of it will apply. (Fine, 2005, p. 283)

The fact that the usual models of thinking will not do for this new conception might partly explain why it was so notoriously diﬃcult to capture what the Belnap et al.’s (2001) idea of branching comes down to, and why it has generated so many controversies and inaccurate interpretations. Fine himself observes, with regards to über-reality and its relation to particular realities, that “there is a constant temptation to try to understand it in more intelligible, yet ultimately inappropriate, terms” (Fine, 2005, p. 282). Naïve Branching Realism is one such misunderstanding, since it interprets über-reality as the actual reality. It is a clearly misguided idea. It is equally misguided (even though less clearly) to interpret über-reality as the collection of possible realities.

2.2.4 Are possible histories possible? I have already indicated that it is misleading to call the elements of the Genuine Realist branching structure “possible moments.” I have observed that the notion of possibility belongs to the “internal” vocabulary and it is a category mistake to apply it to elements

23 CHAPTER 2. BRANCHING REALISM of the branching structure when considered from the external perspective. Kit Fine also diagnosed that such terminological choice might have problematic consequences [O]ne might think of über-reality as a manifold of possible or potential realities. But there is no possibility of potentiality without actuality; and so, on this view one of the realities is distinguished as actual, whereas the view is that all are equally real. (Fine, 2005, p. 283) The fragment expresses the view that nothing can be meaningfully described as “possible,” unless it is contrasted it with what is actual. This attitude originates with Aristotle, who introduced actuality and potentiality to his metaphysics as a pair of mutually complementing notions (see Metaphysics 1048a25–1048b9, specifically, “Let actuality be defined by one member of this antithesis, and the potential by the other.”). If possibility and actuality always operate in tandem, then, if we decide to call the elements of the branching structure “possible moments,” we need to be ready to contrast them with the actual moments. If we do that, however, we encounter a conceptual puzzle. On the one hand, all of the moments in Our World are supposed to be “equally real,” but on the other hand, they are “differently real,” because some of them are actual and some are possible. I think that this very problem has surfaced in the debate among branching theorists. Some philosophers and logicians misunderstood the über-reality of branching events as the structure of possible events and intended to supplement it with actual events. Let me quote here just one early example of such an attempt. In their book on temporal logic, Rescher and Urquhart wrote:

Let us represent the world as an inﬁnite tree branching toward the future (. . . ) The actual course of history will be one among the branches of such a tree. (Rescher and Urquhart, 1971, p. 201)

The idea that the treelike world should be supplemented with an actual history recurred a number of times. It was ﬁnally pined down and described as the “Thin Red Line” by Belnap and Green, and later characterized as follows:

One is thereby tempted to continue to represent objective indeterminism by postulating that our world (up to an idealization) is treelike, but to hold in addition that there is a distinguished history, the Thin Red Line (TRL). (Belnap et al., 2001, p. 161)

This metaphysical amalgam sounds very much like the wrong-headed conception described by Fine, that “one of the realities is distinguished as actual, whereas the view is that all are equally real.” It is not entirely clear whether the philosophers that are usually subsumed under the Thin Red Line view did actually hold the precise combination of ideas. They were often more interested in developing some innovative semantic theories, rather then in drawing a detailed metaphysical picture. In any case, at least some of their remarks might suggest the view (for a more detailed exegesis, see section 5.1) The notion of a distinguished actual branch is usually invoked in connection to semantics of a future tense operator. It is argued that the truth value of future-tensed

24 CHAPTER 2. BRANCHING REALISM sentence should be assessed with respect to the actual branch. The whole idea behind the branching structure is, however, that we abstract from what is actual and from what is possible. If all events in the structure are equally real, one cannot appeal to what is actual in an analysis of the future tense. Belnap and his collaborators has repeatedly pointed out the tension in the Thin Red Line theory (see sections 5.3 for details). I am sympathetic to many of their worries and I agree that

To suppose that there is one from among the histories in Our World that is the absolutely actual history is rather like purporting to stand outside Lewis’ realm of concrete possibilia and pointing to the one that is actual. But this is wrong in both cases. (Belnap et al., 2001, p. 163)

As soon as we have accepted a modal neutrality inherent to the treelike representation of the world, we cannot bring back the idea that one of the branches of the tree is objectively actual. It would be a little bit like admitting that the universe contains multitude of vast galaxies, but to insist at the same token that planet Earth has a distinguished metaphysical status.

2.2.5 Indexical actuality Judging by the problems of the Thin Red Line theory, Genuine Branching Realists conclude that the idea of absolute actuality is fundamentally confused. They argue that on the fundamental level, it is a mistake to divide the reality into the actual and the possible. This view is typical to Genuine Realism in general, as David Lewis has famously argued:

If I am right, the ontological arguer who says that his world is special because his world alone is the actual world is as foolish as a man who boasts that he has the special fortune to be alive at a unique moment in history: the present. The actual world is not special in itself, but only in the special relation it bears to the ontological arguer. Other worlds bear the same relation to other ontological arguers. (Lewis, 1970a, p. 187)

Therefore, if we accept modal neutrality and think of all events on the tree as concrete events, ontologically on a par with what happens around us, then it is problematic to add that only “our” concrete situations is absolutely actual. Lewis proposes a more realist friendly understanding of actuality.

I suggest that “actual” and its cognates should be analyzed as indexical terms: terms whose reference varies, depending on relevant features of the context of utterance. The relevant feature of context, for the term “actual,” is the world at which a given utterance occurs. According to the indexical analysis I propose, “actual” (in its primary sense) refers at any world w to the world w.(Lewis, 1970a, pp. 184–185)

25 CHAPTER 2. BRANCHING REALISM

Thus, according to a Genuine Realist the term “the actual world” does not denote a metaphysically distinguished situation any more than the term “I” denotes a metaphysically distinguished person, the term “here” denotes a metaphysically distinguished place, or the term “now” denotes a metaphysically distinguished time. The Lewisian way of thinking about actuality is endorsed by Thomason:

See Lewis(1970a) and substitute “the actual future” for “the actual world” in what he says. That is the view of the thoroughgoing indeterminist. (Thomason, 1984, p. 215, n. 14)

When we try to proceed with the suggested substitution, however, it quickly turns out that it is easier said than done. At this point, the difference between branching and non-branching versions of Genuine Realism turns out to be crucial. Observe that Lewis claims that the relevant feature of the context upon which the reference of the term “actual” depends is “the world at which a given utterance occurs.” But in Genuine Branching Realism worlds (histories) overlap, so it often happens that the given utterance occurs in more than one world. Consequently, the definite description that Lewis uses does not refer in many contexts! There is no easy fix to this defect, since it is generated by reasons of fundamental importance, as Belnap et al.(2001) make clear, “It seems mystery, however, just how the context of our modest speech act could determine the exact course of world history from now on, long past the dissolution of our galaxy. The phrase ‘our history’ does not make sense, unless determinism be permanently true” (Belnap et al., 2001, p. 164). In fact, even Thomason himself opposed the “actual” history of the context

To a thoroughgoing indeterminist, the choice if a branch b through t has to be entirely prima facie; there is no special branch that deserves to be called the “actual” future through t.(Thomason, 1970, p. 215)

This view has been almost universally accepted among the Genuine Branching Re- alists.15 Interestingly, even though Belnap et al.(2001) are aware of the di ﬃculties the indexical actuality generates in the branching context, they still subscribe to this vision:

As Lewis has argued (Lewis, 1970a), this world’s being the actual world does not favor it over any others, but is just a reﬂection of the fact that this is the world at which we are conversing. (Belnap et al., 2001, p. 163)

It is not straightforward to decode the authors’ meaning. They say that the actual world is simply “the world at which we are conversing.” However, we have already seen that they cannot possibly mean “the history at which we are conversing,” since they themselves argue that such a description does not denote in indeterministic contexts. The second natural guess is that “the world at which we are conversing” is simply Our World. But this interpretation is also out of the question, since we would need to conclude that Our World is actual and, hence, that all the possibilities are actual. Thus,

15For an important exception, see Loss(2012) and Sweeney(2015) who have recently suggested innovative ways to combine the history of the context with the Realist ideology.

26 CHAPTER 2. BRANCHING REALISM we would reduce Genuine Branching Realism to Naïve Branching Realism and make “branching time look silly in a way that it surely isn’t silly.” (Belnap et al., 2001, p. 206). Genuine Branching Realists needs to resolve the tension. On the one hand, they find the Lewisian conception of actuality very attractive, but on the other hand, the Lewisian assumption that no two worlds overlap seems to be built into the very heart of his understanding of actuality. The usual reaction of the Branching Realists is to insist that the relevant feature of the context the term “actual” is sensitive to is a moment, rather than a history. Remember that “moment” does not refer to an instant of time, but to a spatially extensive super-event. Branching Realists tend to identify what is actual relative to a moment with what is settled relative to a moment.16 We have seen, however, that the notion of a history is immensely helpful on the semantic ground. In particular, thanks to the history parameter, sentences like, “There will be a sea battle tomorrow and there will not be a sea battle tomorrow,” are never true, which takes some wind out of the argument against branching. Belnap et al. (2001, p. 207) even state that “Omitting the relativization to histories is intolerable.” Hence, Genuine Branching Realists got involved in a long-standing love-hate relationship with the history parameter. On the one hand, the history parameter is very useful for semantic reasons, and they need it as an element of the semantic index. On the other hand, they find it very suspicious for philosophical reasons and they do not want it as an element of the context. Ultimately, they resign themselves to a regrettable necessity of using a history as a part of the semantic index, but even then treat it with considerable distrust. When Arthur Prior first introduced Ockhamism, he called the choice of a history parameter “prima facie” (Prior, 1967, p. 126). Richmond Thoma- son suggested that we can “provisionally posit” a given history, but was was skeptical even about this provisional procedure (Thomason, 1970, pp. 270–271). When Belnap et al.(2001) introduced the history parameter, they referred to it as “auxiliary” (Belnap et al., 2001, p. 147). The story of systematic discrimination is much longer. So, it is clear that as far as the internal standpoint is concerned, Genuine Branching Realists prefer the standpoint of a moment to a standpoint of a moment/history pair, even if it generates a conflict with indexical actuality.

2.2.6 Towards Branching Actualism The essential feature of Genuine Branching Realism is modal neutrality, which can be summarized in Finean terms:

No single history is privileged, the mooded facts that constitute reality are not oriented towards one history as opposed to another.

Due to modal neutrality, Nuel Belnap insist that Our World, the über-reality, should be represented as a branching structure. From our particular, internal point of view, it

16This attitude is reflected in the semantic definition of the operator “actually.” I discuss the definition and its drawbacks in section 6.2. In personal communication (September 2010), Nuel Belnap expressed the view that “ ‘actually’ is just a ‘dummy word.’ ” In linguistics, a “dummy pronoun” is a pronoun which serves a grammatical function, but does not contribute to the meaning of the sentences in which it occurs. Therefore, he might not be so worried with the problems that “actually” generates in the context of branching.

27 CHAPTER 2. BRANCHING REALISM might seem as if the world were to actualize only one of the histories, but it would be a mistake to take the appearance for an absolute fact. The view is well expressed in the following citation:

Each of two events can happen, but it is not possible that both happen. It sounds to the naive ear as if we are saying that at most one of the two events can be part of Our World, but that is precisely wrong. (Belnap, 2002a, p. 5)

Our World contains both these events. From our viewpoint, they “manifest” themselves as possibilities, and it seems as if only one of them could take place, but it is not the ultimate picture. We need to keep in mind that, from their viewpoint, each of these two events consider itself to be actual. Moreover, in accordance with modal neutrality, neither our viewpoint, nor any of their viewpoints is the absolutely correct perspective. We should not get tricked and remember that Fine has warned us that in non-standard realism

Each particular reality presents itself as the whole of reality. It creates the illusion, if you like, that there are no further facts, even though there are many such realities and each is equally real. (Fine, 2005, p. 283)

We shall see that such a metaphysical position significantly influences the attitude towards the semantics of future contingents. In particular, we cannot correlate the truth value of future-tensed sentences with “what will really happen in our world,” since both options are real and they are parts of Our World. To overcome the difficulty, some theorists wanted to distinguish, within Our World, the events that actually take place and events that possibly take place. We have seen already, and will see in much more detail in chapter5, that it is a misguided strategy. As soon as we accept modal neutrality and agree that all moments on the tree are equally real, we cannot also insist that some are differently real. We need to choose between between modal neutrality encoded by the branching world on the one hand, and the absolute distinction between actuality and possibility on the other. Branching Realists firmly stand on the side of the branching world: To the extent that common sense asks (. . . ) for a unique naturally given “actual history” to which a given utterance-event belongs, to that extent, common sense is asking for something it cannot have. (Belnap et al., 2001, p. 206) I will explore the alternative path and assume that there is the absolute difference between actuality and possibility. As a consequence, I shall give up the idea that the world is branching. Nuel Belnap might describe my position as particularism, since I will base my approach on the exact naïvité that he has described. Based on the observation that it is not possible for both events to happen, I conclude that only one of them can be a part our world. I thus reject modal neutrality and assume that mooded facts are indeed oriented towards one history. They are oriented towards the actual history. I call this view Branching Actualism.

28 CHAPTER 2. BRANCHING REALISM

I will thus base my theory on the assumption that the accurate description of the world should not abstract from what is actual and what is possible. I will assume, in contrast, that the conviction that our modal viewpoint is privileged is not an “illusion.” Our mooded facts are the ultimate facts about our world. They do not require any further modal relativization. I might well be confused in this respect; let me point out, however, that I am confused jointly with a large number of other philosophers, as Kit Fine recapitulates:

If we ask in the modal case whether we should be a non-standard realist (and adopt the principle of neutrality), then the answer has seemed to most philosophers to be clear “No.” It has seemed evident that, of all the possible worlds, the actual world is privileged; it is the standpoint of reality, as it were, and the facts that constitute reality are those that obtain in this world. (Fine, 2005, p. 285–6)

Thanks to my rejection of modal neutrality, I can wholeheartedly embrace the view that the branching structure represents possible events. I can do this, because I am willing to contrast them with actual events and be faithful to the Aristotelian idea that there is no possibility without actuality. In my view, only when we describe both these aspects, we give full account of reality. Observe that the idea that the branching tree represents possibilities is in agreement with how the tree is usually introduced. We typically ﬁrst present an uncontroversial story which suggests that in some situation many outcomes are possible (e.g., two possible results of a coin toss, a few possible ways a person might act, or various possible results of a measurement of an electron). For example, Xerxes could have listened to Artemisia and avoid the battle and he could have resisted her arguments. Given that he had decided to attack Salamis, the Persian ﬂeet could have won and could have lost. The structure of alternatives is very naturally pictorially represented by a tree (like a tree on page8. The description of the situation indicates that there are (at least) three possible ways in which the Xerxes-Artemisia debate could have been followed. Thus, it seems natural to assume that the picture represents the temporal dynamics of possibilities. I propose to consider the tree structure as a generalization of our toy-model; it represents all the possible temporal developments of the entire world. Branching actualists are often accused, however, of being determinists in disguise. Branching Realists believe that the actualist notion of possibility is not substantial enough. They argue that unless one accepts that actuality is relative and that possibilities are as real as actuality, one can only purport to believe in the real possibilities. They typically characterize the actualist possibilities as “epistemic,” “doxastic,” “linguistic,” etc. The dismissive attitude is not limited to Genuine Branching Realism. After all, David Lewis referred to the actualist possible worlds as “ersatzes.” I side with the the actualists who refuse to accept this reasoning. I insist that one can believe that an event is really possible without believing that this event is a part of reality comparable to the actual events (in section 6.4.3, I mention a few of realistic accounts of possibility available to actualists). In this view, I follow the notable example of Robert Stalnaker:

29 CHAPTER 2. BRANCHING REALISM

One could accept thesis one—that there are many ways that things could have been—while denying that there exists anything else that is like the actual world. (Stalnaker, 1976, p. 68)

I agree that indeterminism requires that there is more than one possible way for the world to develop. I accept the branching structure of possibilities as a real and important aspect of the world. Nonetheless, since possibility requires actuality, I add that the world does in fact (indeterministically) develop in one and only one particular manner. In the process of development, the world realizes exactly one of the available possibilities. Observe that as long as we assume that the branching structure represents the possible ways in which the world can develop in time, then the actual world is kept completely out of the picture. This observation has important semantic consequences: if all that the structure represents is what might and must happen in any possible situation, then it is going to be difficult to interpret sentences that simply say what will happen. The history of discussion around branching semantics confirms the conjecture. It is reasonably easy to interpret modal sentences, but, as chapter4 reveals, there are numerous difficulties interpreting the “bare” future tense. Here, I reverse the claim of McKim and Davis(1976), who wrote that

[I]n linear time models we are considering only the series of actual states of the world. If we have no means for representing possibilities that are not actualized then it follows immediately that we have been deprived of the semantical resources required to explicate the concept of a modal future tense. (McKim and Davis, 1976, p. 237, emphasis mine)

I claim, in a similar vein, that In branching models we are considering only the series of possible states of the world. If we have no means for representing possibilities that are actualized then it follows immediately that we have been deprived of the semantical resources required to explicate the concept of a factual future tense. If we side with actualism, we retrieve these semantical resources (I do this in chap- ter6). It means that a position in modal metaphysics can inﬂuence the view on semantics, especially the semantics of future contingents. If we side with Branching Realism and opt for modal neutrality, then, when we consider a future-tensed sentence, we cannot resort to absolute actuality to say whether it is true or false. We also cannot resort to the indexical notion of actuality, since the event of utterance is a part of many distinct histories. As a result, the future-tensed sentences are notoriously problematic in the realist setting. In contrast, in actualist setting, we do distinguish between the actual and the possible, so we can always say that a sentence in future tense is true if and only if what it says will actually take place.

30 Chapter 3

Ockhamist semantics

I try to avoid, wherever possible, highly technical arguments. Many of the theories I discuss, however, have a partially formal character. I cannot therefore proceed without introducing some basic technical vocabulary that will accompany us throughout the whole work.

3.1 Branching structure

I have already discussed the metaphysical significance of the branching structure. Let us now look at it from a more formal angle. Definition 3.1 (Branching Structure). A branching structure B is an ordered pair hM, ≤i, where M , ∅ and ≤ is a relation on M satisfying the following conditions: reflexivity ∀m∈Mm ≤ m;

antisymmetry ∀m1,m2 ((m1 ≤ m2 & m2 ≤ m1) ⇒ m1 = m2); transitivity ∀m1,m2,m3 ((m1 ≤ m2 & m2 ≤ m3) ⇒ m1 ≤ m3); backward linearity ∀m1,m2,m3 ((m1 ≤ m3 & m2 ≤ m3) ⇒ (m1 ≤ m2 or m2 ≤ m1)); connectedness ∀m1, m2∃m3 (m3 ≤ m1 & m3 ≤ m2).

If m1 ≤ m2 and m1 , m2, I write that m1 < m2. If it is not the case that m1 ≤ m2, I write m1 m2. I refer to the elements of the set M as “momentary possibilities” or, for short, “moments.” I read m1 < m2 as “moment m2 might follow moment m1” or “moment m1 must precede moment m2.” The appropriate sense of “might” and “must” is explained in section 3.5 below. If m1 ≤ m2, we can also say that m1 admits m2 and m2 requires m1. I will refer to the set of moments that might follow m1 as its future of possibilities. I will refer to the set of moments that must have preceded moment m1 as its settled past. Two moments, m1 and m2, are compatible iﬀ either m1 admits m2 or m2 admits m1 (i.e., m1 ≤ m2 or m2 ≤ m1). A set of momentary possibilities is compatible iﬀ every two elements of the set are compatible.

31 CHAPTER 3. OCKHAMIST SEMANTICS

Let me comment on a few properties of the ordering relation. Firstly, momentary possibilities do not repeat. Two moments might be indistinguishable, when taken in isolation, but if they occupy a different place on the tree, they are different.1 Secondly, the settled past of a moment is compatible. In contrast, the future of possibilities of a moment might not be compatible, i.e., there might be momentary possibilities in the future of possibilities such that neither follows the other. It is philosophically the most important aspect of the branching structure, since it formally encodes the modal asymmetry between the settled past and the open future. Lastly, the possibilities are “inte- grated”; it means that for any two momentary possibilities, there is a single possibility that they might follow. In the actualistic setting, it can be justified by the observation that all elementary possibilities are “derived” from a single, actual reality (I explore this idea in section 6.4.3). Connectedness reflects the assumption that we study possibilities inherent in our world. To use the branching structure for semantic purposes, we need the notion of a history. A history is a maximal compatible subset of momentary possibilities.2 Thus, every two moments in a history are compatible and, furthermore, every moment not in a history is incompatible with at least one moment in the history. Histories, in contrast to moments, are temporally “thick” possibilities; they depict a whole course of events, from the dawn to the dusk of time. I will use the symbol h as a metavariable ranging over histories. In a branching structure B, a history can be identified with a maximal subset of W, linearly ordered by ≤, i.e., ∀m1,m2 ∀h(m1, m2 ∈ h ⇒ (m1 ≤ m2 or m2 ≤ m1)) 0 and ∀m∀h(m ∈ W\h ⇒ ∃m0∈h(m m & m‘ m)). The set of all histories is denoted by Hist. When m ∈ h, I say that history h passes through moment m. The set of all histories passing through m is denoted by Hm (i.e., Hm B {h|m ∈ h}). Introduction of histories into semantic analysis was one of the great conceptual achievements which allowed Arthur Prior to improve on the simplistic semantic model, originally suggested by Saul Kripke. It allowed Prior to study a language with clearly distinguished modal and temporal operators (I use the term “modal” in a narrow sense encompassing only the historical modalities of possibility and necessity). Thanks to this device, he was able to disentangle the modal and the temporal component implicit in the semantics of future contingents.

3.2 Ockhamist truth

The branching structure originated as a technical tool, devised to interpret a tempo- modal language and to elucidate the more intricate pieces of reasoning on time and modality. It has been used in a variety of diﬀerent ways. One of the most successful theories based on the branching structure, proposed already by Arthur Prior(1966, 1967), is the so-called Ockhamism. We shall see that it has numerous desirable formal

1We can justify the claim in the spirit of Genuine Realism and say that they are distinct, concrete events, so, to use Aristotelian vocabulary, even if they have identical “form,” they are distinguished by their “matter.” An alternative justification says that momentary possibilities are partially individuated by their settled past. Then, even if two internally indistinguishable possibilities have different settled pasts, they are different possibilities. 2The nomenclature in the field is not homogeneous. What I call histories is sometimes called routes, chronicles, or branches. The general proof of existence of histories requires Kuratowski-Zorn Lemma.

32 CHAPTER 3. OCKHAMIST SEMANTICS properties. Furthermore, it is presupposed by many theories discussed in this work, so I will discuss it first. Ockhamism requires a very simple, sentential language containing a countable, infinite set of sentential variables Atom, the left and right parentheses, the classical sentential operators of negation (¬) and conjunction (∧), two one-argument temporal operators, “it will be the case that” (F) and “it was the case that” (P), and a one-argument operator of historical modality “It is possible that” (^). The complex sentences are constructed out of atoms using the standard recursive procedure. The temporal and modal operators have natural duals: G stands for “it is always going to be the case that” (G B ¬F¬), H for “it has always been the case that” (H B ¬P¬), and abbre- viates “it is settled that” ( B ¬^¬). The classical connectives →, ∨, ↔ are defined in the standard manner. I reserve these symbols for the object language connectives. In the metalanguage, I mostly use English and occasionally help myself with the language of set theory (the symbols & and ⇒ stand for metalinguistic conjunction and implication, respectively). A valuation function V assigns a set of moments to every sentential constant, V : Atom 7→ P(M).3 A branching model M, based on a structure B, is a pair M B hB, Vi. The sentences are evaluated in a model, at an index. The exact shape of the index depends on the type of operators included in a language, since the role of an operator is to shift a parameter of the index (therefore, the exact shape of the index will fluctuate throughout the book, tracing the changing linguistic resources). In Ock- hamism, index contains two parameters: a moment parameter is shifted by temporal operators, while a history parameter is shifted by modal operators. In any index hm, hi, m ∈ h. To indicate this property in notation, I write m/h, rather than hm, hi. Con- sequently, sentences are evaluated at triples hM, m/hi. The Ockhamist truth (|=) of a sentence in a model at an index is inductively defined along the following procedure: Definition 3.2 (Sentence φ is Ockhamist true in model M, at index m/h).

5. M, m/h |= Fφ iﬀ ∃m0(m < m0 & m0 ∈ h & M, m0/h |= φ);

6. M, m/h |= ^φ iﬀ ∃h0(m ∈ h0 & M, m/h0 |= φ). Thus, the temporal operators shift the moment of evaluation up and down within the history of evaluation. It is a part of the reason why histories are required as parameters of truth. Thanks to histories, F behaves like “will,” rather than like “might” (as it did in the original Kripke’s model). The second crucial role of the history is to interpret the historical possibility. The sentence, “It is possible that φ,” is true at a history and

3Alternatively, we can say that V : Atom × M 7→ {T, F}. After all, every set can be identiﬁed with its characteristics function.

33 CHAPTER 3. OCKHAMIST SEMANTICS a moment iff φ is true at some history passing through the moment. Observe that if the sentence is about the past or present, then shifting the histories makes no difference (after all, there is no backward branching, so all histories passing through m are identical until m). Therefore, if the sentence is about the past, truth conflates with necessity and possibility (^φ ↔ φ and φ ↔ φ). The history parameter makes a difference only when a sentence is about the future. When the future of possibilities of a moment is incompatible, there are sentences which are true relative to some histories, but false relative to other histories. Thus, if a sentence is about the future, then settled truth implies truth (φ → φ) and truth implies possible truth (φ → ^φ), but not conversely. The asymmetry reflects the idea that the future is modally distinct from the past. The future is “open,” while the past is “closed.” To briefly examine behavior of this semantics, let us once again use a simple model M,4 depicted in figure 3.1.

h h h1 2 3

Greeks (q) Persians (r) win win

no sea battle (¬p) m2 sea battle (p)

m1 Decision whether to ﬁght

Figure 3.1: Sea battle.

Let p stand for “There is a sea battle,” q for “Greeks win,” and r for “Persians win,” and let us consider a few interesting cases.

1. m1/h2 |= F p 6. m2/h3 |= p ∧ HF p

2. m1/h1 |= ¬F p 7. m2/h2 |= HF p ∧ P¬F p

3. m1/h1 |= F p ∨ ¬F p 8. m2/h2 |= PFq ∧ ¬PFq

4. m1/h1 |= ^F p ∧ ^¬F p 9. m1/h3 |= F(p ∧ ^Fr ∧ ^Fq)

5. m1/h3 |= F p ∧ ^¬F p 10. m1/h3 |= F(p → ^Fq) Example8 is of a particular, historical importance. It fact, it explains why the semantics is called Ockhamism5. William of Ockham struggled with an argument that

4I will usually omit symbol of the model in the deﬁnition of truth. I bring it back on the few occasions when I study general logical properties like validity or consequence. 5However, the historical accuracy of the nomenclature is sometimes undermined. For an argument that the semantics does not fully reﬂect Ockham’s original intentions see (Øhrstrøm, 1984).

34 CHAPTER 3. OCKHAMIST SEMANTICS threatened compatibility of divine foreknowledge with human freedom. I will not outline all the details of the argument here (for a detailed exegesis see e.g., Øhrstrøm and Hasle, 1995; Fischer and Todd, 2015; Zagzebski, 2016). Let me just point out that the crucial assumption of the argument is that whatever is past is necessary (settled, inevitable). Should it apply to divine cognitive states, it would imply that if God knew something yesterday, it is now settled that he knew it. Nonetheless, among the things that God knew yesterday, there was that which would happen two days later. It seems to follow that what will happen tomorrow is necessary today. A secular version of the argument has been sketched already by Aristotle:

Again, if it is white now it was true to say earlier that it would be white; so that it was always true to say of anything that has happened that it would be so. But if it was always true to say that it was so, or would be so, it could not not be so, or not be going to be so. (. . . ) Everything that will be, therefore, happens necessarily. So nothing will come about as chance has it or by chance; for if by chance, not of necessity. (De Interpretatione, 18b10–18b16)

To get around this argument (at least in the reconstruction of Arthur Prior, 1967, ch. VII), Ockham rejected the premise that every sentence in past tense is necessary. For example, the sentence, “Yesterday, it was true that I would smoke two days later,” seems to be about the past, but it really is about the future and the principle of the past necessitation does not apply to it. The example8 proves that Ockhamism conﬁrms Ockham’s diagnosis. The sentence, “It was the case that the Greeks would win,” evaluated before the battle is resolved, seems to be about the past, but it really is about the future, so it is not necessary. It means that the principle Pφ → Pφ is not valid in Ockhamist semantics. The theological version of the argument is harder to dispel, however, as it involves mental states which seem to subsume under the principle of past necessitation much more than truth does.6 It is dubious whether the sentence, “Yesterday, God believed that the Greeks would win in two days,” is really about the future. To get his point across, Ockham had to insist that it is, but, as Øhrstrøm and Hasle recount, he also had to admit “that it is impossible to express clearly the way in which God knows future contingents” (Øhrstrøm and Hasle, 1995, p. 98). The theory depicted above is the core of the classical Ockhamism. Nonetheless, it will be useful for many future purposes to slightly extend the repertoire of the logical vocabulary. In particular, it will be useful to introduce metric version of tense operators and “date” operators. Both of them require, however, that we enrich the notion of the branching structure. We need a way to “coordinate” the momentary possibilities, i.e., to determine which of them happen at the same time. The usual procedure is to “cut” the branching tree horizontally into so-called instances. Intuitively, an instant is a set of alternative moments which take place at the same time. I repeat the deﬁnition of Belnap et al.(2001) verbatim:

6More on this issue in (Prior, 1968, ch. 4). For a particularly relevant commentary, see (Belnap et al., 2001, sec. 2B.10).

35 CHAPTER 3. OCKHAMIST SEMANTICS

Deﬁnition 3.3 (Instants). Partition. Instant is a partition of Tree into equivalence classes; that is, Instant is a set of nonempty sets of moments such that each moment in Tree belongs to exactly one member of Instant.

Unique intersection. Each instant intersects each history in a unique moment; that is, for each instant i and history h, i ∩ h has exactly one member.

Order preservation. Instants never distort historical order: Given two instants i1 and 0 i2 and two histories h and h , if the moment at which i1 intersects h precedes, or is the same as, or comes after the moment at which i2 intersects h, then the same 0 relation holds between the moment at which i1 intersects h and the moment at 0 which i2 intersects h . (Belnap et al., 2001, pp. 194–5) The collection of all instants of a structure will be denoted by I. The instant containing moment m is denoted by im. The term m(i,h) refers to the moment in history h taking place at instant i. It will be useful to supplement instants with coordinalization function T, which is an isomorphism between I and R (T : I 7→ R; I assume, for simplicity, that cardinality of I equals cardinality of reals). Thanks to the coordinalization function, instants can be numerically represented. We can also define distance between moments in the structure: dist(m1, m2) = x iff |T(im1 ) − T(im2 )| = x. We can now augment our language with a collection of metric operators, Fx and Px, where x refers to a real positive number. Operator Fx stands for “in x units of time, it will be the case that,” and Px for “x units of time ago, it was the case that.” We can also add date-operators Att, where t refers to a real number. Att stands for “at instant t, it is the case that.” Let M be a model supplemented with I and T. The semantics of these operators is given by the following definition:

Deﬁnition 3.4 (Metric and date operators).

0 0 0 0 1. M, m/h |= Fxφ iﬀ ∃m0 (m > m & m ∈ h & dist(m, m ) = x & M, m /h |= φ);

0 0 0 2. M, m/h |= Pxφ iﬀ ∃m0 (m < m & dist(m, m ) = x & M, m /h |= φ);

3. M, m/h |= Attφ iff M, m(T −1(t),h)/h |= φ. I will translate English into our language in a rather loosely fashion. For example, I will let the context decide what is the intended unit of time (a second, a minute, an hour, a day, a year). Also, when it causes no ambiguity, I will allow F1 to stand for “tomorrow” although it is not an indexical expression and would be better represented as “one day later” (or even more appropriately: “exactly 24 hours later”). Furthermore, even though t refers to a specific instant of time, I will allow Att to encode phrases like “on Monday,” “in year 1933,” “in April,” etc. Any attempt to introduce more precision would require a lot of detailed investigations, while their benefits would be marginal for my purposes.

36 CHAPTER 3. OCKHAMIST SEMANTICS

3.3 A few remarks on the logic of Ockhamism

The general logical properties of Ockhamism are highly appealing. Firstly, it has suﬃ- cient conceptual resources to interpret both modal and temporal operators. Secondly, it validates the truths of logic of linear time and truths of modal logic S5, which “is good for those who (like me) are not determinists, but feel that these validities are intuitively plausible” (Thomason, 1984, p. 215). Thirdly, it elegantly models the mutual relation- ships of time and possibility, and is able to express the temporal dynamics of the notion of historical possibility. Moreover, this semantics has proven its worth in numerous applications in metaphysics, deontic logic, doxastic logic, logic of agency, linguistics, robotics, or software engineering. One might question some technical details, but the core of Ockhamism is sturdy. It is a very good starting point for investigations of time and possibility. Furthermore, we shall see, over and over again, that as soon as one departs the safe ground of Ockhamism, one generates some highly questionable results with respect to interrelations of tense and modality, which is an indirect argument in favor of this semantics. Moreover, although Ockhamism does not validate Pφ → Pφ, it still vindicates the modal asymmetry between the past and the future. Consider a pair of mirror principles:

• If something was once settled, then it is settled now (Pφ → Pφ). • If something will once be settled, then it is settled now (Fφ → Fφ).

The second principle is violated even in our simple model: m1/h2 |= Fp ∧ ¬F p. The first is valid in Ockhamism, because the branching model presupposes backward linearity. Thanks to this assumption, once something gets settled, it remains settled forever. The procession of time closes some future possibilities which now are open, but it does not open new past possibilities which are now closed. It is a matter of some controversy, whether we should go one step further and validate Aristotle’s principle that “with regard to what is and what has been it is necessary for the affirmation or the negation to be true or false” (De Interpretatione, 18a29– 18a30). Since we are dealing with tense logic, it is most natural to assume that atomic sentences are simple, present tensed, declarative sentences. It is more arguable, however, if we can assume that these present tensed sentences have no “trace of futurity” (Prior, 1967, p. 124). Among the present tensed sentences that do have a trace of futurity, one can name e.g., “I am having the last cigarette,” “I am choosing to stay home,” or “Pope Francis is beginning a very long pontificate.” To accommodate such examples, Prior(1967) introduced a syntactic distinction between two kinds of atomic sentences.7 As soon as we assume that the atomic sentences are “wholly about the present,” we can conclude that for any propositional constant p, the principles p → p and Pp → Pp are valid. This gives justice to Aristotle’s principle of necessity of the past and the present. More generally, if F does not occur in φ or if it occurs only in the scope of a modal operator, we have it that φ → φ. I have to admit that I have already built in the assumption that the atomic propositional variables are “chronologically pure” into the definition of the model. If valuation

7One should hope that the trace of futurity can be ultimately traced to some more ﬁne-grained, temporal modiﬁers applied on the sub-sentential level.

37 CHAPTER 3. OCKHAMIST SEMANTICS function V maps atoms into subsets of M and we have that m ∈ V(p), then (by point 1 of definition 3.2) ∀h(m ∈ h ⇒ m/h |= p), which means (by point6) that m/h |= p for any h ∈ Hm. Therefore, for any m/h, m/h |= p ⇒ m/h |= p, which means that I implicitly assumed that the atomic sentences of our language have no trace of futurity. I share this assumption, among others, with Prior(1966), Thomason(1984), and Reynolds(2003). However, the assumption is dispensable. We can avoid it if we accept (together with Burgess, 1979; Zanardo, 1996; Belnap et al., 2001) an alternative, history-dependent notion of valuation: V : Atom 7→ P(M/Hist) (where M/Hist B {hm, hi|m ∈ h}). If we do so, however, the underlying branching structure is no longer reflected in the evaluation of sentences. For example, if we encode the sentence, “John is meeting Paul in the Old Town square in Warsaw,” as an atomic sentence q, then, as far as formalism is concerned, they might well be meeting at m in history h1 and not meeting at the same m in history h2 (even though h1 and h2 identical at m). As a result, the question, “Did John meet Paul in the Old Town square in Warsaw yesterday noon,” has no straightforward answer. In some histories leading to the present, they did; in some others, they did not. To avoid the consequence, we need to accept an extra-logical assumption that in case of this particular sentence q, q → q. The history-dependent valuation has a formal advantage, as it preserves the rule of substitution. Observe that in models with history-independent valuation, p → p is a validity, but if we substitute F p for p, we arrive at a sentence F p → F p, which evidently is not valid. In any case, throughout this entire work, I will focus on the atomic sentences which have no trace of futurity about them. Therefore, the assumption that the valuation function maps sentences to moments rather than moment/history pairs simplifies a lot of my arguments. Another controversy of Ockhamism concerns the notion of the history. It is disputable whether all the maximal linear subsets of the underlying branching structure should be regarded as histories relevant for semantic evaluation. We might want to exclude some maximal linear subsets. However, when we prune the histories, we need to be careful to assure that the limited set of histories, Hist− ⊆ Hist, satisfies the con- − dition that ∀m∈M∃h∈Hist− m ∈ h. The structure hM, ≤, Hist i is called a bundled tree in the literature. The bundle Hist− might be very limited with respect to the original set Hist. In some cases, the original set is uncountable, while we can “cover” all the moments in the structure by its countable subset. Belnap et al.(2001) discuss formal and philosophical reasons why we should prefer the whole set Hist as our bundle, i.e., why we should prefer complete bundled trees. Incidentally, this renders the study of metalogical properties of Ockhamism much more difficult. Ockhamism is a relatively intuitive semantic theory, but it has been surprisingly resistant to attempts of syntactic characterization. While the sixties and the seventies witnessed a real boom of completeness proofs for modal logics, Ockhamism has proven to be particularly difficult to handle. The first attempts to axiomatize it were made by Prior(1966, 1967). In 1970, Richmond Thomason codifies the semantics and writes that:

I mean to present an axiomatization of the theory discussed in Section 7 [Ockhamism] in a forthcoming paper. (Thomason, 1970, p. 279, n. 15)

38 CHAPTER 3. OCKHAMIST SEMANTICS

Over a decade later, he was forced to admit that Since the paper has never appeared, this intention was evidently premature. (Thomason, 1984, p. 223, n. 24) He reports a several (unsuccessful) attempts made in the meantime, for example: In (Burgess, 1979), it is claimed that Ockhamist validity is recursively axiomatizable, and a proof is sketched. Later (in conversation), Kripke challenged the proof and Burgess has been unable to substantiate all the details. (Thomason, 1984, p. 223) A significant step forward was made by Alberto Zanardo, who proved the completeness theorem with respect to bundled frames. The set of “bundled” validities is smaller than the set of Ockhamist validities, however. Nishimura(1979) was the first to point it out. Thomason(1984, pp. 221–2) discusses a few simple examples that are valid with respect to complete bundled trees, but not bundled trees in general. A breakthrough was made by Mark Reynolds. He first proved (Reynolds, 2001) the completeness theorem for CTL∗ (Full Computation Tree Logic∗, which is a simplified version of Ockhamism studied by computer scientists) and then, in 2003, announces that: Despite this effort and interest, technical difficulties have left the presentation of a sound and complete axiom system of Prior’s most basic original Ockhamist logic of historical necessity as an open problem. In section 6 below we present a complete axiom system for this logic. (Reynolds, 2003, p. 356) The system is indeed presented, but the author adds that: This conference paper also gives a brief sketchy overview of the long, complex and quite interesting completeness proof. (. . . ) A full version of the proof (of over 100 pages) is in preparation. (Reynolds, 2003, p. 356) As far as I know, the full proof has not appeared in print as of now.

3.4 Future tense operator

I have mentioned that operator F is meant to stand for “it will be the case that.” The linguists noticed, however, that “will” comes in different flavors. Mikhail Kissine(2008) groups them in five crucial categories, illustrated by the following examples: (1) Mary will come. [future/prediction] (2) Oil will float on water. [generic] (3) Mary will be at the opera now. [epistemic] (4) In winter, Mary will always wear a green coat. [habitual/dispositional/ volitional] (5) You will leave tomorrow by the first train. [deontic] (Kissine, 2008, p. 130)

39 CHAPTER 3. OCKHAMIST SEMANTICS

Ockhamism focuses on the first, “predictive” meaning of “will.” The exact account of this notion has generated some controversy in the linguist community. Kissine recounts that there is a consensus among linguists that cases (2)–(5) have some sort of modal component build into their meaning (but the purpose of his paper is to shake the consensus). There is a considerable disagreement, however, regarding the modal status of (1). Some authors are inclined to the view that predictive uses of “will” involve a form of necessity, while others believe they have a non-modal, “factual” reading (for references, see Kissine, 2008, p. 130). Ockhamism sides with the second group of linguists. In this setting, operator F is relative to a history, but it does not quantify over histories (or other sort of mutually incompatible possibilities). It only shifts the moment of evaluation up the given history and the sentence Fφ is true in a history iff φ is true later in this history. In my final theory, I add that the “default” history on which the operator F operates is the actual history and it can be shifted to another history, only when prompted by a modal operator. Thus, it is assumed within Ockhamism that tenses are purely temporal concepts. They can interact with modal operators, but they are semantically “orthogonal.” Interestingly, when Prior(1967) discusses branching in Past, Present, and Future, he contrasts the Ockhamist, factual reading of F with the “Peircean,” modally loaded reading (I briefly discuss Peirceanism in section 4.3). Thus, the indecision witnessed in the community of linguists is paralleled in Prior’s early studies. One more comment is due. I encode future tense as a sentential operator. In this, I join the theorists of branching who practically unanimously followed Arthur Prior and encoded English future tense as such an operator. Thus, my work would be less intelligible if I broke ranks. There is a general worry, however, regarding this modeling technique:

It is important to be clear at the outset that the claim that tenses are operators that shift features of the index of evaluation is an empirical claim about natural language. It is a claim to the eﬀect that in the best syntax and semantics for natural language, tenses will be treated syntactically and semantically as such operators. I shall argue that given the available evidence, this is an implausible empirical claim. (King, 2003, p. 215)

King then lists a number of examples of English sentences that are more conve- niently modeled if tenses are captured in terms of object language temporal quantifiers, rather than sentential operators (some more examples can be found in Stanley, 2000). He also invokes the authority of numerous linguists who preferred quantifiers over operators. I should mention right away that King’s objection is methodological in character. He acknowledges himself that Max Cresswell(1990) has proved that if we extend the repertoire of tense operators sufficiently and simultaneously extend the semantic index, we achieve the expressive power of the language with quantifiers ranging over instants of time (I shall make use of Cresswell’s technique in sections 6.3.6 and 7.7). King’s methodological complaint focuses on the observation that an analysis in terms of quantifiers “(i) allows for a simpler, more elegant, less ad hoc treatment of tenses (. . . ); and (ii) allows for a more plausible account of the relation between the surface structures

40 CHAPTER 3. OCKHAMIST SEMANTICS of English sentences and the syntactic representations of those sentences at the level of syntax that is the input to semantics” (King, 2003, p. 221). Within my limited expertise, I tend to agree with this statement. My own research in the semantics of tensed expressions embedded in counterfactual constructions convinced me that it is useful to analyze tenses as quantifiers. Nonetheless, I decided to stick with the operator analysis to assure uniformity with the existing philosophical and logical literature. Moreover, the examples I will study are usually so simple that the choice of a particular set-up hardly makes any difference (in fact, in the simplest cases, operators appear to be more plausible than quantifiers). Anyhow, I do not think that anything substantial hangs on this. All the theories presented in my work could be easily translated into the vocabulary of quantifiers. In particular, as far as I can see, interpretation of future contingents is not any easier if we replace operators with quantifiers. Let me mention that King’s reason to privilege quantifiers over operators extends syntactic convenience. There is no place here to recapitulate the whole intricate argument, so let me summarize merely the most important points. King believes that quantifiers are more appropriate devices of semantic analysis than the operators, since the latter, but not the former, require time as an element of semantic index. As a result, the semantic value of a tensed sentence analyzed in terms of operators resembles a temporalist proposition that changes its truth value from one time to another. King believes that “ordinary” propositions do not behave like that, so the semantic value of an operator-analyzed sentence cannot be identified with the proposition expressed by the sentence. This mismatch undermines the operator analysis. I am not convinced by this line of argument, however. Let me limit my point to the most rudimentary observation that King neglects a significant aspect of the quantifier analysis of tenses. As we have learned from Tarski, an appropriate analysis of quantifiers requires that we enrich the semantic index with an assignment of values to variables (or a sequence of objects, or some other analogous device). In case of temporal quantifiers, we need to enrich the semantic index with an assignment that maps temporal variables into instants of time. Then, just as the truth of a tensed sentence depends on the temporal parameter in the operator-analysis (o-analysis), it depends on an assignment function in the quantifier analysis (q-analysis). Just as the sentential operator shifts the temporal index, the temporal quantifiers shift the assignment function. And—most importantly—just as the truth value changes from one time to another on the o-analysis, it changes from one assignment to another on the q-analysis. Thus, the semantic value required by the q-analysis also resembles the temporalist, time-neutral proposition which King finds so suspicious. Thus, this reason alone cannot privilege one analysis over the other.

3.5 Modal operator

Given his prodigious talent and impressive early achievements, in 1908, it was possible for Wittgenstein to become a very successful engineer and a world class specialist in aeronautics. This possibility was signiﬁcantly diminished, however, due to an unfor- tunate exposure to the works of Frege and Russell. After the personal encounter with both authors, Wittgenstein’s failure in engineering was inevitable. When he moved to

41 CHAPTER 3. OCKHAMIST SEMANTICS study in Cambridge, his fate as an intellectual was already settled. The modal operators of Ockhamism, ^ and , are meant to mimic the notions of “possible” and “inevitable/settled” used in the story above. It is often called historical possibility. I distinguish three crucial aspect of this notion of possibility. Namely, it is circumstance dependent, metaphysical, and temporally asymmetric. Circumstance dependence means that what is possible depends on which conditions prevail in a given situation. It implies that the range of possibilities might change if conditions change. Thus, Wittgenstein’s illustrious career in engineering is possible, when he is a young researcher studying jet engines in Manchester. It is not possible, however, in 1937, when he is a mature philosopher busy with his work on Philosophical Investigations; these conditions do not bode well for new technical inventions. The historical notion of possibility is also intended to be metaphysical, which means that possibilities depends on how things are in a given situation.8 Whether it is possible in 1908 that Wittgenstein will participate in the invention of the next generation of jet engine depends on Wittgenstein’s capacities and the resources of the Uni- versity of Manchester. For this reason, historical possibility is sometimes referred to as a real possibility. It should be contrasted with logical possibility. Presumably, not everything that is logically consistent is really possible; in particular it was not really possible for Wittgenstein to land on the Moon. We should also contrast it with epistemic possibility. If some traits of Wittgenstein’s character were essentially at odds with the laborious work as an engineer, then it was not really possible for him to become a world-class designer even if, for all we know, it was possible. Historical possibility is also distinct from doxastic possibility. Even if psychological and physical theories that we currently believe in were to exclude Wittgenstein’s success in engineering, it might, in fact, have been possible anyway. The historical notion of possibility is also temporally asymmetric. The future is open to numerous, mutually incompatible continuations (given they are admitted by how things are in the given circumstances, of course), while the past is settled and immutable. It also means that the range of possibilities gradually diminishes as time goes by and no new possibilities pop into existence in the procession of events. In 1908, it was possible that Wittgenstein would in 1925 pilot the plane of his own design, but it is now settled that he did not pilot the plane in 1925, and it will never become possible that he did. On the one hand, the notion of historical possibility is closely related to physical or natural possibility. One can even risk the statement that an event is historically possible in circumstances c iff it is physically possible, given what the world is like at c. One can go one step further and say that an event is really possible in circumstances c iff the laws of nature and circumstances c admit the event. However, the proponents of real possibility tend to be skeptical of such characterization (see e.g., Belnap et al., 2001, p. 137). Firstly, this definition might not guarantee temporal asymmetry. Many of the current physical theories are time-reversible and if they admit future possibilities, they also admit past possibilities. So, the laws of nature might not ensure that the past is settled. The problem can be easily amended if we explicitly “rigidify” the past and

8“Things” are conceived very broadly here. Depending on one’s choice of metaphysics, they may be thought of as objects, properties, events, facts, processes etc.

42 CHAPTER 3. OCKHAMIST SEMANTICS say that an event is really possible in circumstances c iff it is admitted by the laws of nature, given how the world has been until c. One might argue, however, that this proviso is entirely ad hoc. Secondly, and more importantly, it is not entirely clear if the laws of nature are sufficiently “metaphysical.” The philosophers of real possibility are particularly skeptical regarding the Humean tradition of thinking about the laws of nature—in terms of appropriately chosen regularities and frequencies. Such conceived laws do not involve the idea of necessary connection between elements of reality, so the possibilities generated by these laws are usually taken to be not “substantial” enough. On the other hand, historical possibility is closely tied to what might be called “practical possibility.” An event is practically possible iff it is in human power to influence it. This explains the temporal asymmetry of possibility, which is well phrased by Stephan Torre: I think that our notion of an asymmetry in openness between the past and the future is tied to an asymmetry in what we can affect or have power over. We take ourselves to have power over the future, yet lack power over the past. There is no use crying over spilt milk because once it has happened, there is nothing we can do about it (except clean it up of course). In contrast, we take it to be (partially) within our power whether or not future milk is spilled. (Torre, 2011, p. 361) The verdict is partially supported by Aristotle, who rejects determinism with a simple observation that “we see that what will be has an origin both in deliberation and in action.” Nonetheless, he does not think that it is the whole story since he adds that “in general, in things that are not always actual there is possibility of being and of not being; here both possibilities are open” (De Interpretatione, 19a8-19a11). Thus, the real possibilities seem to extend human capacities. It is in agreement with common sense. If there are indeterministic processes in the core of the Sun, there certainly are many possibilities regarding how these processes will eventuate, nonetheless, it is hardly in the human power (even “in principle”) to affect these processes. Besides, one might convincingly argue that the notion of human agency presupposes the notions of real possibility and open future, so if we tried to define real possibility in terms of human actions, our account of real possibility would be circular. In face of the last remark, we can consider an assumption that the notion of historical possibility is conceptually primitive and that it can be used to understand the “practical” possibility or the “natural” possibility, and not the other way around. Then, branching setting could serve as a general set-up spanning these two accounts of possibility. It could offer a platform that would help to understand the relationship between human and nature. This view is argued for by Thomas Müller:

The theory should lend itself to applications in both of the great theoretical endeavors in which we engage: coming to grips with the world in which we live in terms growing out of our lived experience, as well as in terms provided by science. In bridging that gap, BST [Branching Space Times] would help to establish the “humanistic” and the scientiﬁc respectability of the concept of indeterminism. (Müller, 2010, p. 396)

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For an extended discussion of the notion of possibility embodied by the branching structure, see (Xu, 1997, p. 144–5) and (Belnap et al., 2001, sec. 6A.3). I oﬀer some remarks on the actualist friendly possibility in section 6.4.3.

3.6 Sentences and propositions

In the semantic exposition of Ockhamism, I follow the tradition popularized by Tarski and ascribe truth values to (interpreted) sentences at semantic indexes. One could argue, however, that propositions are more ﬁtting to serve as truth bearers. There is a relatively easy way to introduce propositions into the Ockhamist setting. Actually, there are two relatively easy ways, mentioned by Belnap et al.(2001, pp. 189–90) and later developed by MacFarlane(2014, p. 207). These two ways represent the “temporalist” and “eternalist” propositions, respectively.9 Both these kinds of propositions are “modalist,” which means that they are “history- neutral” and change their truth status from one history to another. The temporalist propositions are additionally “time-neutral.” The temporalist proposition that it is summer on the southern hemisphere does not specify any particular instant of time and it changes its truth status from one time to another. It is true at all and only those times at which it is summer on the southern hemisphere.

Definition 3.5 (Temporalist proposition). Let φ be a sentence. The temporalist proposition expressed by sentence φ, |φ|T , is the set of moment/history pairs, m/h, such that m/h |= φ, i.e., |φ|T B {m/h|m/h |= φ}. For example, the proposition expressed by an atomic sentence p is the set {m/h|m ∈ 0 h&m ∈ V(p)} and the proposition expressed by the sentence F p is the set {m/h|∃m0 m > m & m0 ∈ h & m0 ∈ V(p). We say that the temporalist proposition A is true at a m/h iff m/h ∈ A. Things get more complicated, when we extend the language with indexical expressions. After all, the sentence, “I am hungry,” uttered by me and you—even if they have the same linguistic meaning—express different propositions. One expresses the proposition that I am hungry and the other that you are hungry. I will introduce temporal indexical Now into our language, so the same argument applies in our case. For example, the proposition expressed by the sentence, “It is summer now,” uttered by me now (i.e., in April) is false not only now, but also in the middle of the summer. There- fore, which proposition is expressed by a sentence depends on the context in which the sentence is used. We are also going to need a context of use to express an eternalist proposition with a temporalist sentence. The eternalist proposition is time-specific. It does not change its truth status from one time to another (within a history). The context allows to “fix” the relevant time of reference and evaluate the proposition at the same instant in all histories. Let us call instant ic, the time of the context.

9Both these ways oﬀer rather simplistic accounts of propositions. They do not consider whether propositions are structured, centered, mentally graspable, etc.

44 CHAPTER 3. OCKHAMIST SEMANTICS

Definition 3.6 (Eternalist proposition). Let φ be a sentence. The eternalist proposition E expressed by sentence φ in context c, |φ|c , is the set of histories h such that for m ∈ ic∩h, E m/h |= φ, i.e., |φ|c B {h|m/h |= φ & m ∈ ic}. We say that an eternalist proposition A is true at a history h iff h ∈ A. The semantic interpretation of sentences in Ockhamism is “temporalist.”10 Nonetheless, we can express eternalist propositions with temporalist sentences. The mere fact that sentences change their truth values from one moment to another in Ockhamism does not imply that propositions inherit this kind of variability. Whether they do needs to be decided on independent grounds. I do not intend to settle the issue and all the investigations in this book are meant to be compatible with both resolutions. Remember that the propositions are meant to be not only the semantic values of sentences, but also the objects of propositional attitudes. Therefore, the nature of propositions needs to be decided, at least in part, in broadly conceived philosophy of mind. The nature of propositions partially depends on whether the best theory of propositional attitudes takes their objects to be temporalist or eternalist entities. In any case, deciding the nature of propositions would not solve the problem of future contingents. Let moment m1 in the model depicted on page 34 be a moment of the context of use of the sentence, “There will be a sea battle.” Is the proposition expressed by this sentence true at m1? Consider first the temporalist proposition A— that there will be a sea battle. Clearly m1/h2 ∈ A, while m1/h1 < A. Thus, it is difficult to decide the truth value of A at m1. Let us now consider the eternalist proposition B—that there will be a sea battle later than m1. We have that h2 ∈ B and h1 < B. But m1 ∈ h1 ∩ h2, so it is equally difficult to say whether B is true at m1. The observation points to the most problematic feature of Ockhamism.

3.7 From semantics to postsemantics

In Ockhamist semantics, the truth value of a sentence (or a proposition) is relative to a modal parameter—a history. There is formally nothing wrong with such a relativization, but it creates an obstacle for application of Ockhamism to actual uses of future contingents. Let us take the sentence (S), “There is going to be no world war in the 21st century,” as used during 2014 Wales Summit of the NATO. How should we use the Ockhamist semantics to evaluate this sentence? Well, we need to check if it is true at a moment/history pair. So far so good, but exactly which moment and, more importantly, which history to use? After all, the Wales Summit has many possible continuations. Here the trouble begins. In Ockhamism, we use a technical notion of truth at an index (or truth at a context and an index if we include the indexical expressions). The index needs to contain a moment and a history to guarantee compositionality of semantics, to generate reasonable validities, and to induce a desirable interaction of temporal and modal operators.

10 Unless every occurrence of a sentential variable is placed in scope of a date operator Att, or in scope of operator Now, which “eternalize” the sentence.

45 CHAPTER 3. OCKHAMIST SEMANTICS

Nonetheless, we need to go one step further to apply Ockhamism to the sentence (S) used in Wales. As MacFarlane explains:

We are now deﬁning truth at a context and index (. . . ). At the end of the day, though, what we care about is truth at a context, since it is this notion, not the technical notion of truth at a context and an artiﬁcial sequence of coordinates, that has direct pragmatic relevance. (MacFarlane, 2014, p. 57)

Thus, we need to somehow relate the pragmatically relevant notion of truth of a sentence used at a particular context to a technically relevant notion of truth of a sentence at an index.

Our Ockhamist semantics gives us a deﬁnition of truth at a context and index (world/time pair) for arbitrary sentences in our language. But how can we move from this to the pragmatically relevant notion of truth at a context? A parallel problem arises for propositions. We have an account of truth relative to a world for the propositions expressed by arbitrary sentences in context. But what is it for such a proposition to be true at a context? (MacFarlane, 2014, pp. 207–208)

The easiest way to connect the truth at a context and truth at an index is to distinguish the index designated by the context (it is what I will eventually advocate in chapter6). Nonetheless, as we shall see in the next chapter, the branching theorist almost unanimously reject the easy road. They argue that the context does designate a moment, but it does not designate a history (see especially Belnap et al., 2001, pp. 151– 2, 231–3). Therefore, they face what I call the initialization failure. The Ockhamist semantics requires that the process of semantic evaluation begins at some speciﬁc index, but the context does not initialize the relevant index. To address this problem, MacFarlane(2003) introduces a new level of semantic analysis—postsemantics. Its role is to bridge the gap between the context and the index. Postsemantics dictates how to use Ockhamism to ascribe truth status to sentences at contexts. To avoid confusion, when I formally represent various positions, I use two diﬀerent symbols: ||− to indicate truth-at-context and |= to indicate truth-at-index. The simple |= is reserved for Ockhamist truth throughout my entire work. Other notions of truth at an index will be distinguished in notation.

46 Chapter 4

Semantics of Branching Realism

4.1 Metaphysical constraint of semantics

If a world is a tree, let us step into the shoes of the tree-dwellers and see how (and if) we can talk about the future. One problem with an interpretation of future tense operator is straightforward. Let us take the sentence, “There is going to be no world war in the 21st century,” as used in a branching world during the 2014 Wales Summit of the NATO. There are numerous alternative branches growing upwards from the event of the summit and each of them, on face value, is equally well-suited for the purposes of semantic evaluation and none is preferable to any other. The Ockhamist semantics requires, however, that one speciﬁc history should be chosen. It is the gist of the initialization failure. The standard method to overcome the failure was stated at least as early as in 1900 by Kazimierz Twardowski: “Circumstances accompanying speaker’s words complement what the words do not express” (Twardowski, 1900, p. 6).1 The circumstances which Twardowski refers to are usually called the context in contemporary semantics. The idea behind Twardowski’s claim is fairly simple: the context provides the appropriate circumstances of evaluation. The author discusses the sentence as an example: “It is raining.” Such a sentence is true, when used in certain times and places, but false, when used in other times and places. In most cases, however, we do not hesitate which time and place are relevant for the truth ascription. We just check if it is raining at the time and place “provided” by the context at which the sentence is uttered. Generally, to determine whether a sentence is true at a particular context, we need to check whether it is true at the circumstances provided by the context of its use.2 Why not to apply this idea to solve the initialization failure? To interpret a sentence at a context just use the circumstances provided by the context. That is, use the context

1Translation mine, original: “Okolicznosci,´ towarzysz ˛acesłowom mówi ˛acego, uzupełniaj ˛a,czego one nie wyra˙zaj˛a.” 2Actually, Twardowski preferred a slightly different procedure. He used the information provided by the context to replace the context-sensitive expression with a non-sensitive judgment which specifies all the relevant parameters (like time, place, etc.). Wolenski´ (2011, p. 39) compares the former with open and the latter with closed formulas of first-order logic.

47 CHAPTER 4. SEMANTICS OF BRANCHING REALISM to initialize the moment/history pair relevant to semantic purposes. The Branching Realists strongly object against this notion. I consider the objection to be significantly influenced by metaphysical considerations. Let us assume, together with Branching Realists, that the world is best characterized as a branching structure of concrete events. Now, let us take a specific utterance. It occurs at some specific moment on the tree. It is most natural to assume that this moment is the moment initialized by the context. We have thus solved half of the initialization problem—the moment at which the evaluation procedure should begin is initialized by the context. What about the second semantic parameter, the history? The issue is decidedly more tricky in this case. We should select one particular history, but finding a principled reason to designate any one of them poses a problem. After all, the utterance is a concrete event and it is a part of many overlapping, concrete courses of events, all of them equally real. Unlike worlds, histories overlap, so that a single speech act will typically belong to many possible histories. (Belnap et al., 2001, p. 152) [A] single utterance, together with all the most distant “facts,” belong to many histories (Belnap et al., 2001, p. 233) [T]he utterance takes place in many worlds. (MacFarlane, 2008, p. 85) Consider a concrete case in which a sentence is used. (. . . ) There will be many worlds, in general, that represent the very same past and present happenings. (. . . ) The concrete episode of use takes place in all of them. (MacFarlane, 2014, p. 208) Since the concrete act of utterance is a part of many distinct histories, we cannot select the history of a context as the history in which the utterance happen. The Branching Realists add that if an utterance is a part of many distinct courses of events, then it is unwise to distinguish just one of these as absolutely actual (see section 5.3 of chapter5 for an extended discussion of the arguments). They conclude that the history parameter simply is not initialized by the context of use of the sentence (see e.g., Belnap et al.(2001, pp. 151–152, 163–164, 232–233); John MacFarlane (2003, p. 323; 2014, p. 208); Tomasz Placek (2011, p. 756); or Thomas Müller (2014, p. 350). At this juncture, the metaphysical presuppositions of Branching Realism have a direct impact on the process of semantic analysis. Hence, the simple procedure suggested by Twardowski does not work. The context of a sentence is not sufficient to designate the appropriate circumstances of evaluation of the sentence. The content of the sentence does not seem to do the job either. The meaning of, “There is going to be no world war in the 21st century,” does not indicate which history is being referred to. However, if neither the context, nor the content initializes a history, then how shall one use the Ockhamist semantics? The situation is imperfect. The immediate route from the truth of a sentence at a context to the truth of a sentence at a semantic index is blocked. Therefore, the realists are forced to find another, less direct way to relate the notion of truth at a context and truth at an index. We shall see that the initialization failure is not fatal for Branching Realism and a number of postsemantic strategies to avert the crisis has been proposed.

48 CHAPTER 4. SEMANTICS OF BRANCHING REALISM

Some of them are quite radical. In the face of initialization failure, they abandon the safe ground of Ockhamism and introduce a modified semantics, where a world/history does not feature as an element of the semantic index (see sections 4.2, 4.3, 4.4, and 4.8). If history is not an element of the semantic index, then there is no need to initialize a history and, consequently, there is no initialization failure. These attempts are usually easier to discredit. They fiddle with the intuitive definitions of temporal and modal connectives offered by Ockhamism and, as a result, they usually generate controversial semantic consequences. In particular, one usually finds a rather unexpected specimens among the validities of these semantics (or one fails to find the expected validities). These proposals rarely turn out to be consequently convincing. If the results of a semantic theory diverge from the common sense, we tend to trust the latter, rather than the former. Of course, in a case of a very complex sentence or reasoning our intuition is often lost and then it is very useful to have a rigorous semantic machinery to set our thinking straight. However, if semantics fails to meet common sense in the basic cases, the usual objection is that the semantics models some technical notion of “will,” “was,” or “possible,” rather than their English counterparts. Therefore, it is no longer clear if the semantics can be used to elucidate the everyday notions of past, future, or historical possibility. Other theorists rely on Ockhamism and try to figure our some other way around the initialization failure (see sections 4.5, 4.7, 4.6). They usually use the semantics, but in a less straightforward manner. They resort to postsemantics to settle the relation between truth at a context and truth at an index. On the methodological level, the (post)semantic theories might be distinguished depending on their attitude to Ockhamism (acceptance or denial). On the conceptual level, they can be divided according to their attitude to relativity of truth. Some theories insist that even if the context does not initialize a single history, it is still sufficient to determine the truth status of a sentence used in the context. I divide these theories into the following categories: extremism, modalism, many-valued semantics, and supervaluationism (sections 4.2–4.5). Other kind of theories deny this. They claim that a sentence plus the context do not determine the truth status of a sentence. According to these theories, the truth value of the sentence is relative to some extra factor. A sentence at a context may be true relative to one factor, false relative to another, and truth-valueless relative to still another. I discuss the relativist theories in sections 4.6— 4.8.

4.2 Extremism

Let me begin with the most radical approach to the semantics of future contingents. It is a common view among the Branching Realists that the notion of the actual future is bogus. One needs to be careful with such statements, however, as they are likely to result in overreaction. One such, as I shall argue, inadequate reaction can be found in a recent paper by Patrick Todd(2015a). In his project, he attempts to motivate a novel treatment of future contingents which would render them all false. He oﬀers a surprisingly simple rationale for his semantic endeavor: a sentence in future tense is true, if and only if what it says happens in the actual future. But if there is no

49 CHAPTER 4. SEMANTICS OF BRANCHING REALISM actual future, then nothing happens in the actual future and any sentence in future tense is false. It certainly is an original semantic proposal, well-grounded in existing philosophical views. I am going to argue, however, that it ultimately fails. Upon closer examination it turns out that Todd’s definitions generate numerous difficulties which are hard to accept (or overcome). Many of the arguments presented below apply to more conventional accounts of future contingents. Let me first briefly recapitulate metaphysical considerations that influence Todd’s semantic decisions. The future, following the author’s argument, is contingent, if and only if it is not causally determined. Todd prefers to describe the contingent, undetermined future as the “open future.” Then, he imposes a very specific condition on the notion of the open future:

(OF) If the future is open, then there is no actual future.

The author does not motivate the condition, only stipulates it as the starting point of his investigations. One needs to be careful with claims like (OF), however, because, under one natural understating of the claim that there is no actual future, it says that nothing will actually happen—that is, that all the world’s days are run. But then, (OF) would allow us to infer that the world has just come to its end from the assumption that the future is open. It is surely not a valid inference, so (OF) must be wrong (an argument along this lines has been presented by David Lewis, 1986, p. 207). It indicates that the claim that there is no actual future requires clarification since, when taken at face value, it might lead to unwelcome consequences. I should stress that Patrick Todd does not fully embrace (OF) in the paper I discuss. At some point, he even admits that “[t]he motivation for having an open future view may be dubious” (p. 4). He just stipulates (OF) and proceeds to construct a hypothetical argument: If one wished to accept such a notion of the open future, then one could, or even should, accept the semantic proposals he puts forward. Therefore, I will assume for the purposes of reconstruction such a notion of the open future. Todd offers two theories. Let me first reconstruct Todd’s initial, ingenious proposal (I will refer to it as F1).

1. Fφ is true iﬀ 2. In the actual future, φ iﬀ 3. There is a unique actual future and in the future, φ.

If the future is open and (OF) is true, then there is no actual future. Then,3 is false in virtue of falsity of the ﬁrst conjunct. Therefore, in face of the open future, every sentence in future tense is false. Todd is ready to admit the consequence:

On the relevant semantics for “will”, something “will” happen (as a ﬁrst approximation) if and only if “the unique actual future” features the thing happening. But if there is no “unique actual future,” as open futurists contend, then (on a Russellian analysis) such a proposition simply comes out false. (Todd, 2015a, p. 2)

50 CHAPTER 4. SEMANTICS OF BRANCHING REALISM

Let me note that the future might well be open, even if some truths are settled. For the future to be open, it is sufficient that some truths are not settled and Aristotle once sanely noticed that “it is necessary that he who lives shall one day die (. . . ) But whether he dies by disease or by violence, is not yet determined” (Metaphysics 1027b, 10–14). In case the world is not completely deterministic, the future is open, and therefore it does not exist.3 Therefore, if the future is open, Todd’s claim extends to settled truths. It is settled that I will die, but the sentence which states that I will die is false according to (F1). The result of all sentences in future tense being false might look unappealing at first, but Todd argues that if it is where our philosophical precepts take us, we can do nothing but follow. I admire the Puritan spirit of this proposal. As Todd puts it himself: “On my view, when we try to talk about ‘what will happen’, we presuppose a metaphysical picture of time and the world that philosophical reflection ultimately recommends that we reject” (Todd, 2015a, p. 23). Todd describes his project as a “Russellian” approach to future contingents and he contrasts it with a “Strawsonian” approach. He alludes to the famous debate regarding the analysis of definite descriptions. On the one hand, according to Russell (1905), every affirmative sentence with a non-denoting definite description as a subject is false. On the other hand, Strawson (1950) argues that such a sentence should be considered neither true nor false. Patrick Todd(2015a) argues that the same argument can be restated for sentences which talk about the future. If there is no actual future, then every sentence of the form “The future features φ” should be considered false, on the Russel- lian approach, and indeterminate, on the Strawsonian. The supervaluational proposal of Thomason (1970) (which I discuss in section 4.5) might be seen as a paradigmatic Strawsonian account of future contingents, whereas Todd proposes a Russellian alternative. The elegant simplicity of his theory is impressive. In a sense, it is astounding that it had to wait until 2015 to be presented. Well, this is not quite true. I have identified two sources, where a view like (F1) is mentioned. Firstly, it is considered (but instantly discarded) by David Lewis:

It is false that the future holds a sea ﬁght; because “the future” is a denota- tionless improper description. (. . . ) But [if we go this way], our customary thought about “the” future is in bad trouble. (Lewis, 1986, p. 207)

A sketch of a theory like (F1) can be also identiﬁed in a few remarks by Michael Tooley(2012):

First of all, there is what might be called “extreme” presentism, where this is the view that any positive proposition that is expressed by some statement about the past, or about the future, however that statement is interpreted, is false. (Tooley, 2012, p. 26)

Tooley does not develop this view at any length, only brieﬂy comments:

3One might be inclined to say that “parts” of the future that are determined do exist. The insight is incorporated into the second of Todd’s theories.

51 CHAPTER 4. SEMANTICS OF BRANCHING REALISM

[E]xtreme presentism with its radical view that all positive propositions about the past (and about the future) are false, has not recommended itself to many philosophers. (Tooley, 2012, p. 26)4

(F1) is not, strictly speaking, a case of extreme presentism in Tooley’s sense, as Todd limits his investigation to future tense. Anyhow, it is fair to describe his initial proposal as extremism. Not only does it partly coincide with Tooley’s extreme presentism, but, more importantly, advocates a quite extreme revision of the future talk of English (it might be what Lewis had in mind, when he warned against “bad trouble”). Russell rightly suggested that “[a] logical theory may be tested by its capacity for dealing with puzzles, and it is a wholesome plan, in thinking about logic, to stock the mind with as many puzzles as possible, since these serve much the same purpose as is served by experiments in physical science” (Russell, 1905, pp. 484–485). In what follows, I am going to present a short list of puzzles that extremism should solve. In my opinion, these “logical experiments” jointly constitute a serious challenge for this view. To examine the details, let us assume then that the future is open (and thus, all F-sentences are false) and study how sentences in future tense behave in various contexts.

1. In (F1), sentences resembling tautologies come out false. Let us take the sentence:

I will have a cup of coﬀee tomorrow or I will not. (F1 p ∨ F1¬p) Patrick Todd realizes that this kind of cases might be problematic for his proposal. Therefore, he treats them with special care. In the end, he adopts the tactics originally devised by Russell himself. He argues that the sentence above is ambiguous and it is false only under one of its readings (just as the sentence, “The present king of France is bald or not,” is false under one of Russellian readings). “I will have a cup of coﬀee tomorrow or I will not” is false, only if we understand it as F1 p ∨ F1¬p. Unfortunately for Todd, elsewhere (p. 6) he assumes that F1¬ is the default reading of “will not.” Therefore, according to Todd, we should naturally take the sentence above to be false, while most of us would consider it true. It also implies, by De Morgan’s law, that we need to accept as true a sentence that sounds pretty much like a contradiction:

It is not the case that there will be a sea battle, but it is not the case that there will be no sea battle (¬F p ∧ ¬F¬p).

Todd rightly stresses (pp. 20–21) that, given the logical form he proposes, “will φ”(Fφ) does not contradict “will not φ”(F¬φ). So, strictly speaking, Fφ ∨ F¬φ is not a case of the law of excluded middle and it is logically consistent to say, as he does, that both elements are false. To defend his account, Todd could in principle argue that the default reading of “will not” is overruled in the example above and that for some reason the logical form of the sentence considered above

4Unfortunately, Tooley does not state the names of the precious few philosophers to whom extreme presentism recommended itself.

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is F p ∨ ¬F p. This sentence is always true in Todd’s semantics and it ﬁts our intuitions. However, it is crucial for this strategy that F¬φ and ¬Fφ are carefully distinguished, which brings me to the following problem. 2. According to (F1), the English phrase “will not” is ambiguous. More speciﬁcally, there is a syntactic scope ambiguity of the sentences like

(a) I will not drink coffee tomorrow. It can be understood in either of these two ways: (b) It will be the case tomorrow that I do not drink coffee. (c) It is not the case that it will drink coffee tomorrow.

However, as John MacFarlane notices, this kind of ambiguity does not seem to be present in the English “will not”: It is striking, though, that although we can mark the syntactic distinction by resorting to cumbersome circumlocutions, as in (b)–(c), these variants seem like different ways of saying the same thing. If you ask somebody who utters (a) whether they meant (b) or (c), you are likely to be met with a blank stare. (MacFarlane, 2014, p. 216) Therefore, if MacFarlane is right, the users of English do not recognize the two meanings of “will not” that Todd is forced to stipulate. It seems, however, that the difference should be easily detectable, given that every sentence of the form F¬ is false, while every sentence of the form ¬F is true. 3. Let us reflect for a moment on the temporal operator “it is always going to be the case” (G). It is commonly introduced as a dual of F (G = ¬F¬). But then, since every sentence of the form Fφ is false, then every sentence of the form ¬Fφ is true. Therefore, (F1) implies that every sentence of the form “It is always going to be the case that φ” is true. In a sense, it should not come as a surprise. If Fφ means something like “some moment in the actual future features φ,” then Gφ means something like “all moments in the actual future feature φ.” However, since there are no moments in the actual future (because there is no actual future), then the latter claim is vacuously true. Thus, according to (F1), it is false that I will drink another coffee. (F p) But it is true that

I am always going to be drinking coffee. (Gp) Also, a short reflection is sufficient to realize that, if we use operators to encode tense, then ¬F is more plausible than F¬ as the form of English “will not.” When I say that I will not eat meat, I mean that at no future occasion will I eat meat

53 CHAPTER 4. SEMANTICS OF BRANCHING REALISM

rather than that at some future occasion I will not eat meat. (Also, Todd should endorse this reading of “will not” to solve problems1 and7.) But if this is right, then on (F1) all the “will not” sentences come out true. Then, it is true that I will not eat meat and it is also true that I will not eat anything else. 4. (F1) falsifies the basic principle of the majority of tense logics: φ → HFφ. The operator H stands for “It has always been the case that.” Observe that in (F1), even if φ is true now, the sentence Fφ was once false (assuming that what has in fact happened, might not have happened). Therefore, even if φ is true, HFφ is not. According to (F1), it means that, even if I do drink coffee right now, it was false to say that I would. This stands in conflict with how we usually assess our predictions. If you said that I would drink coffee today and I do, then it seems that I am entitled to say that what you said was true (or, more commonly, that you were right).5 Also, the principle φ → HFφ, together with its dual φ → GPφ, are meant to grasp an elementary symmetry between temporal concepts. They encode the idea that the present is in the past of the future and in the future of the past (in relational semantics, these two sentences guarantee that the accessibility relation for operator F is the converse of the relation for operator P). (F1) violates this idea in at least one direction—the present is no longer in the future of the past. 5. (F1) also generates problems for analysis of speech acts. Many of them seem to be systematically related with the truth value of the sentences used in the acts. The paradigm example is the act of betting. If α bets that φ, then α wins, only if φ is true. (For a detailed discussion of speech acts in the context of open future, see Belnap, 2002b). If this understanding of betting is sound, then, if (F1) is accepted, then no bet can ever be won. After all, if the future is open, whenever you say “Eclipse will come first,” your sentence is false (and it will remain false forever). Therefore, even if you bet on Eclipse and Eclipse does in fact come first, you still stand no chance, when bargaining with the bookie about your payoff. Actually, Todd refers (p. 9) to an argument by Arthur Prior to the effect that betting is problematic for an open futurist. He uses this argument to distinguish his version of open futurism from Prior’s. However, he does not restate this argument against his own proposal, nor does he make any attempt to rationalize betting behavior within his account of future tense.

6. Let us introduce one more connective, “it is determined that” ( D ) expressing causal necessity. We say that D φ is true, iﬀ φ is true in all causally possible futures. Remember that even if D φ is true, the future might still be open. It is enough that some other aspect of the future is indetermined. Then, if the future is open, (F1) breaks the natural connection of “will” (F) and “determined” ( D ). For example, even if φ is causally necessary to happen, it is still false that it will happen. So, the following implication is false:

5I discuss the so-called retrospective accuracy assessment in sections 4.5 and 4.6.

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If it is determined that I will drink a cup of coffee tomorrow, then I will. ( D p → F p) 7. Since all sentences in future tense are false, then “I will drink a cup of coffee tomorrow” and “I will not drink a cup of coffee tomorrow” are both false. Todd is well-aware of that and he is ready to accept it. However, he might have not noticed that it means that these two are (materially) equivalent. Thus, if the future is open, then it is also true that: I will drink coffee tomorrow, if and only if I will not. (F p ↔ F¬p) (Of course, the alleged scope ambiguity might come to the rescue here.)

8. Every (material) implication with an antecedent in future tense is true. Therefore, it is true that

• If I drink coﬀee tomorrow, then I will be the king of France. (F p → Fq). • If I drink coﬀee tomorrow, then it is determined that I will not. (F p → D ¬p).

Perhaps none of the arguments above is decisive. When accumulated though, they constitute a rather strong case against (F1) as a semantics of the future tense. This result might serve as a warning to take cum grano salis the slogans like “there is no (actual) future.” To my mind, a proponent of (F1) has two solutions open to him. He can bite the bullet, endorse the error theory and insist that people are massively confused when they use the future tense. This strategy has not recommended itself to many philosophers indeed. Even Todd admits that “[t]his is not an easy philosophical road to walk” (p. 23). This is not to say that a philosopher has nothing to say about the most fortunate semantics of future tense. It might be that our ordinary attitude towards “the future” is tangled and a philosopher can help to clarify, or even regiment, our way of talking. However, when a philosophical theory (and a semantic theory in particular) gets as remote from common usage as (F1) does, then it is most likely that the concept of the future encoded by this theory is very distantly related to our every-day concepts. It leaves us with the other solution, which is to admit that (F1) models a technical sense of “will” used, for example, by extreme presentists. Then, (F1) might be useful, if the purpose was to study the linguistic niche of these philosophers. However, it will not teach us much about how people actually do, or should, think about notions like “the future.” None of these two solutions seems particularly appealing and both might discourage one from endorsing (F1) (they certainly discourage the person writing these words). However, I still believe that any of these two solutions are better than the route described below that Patrick Todd has actually chosen. In the face of the mounting technical diﬃculties,6 Todd has relaxed his view. He gave up some of his philosophical chastity and acknowledged that some sentences

6I am not sure which of the problems I have discussed above presented themselves to Todd. In his paper, he writes (pp. 18–19) that he was particularly discouraged by the result that I described in point6.

55 CHAPTER 4. SEMANTICS OF BRANCHING REALISM about the future should be true, even if there is no future. Specifically, he makes an exception for the sentences about the future events which are causally determined to happen. For example, he admits that the sentence like “I will die” is true. To convey this insight technically, the author proposes a modification of the definition of future tense (F2):

It will be the case that p iﬀ there exists a unique actual future, and that future features p, OR p is true in all causally possible futures. (Todd, 2015a, p. 19)7

Since all causally possible futures feature my death, the sentence “I will die” is true. Thus, the semantic maneuver allows the author to generate the result he desired. The semantic shift suggested by Todd marks a transition from “extreme” to “austere” form of semantics, to use Tooley’s (2012) terms once again. In the austere version of presentism, we can reasonably talk about the future as long as it is in some sense present, for example, if it is “present in its causes.” I understand the “linguistic” motivation to escape from the dubious theory (F1). However, the ﬁx proposed by Todd reduces rather than increases the philosophical allure of his position. To be fair, (F2) does avoid some of the concerns pointed out above. To be exact, the problems3 and6 do not threaten this position. However, the remaining problems are not answered by this semantic change (these problems apply to any φ that is not causally determined to happen). Worse even, the transition from (F1) to (F2) generates new semantic oddities. For example:

9. If it is causally determined that I will have one more cup of coﬀee today, but it was not causally determined two hours ago (e.g., I could have taken a nap an hour ago), then it is true to say:

I will drink a cup of coﬀee today but an hour ago it was false that I would. (Fφ ∧ P¬Fφ) In fact, Todd does not ﬁnd this sentence as peculiar as I do. He even tries to justify why we should consider it true in a recent paper (Todd, 2015b).

10. Whenever it is determined that I will have either a cup of coffee or a cup of tea, but it is indetermined which one, I can truly say that: I will drink coffee or tea, but it is not the case that I will drink coffee or that I will drink tea. (F(φ ∨ ψ) ∧ ¬(F p ∨ Fq))

To my ear, the sentence above sounds very much like a contradiction. The intuition becomes even clearer, when restated as: I will drink coﬀee or tea, but I will not drink coﬀee and I will not drink tea. (F(φ ∨ ψ) ∧ ¬Fφ ∧ ¬Fψ)

7Incidentally, Malpass and I (see Malpass and Wawer, 2012, p. 132) have proposed a very similar deﬁni- tion of the future tense operator. I present it in section 5.3.5.2. The argument presented below does not apply to our view.

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I ﬁnd it utterly bizarre that this sentence should ever be true. However, I have to admit that Todd may attempt to explain away the truth of the last sentence by once again evoking the scope ambiguity. 11. The next problem with (F2) is that nothing is ever true in this theory without being causally determined. It means that it is never true that:

It is not determined that John will drink another coﬀee today, but he will. (¬ D p ∧ F p)

It is a matter of some controversy whether these kind of sentences should ever be considered true (see MacFarlane, 2014, pp. 215–216). However, it is worth noticing that they always come out false in (F2). (In fact, they come out false also in (F1), but for diﬀerent reasons.)

On top of all of all of those, there is a methodological problem with (F2). Namely, the second disjunct in the deﬁnition of “will” seems to be added entirely ad hoc, just to explain away some controversial consequences. I hope that by now it is clear that the switch from (F1) to (F2) is not as good a deal as it might seem.8 We do get rid of two controversial cases, but we generate two new ones. Even if this new semantics is slightly better, it not much better than the previous one. Thus, the meta-philosophical reasons that might discourage us from accepting (F1), apply to (F2) as well. Things, however, get even worse. . . Remember that Todd has accepted (OF), which says that if the future is open, then there is no actual future. Bearing this in mind, let us muse a little more on the modiﬁed truth condition of the future tense operator:

Fφ is true iﬀ there exists a unique actual future, and that future features φ, OR φ is true in all causally possible futures.

I shall encode this condition symbolically:

Fφ iff (@Fφ or D φ) Let us now consider two scenarios. First, assume that the future is open. This implies that there is no actual future. It implies in turn that the first disjunct in the definition above is false and Fφ inherits the truth value of the second disjunct. Thus, if the future is open, we arrive at a simplified definition of truth:

Fφ iff D φ Now, let us assume that the future is not open. Then, there is only one possible way in which the world can develop. Todd admits that in this case the actual future exists and it is simply identical to the unique, causally possible future (see Todd, 2015a, p. 11). But then, doubtlessly, whatever the actual future features, it is determined to happen. Thus, in case the future is not open, we arrive at the same simplified definition:

8It deﬁnitely seemed a good deal to Todd, who writes “I am deeply grateful to Andrew Bailey for sug- gesting this disjunctive approach, thereby saving me from a great many complications” (Todd, 2015a, p. 19, n. 28).

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Fφ iff D φ Therefore, no matter if the future is closed or open, the truth conditions for the future tense operator are exactly the same. The first disjunct in definition (F2) makes no difference and Fφ and D φ are necessarily equivalent. The equivalence can be derived on the basis of the definitions (OF) and (F2), so the identity of F and D is not only necessary, but also analytic. We can put this point differently, if we notice that on the open future view the following implication holds: ¬ D φ → ¬@Fφ. It says that if φ is not determined to happen, then the actual future does not feature φ. Todd admits that much when he says that “given indeterminism, there does not now exist a complete ‘story of the future’ ” (Todd, 2015a, p. 2). We can now formally restate the previous argument as:

Fφ ↔ (@Fφ ∨ D φ) (F2 deﬁnition of “will”) ¬ D φ → ¬@Fφ (from the stipulated meaning of OF) Fφ iﬀ D φ

Thus, it is clear that the first disjunct is just a smokescreen. It is redundant for any open futurist in the style of Todd. The classical logic and Todd’s definitions are sufficient to establish equivalence of “will” and “determined.” Therefore, we can safely conclude that in (F2), “It will be the case” means “It is determined to be the case.” However, if we identify these two, we simply end up with the good old Peircean semantics of Arthur Prior (to be discussed in the following section). Of course, it is not a sin to adopt (even unknowingly) a semantic devised by such an illustrious philosopher. Todd’s major problem is not that the semantics he devised is not entirely novel, but that has himself wants to distinguish his view from Priorian Peirceanism:

Hartshorne and Prior showed that one could have an open future without denying bivalence, given (at least what most will regard as) a rigged, causally-loaded semantics for the future-tense “will,” according to which to say that something will happen is (roughly) to say that it is determined to happen. However, I aim to show that one can have such an open future without adopting these semantics. (Todd, 2015a, p. 3)

Given the argument above, Todd fails to fulfill his promise. Doubtless, his initial proposal (F1) does achieve his aim. It does distinguish “plain” future tense from “causally-loaded” future tense. In Todd’s initial proposal, the sentence “I will die” is false, while it is true in Peirceanism. However, Todd decided to give up (F1) as an analysis of “will” and accept another semantics—(F2)—that he finds less problematic. Closely examined, however, (F2) turns out to be nothing but Prior’s Peirceanism. Ultimately, the author has sacrificed the novelty of his proposal for the sake of a slightly more intuitive explanation of a few controversial cases. At the same token, he failed to achieve his primary aim, which was to propose a semantics which would (a) render all future contingents false, and (b) distinguish “plain” future tense from “causally loaded” future tense. Therefore, I would suggest Todd to reject the trade-offs and stick to (F1). In fact, I believe that he could philosophically support the result that scared him away from (F1). After all, if there is no actual future, the actual future does

58 CHAPTER 4. SEMANTICS OF BRANCHING REALISM not feature my death, so the full-ﬂedged Russellian open-futurist in the style of Todd can easily explain, why it is false that I will die. To sum up, none of the semantics of “will” recently proposed by Patrick Todd is a reasonable analysis of the English future tense. Moreover, I claim that the proposal he ultimately recommends, on closer inspection, turns out to be a version of the semantics which he wants to reject. In light of these results, it would be better to either abandon his semantic project altogether, or stick to his initial proposal, as it might be useful for some theoretical applications in philosophy of time.

4.3 Modalism

I will now discuss a more moderate approach to the idea that there is no actual future. The idea is to reinterpret the “plain” future in modal terms, (“in an indeterministic context, factual statements are covertly modal” McArthur, 1974, p. 283). If so reinterpreted, the sentence no longer refers to actual future and we can ascribe to it a history-independent truth value. I will use modalism as an umbrella term to cover a family of such theories. McArthuro ffers two modalist interpretation of the operator F: • It may be the case • It must be the case Both these translations share the common feature: the truth value of so understood future operator is independent of the choice of a history and the initialization failure never arises. Let us consider the first of McArthur’s proposals and identify “will” with “may” understood as “possibly-will.” I call this view “possibilism.” Definition 4.1 (Possibilism). 0 0 m|=Fφ iff ∃m0 (m < m & m |= φ) 0 0 m|=Pφ iff ∃m0 (m < m & m |= φ) In a sense, possibilism is the most natural choice from the semantic viewpoint, since it treats the branching structure hM, ≤i as a proper Kripke frame of temporal logic. In ordinary temporal frames, we take the relation which orders the structure to be a relation of temporal precedence and we treat this relation (and its converse) as accessibility relations of temporal operators. Nonetheless, if we continue to use the tree to represent modal reality, we arrive at the conclusion that whatever may happen, will happen.9 The consequences of such identification are intuitively unacceptable. It was recognized already by Arthur Prior (1967, pp. 53–54), who was cautious not to understand the possibilist future operator f as the future tense of English. To exemplify but a few problems with this semantics, let us take p to stands for “There is a sea battle” and q for “There is a land-battle.” Then, under the possibilist reading of “will” we can conclude that at some moments of some models the following sentences are true:

9Prima facie, it resembles the so-called Diodorean deﬁnition of possibility, see (Rescher, 1968), but the idea is substantially diﬀerent.

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1. There will be a sea battle tomorrow and there will not be a sea battle tomorrow (F1 p ∧ F1¬p, thus the implication Fn¬p → ¬Fn p is not valid). 2. There is a sea battle, but yesterday, there was going to be no sea battle today (p ∧ P1F1¬p). 3. There was going to be a sea battle, but it has never happened and never will happen. (PF p ∧ ¬Pp ∧ ¬p ∧ ¬F p). 4. Once, an endless sea battle will begin and once, an endless sea-peace will begin (FGp ∧ FG¬p). 5. There will be a sea battle and once, there is always going to be the case that there has been no sea battle (F p ∧ FG¬Pp). 6. There will be a sea battle and there will be a land-battle, but the sea battle will be neither in the past nor in the future nor in the present of the land-battle (F p ∧ Fq ∧ ¬F(p ∧ Fq) ∧ ¬F(q ∧ F p) ∧ ¬F(p ∧ q)).

I hope that these few examples demonstrate conclusively that the possibilist “will” has little to do with the future tense of English. Possibilism may be used, perhaps, to study the temporal logic of Naïve Branching Realism or a simplified logic of the “trousers-like” spacetimes of general relativity that Earman writes about, but it does not grasp the common sense temporal relations. In the previous section, I mentioned the more promising modalist alternative— Peirceanism. This semantics has been proposed by Arthur Prior(1967) to grasp the notion of the future he reconstructed from the writings of Charles Sanders Peirce This corresponds (. . . ) to C.S. Peirce’s description of the past (with, of course, the present) as the region of the “actual,” the area of “brute fact,” and the future as the region of the necessary and the possible. (Prior, 1967, p. 132) The gist Prior’s the semantic definition is easy enough, a sentence in future tense talks about what is predetermined to happen or, as he concisely puts it, “ ‘Will’ here means ‘will definitely’ ” (Prior, 1967, p. 129). Definition 4.2 (Peirceanism). P 0 0 0 P m|=Fφ iff ∀h(m ∈ h ⇒ ∃m0 (m ∈ h & m < m & m |=φ)); P 0 0 0 P m|=Gφ iff ∀h(m ∈ h ⇒ ∀m0 (m ∈ h & m < m & m |=φ)); P 0 0 P m|=Pφ iff ∃m0 (m < m & m |=φ). The operators F and G need to be independently introduced, since they are not duals.10 It means that the equivalence F p ↔ ¬G¬p is not valid. If p happens in some, but not all continuations, then F p is false, while ¬G¬p is true. In contrast, its mirror

10The operator dual to F is sometimes called g. The sentence gφ is true at m iff φ is true at all moments later than m in some history passing through m. The dual of G is f and f φ is true at m iff φ is true at some moment later than m of some history passing through m. The operator f can be identified with the possibilist sense of “Will.”

60 CHAPTER 4. SEMANTICS OF BRANCHING REALISM image, Pp ↔ ¬H¬p, is valid, because if something happened in one past, it happened in all pasts. It means that Peirceanism sharply distinguishes the past from the future and it has a way of expressing that the future, contrary to the past, is open. Nevertheless, Peircean semantic is open to numerous objections. First of all, I agree with Todd that Peirceanism relies on intuitively suspicious, “rigged, causally- loaded semantics for the future-tense.” We clearly distinguish the future from the necessary future and “will” follows the former rather than the latter.11 Due to the spurious identification, Peirceanism generates a number of unsettling results. Specifically, the arguments 1, 2, 4, 5, 7, 8, 9, 10, 11 discussed in the previous section, all apply to Peircean semantics. In fact, many of them have already been noticed by Prior (1967), but he was so hostile to the notion of the actual future that he was nonetheless inclined to accept Peirceanism over Ockhamism. In my opinion, these arguments combined seriously undermine the feasibility of Peirceanism. In fact, I found only one defense of this semantics, as applied to English, in a paper by Alan Rhoda(2006). However, the author does not rebut a majority of the linguistic oddities mentioned above. His argumentative strategy focuses on the metaphysically motivated reasons to embrace Peirceanism (he supports the move with the contention that there is no future). Moreover, the author seems to mistake the pragmatic phenomena accompanying the speech act of assertion for the semantic phenomena indicating the meaning of future tense. It is a common tendency in the literature on future contingents, so let me briefly comment on the mistake. An act of assertion does indeed seem to require more than simple truth to be correct. In particular, the assertor should have solid grounds to make an assertion. One might even argue that the grounds must be so solid that they guarantee that the sentence asserted is true and they exclude any alternative possibility.12 If this was correct, then one could assert “There will be a sea battle tomorrow” only if one were in a position to also assert “It is settled that there will be a sea battle tomorrow.” I believe, however, that this phenomenon should be explained on the level of pragmatics of asserting rather than on the level of semantics of tenses. Even if the separation is frequently difficult, the pragmatic and semantic considerations should be separated. It means that an extensive focus on assertion might be misleading. First of all, examples 1–12 in the previous section indicate that the future tense does not have the strong modal meaning when embedded in scope of other connectives (for example, it is not modally loaded in construction like F1φ ∨ F1¬φ). Surprisingly enough, even Michael Dummett concludes that “this compels us to make the sharpest possible distinction between the condition for the truth of a sentence and that which entitles a speaker to make an assertion” (Dummett, 1976, p. 52). Second of all, “will” does not function like “necessary-will” when used in acts other than assertion (see e.g., Belnap et al., 2001, p. 160). I touched upon this issue discussing problem 5 of the preceding section. Consider Eclipse, who is about to run in a horse race. She is a well-known underdog but, being a risk taker, I say:

11Interestingly, a similar kind of reasoning persuaded theoretical computer scientists to abandon Peirceanism in favor of Ockhamism, (cf. Gabbay et al., 1994, pp. 5–6). 12Nevertheless, if we required such solid grounds, hardly anyone could hardly ever make any assertion, about the future or otherwise.

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1. I bet that Eclipse will come first. and now compare my bet with its modalistic variants: 2. (Peirceanism) I bet that it is settled that Eclipse will come first. 3. (Possibilism) I bet that it is possible that Eclipse will come first. It is clear that bet 1 is significantly different from the remaining two. If, against all odds, Eclipse in fact does come first, then whomever made bet 1 wins, while whomever made bet 2 looses. If Eclipse does not overcome the odds and comes third, then one looses bet 1, but one might still win bet 3.13 No bookie would be much impressed if I demanded the payoff after the lost race on the ground that it was possible that Eclipse would win. He would be as unimpressed as I would be upset if he denied me the money after Eclipse’s victory on the ground that the victory was not necessary. It means that neither I nor the bookie understand “will” as “possibly-will” or “necessarily-will.” A similar kind of objections can be rephrased in terms of other propositional attitudes like hopes, desires, fears, guesses, or conjectures. For example, when I hope that Eclipse will come first, I hope for something very different than when I hope that it is settled or possible that she will come first. It seems that assertion is to a large degree exceptional. There is one more reason to distinguish F from F and ^F. Compare three sentences uttered just before a fair coin is being tossed: (i) The coin will land heads up. (ii) The coin will possibly land heads up. (iii) The coin will necessarily land heads up. Let us take John who knows that the coin is fair (equally likely to land both ways). Consider how strongly John believes in what is expressed by sentences above. It is quite clear to me that on a scale from 0 to 1, John’s degree of belief that (i) is about 0.5, John’s degree of belief that (ii) is almost 1, and John’s degree of belief that (iii) is almost 0. It shows that John’s cognitive attitude towards what is expressed by (i)–(iii) is different in each case, which indicates in turn that he does not equate the meaning of these four sentences. Neither do I. I close the discussion of modalism convinced that it is not a viable candidate for a semantic analysis of future tensed expressions. It prevents initialization failure, but the benefit is far-outweighed by the costs incurred.

4.4 Many-valued semantics

Another way to tackle the initialization problem is to incorporate Łukasiewicz’s formal insight into the branching setting and introduce yet another history-independent semantic theory. Łukasiewicz has developed his formal machinery long before the treelike

13Also, the epistemic procedures required to settle these bets are crucially different. In case of the first bet, it is enough to wait and see which horse comes first, while the remaining two require different kind of investigation altogether.

62 CHAPTER 4. SEMANTICS OF BRANCHING REALISM model was first used for semantic purposes, so let me briefly sketch his original proposal, before I adopt his theory to suit the branching representation. Łukasiewicz got interested in the foundational issues in the philosophy of logic quite early in his philosophical career. In 1910, he wrote a book entitled On the Prin- ciple of Contradiction in Aristotle and an article devoted to the law of excluded middle (see Łukasiewicz, 1910a,b). Part of the reason why Łukasiewicz was so interested in these fundamental logical principles might have been his conviction that they imply determinism. He expressed such a conjecture in 1910 (Łukasiewicz, 1910b) and explicitly argued this point in 1913 (Łukasiewicz, 1970a, pp. 35–37). The notion that the classical logic has deterministic consequences has probably been instilled in Łukasie- wicz by (his reception of) Aristotle’s De Interpretatione. Nonetheless, Łukasiewicz was unwilling to accept the deterministic worldview. He was driven by a firm incompatibilist belief that determinism precludes creativity and free human action. He passionately defended human freedom against the threat of physical and logical necessity in 1922, in his famous lecture “On determinism” (first published almost four decades later). The incompatibilist position, conjoined with his conviction that classical logic presupposes determinism, led Łukasiewicz to question the foundations of classical logic, especially bivalence. He preferred to sacrifice the basic principles of logic than human creativity and freedom. He was not the only logical revolutionary living in Lvov during that period. His student, Tadeusz Kotarbinski´ , wrote an article arguing that the law of bivalence is not universal (Kotarbinski´ , 1913). He offered two arguments to the effect that the law does not apply to propositions regarding the portion of the future that might be influenced by human actions. The similarity between ideas presented by Kotarbinski´ and those later defended by Łukasiewicz is striking. It is a matter of some controversy who inspired whom in this respect. Doubtlessly, Kotarbinski´ has been influenced by Łukasiewicz’s critical assessment of classical logic in general, but it is unclear how many of the specific arguments presented in (Kotarbinski´ , 1913) had earlier been advocated by Łuka- siewicz.14 However, Kotarbinski´ soon abandoned the idea (convinced by Lesniewski’s´ critique, I briefly recapitulate their debate in section 6.5), while Łukasiewicz took it one step further. In 1917–20, he transformed the general notion that the future contingents are neither true nor false into a specific formal system. A system that postulates three 1 logical values: the true (1), the false (0), and the possible ( 2 ). Then, he interpreted logical connectives as functions defined on the 3-element set. His definitions can be represented by table 4.1.

1 → 0 2 1 ¬ 0 1 1 1 1 1 1 1 2 2 1 1 2 1 1 0 2 1 0 Table 4.1: Łukasiewicz’s 3-valued logic.

14 Grodzinski´ (1989, p. 39) and Malinowski(2007, p. 17) suggest that Kotarbi nski´ ’s ideas were a source of inspiration for Łukasiewicz, while Surma(2012, p. 101) o ﬀers a range of arguments to the contrary. The material on the subject is too scarce for the matter to be decisively resolved. See (Wolenski´ , 1990, pp. 194–5) for a detailed, historically plausible reconstruction.

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Łukasiewicz does not elaborate on his choice of these particular truth functions as semantic interpretations of implication and negation. He just mentions that his decision was based on “detailed considerations, which were more or less plausible to me” (Łukasiewicz, 1970d, p. 166). An interesting rationalization of the table has been offered by Alasdair Urquhart(2001), who argues that the truth table is natural, if we think 1 about Łukasiewicz’s truth values as sets of classical truth values: 1 = {T}, 2 = {T, F}, and 0 = {F}. Each set represents a range of classical truth values that a given proposition can take in the future. Then, to compute the truth value of a complex proposition, we examine all classical combinations of the truth values of its components, e.g., if φ has the truth value {T, F} and ψ, the truth value {T}, then the truth value of φ → ψ is {T} because a classical implication with a true consequent is true regardless of the truth value of the antecedent. The truth table computed according to this method (see table 4.2) closely resemble the table above.15 In fact, the only diﬀerence is in the center of

1 → 0 2 1 ¬ 0 1 1 1 1 1 1 1 1 2 2 2 1 2 1 1 0 2 1 0 Table 4.2: Urquhart truth table. the truth table of implication. According to Urquhart, the implication of two possible sentences should be possible. Let us consider a specific example. Let us say that I eat either a doughnut or a croissant and I drink either tea of coffee every morning. The choice of food does not strongly correlate with the choice of beverage; each of the four combinations happens every now and then. Let us now consider a sentence “If I drink coffee tomorrow, I will eat a doughnut.” According to Urquhart’s procedure, this sentence is possible. It might turn out true (if I have tea or coffee+doughnut), but it also might turn out false (if I have coffee+croissant). 1 1 Why would Łukasiewicz disagree and require that the implication from 2 to 2 should be true? Observe that under Urquhart’s semantics, arbitrary sentence expressed by means of negation and implication, whose all basic components have the truth value 1 1 2 , also has the truth value 2 . This implies that no sentence of this language is true regardless of the valuation, i.e., the set of validities is empty. Urquhart conjectures that the Polish logician wanted to avoid this consequence. In particular, Łukasiewicz seem to have been convinced that the sentence φ ↔ φ should always be true. There is another reason that might have supported Łukasiewicz’s decision. In 1922 (see Łuka- siewicz and Tarski, 1970, p. 140, n. 15), he extended his semantics to account for an infinite and all finite-valued logics. Let v(φ) be the truth value of φ. Łukasiewicz’s idea can be expressed by the following, general pattern:  1, if v(φ) ≤ v(ψ),  v(φ → ψ) =  v(¬φ) = 1 − v(φ).   (1 − v(φ)) + v(ψ), otherwise. This pattern induces Łukasiewicz’s 3-valued table as a special case. Therefore,

15Incidentally, it is the truth table of Kleene’s strong implication.

64 CHAPTER 4. SEMANTICS OF BRANCHING REALISM theoretical uniformity supports Łukasiewicz’s choice over Urquhart’s. The remaining standard operators can be deﬁned in Łukasiewicz’s theory as abbre- viations: (φ ∨ ψ) ↔d f ((φ → ψ) → ψ), (φ ∧ ψ) ↔d f ¬(¬φ ∨ ¬ψ), (φ ↔ ψ) ↔d f ((φ → ψ) ∧ (ψ → φ)). When, we compute the functions induced by these deﬁnitions in 3-valued case, we arrive at the functions represented in table 4.3.

1 1 1 ∨ 0 2 1 ∧ 0 2 1 ↔ 0 2 1 1 1 0 0 2 1 0 0 0 0 0 1 2 0 1 1 1 1 1 1 1 1 1 2 2 2 1 2 0 2 2 2 2 1 2 1 1 1 1 1 1 1 0 2 1 1 0 2 1 Table 4.3: Truth tables for disjunction, conjunction, and equivalence.

It is crucial to stress that all the connectives of this logic are entirely extensional, i.e., the truth value of the composed sentence functionally depends on the truth values of its components. Łukasiewicz never questioned this property, even when he was fully aware of the intentional systems of C. I. Lewis or H. von Wright.16 Łukasiewicz does not abandon extensionality, even when he extends his logical system to include modal notions like “possible” and “necessary.” In fact, as he explains in (Łukasiewicz, 1970d, pp. 154–164), he introduced the third truth value, partly because he realized that no truth function defined on the set of only two values can interpret an operator that would verify some basic modal claims (i.e., ¬^φ → ¬φ, ¬φ → ¬^φ [sic], and ∃φ(^φ ∧ ^¬φ)). The third truth value allowed him to introduce a truth function that interprets modalities considerably better than any of the two-valued truth functions. Łukasiewicz, inspired by Alfred Tarski’s remark from 1921, defines the notion of possibility in terms of implication and negation as ^φ ↔d f (¬φ → φ). Then, he introduces as the dual of ^, := ¬^¬ (Łukasiewicz used M and L to indicate possibility and necessity, respectively). This allows modalities to be defined in terms of the truth functions illustrated by table 4.4.

φ ^φ φ 0 0 0 1 2 1 0 1 1 1

Table 4.4: Truth tables for modalities.

Notice that the modalized fragment of the language is bivalent. It sounds reason- 1 able, if the truth value of φ is 2 , i.e., the possible, the truth value of ^φ should be 1, and the truth value of φ should be 0.17 Nonetheless, defining modalities in extensional terms sounds almost paradoxical from the contemporary perspective. Łukasiewicz’s idea is much more comprehensible, 16Wolenski´ (1989, p. 271–2) explains that such an attitude was common among the key figures of the philosophical Lvov-Warsaw school. Lesniewski´ was particularly hostile towards intensional contexts and he probably influenced Łukasiewicz, Kotarbinski,´ and Ajdukiewicz (Wolenski´ , 1989, p. 145). 17Interestingly, this intuitive result does not generalize to more-than-three valued logics build according to Łukasiewicz’s pattern. Let us take an infinitely valued logic whose truth values are the rational numbers in [0,1]. Given that ^φ is an abbreviation for ¬φ → φ, we can compute the semantic value of ^φ as follows:

65 CHAPTER 4. SEMANTICS OF BRANCHING REALISM however, if we remember that he thought about the truth values themselves in modal terms. He repeats on numerous occasions, from 1913 (Łukasiewicz, 1970a) through 1956 (Łukasiewicz, 1957), that a sentence is true only if what it says is necessary and it is false only if what it says is impossible. Hence, in Łukasiewicz’s thought, truth and falsity are modally charged. The third truth value also has a modal character. After all, Łukasiewicz even calls it “possibility” (in Łukasiewicz, 1970b, he referred to it as “indeterminacy”). It represents the modal territory stretching between the impossible and the necessary. The truth tables presented above partially conﬁrm the thought. Observe that φ is true if and only if φ is true and φ is false if and only if ¬^φ is true. Nonetheless, truth and necessity are not synonymous, as φ is false in diﬀerent conditions than φ. Consequently, the equivalence φ ↔ φ is not valid. Interestingly, also the implication 1 1 φ → φ is not always true in Łukasiewicz’s semantics (it is 2 , when φ is 2 ), while one of his reasons for introducing the third truth value was to validate this implication. Łukasiewicz observes that a similar implication φ → (φ → φ) is indeed valid in his system and, as a result, φ can be properly inferred from φ. Clearly then truth and necessity, just as falsity and impossibility, are very closely related in Łukasiewicz’s theory. This might seems surprising. After all, we would not naturally say that if something does not happen, it is impossible. To refute this controversy and to defend Łukasiewicz’s modal logic, Arthur Prior(1953) stressed that the notion of possibility that Łukasiewicz had in mind was not logical possibility, but the kind of temporal possibility that Aristotle considered when he discussed future contingents. It is the notion of possibility that allows only the contingent future to be open to alternative options. The past and the present are, in this sense, necessary. If we conjoin this claim with a view that a sentence can be true only in virtue of what is past or present, then we need to conclude, as Łukasiewicz did, that whatever is true, is necessarily true. Prior’s rationalization largely coincides with Łukasiewicz’s own views; the latter essentially opted for what I call a “temporally local” notion of truth (see sec. 6.5). In his view, a sentence is true at time t if and only if there exists at time t something that grounds the truth of the sentence. In particular, presently existing causes which necessitate what a sentence predicts are the only ground for the truth-at-now of the sentence.

The cause of the future fact, which the sentence “p” states and which exists at instant t, is an actual correlate of the sentence “it is the case at instant t that p.” (Łukasiewicz, 1970b, p. 122)

He repeats essentially the same argument over four decades later:

 1, if v(¬φ) ≤ v(φ) iﬀ 1 − v(φ) ≤ v(φ) iﬀ v(φ) ≥ 1 ,  2 v( φ) = v(¬φ → φ) =  ^   (1 − v(¬φ)) + v(φ) = (1 − (1 − v(φ))) + v(φ) = 2v(φ), otherwise. 1 As a result, if the truth value of φ ranges from 2 to 1, then ^φ is true, but if the truth value of φ is less than 1 2 , then the truth value of ^φ is not 1, but twice the truth value of φ. If we interpret, as Łukasiewicz(1970d, p. 173) suggests, the truth value of φ as its “degree-of-possibility” (analogous to probability), then we need 1 to conclude that even if the degree of possibility of φ is 3 , then the degree of possibility of ^φ is not 1, but 2 3 .

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I mean that there is nothing that is real today and that would cause there to be a sea-fight tomorrow, not yet anything that would cause there not to be one. Hence, if truth rests on conformity of thought with reality, the proposition “The sea-fight will happen tomorrow” is today neither true nor false. (Łukasiewicz, 1957, pp. 155–156) In case of sentences referring to the past, the only things that can ground their truth are presently existing effects or traces of what the sentences represent (see Łukasiewicz, 1970b, p. 112, 128). In any case, a sentence φ is true at time t iff what exists at t makes it unpreventable that φ takes, took, or will take place.18 Due to this feature of Łukasiewicz’s semantics, Borkowski(1981) claims that it is misleading to say that he rejected bivalence. He argues that it is more accurate to say that Łukasiewicz replaced the classical notions of “truth” and “falsity” with an alternative (time-indexed) triad which is best characterized as • true-at-t = determined-at-t, • false-at-t = precluded-at-t, • contingent-at-t = neither-determined-nor-precluded-at-t.19 I am in broad agreement with Borkowski’s observation. I return to this issue in section 6.5. The three-valued approach has never been widely popular in the branching community. It is worth noting, however, that it has recently resurfaced in a book by John MacFarlane(2014). To incorporate Łukasiewicz’s insight into the semantics of branching, MacFarlane ascribes one of three truth values to sentences evaluated at moments. I will use v to denote a function that maps the Cartesian product of the set of all sentences 1 1 and the set of all moments to the set {0, 2 , 1} (v: S ent × M 7→ {0, 2 , 1}). Unfortunately, MacFarlane does not specify how to evaluate atomic sentences, so let me assume that v maps them to the set {0, 1}. Thus, I assume that the atomic sentences are “wholly about the present,” while Łukasiewicz makes clear that the present is not open to alternative 1 possibilities. Therefore, no sentence about the present should have the truth value 2 . The function v needs to satisfy the requirements for the connectives ¬, ∨, ∧, and ^, as illustrated by Łukasiewicz’s tables above. MacFarlane enriches Łukasiewicz’s language with a temporal indexical Tomorrow. To avoid the problem of context-sensitivity, let me modify his definition and introduce a non-indexical, temporal operator F:  0 0 1, if ∀ 0 ∃ 0 m > m & v(φ, m ) = 1,  h ∈Hm m ∈h   0  , if ∀ 0 ∀ 0 v φ, m , v(Fφ, m) =  0 h ∈Hm m >m ( ) = 0    1 2 , otherwise.

18This conception also explains why the modal fragment of the language should be bivalent. For any φ, either the necessitating causes/effects of φ presently exist or not. If they do, φ is true and ^¬φ is false, if they do not φ is false and ^¬φ is true. Given the present state of the world is fully determinate, every modal sentence can be evaluated as true or false based on the present state of the world. 19Essentially the same definition of the three truth values was given by Słupecki(1964), who o ffered an algebraic interpretation of Łukasiewicz’s logic (see Malinowski, 2007, pp. 20–22 for a short summary of Słupecki’s ideas).

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1 MacFarlane’s theory guarantees that 2 is ascribed to a sentence only if it contains operator F. Therefore, it adequately conveys the Aristotelian idea that only the future is the source of contingency.20 MacFarlane’s theory significantly departs, however, from the historical Łukasiewicz. As I have already mentioned, the Polish logician held sacrosanct the extensionality of logical connectives, while MacFarlane’s temporal operator violates this requirement. The truth value of Fφ at m does not functionally depend on the truth value of φ at m. Let me remedy the problem and introduce a fully extensional, 3-valued semantics to the branching setting. The easiest way to do this is to conceive of English sentences like “It is sunny” as somehow incomplete, missing an essential reference to an instant of time. Following Twardowski(1900), Łukasiewicz endorses this claim: “Statements of facts are singular and include an indication of time and place” (Łukasiewicz, 1970b, p. 118).21 To grasp Łukasiewicz’s idea, let me assume that the atomic sentences of a language are tenseless and they have the form pt which, as he explains, stands for the English “it is the case at instant t that p”(Łukasiewicz, 1970b, p. 112). I will call the set of such atomic sentences AT. The only logical connectives required are ¬ and →, the remaining connectives, including the modal connectives, can be defined along the procedure described by Łukasiewicz. To interpret this language in the branching structure (with instants), we need a proto-valuation function V that maps AT to the power-set of Hist (V : AT 7→ P(Hist)) which satisfies an extra requirement ∀t∀h1,h2∈Hist((h1 ∩ t = h2 ∩ t) ⇒ h1 ∈ V(pt) ⇔ h2 ∈ V(pt)

It means that if two histories overlap at time t (that is, they share a moment at instant t), then all the atomic sentences referring to time t have the same truth value in both these histories. Observe that function V is, so to speak, fully “bivalent.” For any sentence pt and any history h, either h ∈ V(pt) or h ∈ Hist\V(pt), there are no “halfways.” It reflects the fact that in each specific possibility, either p happens at t or not. The indeterminism is encoded by multiplicity of possibilities, not by indeterminacy of those possibilities. Using proto-valuations, we can define the proper valuation function v that assigns one of three truth values to sentences of our language, relative to moments (v: S ent × 1 M 7→ {0, 2 , 1}). Remember that Hm = {h ∈ Hist|m ∈ h}. For atomic sentences AT, the function works as follows:  1, if H ⊆ V(p ),  m t   v(pt, m) =  0, if Hm ∩ V(pt) = ∅,    1 2 , otherwise.

20It does not cohere well, however, with Łukasiewicz’s view that both the future and the past can bring about indeterminism. Łukasiewicz’s idea is notably diﬃcult to square with the standard branching model, so I will leave it aside. 21This statement goes against Prior’s deep conviction that the sentence “It is sunny” expresses a complete, “temporalist” proposition which changes its truth value from one time to another. Clearly, the two great logicians would disagree at this point, since Łukasiewicz expresses the “incompleteness” thesis even in his posthumously published (Łukasiewicz, 1957, p. 147).

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It intuitively means that an atomic sentence referring to time t is true at moment m iff what the sentence says happens at t in all the histories passing through m. It is false iff it happens at t in no history passing through m, and it takes a third truth value iff it happens at t in some, but not all histories passing through m. I prefer this approach to atomic sentences, because it explains why an atomic sentence takes a truth 1 value 0, 1, or 2 at a moment m, whereas in Łukasiewicz’s theory the ascription of truth values to the basic sentences is just “a given.” It also clearly indicates that the 1 truth value 2 arises from, and only from, future-oriented indeterminism encoded by the branching histories. To extend v to the set of all sentences, we need to require that the valuation function obeys the requirements encoded by the three-valued truth functions depicted by tables 4.1, 4.3, and 4.4. The validities generated by the semantics I have just described coincide with the validities generated by Łukasiewicz’s semantics. This should come as no surprise as the the very same truth functions are used to interpret the operators. The only difference is that the truth values of the atomic sentences are “grounded” in the properties of the branching structure. Such semantics has the interesting characteristics that the truth value of a sentence 1 at a moment depends on a temporal dimension. In particular, if v(pt, m) = 2 , then 0 0 00 00 v(pt, m ) = 1, for some m > m and v(pt, m ) = 0 for some m > m. This naturally encodes the idea that a sentence which is initially possible is later true in some possibilities and false in other. The phenomenon shows that there is no straightforward relation between the fact that a sentence is tenseless, on the one hand, and that its truth value is time-independent, on the other. The logic just described is not a temporal logic as there are no temporal operators in the language I propose to account for Łukasiewicz’s finding in branching structure. The whole temporal aspect is build into the semantics (and syntactics) of atomic sentences. Hence, the result is a time-sensitive, non-temporal logic. It might be also described as time-sensitive modal logic. A downside of the semantics is its relative limitation. In particular, it offers no means of judging if what is the case at m has always been inevitable to be the case. For example, it is true that you read these words at your present time t. It is true regardless of whether it is contingent or necessary that you would read them at time t0 when I write them. Thus, the truth-value-at-t of the sentence “It is necessary at t0 that pt” does not functionally depend on the truth-value-at-t of pt. It means that this semantics can assess what is possible, but it has no means of assessing what was and will be possible. Therefore, it does not exploit the full potential of the branching structure. I doubt that the problem could be easily remedied without violation of extensionality requirement. It is then possible to introduce a fully extensional, three-valued semantics to the branching setting. Let me discuss, however, the main reason why this idea has not gained much popularity. I have already touched on this delicate point. Remember the controversy regarding the truth value of an implication whose antecedent φ and 1 consequent ψ have the truth value 2 . According to Łukasiewicz it should be true, while according to Urquhart it should be possible. Which is it? The answer is: it depends on what φ and ψ mean. Evidently, if φ is synonymous to ψ, then φ → ψ comes down to φ → φ and it should be true, as Łukasiewicz requires. However, if φ and ψ are logically and causally independent, then we have the doughnut-coffee situation and it 1 should have the truth value 2 . Therefore, one cannot determine the truth value of an implication solely on the basis of the truth values of its components. One needs to take

69 CHAPTER 4. SEMANTICS OF BRANCHING REALISM into account the meaning of the components as well.22 An example usually used to argue against 3-valued semantics appeals to disjunction 1 rather than implication. Observe that if φ and ψ have the truth value 2 , then φ ∨ ψ also 1 has the value 2 . In some cases, this is desirable. If φ stands for “I will drink coffee 1 tomorrow” and ψ for “I will eat croissant tomorrow,” then φ ∨ ψ should be 2 . Notice, 1 however, that if φ has the truth value 2 , then ¬φ also has this truth value. As a result, the sentence φ ∨ ¬φ is not true, but possible, which means that the sentence “I will drink coffee tomorrow or not” is not true. The result struck many people as decisive argument against 3-valued approach. In fact, even Łukasiewicz required that the alternative composed of these sentences, “either John will be at home tomorrow noon or John will not be at home tomorrow at noon,” must be true in accordance with the principle of the excluded middle. (Łukasie- wicz, 1970b, p. 124)23 We can construct a similar argument to demonstrate that an apparently self-contradictory sentence, “I will drink coffee tomorrow and I will not” is not false. A possible rationalization of this controversial result is to impose a verificationist- like reading of disjunction and conjunction. We could insist that a disjunction can be true only in virtue of one of its disjuncts being true and a conjunction can be false only in virtue of one of its conjuncts being false, which is is based on the idea that we should not trust the principles of classical logic as such to be the ultimate ground for truths. Interestingly, a trace of such approach can be found in Łukasiewicz’s early book On the Principle of Contradiction in Aristotle, where he writes

The propositions about which we do not know whether they are true or false have no logical value, until their truth can be stated; they are logically valueless propositions. The principle of contradiction is among such propositions, if we apply it to the being in general, especially to the real being. (Łukasiewicz, 1910a, p. 135, translation mine)

At that period, he treated the principle of excluded middle along similar lines (Łu- kasiewicz, 1910b). He argued that these principle cannot be demonstrated to be true. Pragmatic considerations might convince us in favor of these principles, but pragmatic utility should not be confused with indubitable truth.24 If the truths of logic are so understood, then, if φ is a future contingent, it is unreasonable to expect that φ ∨ ¬φ should be true. If logic cannot serve as the ground for its truth, it needs to be grounded either in the truth φ or in the truth of ¬φ. However, neither of these two is true, given that φ is a future contingent. If you are unsatisﬁed with this rationale, you will probably share Arthur Prior’s impression regarding future contingents that “The truth-functional technique seems

22This observation is attributed to Gonseth(1941) (see, e.g., Urquhart, 2001; Surma, 2012). 23Incidentally, Łukasiewicz explicitly states in a paper published in 1920, two years before the lecture “On determinism,” that the formula φ ∨ ¬φ is not valid in his system (Łukasiewicz, 1970c, p. 88). The supervaluational system that I discuss in the next section seems to be closer to Łukasiewicz’s informal motivation than the 3-valued system he introduced. 24Jan Wolenski´ (2014, pp. 9–10) has further strengthened the case for classical logic as he provided an extensive list of pragmatic considerations favoring bivalence.

70 CHAPTER 4. SEMANTICS OF BRANCHING REALISM out of place here” (Prior, 1967, p. 135). John MacFarlane is convinced by Prior’s argument and restates that “there is no way for a truth-functional semantics to give all future contingents the value i without also assigning i to sentences like ‘Either it will be sunny tomorrow or it won’t be sunny tomorrow’ ” (MacFarlane, 2014, p. 221). This consequence dissuaded Prior from Łukasiewicz’s approach to temporal modalities. He viewed his Ockhamism and Peirceanism as significant improvements on the three-valued logic. Also, by this time extensionality had lost a lot of its allure. The birth of formally rigorous, relational semantics for modal logics, which Prior helped to initiate, gradually included the intensional operators in the realm of legitimate logical tools.25 As a result, the three-valued approach to modalities has never been widely popular in the branching community. I do not intend to suggest that Prior wanted to entirely diverge from Łukasiewicz’s ideas. On the contrary, he felt a strong affinity with his thought. It should not be surprising, as both were strongly inspired by classical philosophy and logic, both took metaphysics seriously, both appreciated the temporal dimension in logical reasoning, both stressed the inherent link between time and modality, both highly valued precision and formal methods, both even used the Polish notation invented by Łukasiewicz. There is no doubt that Prior looked up to Łukasiewicz as his master. In fact, he expressed this view himself in the preface to Time and Modality: “And while I differed radically from the late Professor Łukasiewicz on the subject of modal logic, my debt to him will be obvious on almost every page” (Prior, 1957, pp. vii-viii). To be fair, even Łukasiewicz himself seemed to have been discouraged by the results mentioned above. In his later works, he did not try to provide an intuitive interpretation of many-valued systems. He studied them in isolation, as abstract mathematical constructions—and finally found a way to answer the problem. In 1953, Łukasiewicz offers a truth-functional semantics, which renders a disjunction of two possible sentences like “I will have a coffee or a doughnut” neither true nor false, while it also renders the sentence “I will have a coffee or not” true. The achievement of the logician is not widely known, so let me quickly summarize it. Łukasiewicz presents a four-valued modal logic—Ł4 in my terminology—in which the distinguished truth value is the truth (i.e., the truth is used to determine the set of validities, and is symbolized by Łukasiewicz as 1). The semantics also includes falsity (denoted by 4) and two “middle” values—2 and 3—about which Łukasiewicz comments that they “represent one and the same possibility in two distinct shapes” (Łukasiewicz, 1953, p. 290). Łukasiewicz introduces three primitive operators: →, ¬, and ^ (possibility is not definable in terms of other connectives). The operators of disjunction, conjunction, and equivalence can be defined in the classical manner and the necessity operator , as ¬^¬. The truth tables of these operators are easily understood if we represent the truth values as ordered pairs of classical truth values, i.e., 1 B hT,Ti, 2 B hT,Fi, 3 B hF,Ti, 4 B hF,Fi (for simplicity sake, I will omit the square brackets and the comma). To arrive at the truth functions for the “classical” connectives, the classical, two-valued

25Łukasiewicz died at the time when the relational semantics had just germinated, so it would be hard to predict how he would have reacted to the new program. The correspondence that Prior and Łukasiewicz exchanged in the 1950s could shed some light on Łukasiewicz’s initial attitude.

71 CHAPTER 4. SEMANTICS OF BRANCHING REALISM functions are applied axis-wise. For example, TT→FT=FT, since the classical implication from T to F is F, while the implication from T to T is T. The possibility operator applied to XY leaves X untouched and raises Y to T. These procedures result in the following table :

→ TT TF FT FF ¬ ^ TT TT TF FT FF FF TT TF TF TT TT FT FT FT TT TF FT TT TF TT TF TF FT FF FF TT TT TT TT TT FT FF

Let us return to the problem with Łukasiewicz’s three-valued logic. Consider two equivalent sentences φ and ψ, such that neither of them is true and neither is false. If φ ↔ ψ is TT and φ is neither TT nor FF, then both φ and ψ have the same possible truth value, either TF or FT. Then, if you apply the classical disjunction axis-wise, you will easily compute that φ ∨ ψ has the same truth value as both disjuncts, i.e., it is either TF or FT. Hence, the disjunction of two equivalent possible sentences is also possible. By contrast, if φ is neither true nor false, for example it has the truth value TF, then ¬φ has the truth value FT and, as a result, φ∨¬φ has the truth value TT. Therefore, the disjunction of two contradictory, possible sentences is true. Thus, contrary to MacFarlane’s claim, there is a truth functional mechanism which assigns truth to a disjunction of two contradictory sentences that are neither true nor false. By an analogous argument, φ∧¬φ is false, while φ∧ψ is neither true nor false. Also, if φ and ψ are neither true nor false, and they are not equivalent, then the implication φ → ψ is neither true nor false, while both implications φ → φ and ψ → ψ are true. Consequently, the four-valued, truth-functional semantics can cope with the typical objections raised against 3-valued semantics. The fact is hardly recognized in the literature on future contingents. It is pointless to look for controversial results in the non-modal fragment of the language since, as Font and Hájek(2002, p. 162) note, Ł 4 is a conservative extension of the classical, propositional logic. The diﬃculties arise, however, when we study the modal fragment of the language. Some of the modal tautologies of Ł4 are familiar from other modal systems. For example, we have that:

• φ → ^φ • φ → φ • (φ ∧ (φ → ψ)) → ψ • φ ↔ φ • (φ ∧ ψ) ↔ (φ ∧ ψ) The system, however, generates a range of rather unusual eﬀects. For example, if you consult the truth tables above, you will realize that no sentence of the form φ is ever true and that no sentence of the form ^φ is ever false. It means, in particular, that the sentence (p ∨ ¬p) is not true and the sentence ^(p ∧ ¬p) is not false. These results seem peculiar from the modern reader’s viewpoint. Łukasiewicz did not share

72 CHAPTER 4. SEMANTICS OF BRANCHING REALISM the contemporary sentiment, as he claimed (in his discussion of classical tautologies) “that true propositions are simply true without being necessary, and false propositions are simply false without being impossible. This certainly does not hurt our logical intuitions, and may settle many controversies” (Łukasiewicz, 1953, p. 377). In fact, he had a positive reason to accept the result. He observes (in Łukasiewicz, 1957, pp. 149–151) that if we assumed that (x = x) is true, then we could infer (x = y) from x = y, “That means, any two individuals are necessarily identical, if they are identical at all” (p. 150). While Kripke took this principle as one of the basic tenets of Naming and Necessity, Łukasiewicz considered it “obviously false” (p. 150). He based his rejection on Quine’s example of the number of planets being contingently equal to nine. Since Łukasiewicz refused to limit the substitution in modal contexts (once again standing ﬁrmly on the side of extensionality), he was forced to conclude that ¬(x = x), i.e., “we are compelled to assume that no analytic proposition is necessary” (Łukasiewicz, 1957, p. 151). Another controversial property of Ł4 is that it validates

• (^φ ∧ ^¬φ) → ^(φ ∧ ¬φ) which seems wrong. If φ is a contingent statement, it is natural to expect that both ^φ and ^¬φ are true, but to derive on this basis that a contradiction is possible is dubious. Łukasiewicz defends the sentence in an astonishing way (Łukasiewicz, 1953, p. 378). He discusses C.I. Lewis’ example of an (apparent) future contingent, “A reader will see it” and subsequently presents the following line of reasoning:

1. Either a reader will see it or he won’t. 2. If a reader will see it, then it’s true that a reader will see it.

3. If it’s true that a reader will see it, then it’s impossible that a reader won’t see it. 4. If a reader won’t see it, then it’s true that a reader won’t see it. 5. If it’s true that a reader won’t see it, then it’s impossible that a reader will see it.

6. Therefore, it can never be jointly proven that it’s possible that a reader will see it and that it’s possible that a reader won’t see it.

I ﬁnd this line of defense very surprising, as Łukasiewicz seems to go against his entire life project and assume that out of two contradictory sentences (even about a contingent future) one needs to be true and the other needs to be false. As a result, he seems to conclude that out of any two contradictory sentences only one is possible. The proviso that it can never be “proven” that both ^φ and ^¬φ slightly weakens Łukasiewicz’s conclusion, but he must have realized that to reject (^φ ∧ ^¬φ) → ^(φ ∧ ¬φ), we do not need to prove ^φ and ^¬φ, but only need to assume that ^φ and ^¬φ are consistent. Łukasiewicz should be more than willing to accept this assumption on the philosophical grounds. To further reinforce the impression that his formula is valid, Łukasiewicz investigated the following example:

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If it is possible that n is even and it is possible that n is odd, then it is possible that n is even and n is odd. He then proceeds to demonstrate that the implication holds good, because the two conjuncts in the antecedent can never be jointly true, so the whole implication is vacuously true. Nonetheless, there is a great diﬀerence in the modal status of the sentence “n is even” and the sentence “A reader will see it,” and I ﬁnd it extremely surprising that Łukasiewicz should be a person to miss it. Łukasiewicz is obviously formally correct. If you inspect his semantics, you will clearly realize that the pair of sentences ^φ and ^¬φ cannot be jointly true, which seems to imply that nothing is contingent. We can alleviate the problem, if we use a “twin” modality, , introduced by Łukasiewicz in form of the following table: TT TT TF TF FT TT FF TF

Łukasiewicz comments that ^ and “are like twins who cannot be distinguished when met separately, but are instantly recognized as two when seen together” (Łukasiewicz, 1953, p. 370).26 If we have both these modalities at our disposal, we can restore contingency in our system, since if φ is TF, then ^φ ∧ ¬φ is true, and if φ is FT, then φ ∧ ^¬φ is true. Interestingly, neither of the two: • φ ∧ ^¬φ → ^(φ ∧ ¬φ) • φ ∧ ^¬φ → (φ ∧ ¬φ) is valid in Ł4. I cannot imagine how Łukasiewicz would react to this observation in face of his fervent defense of (^φ ∧ ^¬φ) → ^(φ ∧ ¬φ). Another oddity of Łukasiewicz’s system has been observed by Arthur Prior(1957, p. 3), who noticed that (φ → ψ) → (φ → ψ) is valid in Ł4. He undermined the truth with the following example. It is necessary that if someone is a logician, then someone is a logician, but it is not the case that if someone is a logician, then it is necessary that someone is a logician. Łukasiewicz’s response to these problems would probably point to the fact that no sentence of the form φ is true in his system. Therefore, no counterexamples with a necessary antecedent poses a serious threat. Another property of Ł4 that is rather unusual in the realm of modal logics is what Łukasiewicz himself calls “laws of extensionality in a wider sense”:

• (φ → ψ) → (^φ → ^ψ) • (φ → ψ) → (φ → ψ)

26By which he means that if you systematically replace every occurrence of ^ with in a validity that contains just ^, the resulting sentence will also be valid. They are not the same, however, because, e.g., ^^φ ↔ ^φ is a valid, while ^φ ↔ ^φ is not.

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Both are valid in his system. Łukasiewicz justiﬁes these results as a faithful representation of “Aristotle’s most important and—as I see it—most successful attempt to go beyond basic modal logic” (Łukasiewicz, 1957, p. 138). Łukasiewicz provides an extensive exegesis of a fragment of Aristotle’s Prior Analytics, where these principles seem to be endorsed (Łukasiewicz, 1957, §39–§42). However, Aristotle’s claims are vague enough to allow for alternative interpretations. In particular, Łukasiewicz(1957, p. 147) himself noted that they might be given a weaker reading, acceptable for a wide variety of modal notions:

• (φ → ψ) → (^φ → ^ψ) • (φ → ψ) → (φ → ψ) Nonetheless, a detailed analysis of Aristotle and his commentators convinced Łukasie- wicz that the stronger reading is more accurate. Interestingly, Łukasiewicz noticed that one of the ancient interpreters of Aristotle—Alexander—stressed the stronger, modal reading of the principle. In response, Łukasiewicz argues that the modality in the antecedent should be read in a temporal fashion, (φ → ψ) means that it is always the case that if φ, then ψ. Then, he argues that it is consistent with his stronger understanding, since “a true material implication must be, of course, always true” (Łukasiewicz, 1957, p. 147). I would not expect to hear such a declaration from Łukasiewicz. For example, I had a doughnut and tea this morning (at time t), consequently, the material implication, “If I eat a doughnut at t, then I drink tea at t,” is true, but does Łukasiewicz really want to conclude that this sentence has always been true, even yesterday, when it was undetermined what I would eat the next day? Certainly, an important fact about Łukasiewicz’s four-valued logic is the solution it provides to the problem that Prior considered unsolvable. However, just the modest list of objections I have presented undermines accuracy of Ł4 as a faithful representation of the notion of possibility. Doubtlessly, most formal systems diverge from the ordinary usage to some degree, but Łukasiewicz’s system is extreme in this regard. Hughes and Cresswell underline that: If by a “modal logic” we mean a logic of possibility and necessity, this system takes us to the limit of what we should regard as a modal logic at all. (Hughes and Cresswell, 1968, p. 310) Once again, Arthur Prior has proved to be Łukasiewicz’s most able attorney and at- tempted to explain some of the apparently counter-intuitive consequences of Ł4. Prior (1954, 1957) argued that there is a lower and an upper limit of what an (alethic) modality can mean. The weakest reading of φ is that φ is equivalent to φ, while the strongest reading of φ is that φ implies everything (i.e., it is equivalent to φ ∧ ¬φ). Regarding ^φ, its strongest reading is the weakest reading of φ, while its weakest reading is that ^φ is implied by anything (i.e., is equivalent to φ ∨ ¬φ). A formula of Ł4 is true if and only if it is true in the classical, 2-valued logic, under the strong and under the weak reading of the modalities (systematically applied to all their occurrences in the sentence). Prior does not advocate that it is a common usage of the modals, but observes that it is not entirely outrageous: “Sometimes when a man says ‘Possibly p’ it does look as if he is trying to convey to some people the idea that he is

75 CHAPTER 4. SEMANTICS OF BRANCHING REALISM assenting to the proposition p, and to others that he is not really committing himself to anything at all” (Prior, 1957, p. 5). A sentence is Ł4-true iff both theses groups would agree that what the man said is true. The validities of Ł4 are easily explainable under this interpretation. Clearly, no sentence of the form φ is valid, because under the strong interpretation of φ, it is equivalent to φ ∧ ¬φ which is a classical counter-tautology. The sentence φ → ^φ is valid, because both φ → φ and φ → (φ ∨ ¬φ) are classical validities. This reading also explains why ^^φ ↔ ^φ is a validity, while ^φ ↔ ^φ is not. The usage of the two different weak modals allow us to substitute two distinct readings of “possible” within a single sentence. I leave it to the reader to verify that all the examples discussed above agree with the interpretation proposed by Prior. In any case, this usage of the modals remains highly unusual. Hence, Ł4 is interesting mainly from the formal point of view and it has indeed been studied formally, both on the syntactic and the semantic level (for an overview, cf. Font and Hájek, 2002). It has not found its way, however, to philosophical applications. Even historians of modal logic tend to forget about Łukasiewicz’s achievements. Font and Hájek(2002, p. 173) offer a list of influential publications on modal logic that do not even mention Ł4 (their list could be extended with (Blackburn et al., 2001) and (Goldblatt, 2006), who also neglect this work). It is symptomatic, perhaps, that it is discussed in Hughes and Cresswell’s An Introduction to Modal Logic (1968), while it is omitted in their A New Introduction to Modal Logic, published nearly three decades later (Hughes and Cresswell, 1996). In this manner, the history has not vindicated Łukasiewicz’s hope that his system “is remarkably important both from the philosophical and logical viewpoint” (Łuka- siewicz, 1953, p. 284). As I have already mentioned, a part of the reason was Łukasie- wicz’s attachment to extensionality. In his final book, he writes that: [The truth of the laws of extensionality for modal operators] seems to be perfectly evident, unless modal functions are regarded as intensional functions, i.e., as functions whose truth values do not depend solely on the truth values of their arguments. But what in this case the necessary and the possible would mean is for me a mystery as yet. (Łukasiewicz, 1957, p. 140) At that time, modal logic was taking a sharp turn towards intensionality. The raise of relational semantics paved the way for intensional notion to the mainstream of formal logic. Unfortunately, Ł4 does not fit very well into this set-up. It is probably the main reason why Łukasiewicz’s work on modal logic has been largely forgotten.

4.5 Supervaluations

Let me turn now to the ﬁrst truly postsemantic response to the initialization failure. The response is the one recognized by John MacFarlane in Richmond Thomason’s (1970) supervaluationism. This view in many respects resembles three-valued semantics. Most notably, future contingents are neither true nor false in this theory. The key

76 CHAPTER 4. SEMANTICS OF BRANCHING REALISM difference is, however, that it is not assumed that they posses some other truth value. In this setting, future contingents have no truth value. To achieve this effect, Thomason applied van Fraassen’s (1966) technique of supervaluations to branching setting. He distinguished two distinct notions of truth, which I shall identify with truth at a context (||−) and truth at an index (|=). The supervaluational postsemantics relates these two notions to one another along the following lines: Definition 4.3 (Supervaluationism). m||−S φ iff m/h |= φ for every h such that m ∈ h. In supervaluationism, a sentence is true at a context iff it is true at each history passing through the context. In this framework, a sentence is false at a context iff its negation is true at the context. As a result, the fact that a sentence is not true does not imply that it is false. In particular, future contingents are neither true nor false. If the sentence φ is true at some histories passing through m and its negation is true at others, then m||−/S φ and m||−/S ¬φ. The distinction between the two notions of truth allows supervaluationism to bypass the initialization failure at a relatively low cost. We can ascribe truth status to sentences used at contexts without specifying a history parameter. At the same token, we do not need to give up the attractive features of Ockhamism, since we use it as the base semantics proper. In particular, we do not need to modalize the meaning of the future tense or introduce the third truth value. Nevertheless, the notion of truth is closely related to necessity in supervaluationism. Just as in Peirceanism and in three-valued semantics, a sentence is true at a contexts S S iff it is historically necessary, i.e., ∀m(m||−φ ⇔ m||−φ). These two are not identified, however; if φ is a future contingent at context m, m||−/S φ ↔ φ.27 A short exegetic remark is in place here. Although MacFarlane (2003; 2008; 2014) identifies supervaluationism as an exemplary postsemantic theory, such categorization requires significant addition to Thomason’s own exposition of the theory. Thomason (1970, pp. 273–274) does distinguish two notions of truth of a formula in a model—the truth at a moment/history pair and the truth at a moment. However, he does not identify, at least not openly, the former with the truth at an index and the latter with truth at a context.28 A more suggestive description is provided in (Thomason, 1984), where the author distinguishes between history-relative truth and absolute truth at a moment (p. 215), but the connection of these notions of truth to truth at an index and a context is also unclear. I am not going to settle these interpretative controversies. I am satisfied with the observation that there doubtlessly is an explicit distinction between two notions of truth in the papers of Thomason(1970, 1984) and, thanks to MacFarlane’s ingenuity, we can use the distinction to address the initialization failure. The logic generated by supervaluations fares better than logics of the theories discussed above, i.e., modalism, extremism and three-valued semantics. In fact, if we concentrate on the set of supervaluational truths, i.e., sentences true at every context,

27We can recognize the diﬀerence on the postsemantic level if we inspect the falsity conditions of these two sentences: m||−/S ¬φ, but m||−S ¬φ. 28Probably due to the fact that the distinction was popularized largely by David Lewis’ paper (1970b) which appeared in the same year as Thomason’s. Thomason ascribes truth to “statements” (p. 265), “predictions made” (p. 270), or “assertions” (p. 279).

77 CHAPTER 4. SEMANTICS OF BRANCHING REALISM then we end up exactly with the Ockhamist tautologies. It is instructive to discuss two important cases:

1. In contrast to three-valued semantics F1φ ∨ ¬F1φ is true at context m, even if neither F1φ nor ¬F1φ is true. The disjunction is true at every context, because the sentence F1φ ∨ ¬F1φ is Ockhamist-true at m/h, no matter which history h passing through the moment of context m we choose, . Incidentally, it implies that the “classical” connectives are not (postsemantically) truth-functional. We have just observed that a disjunction can be true at a context at which neither of the disjuncts is. However, if F1φ is neither true nor false at m, then ¬¬F1φ is also neither true nor false. Therefore (since F1¬φ ↔ ¬F1φ), F1φ ∨ ¬¬F1φ is neither true nor false. So in this case the disjunction of two truth-valueless sentences lacks a truth value. In consequence, we cannot establish the truth status of a disjunction in a context, basing it solely on the truth values of its disjuncts in the context (the same applies to conjunction, implication and equivalence). It is the cost which Łukasiewicz was not ready to accept. He preferred to sacriﬁce the universal truth of the sentence φ ∨ ¬φ, rather than the truth-functionality of disjunction. Supervaluationism makes an alternative decision, but it is important to realize that it comes with a price.

2. Let us explore another signiﬁcant example. In supervaluationism, the sentence φ → HFφ is valid. Let us study this example in detail on a very simple model.

S S m1||−It was going to be sunny. m2||−It was going to be rainy. h1 h2

m1: m2:!

S S m0||−/ It is going to be sunny. m0 m0||−/ It is going to be rainy.

First, let us take the sentence, “It is going to be sunny,” (F p) used at context m0. It is neither true nor false at the context. However, at a later context m1, the sentence, “It was going to be sunny” (HF p) is true since for arbitrary history h passing through m1, m1/h |= HF p. This eﬀect looks even more interesting if we express it in the language with dates. Let us say that both m1 and m2 occur at instant t. Now, let us study the sentence, “At instant t, it is sunny” S S S (Att p). We can conclude that m0||−/ Att p, while m1||−Att p and m2||−¬Att p. Thus, the very same sentence, which has the very same semantic value, changes its

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truth status from one context to another (such a behavior is characteristic to what MacFarlane(2009) calls non-indexical contextualism). The e ﬀect represents the claim that sentences which are initially truth-valueless become true or false with the procession of events in time (in Branching Realism, they become true in some parts of the world and false in others).

In general, as far as the set of supervaluational truths is concerned, there is nothing to worry about, since it coincides with the set of Ockhamist truths. The pretty parallel is lost, however, if we focus on preservation of truth rather than on truth itself. Timothy Williamson(1994, 151–152) pointed out that truth at a context is not preserved by very natural rules of inference. It means that the consequence relation for truth-at-context has questionable features under supervaluationism. Let me use the symbol φ||−S ψ to S S indicate that ∀m(m||−φ ⇒ m||−ψ), i.e., that inference from φ to ψ preserves truth at a S S context; and symbol ||−φ to indicate that ∀mm||−φ, i.e., that φ is true at every context. Williamson notices that supervaluational consequence relation violates:

Contraposition φ||−S φ, but ¬φ||−/S ¬φ.

Conditional proof φ||−S φ, but ||−/S φ → φ. Argument by cases φ||−S φ ∨ ¬φ and ¬φ||−S φ ∨ ¬φ, but φ ∨ ¬φ||−/S φ ∨ ¬φ. Reductio ad absurdum φ ∧ ¬φ||−S φ and φ ∧ ¬φ||−S ¬φ, but ||−/S ¬(φ ∧ ¬φ) (observe that ¬(φ ∧ ¬φ) is equivalent to φ → φ, which lacks a truth value if φ is a future contingent).

Let us study the ﬁrst case only. The fact that φ||−S φ is easy enough to understand. After all, if c||−S φ, then φ is true at every history passing through c, which means that S S c||−φ. To see that ¬φ||−¬φ does not hold, let us substitute F1(sunny) for φ. If you S consults the model depicted on page 78, you will clearly see that m0||−¬F1(sunny), S S while m0||−/ ¬F1(sunny). Therefore, ¬φ||−/ ¬φ. The results above are problematic for supervaluationism, because we have a clear sense that each (or at least some) of the rules of reasoning mentioned above are valid. Therefore, we need to conclude that validity of reasoning does not coincide with preservation of truth of a sentence at a context. I consider this to be a bad result, since the notion of truth at a context was supposed to be a “down-to-earth” notion of truth, which is most familiar and upon which we build our intuitions upon. Therefore, if the intuitive, “down-to-earth” notion of valid inference diverges from the preservation of truth- at-context, then it turns out that our basic intuitions regarding valid, truth-preserving reasoning are not connected to preservation of truth-at-context. Supervaluationism generates problematic results, since the indexes on which the operators are deﬁned, i.e., moment/history pairs, are not the appropriate candidates for contexts. Hence, the notions of truth at an index and truth at a context diverge. However, only the former is required to behave in accordance with the standard rules of logic. It comes as no surprise, therefore, that truth at a context is not a logically orderly notion. For example, let us say that extensionality fails for the sentence F1 p ∨ F1¬φ at context m. The sentence is true at context m due to its truth at all pairs m/h.

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Nonetheless, no pair m/h is an appropriate context. Thus, the truth-at-context of F1 p ∨ F1¬φ cannot be explained in terms of truth-at-context of F1φ or F1¬φ. Also, all the rules of reasoning discussed above preserve truth at an index. If only every index was an available context, none of the problems mentioned above would arise. It also means that there are context at which the sentence ^(F p ∧ ¬F p) is true, even though there is no context in which the sentence F p ∧ ¬F p is true. So, a supervaluationist can truly say that it is possible that a non-necessary sea battle will take place, but it is not possible for her to truly say that a non-necessary sea battle will take place. This last observation was transformed into a more general argument by Tweedale (2004), who noticed that φ, ^ψ||−S ^(φ ∧ ψ) is a valid inference rule in standard supervaluationism. It is a bad result, since if we substitute F p for φ and ¬F p for ψ, we obtain that F p, ^¬F p||−S ^(F p ∧ ¬F p). Namely, we can infer a logical impossibility from logically possible set of assumptions.

A problem with retrospective accuracy ascription Supervaluationism has its problematic features, especially concerning the formal properties of the notion of truth-at-context (truth value gaps, non-extensionality of classical connectives, consequence relation). In addition to these John MacFarlane(2003; 2008; 2014) has distilled one more argument that undermines supervaluationism. It is based on the Retrospective Accuracy judgment. Generally, MacFarlane is greatly impressed by the results that supervaluationism oﬀers for future contingents (see MacFarlane, 2014, p. 226). He thinks, however, that it needs to be updated to be entirely faithful to our linguistic intuitions. The leitmotiv of MacFarlane’s theorizing about future contingents is the clash which he observes between two attitudes towards the future oriented talk: “ante factum” indeterminacy intuition and “post factum” determinacy intuition. In his ﬁrst paper on the subject, he describes the clash in the following manner:

Suppose that the world is objectively indeterministic. In some possible futures, there is a sea battle tomorrow. In others, there is not. How should we evaluate an assertion (made now) of the sentence “There will be a sea battle tomorrow”? The question is difficult to answer, because we are torn between two intuitions. On the one hand, there is a strong temptation to say that the assertion is neither true nor false. After all, there are possible future histories witnessing its truth and others witnessing its falsity, with nothing to break the symmetry. I shall call this “the indeterminacy intuition.” On the other hand, there is a strong temptation to say that the assertion does have a definite truth value, albeit one that must remain unknown until the future “unfolds.” After all, once the sea battle has happened (or not), it seems quite strange to deny that the assertion was true (or false). I shall call the thought that the assertion does have a definite truth value “the determinacy intuition.” (MacFarlane, 2003, p. 321).

In 2003, he argued that supervaluationism gave justice only to the indeterminacy

80 CHAPTER 4. SEMANTICS OF BRANCHING REALISM intuition.29 A few years later, MacFarlane(2008) softened his criticism. He realized that supervaluationism actually had a way to capture determinacy intuition. I will not discuss all the details of his argument, let me just note its core. Observer that in the middle of the sea battle a supervaluationist can truly say, “It was true yesterday that there would be a sea battle today.” To reconstruct the observation, I need to augment our language with a truth operator.30 Since we are dealing with two notions of truth in supervaluationism, the decision about the semantics of operator Tr is not straightforward. Both Thomason(1970) and MacFarlane(2008, 2014) decide to link the truth operator with a history-relative notion: Definition 4.4 (Truth and falsity operators). m/h |= Trφ iff m/h |= φ; m/h |= Flφ iff m/h |= ¬φ. Let us now analyze the sentence, “It was true yesterday that there would be a sea battle today,” uttered in the middle of the sea battle.

S 1. m1||−P1TrF1 p iﬀ (by def. 4.3)

2. ∀h(m1 ∈ h ⇒ m1/h |= P1TrF1 p) iﬀ (by semantics of P1)

3. ∀h(m1 ∈ h ⇒ m0/h |= TrF1 p), where m0 is the moment one instant earlier than m1, iﬀ (by def. 4.4)

4. ∀h(m1 ∈ h ⇒ m0/h |= F1 p)

Remember that there is a sea battle at m1, so in all the histories passing through m1, moment m0 is followed by a sea battle (by analogy, at context m2 where there is no sea battle, one can truly say “It was false yesterday that there would be a sea battle today”). Let me mention one concern regarding this line of defense of supervaluationism. If post factum determinacy is explained in terms of the truth (at m1) of the sentence “It was true yesterday that there would be a sea battle today,” then ante factum indeterminacy should be expressed, at face value, in terms of truth (at m0) of the sentence, “It is not true that there will be a sea battle” (¬TrF1 p). Nonetheless, a supervaluationist cannot truly say ¬TrF1 p a m0 even though the sentence is not true at context m0. In fact, she can truly say at m0, “It is true that there will be a sea battle or it is false that there will be a sea battle” (TrF1 p ∨ FlF1φ). It seems, therefore, that if a supervaluationist can express post factum determinacy, she can no longer express ante factum indeterminacy. MacFarlane decided to modify his criticism of supervaluationism. Rather than phrasing it in terms of truth of things asserted, he presents the determinacy/indeterminacy conﬂict in terms of accuracy of acts assertions:

This, then, is the puzzle:

29Let me note that not everyone shares both MacFarlane’s intuitions. Some are willing to accept ante factum determinacy, while others the post factum indeterminacy. I will argue in sec. 4.6 (pp. 90, ﬀ.) that the apparent conﬂict is based on equivocation of two notions of “assertion.” 30I assume that the necessary precautions have been taken to avoid the semantic paradoxes.

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• present assertions concerning the future can be shown to be inaccurate by a proof of present unsettledness, but • past claims concerning the present cannot be shown to have been inaccurate by a proof of past unsettledness. (MacFarlane, 2014, p. 226).

It is unclear, at this point, how accuracy of acts of assertion can aﬀect the postsemantic theory of truth at a context. To establish the connection, MacFarlane postulates the following norm:

Suppose that at c0 (on Monday) Jake asserts, (3) Tomorrow Berkeley will be sunny. (. . . ) the assessor should take Jake to have spoken accurately just in case (3) is true at c0.(MacFarlane, 2014, p. 210)

We can extract from this fragment the general norm of assertion: Deﬁnition 4.5 (Supervaluational truth norm). If an act of assertion is accurate, then the sentence asserted is true in the context in which the act takes place.31

The most important consequence of the truth norm is that if the sentence is not true-at-a-context, then the assertion which uses this sentence as a vehicle is not accurate. The inaccuracy of an act of assertion is inherently connected to the retraction obligation: if an act of assertion is not accurate, then the assertor who made it should retract it, or “take it back.”32 With the truth norm at our disposal, we can use the accuracy and retraction judgments to test diﬀerent postsemantic theories. In particular, when the truth norm is applied to supervaluational postsemantics, we arrive at the norm which says that an act of assertion made at context m is accurate only if the sentence is settled true at c. The consequence vindicates ante factum indeterminacy intuition. As an example, let us take a look at the tree depicted on page 78. At moment m0, at which the future weather is objectively undetermined (which can be attested by the Bureau of Quantum Weather Prediction), Jake makes the assertion, “It will be sunny tomorrow.” According to supervaluationism, the sentence asserted is not true at context m0, which implies, by the truth norm of assertion, that Jake’s act of asserting is not accurate. Moreover, if the director of the Bureau confronted Jake with the proof of indeterminacy, Jake should retract his claim.

31Which could just as well be stated in terms of propositions: [T]he assertion under discussion is accurate just in case the proposition denoted by “what was asserted” is true at all the circumstances compatible with the context of the assertion. (MacFarlane, 2014, p. 224).

32Of course, the obligations resulting from the truth norm can be overruled by other consideration. For example, people are obliged to protect their families, which may imply that in some circumstance (for example, when interrogated by a member of an enemy army), they should not tell the truth and should not retract inaccurate assertions.

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Let us look at the situation from the perspective of context m1. In the middle of sunbathing, we reﬂect on the act of assertion that Jake made the day before. MacFar- lane argues that we should accept it as accurate, “after all, the assertor has only to feel the sun on her skin to know that Jake’s assertion was accurate” (MacFarlane, 2014, p. 210). However, it is not the prediction that supervaluationism generates. According to supervaluationist postsemantics conjoined with the truth norm of assertion, no matter what we feel on our skins, the sentence asserted at m0 is not true at m0, so the act of assertion is inaccurate and should be retracted.

And that seems wrong. For assessors at m1, the fact that rain was still a possibility when Jake made his assertion isn’t relevant to its accuracy. (. . . )

To see how strange the supervaluationist’s verdict is, suppose that at m1, the Director of the Bureau of Quantum Weather Prediction oﬀers Jake an irrefutable proof that, at m0, it was still an open possibility that it would not be sunny on the next day. Should such a proof compel Jake to withdraw his assertion? Clearly not. (p. 225 MacFarlane, 2014, I inserted my names of the moments). Therefore, supervaluationism violates determinacy intuition for retrospective accuracy ascription. It is the core of MacFarlane’s Retrospective Accuracy argument against supervaluationism. The very same argument can be raised against extremism, Peirceanism, and three-valued semantics discussed before.

4.6 Assessment relativism

MacFarlane’s sets for himself the task of improving supervaluationism and his work results in assessment relativism. Let me first note that the semantic theories I discussed so far are not completely absolutist either. If only the state of the weather is a chancy affair, then each single one of them allows for the same tenseless sentence “It is sunny in Lvov on March 1, 1900” to be not true at some context earlier than March 1, true at some context later than March 1, and false in other contexts later than March 1. Nonetheless, each of these theories hold that once the context is fixed, then the truth value of a sentence (or the lack of it) is absolute. The problem with supervaluationism, MacFarlane diagnoses, originates in this last assumption. Therefore, he has proposed a modification of supervaluationism which takes into account another aspect—the context of assessment. He argues that the truth value of a sentence can be determined in an absolute manner only if we also take into account the circumstances from which the truth value is assessed. A premonition of such an idea can be traced back to (Thomason, 1970), who suggests that: [R]ather than making formulas true or false with respect only to the times at which they are true or false, we make their being true or false relative to subsequent times as well. (Thomason, 1970, p. 268). The particular technical realization of the idea proposed by Thomason was dis- satisfactory even to its author. It was later revived by Nuel Belnap(2002b), under the

83 CHAPTER 4. SEMANTICS OF BRANCHING REALISM name of “double-time reference.” Belnap did not use the technique, however, to assess the truth value of a sentence, but to provide satisfaction conditions for assertion and other speech acts. Finally, John MacFarlane used Belnap’s technical apparatus to formalize the double-relativized notion of truth in form of “double-time reference postsemantics” (MacFarlane, 2003, p. 331). The postsemantic has been later incorporated into a more general theory of assessment relativism (MacFarlane, 2014). The formal idea of assessment relativism, as applied to branching, is such that when we assess a truth value of a sentence used in one context from the perspective of another context, we should check if the sentence assessed is true at the context of use with respect to histories passing through the context of assessment.33 To state the relativist postsemantics, we need an auxiliary notion of a set of histories passing through a pair of moments  H ∩ H , if m ≤ m ,  m1 m2 1 2 Deﬁnition 4.6.  Hm1|m2 =   Hm1 , otherwise. We can now state assessment relativism as follows

R Deﬁnition 4.7. mu, ma||−φ iﬀ M, mu/h |= φ for every history h ∈ Hmu|ma .

A sentence is true at a pair of contexts mu, ma iﬀ it is true at moment mu in all histories passing through ma (or all histories passing through mu if mu ma). The assessment relativism truly deserves its name, since the very same sentence used in a single context can be true when assessed from one perspective, false when assessed from another perspective, and neither true nor false when assessed from still another perspective.34 Let us study relativist postsemantics with a particular example:

h1 h2

m1: m2:!

It will be sunny tomorrow. m0 m3:

33In particular, when the context of use is identical with the context of assessment, we should use all the histories passing through the context of use. 34In terms of propositions, assessment relativist insists that the proposition expressed by a sentence in a given context changes its truth status relative to the context of assessment. The term ﬁts well with the notion of relativism introduced by Kazimierz Twardowski(1900). According to Twardowski’s absolutism, if we take a unambiguous sentence and specify the context-dependent parameters (he mentions: time, place, and the speaker), then the sentence is true or false. Assessment relativism reject this notion.

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In such a model, we have that: R R • m0, m0||−/ F1(sunny) • m0, m2||−¬F1(sunny)

R R • m0, m0||−/ ¬F1(sunny) • m0, m3||−/ F1(sunny)

R R • m0, m1||−F1(sunny) • m0, m3||−/ ¬F1(sunny) So, at contexts m0, m0, the sentence “It will be sunny tomorrow” is neither true nor false, at m0, m1, it is true, and at m0, m2, false. At m0, m3, it is neither true nor false. As for the last case, MacFarlane gives very little explanation. He does not say what it means to assess a truth value of a sentence used in a different possibility.35 With relativist postsemantics at our disposal, we can address the problem of supervaluationism, but first, we need to adjust the truth norm of assertion, to take into account the extended context: Definition 4.8 (Relativist truth norm). If an act of assertion is accurate, then the sentence asserted is true at a context of use and a context of assessment pair.

Let us have another look at Jake’s act of asserting “It will be sunny tomorrow,” performed at m0. When assessed from the perspective of m0, it is not accurate, since it is not true at the pair of contexts m0, m0. However, when assessed from the perspective of m1, it might well be accurate, since it is true at m0, m1. Thus, an act of assertion can be accurate even if the sentence uttered is not settled true at the moment at which the utterance happens.

When m0 is in the past of m1, Hm0|m1 = Hm1 , so an assessor at m1 should take an assertion made at m0 to be accurate just in case its content is true at all the worlds overlapping at m1. That is why a proof of past unsettledness is not sufficient to compel retraction. But when m0 = m1, the assessor should take the assertion to be accurate just in case its content is true at all the worlds overlapping at m0. That is why a proof of present unsettledness is sufficient to compel retraction. (MacFarlane, 2014, p. 227, notation modified).

Hence, thanks to relativization of truth and accuracy to the context of assessment, MacFarlane was able to overcome the diﬃculties he diagnosed in supervaluationism and wed determinacy and indeterminacy intuitions.

Some controversies regarding assessment relativism I am going to question both MacFarlane’s diagnosis and the cure he prescribes. I think that he misinterpret certain linguistic phenomena, which, when we properly understood, do not require relativism. Firstly, I demonstrate that all the results desired by MacFarlane can be achieved without relativism. Secondly, I argue that his desired results are not necessarily the results we actually want. I think that relativism introduces

35One might surmise that assessment of a sentence used in a diﬀerent possibility comes down to assessment from the actual context, whether the sentence would have been true, had it been used. However, at another juncture, MacFarlane(2014, p. 228) claims that counterfactual constructions shift the context of assessment.

85 CHAPTER 4. SEMANTICS OF BRANCHING REALISM a rather peculiar normative theory of language. I also explain the appeal of relativism, pointing to equivocation easily introduced by the term “assertion.” Moreover, in appendix 7.7, I present a more technical result that relativism generates wrong accuracy judgments for counterfactual accuracy ascription, under a plausible formal reconstruction.

Relativist results without relativism First, I will formally reconstruct the retrospective accuracy problem, which MacFarlane identified in supervaluationism. Then, I will show how it can be overcome on supervaluational ground without evoking the context of assessment. To explain MacFarlane’s argument, let me introduce a slightly convoluted connective to our language, “It is accurate to assert that.” For simplicity’s sake, I take it to be a sentential connective, rather than a predicate operating on names of sentences. In other words, I focus on constructions like “It is accurate to assert that it will rain” rather than “The assertion of a sentence ‘It will rain’ is accurate.” To indicate that the operator is sensitive to the supervaluational truth at a context, I will call it AccS . Definition 4.9 (Supervaluational accuracy). If m/h |= AccS (φ), then ∀h(m ∈ h ⇒ m/h |= φ). I state the definition in form of implication, and not equivalence, since the truth at a context is a necessary condition of accuracy, but might be not sufficient. With the accuracy operator at our disposal, let us return to our meteorological model depicted at page 78. We can establish that it is not accurate to assert at m0 that on the following S day, it would be sunny, i.e., m0||−/ AccS F1(Sunny).

S 1. m0||−AccS F1(Sunny) iﬀ (by supervaluational postsemantics, def. 4.3)

2. ∀h(m0 ∈ h → m0/h |= AccS F1(Sunny), then (by def. 4.9)

3. ∀h(m0 ∈ h → m0/h |= F1(Sunny)) iﬀ

4. m1/h1 |= Sunny and m2/h2 |= Sunny.

As it is not sunny at m2, it is not accurate to assert at m0 that it would be sunny on the next day. MacFarlane’s objection to supervaluationism can be reconstructed as well. Let us study the sentence “Yesterday, it was accurate to assert that it would be sunny today” (P1AccS F1(Sunny)), uttered at m1, in the middle of a sunny afternoon

S 1. m1||−P1(AccS (F1(Sunny))) iﬀ (by def. 4.3)

2. ∀h(m1 ∈ h ⇒ m1/h |= P1AccS F1(Sunny)) iﬀ (by semantics of P1)

3. ∀h(m1 ∈ h ⇒ m0/h |= AccS F1(Sunny)), then (by def. 4.9)

4. ∀h(m0 ∈ h ⇒ m0/h |= F1(Sunny)) iﬀ (be def. of F1)

5. m1/h1 |= Sunny & m2/h2 |= Sunny

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As 5 is false (due to the rain at m2), one cannot say at m1 that it was accurate to assert a day before that it would be sunny on the next day. This very claim is the core of MacFarlane’s criticism of supervaluationism. There is, therefore, a sharp contrast between accuracy ascription and truth ascription:

S Prospective truth ascription m0||−/ TrF1S unny

S Prospective accuracy ascription m0||−/ AccS F1S unny

S Retrospective truth ascription m1||−P1TrF1S unny

S Retrospective accuracy ascription m1||−/ P1AccS F1Sunny

Thanks to this observation, we can easily understand MacFarlane’s claim that in supervaluationism “monadic truth ascription can come apart from accuracy judgments and retraction obligations,” (MacFarlane, 2014, p. 224). The truth and accuracy ascription coalesce at moment m0, but they diverge at the later moment m1. MacFarlane approves of the first three verdicts of supervaluationism, while he dis- approves of the fourth. The simplest way to rescue supervaluationism would be to identify AccS with Tr. In fact, I consider it to be a very reasonable strategy. In my view, post factum determinacy intuition is solid only if we understand “accurate” as “true.” I will promptly return to this issue. Nonetheless, MacFarlane does not share my opinion. He thinks that accuracy ascription should be intrinsically connected with truth at context(s) and, thus, he wants to connect accuracy ascription with the truth at a pair of contexts along the relativist lines: “When m0 is in the past of m1, (. . . ) an assessor at m1 should take an assertion made at m0 to be accurate just in case its content is true at all worlds overlapping at m1”(MacFarlane, 2014, p. 227, notation modified). It turns out, however, that we do not need to evoke relativism and introduce the context of assessment to achieve this result. It is enough to assume that accuracy ascription is an expression sensitive to the context of use: Definition 4.10 (Relativist accuracy).  ∀h(m ∈ h → m/h |= φ), if m ≤ m  c c If m , m/h |= Acc (φ), then  or c R   ∀h(m ∈ h → m/h |= φ), otherwise. Given this definition (inspired by relativist postsemantics), the accuracy operator is sensitive to the context in which the accuracy judgment is made. If a supervaluationist subscribes to the context-sensitive notion of accuracy, she can generate the results desired by MacFarlane

S 1. m1||−P1AccRF1(sunny) iﬀ (by def. 4.3)

2. ∀h(m1 ∈ h ⇒ m1, m1/h |= P1AccRF1(sunny) iﬀ (by def. of P1)

3. ∀h(m1 ∈ h ⇒ m1, m0/h |= AccRF1(sunny) iﬀ (by def. 4.10)

4. ∀h(m1 ∈ h ⇒ m1, m0/h |= F1(sunny) iﬀ (by def of F1)

87 CHAPTER 4. SEMANTICS OF BRANCHING REALISM

5. ∀h(m1 ∈ h ⇒ m1, m1/h |= sunny)

The last statement is true in our meteorological model, so is the ﬁrst. Therefore, a supervaluationist can say, on a sunny day, that she was right the day before asserting that it would be sunny on the following day.

S Retrospective accuracy ascription m1||−P1AccRF1Sunny

Therefore, the context of use alone provides suﬃcient resources to secure the results MacFarlane requires.36 One might argue that the solution I have just described is just a notation variant of the assessment relativist postsemantics, which, in a way, it is. It is faithful to the idea that when we assess accuracy of an assertion, at the back of our minds, we should keep the context in which we perform the assessment. In any case, I ﬁnd this variant revealing, as it demonstrates that the key issue is not the relativity of truth-at-context, but whether the accuracy ascription is context-sensitive. It means that context-sensitivity of accuracy, rather than the mechanism of double-time reference, is the core of relativist proposal. We have just seen that a supervaluationist can account for both determinacy and indeterminacy intuition within their theoretical setting. I am now going to argue, however, that the game is not worth the candle. The the tension between the two intuitions can be explained away. Furthermore, relativism imposes unreasonable set of norms on the language users.

Strange consequences of relativist norms The truth at a context is connected to our linguistic practice through the truth norm of assertion. Consequently, postsemantics poses some constraints on our actions—it dictates when we are entitled to make certain assertions and when we are required to retract them. I think that inspecting the collection of rights and duties imposed by relativism will demonstrate their very limited appeal. Let us think through the implications of the relativist proposal with the help of a speciﬁc example. Our good friend, Jake, asserts at moment m0: (S) “It will be sunny tomorrow.” The sentence is not true at the pair of contexts m0, m0. Therefore, if we assess Jake’s act of assertion at moment m0, relativism gives a straightforward verdict: Jake’s assertion is not accurate. He should not have made it and he should retract it, if urged. So, at m0 it is inappropriate for Jake to assert “It will be sunny tomorrow.” Well, it does happens. After all people sometimes do as they should not. This by itself is not a problem for relativist theory. An unpalatable conclusion follows if we continue the relativist story. Let us say that, per chance, it is sunny on the next day, i.e., we end up at the sunny moment m1. According to relativism, at the pair of contexts m0, m1, the sentence (S) is true. So, the act of assertion made by Jake a day before might be accurate. In fact, MacFarlane argues that even the expert opinion of the Director of the Bureau

36However, to apply the supervaluational ﬁx I have just described, we need to relax the connection between accuracy ascription and truth at context(s). As far as I understand, it is also the upshot of the paper by Brogaard(2008). I imagine that John MacFarlane could object to my strategy since he expressed, in personal communication, concerns regarding such relaxation.

88 CHAPTER 4. SEMANTICS OF BRANCHING REALISM of Quantum Weather (that on the previous day, the state of weather on the following day was undetermined) should not convince Jake to retract his act of assertion. Then, unless some other unusual norm intervenes, Jake has every right to boast at m1 about his perfectly accurate prediction! Weirdly, the pure passage of time somehow absolved Jake from his previous sin. The act he should not have performed (and should, if urged, retract) turned into an act that he had every right to have performed, to be proud of, and resist to retract. This case generalizes to all assertions of future contingents. It turns out that under relativism, Jake can make a (retrospectively) accurate act of assertion about contingent future, only if he previously decides to perform an assertion which, by the same standards, is impermissible. I take it to be a very bizarre normative theory. It implies that only the people who are willing to violate the norms of assertions can be rewarded for having made accurate assertions. Only they, by performing (initially) inaccurate acts, stand a chance to be post factum appreciated for having made accurate predictions. Compare this case with any other norm. For example, let us imagine a norm that says that if one is caught stealing red-handed, one is declared a thief and punished for such an action. However, if one manages to escape with the loot and hide suﬃciently long, not only does one become the owner of the item, but we should also, retrospectively, assess that the act this person performed was not an act of theft! Retrospectively, we can truly say that one just took what was rightfully theirs. To my judgment, it is a rather barbaric and predatory set of norms. A similar problem is discussed by García-Carpintero(2013), regarding epistemic relativism. He discusses the case when Sally has a strong evidence that Uncle Jack is coming to lunch and she asserts “Uncle Jack is coming to lunch.” Later on, however, she receives a new piece of evidence about Uncle Jack: he broke his leg in the morning and needs to stay in the hospital. Having heard the news, Sally retracts her previous assertion. The story is perfectly intelligible and García-Carpintero does not question appropriateness of Sally’s behavior. He does, however, question the interpretation proposed by relativists. The common sense suggests that when Sally said, “Uncle Jack is coming to lunch,” she thought she made an accurate assertion, but she in fact made an inaccurate one (as the further evidence made clear). Nonetheless, according to relativism, she initially performed a perfectly accurate act of assertion, it just so happened that it later turned into an accurate act and that is why Sally needs to retract it. García- Carpintero makes a telling comment:

I cannot see how it can ever be rational to carry out activities governed by a relativist truth norm, and although it is, in principle, possible that we are foolish enough to have instituted an intrinsically irrational practice, I ﬁnd it methodologically advisable not to assume that this is so. (. . . ) What MacFarlane’s account envisages is rather that I can perform an action that is constitutively legitimate—an assertion that meets it constitutive norm—and later be obliged to take it back. One should be excused for not ﬁnding this an intelligible possibility. (García-Carpintero, 2013, p. 24–5).

A similar kind of problem can be identiﬁed in case of future contingents. In the epistemic case, an accurate act turns into an inaccurate act under the inﬂuence of addi-

89 CHAPTER 4. SEMANTICS OF BRANCHING REALISM tional information. In case of future contingents, the converse is true: the very same act of assertion which initially violates an important linguistic norm, turns later into a perfectly accurate act, in agreement with all the norms. I agree with García-Carpintero’s judgment that “I cannot see how it can ever be rational to carry out activities governed by a relativist truth norm.” Something clearly went wrong. To give proper due to both determinacy and indeterminacy intuitions, MacFarlane introduced the context of assessment as an extra postsemantic parameter and then relativized truth to a pair of contexts. But we have just seen that the notion of accuracy induced by relativism leads to a counter-intuitive account of our linguistic practice. What should we do then? I think that we should recognize that there is no actual conflict between the two intuitions. If we understand them properly, the tension disappears. Let me restate the two conflicting claims: This, then, is the puzzle: • present assertions concerning the future can be shown to be inaccurate by a proof of present unsettledness, but • past claims concerning the present cannot be shown to have been inaccurate by a proof of past unsettledness. (MacFarlane, 2014, p. 226). I think that a large part of the energy that inflates the puzzle resides in the equivocation induced by the ambiguity of English. Observe that words like “assertion,” “claim,” “prediction,” can denote both the act of asserting, claiming, predicting, and the thing asserted, claimed, predicted, i.e., “assertion” can denote either an act of assertion or the content of the sentence asserted.37 In my opinion, indeterminacy intuition is valid for the acts of assertion of future contingents, while determinacy intuition applies to the contents of assertions. Therefore, when we say that an assertion is (in)accurate, we need to be very careful distinguishing between these two meanings of “assertion.” I think that if we focus our attention on one meaning of “assertion” at a time, the tension between two intuitions does not arise. Let us first investigate the action of asserting a future contingent, i.e., the act of predicting. Let us take Jake who, on Mon- day, provides forecast for a national television. He consults the standard meteorological model and makes the prediction: “It will be sunny tomorrow.” Let us further say that it was one of these very rare cases where the standard model was misleading and only a very diligent quantum proof could show that the probability of sunny weather was actually only 20%. Fortunately, just when he picked up the receiver to call the television, Mike—the Director of the Bureau of Quantum Weather—entered Jake’s office with the exact proof that the weather on the next day is unsettled. Mike’s proof shows that the act of Jake’s assertion was inaccurate, so he needs to take it back.38 Jake says when

37Or, to use Kazimierz Twardowski’s (1912) vocabulary, it can denote either the action, or the product of asserting. 38I do not agree with MacFarlane that any degree of indeterminacy invalidates a prediction. For example, if Mike’s proof conclusively established that the probability of non-sunny weather on the next day is 0.000000001%, it would probably not convince Jake to retract his assertion. I think that Jake could easily say that such a marginal probability does not need to be accounted for, even by professional forecasters; i.e., he would deny to increase his epistemic standards to an unreasonably high level.

90 CHAPTER 4. SEMANTICS OF BRANCHING REALISM confronted with the proof, “Oh dear, you are right Mike, my (act of) prediction was not quite accurate, I should not have said what I did, given the data you have just offered.” Jake’s behavior confirms MacFarlane’s indeterminacy intuition. Let us now imagine that Mike got stuck with his proof in a huge traffic jam and did not make it on time to warn Jake. Fortunately for the forecaster, however, the weather turned out to be sunny anyway (after all, there was 20% chance for such an outcome). Mike enters Jake’s office in the middle of the sunny Tuesday and gives him the proof to show him how lucky he was. In my opinion, if Jake is a responsible forecaster, he should react in the exact same way: “Oh dear, you are right Mike, my (act of) prediction was not quite accurate, I should not have said what I did, given the data you have just offered.” The mere fact that Jake got lucky does not make his previous act immune to Mike’s objection. If that is right, it goes against MacFarlane’s determinacy intuition. I think, therefore, that if we focus on the acts of assertion, the indeterminacy intuition prevails, regardless of the context of assessment. How, then, to explain MacFarlane’s argument to the contrary: Should such a proof compel Jake to withdraw his assertion? Clearly not. If he had asserted that it was settled that it would be sunny on Tuesday, he would have to stand corrected. But he did not assert that. He just said that it would be sunny on Tuesday—and it is. (MacFarlane, 2014, p. 225)? I think that the argument goes down only if we understand “assertion” as the proposition asserted. In this case, Mike’s proof is indeed unconvincing. Mike’s proof shows that it was not necessary that it would be sunny on Tuesday, but it cannot undermine the brute fact that it was true (to prove it, you do not need quantum mechanics; it is sufficient to look out of the window). There is no doubt that Jake was right, regardless of Mike’s proof. That Jake was “right” does not mean, however, that he had the right to make the act of assertion that he had, but that the proposition he had asserted was true.39 Post factum determinacy intuition, therefore, is solid only if we focus on the content of assertion. Nonetheless, the ante factum indeterminacy intuition in this case is lost to me. After all, even if Jake was confronted with Mike’s proof on Monday, he could have easily (although irresponsibly) answered:

If I asserted that it is settled that it will be sunny tomorrow, then your proof would prove me wrong. But I don’t assert that! I just say that it will be sunny tomorrow. The weather tomorrow will show if my assertion is accurate or not.40

In my opinion, the legitimacy of Jake’s response proves that when we focus on the truth bearers, then the determinacy intuition prevails. Not only do we have post factum determinacy intuition, which MacFarlane likes to stress, but we also harbor ante factum

39The Czech language might be our guide here. If we wanted to say in Czech that Jake was right, we would say “Jake melˇ pravdu” which literally translates to “Jake had the truth.” 40 MacFarlane himself might have sensed this problem, since he admits that even if the director visited Jake just after he had made his prediction, then Jake “arguably would have had to retract” (MacFarlane, 2014, p. 225, emphasis mine).

91 CHAPTER 4. SEMANTICS OF BRANCHING REALISM determinacy intuition, which he does not seem to notice (or prefers to neglect). Observe that when Jake says: “It will be sunny tomorrow,” not only has he the right to say “My assertion was accurate, (i.e., true)” on the next day, experiencing the sunny sky, but even on the previous day, he can say: “If it will be sunny, then my assertion is accurate. (i.e., true).” The fact that the rain is a possibility for the day after the day of assertion does not shake the determinacy intuition in neither of these cases. Therefore, I think that determinacy and indeterminacy intuition are not really in conflict and that accuracy judgments actually go in pairs. When we focus on the proposition asserted, then judging an assertion accurate post factum, we should say that it was accurate (unbeknown to us) already at the context at which it was made. Otherwise, when we focus on the act of assertion, then if we initially judge an act of assertion inaccurate, we should abide by our judgment post factum. In face of this observation, there are two theoretical choices regarding the postsemantic theory and the notion of truth-at-context. One the one hand, we can let it follow our intuitions regarding the acts of assertion and accept supervaluational postsemantics, i.e., that every act of assertion of a future contingent is inaccurate (regardless of it being assessed before or after the indeterminacy is resolved). On the other hand, the postsemantic notion of truth-at-context can follow the intuitions regarding the content of the sentence asserted and accept a theory which allows some future contingents to be true. In any case, we do not need to succumb to relativism. Incidentally, I think that it is more reasonable to choose the second option. To explain the inaccuracy of assertions of (some) future contingents, we do not need to postulate that sentences asserted are lacking in truth value (or false). Notice that according to the truth norm of assertion, the truth-at-context of the sentence asserted is a necessary, but not a sufficient condition of accuracy for an act of assertion. Therefore, the fact that we sometimes take assertions of future contingents inaccurate can be explained by failure of another necessary condition. Most evidently, it is often very hard (arguably, even impossible) to justify that a contingent state of affairs will in fact take place. Thus, if some epistemic factor influences accuracy of acts of assertion, then we do not need to abandon bivalence to explain why the acts of assertions of future contingents are often inaccurate. They are inaccurate, because the claims asserted are not sufficiently well justified (even if they happen to be true). Moreover, another piece of data suggests that we actually should allow some future contingents to be true. Namely, people are often willing to accept assertions of future contingents as accurate, even if it is not objectively settled that the asserted state of affairs will occur. For example, if my friend checked the timetable and told me that our train departs in 30 minutes, I would have no trouble accepting his assertion as accurate (even though the trains are not physically determined to depart according to the schedule and I am perfectly aware of that). If we choose supervaluational postsemantics, we need to conclude that people are regularly wrong in such cases, i.e., we need to accept a version of an error theory. However, if we agree that some future contingents are true, we can grant that some assertions of future contingents are accurate indeed, thereby explaining why people talk as they do. An elaborate argument along these lines is presented by Besson and Hattiangadi(2014).

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4.7 History relativism

History relativism agrees with assessment relativism that the truth status of a sentence used in a context cannot be determined based solely on the meaning of the sentence and features of a context of use. According to MacFarlane’s relativism, we need to specify one more factor—the context of assessment. Only if another moment is identified as the perspective of assessment can we reasonably ask if the sentence is true, false, or neither. History relativism takes assessment relativism to the extreme. To use Richmond Thomason’s words, in history relativism we are “adopting a whole possible future for α as our perspective, rather than a single time in the future of α”(Thomason, 1970, p. 269. In appendix 7.6, I study the specific sense in which history relativism is indeed an extreme case of assessment relativism). Such attitude is characteristic to Belnap et al.(2001). 41 In their view, unless a specific possible history is specified, a future contingent cannot be evaluated in a given context. The authors express their attitude in the following words:

Then the truth of that sentence (given indeterminism) depends not only on the moment at which the sentence is uttered. It depends in addition on which future course of events—which history—is being considered. (Belnap et al., 2001, p. 225)

Nonetheless, the authors ferociously argue that a context does not initialize a history of evaluation (after all, an utterance is a part of many diﬀerent courses of events). As a result, it is simply meaningless to call the sentence, “There will be a sea battle,” true or false in the context in which the sea battle is contingent. As the authors put it

“M, mc |= Will: the coin lands heads” does not make sense. (Belnap et al., 2001, p. 155)

Only if one independently speciﬁes a continuation of a moment of utterance, one can ask about the truth value of the uttered sentence. Therefore, the history relativist answer to the question whether it is true that there is going to be a sea battle tomorrow is very philosophical: It depends! Relative to a sea-battly continuation, this sentence is true, relative to no-sea-battly continuation, it is false. It is as much as can be said regarding the truth value of a sentence uttered in a context. One can say that history relativists simply capitulate in face of the initialization failure. Given the evident indispensability of the history parameter in the Ockhamist semantic analysis, they simply duplicate it on the postsemantics level. We end up with a theory according to which the truth value not only of a sentence-at-index, but also of a sentence-at-context is relative to a history.

Deﬁnition 4.11 (History relativism postsemantics). m/h||−hφ iﬀ m/h |= φ. A minor problem with such a relativist approach is that it is prone to over-generalization (a similar point is raised by Sweeney, 2015, p. 9). Remember that for the sake

41Their terminology diﬀers from mine. When I write about a sentence being true at a context, Belnap et al. (2001) write about a stand-alone sentence being true at a context-initialized point of evaluation.

93 CHAPTER 4. SEMANTICS OF BRANCHING REALISM of uniformity the Ockhamist semantics always requires history-relativization, independently of whether the sentence is about a contingent future or not. Consider the phrase “M, mc |= Was: the coin lands heads.” Since the history is not speciﬁed, it is not a well-deﬁned expression in the framework of Ockhamist semantics. So, apparently, even when we have witness the result of the coin-toss, we cannot call the sentence, “The coin has landed heads,” true. It means that we can hardly ever call any sentence true in any context. In response to this objection, Belnap et al.(2001) observe that in many cases the truth value of a sentence at a context remains the same regardless of the choice of the history parameter.42 In Belnap et al.’s (2001) terminology, such a sentence is independent of the choice of history parameter and this parameter can be closed by independence (see, pp. 153– 154). The authors give an example of the present tense sentence, “Meg is hungry,” and they claim that the phrase

“M, mc |= Meg is hungry” makes sense (Belnap et al., 2001, p. 155).

Consequently, such a sentence can be ascribed a truth value at a context. It is the truth value which this sentence has for an arbitrary choice of the history parameter. In this respect, history relativism resembles supervaluationism. A sentence can be evaluated as true (false) in a context if and only if it is true (false) at all histories passing through the context-initialized moment. However, the authors distance themselves from supervaluation-like reading of their theory (see Belnap et al., 2001, p. 156). The diﬀerence is that, according to supervaluationism, the sentence which is true at some histories and false at others is described as missing a truth value, while for history relativism, it is meaningless to ask about the truth value at context of such a sentence. The appeal to independence permits ascription of truth values at a context to sentences about the past and the present. The strategy has a controversial consequence, however. It implies that the truth-aptness of a sentence in a context cannot be established a priori. It depends on whether the sentence is settled true or settled false in the context. For example, let us suppose that I play a coin tossing game with a conjurer. Unbeknown to me, the coin which is about to be tossed is double-headed. Then, the history parameter can be closed by independence and, therefore, the phrase

“M, mc |= Will : the coin lands heads” makes sense. However, as soon as the prestidigitator secretly replaces the rigged coin with a fair one, the truth ascription makes no sense again. It is a rather bizarre postsemantic mechanism. In my opinion, it would be preferable if the truth-aptness of sentence in a context depended more systematically on the features of the sentence. For the same reason, before the conjurer tosses the fair coin, one cannot meaningfully assign denotation to the deﬁnite description “The side on which the coin will land.” Nonetheless, the sentence “There is a side on which the coin will land” is true

42Strictly speaking, it is not guaranteed by Belnap et al.’s (2001) semantic deﬁnitions. They assume that truth value of atomic sentences depends both on a moment and a history (see Belnap et al., 2001, p. 227, def. 15). Thus, it is possible that the sentence “Was:the coin lands heads” is true relative to some histories passing though mc and false relative to other histories. We need to independently assume that the meaning of “lands” excludes such valuations.

94 CHAPTER 4. SEMANTICS OF BRANCHING REALISM in this very context. It is true, because it is closed by independence (in every history passing through the context there is exactly one side on which the coin lands). It is slightly peculiar result, however. Since it is true that there is a side on which the coin will land, the uniqueness condition for definite description seems to be satisfied. There- fore, we should be able to attributively43 use the description, “The side on which the coin will land,” to denote the unique side of the coin such that the coin will land on it. The authors claim, however, that the ascription of denotation makes no sense in this context. In my opinion, to avoid the above-mentioned problems, a history relativist should just admit that it is never meaningful to call a sentence true or false in a context (unless a specific history is specified, of course). More precisely, it is never meaningful to ascribe a truth value in a context to a sentence which contains any occurrence of a propositional variable unbound by a (historically) modal operator. It is an unusual approach, but easily understandable within Branching Realism. After all, for a Genuine Branching Realist the world is a modal object. So, the only sentences fitting to describe the world are the sentences which talk about the modal reality—the sentences which say what is possible and what is necessary. These sentences have the “appropriate format” to describe the world and they are truth-apt, while the non-modalized sentences, which simply say what is the case, are somewhat defective, since it is unclear what they refer to in a world “made of possibilities.” Regardless of whether the relativists would agree that the truth value of all (non- modal) sentences is history-relative, it is clear that at least the truth of future contingents is. It implies, however, that under some specifications of the history parameter, the sentence is true and under other specifications, it is false. It means that the truth value of a sentence in a context is highly arbitrary—it depends on something as whimsical as an entirely unmotivated choice of a parameter. It is not how we ordinary think about the truth values of future tensed sentences. We usually think that the truth value of a sentence used in a context should be grounded in something more solid than just an ad hoc decision of a semanticist who needs one history or another to do their job. It is also not entirely clear what this decision should consist of. When the relativists talk in terms of abstract Ockhamist semantics, they say that a possible future needs to be “posited” (Thomason, 1970, p. 271) or “supplied” (Belnap et al., 2001, p. 156). However, when they want to give a more down to earth description of the procedure, they often help themselves with intentional vocabulary. For example, Burgess writes that “The truth value of a future tense statement depends on which branch we think of as representing the course of events which is actually going to turn out to happen” (Burgess, 1979, p. 575, emphasis mine) and Müller(2014) echos that “we normally need to specify which of the equally possible futures we mean to refer to” (Müller, 2014, p. 354, emphasis mine) (we call this procedure “inner baptism” in Malpass and Wawer, 2012, p. 122). However, if all that is required to specify a possible history is an intention of a speaker, then making predictions true or false would be all too easy. Such a procedure has very little in common with everyday usage. If Leszek and Juliusz are about to visit Tomasz and Leszek says, before the visit, “Tomasz will offer white

43See the classical (Donnellan, 1966) for the distinction between referential and attributive uses of deﬁnite descriptions.

95 CHAPTER 4. SEMANTICS OF BRANCHING REALISM coffee,” then no-one can make this sentence true or false by fiat—just by thinking of this or that possible future. What, then, does relativization to a history consists in? To domesticate their proposal, Belnap et al.(2001) refer to certain formal analogy. They compare unmodalized sentences (in particular, those about the contingent future) with sentences of first-order logic containing unbound variables. They argue that the sentence like, “The summer will be hot,” is semantically analogous to the sentence “x is white.”44 Clearly, the context does not initialize the “default” assignment of free variables, and the the sentence, “x is white,” changes its truth value depending on which object is assigned to x. Therefore, the sentence with a free variable is closed neither by context, nor by independence. As a result, we cannot reasonably assess if the sentence “x is white” is true in a given context, unless a particular assignment is explicitly specified. The authors claim (Belnap et al., 2001, p. 155, thesis 6–6) that the future tensed sentences are exactly analogous. In their case, the context also does not initialize the “default” value of the history parameter, so the sentence, “The summer will be hot,” changes its truth value depending on which history is chosen. Therefore, the contingent sentence is closed neither by context, nor by independence. Consequently, we cannot reasonably assess if the sentence, “The summer will be hot,” is true in a given context, unless a particular value of a history is explicitly specified. Notice that we are not worried, when we realize that the truth of the expression, “x is white,” depends on an entirely arbitrary choice of an assignment, because we recognize that this expression is merely a by-product of compositional analysis of the sentences like, “Everything is white,” “Something is white,” or “Most things are white.” Even though the truth of “x is white” does depend on the assignment, the truth of the last three sentences does not depend on a particular choice of assignment parameter. And it is the truth of these three sentences that we truly care about. By analogy, the expression, “The summer will be hot,” might be thought of as a by-product of compositional analysis of sentences like “It is settled that the summer will be hot,” “It is possible that the summer will be hot,” or “Most likely, the summer will be hot,” whose truth value is history-independent. This line of thought is not entirely convincing, however. When we are faced with the expression, “x is white,” we immediately recognize that it is not even a sentence of English. It is not an accident that in the traditional logical vocabulary introduced by Tarski(1933), “ x is white,” it is not called a sentence, but an open formula.45 He chose the vocabulary, since he believed that when we study formal languages, we should remember that: We shall always ascribe quite concrete and, for us, intelligible meanings to the signs which occur in the languages we shall consider. The expression which we call sentences still remain sentences after the signs which occur in them have been translated into colloquial language. (Tarski, 1956, p. 167) Clearly, “x is white” does not look like a sentence of English. The problem with

44To make these cases analogous also on the syntactic level, you can assume that in its deep structure the sentence, “The summer will be hot (in h),” has a free variable ranging over histories which is unbound by any “modal” quantiﬁer. 45Wolenski´ (2003, p. 129, vol. III) explains that Tarski inherited the view from Le sniewski.´

96 CHAPTER 4. SEMANTICS OF BRANCHING REALISM this expression is that we cannot reasonably call it true or false. Tarski was fully aware of the difficulty and for this reason he employed the notion of satisfaction, rather than truth, in his recursive analysis of quantifiers. In contrast, when faced with the expression, “The next summer will be hot,” we immediately realize that it is a full-fledged sentence of English. When we know at which moment it was used, we do not have the nagging feeling that it is somehow defective or it needs to be supplemented with additional information to make sense. It is not likely that its role can be reduced to a by-product of semantic analysis. One could even add (as I will in chapter6) that we understand perfectly well what needs to happen for this sentence to be true. Namely, the next summer needs to be hot. The truth value of this sentence is relative, but only in a trivial sense: it depends on what will in fact happen (just as the truth of past tense statements depends on what in fact happened). Thus, the analogy that Belnap proposes is imperfect. “x is white” is not a sentence, and it is not truth-apt, but satisfaction-apt, while “The next summer will be sunny” is a sentence and it seems to be truth-apt. To parry this argument, Belnap et al.(2001) respond that the di fference between these two cases resides in pragmatics rather than in semantics. Namely, the sentence, “The next summer will be hot,” is assertable, while “x is white” is not (cf. Belnap et al., 2001, p. 157, assertability thesis 6–7). What makes the difference is not the first sentence having a truth value at a context and the second lacking it, but their different “modal profile” . In case of a sentence about the future, it is settled that it will have been settled true or settled false. By the next Autumn, Each history containing the utterance of this sentence will have given the verdict whether it is settled that the sentence was true. (Of course, in case of future contingents, different histories will have given conflicting verdicts, which should be expected from a relativist theory.)46 The difference in the “modal profile” can be used to explain why history-open sentences like “The next summer will be hot” are assertable, while the assignment-open sentences like “x is white” are not. The authors argue that when people assert sentences, they enter the market of linguistic transactions in which they exchange rights and obligations. Assertions are comparable to wagers. The assertor “bets” that the things will have settled as she said they would. In the parts of the world in which things thus settle, the assertor is vindicated, while in the parts of the world in which they settle otherwise, she is impugned. This kind of wager cannot be made using the expression of the type, “x is white,” because it never gets settled whether it is true that x is white. The procession of time does not settle issues like that, whereas the procession of time does settle, in every alternative future, the issue of the next, hot summer. The notion allows the authors to deal with another problematic feature of their theory. Remember that the concept of truth at a context was meant to serve an important theoretical goal. It was meant to serve as an intermediary between the technical apparatus of semantic analysis and the pragmatics of asserting. After all, the accuracy of an act assertion was supposed to depend, through the truth norm, on the truth-in-context of the sentence asserted. However, Belnap et al.(2001) argue that it makes no sense to say that a sentence at a context is true or false. To overcome the obstacle, Belnap

46To explicate the idea that what is said at moment m is settled true at a later moment, Belnap(2002b) develops the formal apparatus of double-time reference.

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(2002b) “postpone” the assessment of accuracy of an act of assertion until what the sentence says gets settled. Belnap argues that as soon as the sentence asserted gets settled true, the assertor is vindicated (her assertion turns out to have been accurate), as soon as it is gets settled false, the assertor is impugned (her assertion turns out to have been inaccurate), and as long as the sentence asserted is neither settled true, nor settled false, the assertor is neither vindicated, nor impugned. So, the vindication of an act of assertion made at one moment depends on its modal status at another moment (i.e., whether it is settled at another moment). And the same act of assertion can be vindicated from the perspective of one moment and impugned from the perspective of another moment.47 As an example, take an assertion of the sentence, “The coin will land heads.” It is vindicated at the moment at which the coin lands heads, it is impugned at the moment at which it lands tails, and it is neither vindicated, nor impugned at the moment at which the toss has not yet been resolved. Then, the only difference between assertions about the past and about the future is that the acts of assertions about the past are accurate/vindicated “immediately,” i.e., at the same moment at which the assertion is being made, while the acts of assertions about a contingent future are vindicated at later moments. When we associate the notion of accuracy of an act of assertion with Belnap’s notion of vindication, we do not need the notion of truth-at-context to do the job of the middleman between Ockhamist semantics and pragmatics of assertion. In particular, the accuracy of the assertion (about the past, present, or future) does not depend on whether the sentence asserted is true at the context. It depends on whether what the sentence says is settled true. This line of thought can be used to alleviate the problem of arbitrariness of the truth value. Remember that the truth value of a sentence about a contingent future used at a particular context depends on a whim—the entirely unmotivated choice of a particular history parameter. However, given the theory of assertion just sketched, a history relativist can baldly answer: so what! After all, the truth value of a sentence at a context has little to do with accuracy/vindication of the act of assertion made with a use of the sentence. The accuracy of an act of assertion depends on whether what is asserted is necessary or contingent, not on whether it is true or false. And what is necessary and contingent is no longer arbitrary—it is decided by how the world is. Therefore, the arbitrariness of the truth value of a sentence asserted does not affect the practice of assertion. Consequently, the theory of assertion of Belnap(2002b), based on the idea of a double-time reference, has a lot to offer to history relativism. It can be used to explain why we draw a distinction between future contingents and formulas with free variables. It explains what people do, when they seem to be talking about what will actually be the case. It explains how to detach the notion of vindication of an act of assertion from the truth at a context of the sentence asserted and attach it to the settled truth of the sentence. Finally, it tones down the objection of arbitrariness of the truth value of a sentence in a context. The speech act theory so construed can teach us to stop worrying and learn to love history relativism.

47This mechanism clearly resembles the assessment relativism of John MacFarlane.

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4.8 Local relativism

According to Belnap et al.(2001), it is unintelligible to ascribe truth value to a sentence in a context, unless a history is specified. At least a few theorist, including some Belnap’s most faithful followers, observed that even if we grant relativity of truth, it is still hard to believe that such a truth should be relative to histories. After all, even if we so desired, it is unclear how we should specify a history to which we intend to relativize the truth value. We do not have anything like “names” of possibilities, which could be used for the purpose. We also cannot refer to a history directly, by pointing to “this” or “that” continuation. It seams that the only way to specify a possible history is by description. However, histories are very “large” objects (they are complete courses of events, from the dawn to the dusk of time) and to describe a history is not an easy task. In many cases, it might require an infinitely long story to uniquely specify a history. Consequently, even if in some sense a sentence is true at a context, grasping this sense would require abilities well above finite human beings. Therefore, those who look for a “human-scale” notion of truth at a context need to substitute histories with somewhat smaller, more approachable objects. Also, it is very much in spirit of Branching Realism to “go local,” that is, to em- phasize the perspective of a moment that has a somewhat limited vista on the treelike world. Such an attitude has been detectable already in (Prior, 1967) and (Thomason, 1970), but it was most openly stated by Belnap et al.(2001):

It would seem better to begin with a theory about more local incompatible possibilities, such as those available within ten or ﬁfteen minutes, or available within ten or ﬁfteen seconds, or (best) available immediately. (Belnap et al., 2001, p. 197)

A number of authors have transformed this piece of advice into full-blown semantic theories which I will discuss in the four sections below. On the formal level, the four approaches are variations on the theme of Ockhamist semantics. In Ockhamism, we relativize truth to a moment/history pair; the theories below preserve the general mechanism, but modify or limit the notion of a history. The ﬁrst project of Alberto Zanardo does not explicitly address the initialization problem, so the author might not necessarily classify himself as a relativist. However, I decided to incorporate his research in this chapter, since it serves as a natural introduction to subsequent theories. The next idea originates with Peter Øhrstrøm and his collaborators. It is an interesting case, as, otherwise, they are famous anti-relativists in the philosophy of future contingents. Finally, the next two projects of Tomasz Placek and Thomas Müller openly respond to Belnap et al.’s (2001) appeal and device theories “about more local incompatible possibilities.”

4.8.1 Recognized possibilities Let me begin with the theory of Roberto Zanardo(1998). The author replaces the notion of a history with an alternative concept of recognized possibility. Although the name carries an epistemic overtone, the author does not primarily construe of his

99 CHAPTER 4. SEMANTICS OF BRANCHING REALISM models in epistemic terms. Recognized possibilities are clusters of histories grouped together by a factor which the author calls “indistinguishability function” (henceforth, I-function). The set of recognized possibilities (RP) is moment-dependent—different moments might recognize different clusters of histories as possibilities. His paper is technical in nature and the author does not provide any specific interpretation of his notions. All we know is that for each moment m the set RPm is a partition of the set of histories passing through m.48 It is also assumed that I-function, encoding recognized possibilities, has “good memory,” i.e., if the function distinguishes two histories at a given moment, then the histories cannot become indistinguishable at a later moment. In symbols, let M B hM,

Definition 4.12 (undivideness, splitting). Two histories h1 and h2 in Hm are undivided 0 at m, h1≡mh2, iff ∃m0>mm ∈ h1 ∩h2. Two histories h1 and h2 split at moment m, h1⊥mh2, iff h1, h2 ∈ Hm and h1.mh2.

It is immediate to verify that ≡m is an equivalence relation on m and thus it induces a partition Πm of Hm into classes of histories which are undivided at m. Definition 4.13 (Elementary possibilities). Elementary possibilities open at m are the members of the set Πm B {H|H ⊆ Hm and ∀h1,h2 ((h1 ∈ H & h2 ∈ H) ⇔ h1 ≡m h2)}. In his paper, Zanardo proves a remarkable result for a semantics which extends Ockhamism with an extra modal operator sensitive to elementary possibilities. Such a language is more expressive than standard Ockhamism. In particular, it is inexpressible in Ockhamism, and expressible in the enriched language, that “each history undivided with the history we are now considering at this moment verifies φ at some future moment” (cf. proposition 3.1, p. 303). The I-function is not limited by elementary possibilities, however. It can recognize as a single possibility two histories which belong to different elementary possibilities open at a moment (this reading allows for the epistemic notion of indistinguishability). We can also use the I-function to model the idea of histories indistinguishable by a

48The author phrases his deﬁnitions in terms of equivalence relation on the set of histories passing though the moment.

100 CHAPTER 4. SEMANTICS OF BRANCHING REALISM choice of an agent. Then, the I-function is just a choice function in the sense of Belnap et al.(2001). Zanardo(1998) uses I-functions to provide a new semantics for a language with temporal and modal operators. He enriches the notion of a model, so an I-model is a triple hM, <, I, Vi, where I is an I-function. The sentences are no longer evaluated at moment/history pairs, but at pairs consisting of a moment and a possibility recognized at that moment, i.e., m/r such that r ∈ I(m). No single recognized possibility is distinguished as the default parameter of evaluation at a given moment; thus, a truth value of a sentence at the moment is relative to the choice of one of the possibilities recognized at that moment. Hence, unless further developed, Zanardo’s theory can be subsumed under the concept of relativism. The notion of recognized possibility is crucial to semantic evaluation of sentences of the modal language. Zanardo studies a sentential language with four modal operators: ^, P, F, G. Operators F and G are introduced independently, since, just as in Peirceanism, they are not inter-definable. The influence of I-function is detectable already on the basic level of the valuation function, which maps propositional constants into sets of pairs hm, ri, rather than into sets of moments. Thus, the truth value of a propositional constant at a given moment might change from one recognized possibility to another.49 However, the influence of I-function is most explicit in the interpretation of modal operators. Zanardo’s theory is a specific blend of an Ockham-like notion of possibility with a Peirce-like notion of future tense. Let me denote by rh the recognized possibility r containing history h. Then, truth clauses for modal operators can be defined as follows: Definition 4.14.

rp • m/rh|= p iﬀ m/rh ∈ V(p), for p ∈ Atom; • standard deﬁnitions of classical connectives;

0 rp 0 0 0 0 0 0 rp • m/rh|= Fφ iﬀ ∀h ∈rh ∃m |m ∈h & m >mm /rh |= φ;

rp 0 0 rp • m/rh|= Gφ iﬀ ∀h0∈r∀m0|m0∈h0 (m > m ⇒ m /rh0 |= φ);

rp 0 rp • m/rh|= Pφ iﬀ ∃m0

rp 0 0 rp • m/rh|= ^φ iff ∃h ∈Hm m/rh |= φ. Possibility operator shifts the point of evaluation from one possibility recognized at a moment to another, and tense operators basically work as Peircean operators limited to a single recognized possibility.50 The notion of a history is used in Zanardo’s definition, but it does not feature as a semantic parameter of truth. The author shows (proposition 5.3, p. 314) that the truths of his semantics coincide with the truths of Ock- hamism iff I = IO, i.e., iff the I-function distinguishes every history at every moment as

49It is a consequence of Zanardo’s general approach to valuations in the branching setting. Even when he studies Ockhamism, he usually relativize the truth value of propositional constants to moment/history pairs, rather than to moments (see, e.g., Zanardo, 1985, 1996). I discuss this issue in section 3.3. 50Observe, however, that when a temporal operator shifts the moment of evaluation, the rp of evaluation 0 0 might change as well. If m < m , then rh at m might well diﬀer from rh at m .

101 CHAPTER 4. SEMANTICS OF BRANCHING REALISM a distinct possibility. The truths of his system coincide with the truths of Peirceanism iff I = IP, i.e., when the I-function has no distinctive power whatsoever—for every m, I(m) = {{Hm}}. Moreover, Zanardo proves (proposition 5.4, p. 314) that I-structures based on IO and IP are characterizable within the modal language he introduces. A thorough formal investigation of Zanardo’s system, including a proof of completeness, can be found in (Gatto, 2015). The semantics of Zanardo is a remarkable technical development, but I would claim that it is not useful for our project, i.e., to interpret future contingents and investigate interactions of tense with modality. First of all, Zanardo’s theory inherits all the problems of Peirceanism described in section 4.3. Even if the interpretation of temporal operators is relativized to recognized possibilities, these possibilities might still contain many incompatible histories. Since the semantic of future operator requires quantification over all the histories within the chosen recognized possibility, we can reconstruct all the problems of Peirceanism in this semantics. Secondly, Zanardo looses the crucial advantage of Peirceanism (or any other kind of modalism). In Peirceanism, the truth value of a sentence depends uniquely on a moment of evaluation. Therefore, we do not need to introduce history as a parameter of evaluation. Consequently, we do not need to answer the initialization problem. However, Zanardo does introduce an extra parameter of evaluation: recognized possibility. Just as no history is designated in the Ockhamist semantics, no recognized possibility is designated in Zanardo’s semantics. Therefore, as long as Zanardo does not provide an additional story about which recognized possibility (if any) is supposed to be chosen as a default modal parameter of evaluation, the initialization problem returns at full strength. In Consequence, we we loose the intuitive validities of Ockhamism, but preserve the initialization problem resulting from the semantics. Not a very fair deal, as far as the semantics of future tense is concerned. In fact, it gets worse. Thanks to the introduction of recognized possibilities as parameters of truth, Zanardo can introduce a non-trivial modal operator ^ which shifts the point of evaluation from one recognized possibility to another. Then, we can investigate the interaction of this new modality with temporal operators. And the results are far from perfect, if we aim to model interaction of time with possibility. I call the concerns I have identified problems of emerging possibilities, vanishing necessities, and emerging necessities. First, let us look at a sentence expressing the emergence of new possibilities: It will be possible that there will be a sea battle, but it is not possible that there will be a sea battle. (F^F p ∧ ¬^F p) It turns out that this sentence is true under Zanardo’s semantics in a very simple model (fig 4.1) Colors indicate the recognized possibilities generated by the I-function at particular moments, i.e., I(m1) = {{h1, h2}, {h3, h4}}; I(m2) = {{h1}, {h2}}; I(m3) = {{h3}, {h4}}. Letter p indicates the only two points in the model (m4/{h1} and m5/{h3}) where p is rp true. Observe now, that m1/rh2 |= F^F p, since in every history in rh2 (that is in the set

{h1, h2}) there is a future moment (e.g., m2) at which ^F p is true. ^F p is true at m2/rh1 and m2/rh2 , since there is a possibility recognized at m2, i.e., {h1} which makes F p true. In fact, F^F p is true at m1.

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h1 h2 h3 h4 p m4 ¬p p m5 ¬p

m2 m3

Figure 4.1: Emerging possibilities.

At the same token, ¬^F p is true at m1, since there is no such recognized possibility at m1 that within the recognized possibility F p would be true. In the “left” recognized possibility, h2 testiﬁes against the truth of F p at m1, in the “right” possibility, h4 does the same. In fact, it means that ¬^F p is true at m1, thus, we can strengthen the problem of emerging possibilities:

It is settled that it will be possible that there will be a sea battle, but it is settled that it is not possible that there will be a sea battle. (F^F p ∧ ¬^F p)

The problem of emerging possibilities sounds even graver, when phrased in terms of metric operators:

Tomorrow, it will be possible that on the next day there will be a sea battle, but it is impossible that there will be a sea battle the day after tomorrow. (F1^F1 p ∧ ¬^F2 p)

We can also look at the problem of emerging possibilities from the perspective of one of the possibilities that have emerged. For example, at the moment m4, you can truly say:

There is a sea battle, but in the past, it was not possible that there would be one. (p ∧ P¬^F p).

The sentence is true at m4/rh1 , since there is a sea battle (p) raging at that moment, but at a past moment m1, there was no recognized possibility such that the sentence F p would be true in this possibility. The problem of emerging possibilities is paired with a similar problem of vanishing necessities. Consider the model depicted in ﬁgure 4.2.

The diﬃculty with this model consists in the fact that at m1/rh1 , the following two sentences, which sound contradictory, are jointly true:

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h1 h2 h3 h4 p m4 ¬p ¬p p m6

m2 m3

Figure 4.2: Vanishing necessities.

1. It is settled that in two days there will be a sea battle, or there will be no sea battle. (F2 p ∨ F2¬p). 2. Tomorrow, it will not be settled that on the following day there will be a sea battle or there will be no sea battle. F1¬(F1 p ∨ F1¬p).

Moreover, if we shuﬄe the valuation of the previous model slightly, see ﬁg. 4.3, we can also introduce the phenomenon of “emerging necessities” and verify a conjunction which, to my ear, sounds like a contradiction. Namely, the following three sentences are jointly true at m1:

1. It is impossible that tomorrow it will be settled that the sea battle will happen. (¬^F1F p) 2. It is impossible that tomorrow it will be settled that the sea battle will not happen. (¬^F1F¬p) 3. It is settled that tomorrow either it will be settled that the sea battle will happen or it will be settled that it will not happen. (F1(F p ∨ F¬p))

I believe that the few examples mentioned above demonstrate that the operators F and ^, as interpreted in Zanardo’s semantics, do not square well with the ordinary notions of future and temporal possibility. Not only are the questionable results of Peirceanism conﬁrmed, but in addition, the interactions of time with possibility are far from expected. Moreover, it does not provide an easy way to answer the initialization problem. To conclude, Zanardo’s semantics is an interesting and important formal development which might give a good account of some sort temporally dynamic agentive possibility (what is within a power of the agent) or epistemic/doxastic possibility (what is consistent with subject’s knowledge/beliefs which develop in time). It is rather un- helpful, however, if our project is to investigate future contingents and interaction of time and historical possibility.

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h1 h2 h3 h4 p p ¬p ¬p

m2 m3

Figure 4.3: Emerging necessities.

4.8.2 Counterfactual branches The next relativist proposal has an unexpected source. It was introduced in a series of articles co-authored by Peter Øhrstrøm (see Braüner et al., 1998, 2000; Øhrstrøm, 2009), who had defended non-relative truth of future contingents since (Øhrstrøm, 1981). I will base my exposition of this relativist theory on the version presented in (Braüner et al., 2000). The semantics they propose has its roots in their earlier project which explicitly aimed at a bivalent and non-relative treatment of future contingents (I recapitulate the details of the project in section 5.3.4 of the next chapter). They abandoned the earlier version of their semantics, since it encountered certain formal difficulties, in particular, (a) it did not validate the sentence φ → HFφ, which is a basic axiom of almost every temporal logic, and (b) it was difficult to introduce operators of historical modality into their earlier system. The authors tried to overcome both these difficulties with a degree of relativization. The project of Øhrstrøm and his collaborators originates with a very anti-relativist attempt to answer the initialization problem. They start with an idea that every possible moment on the tree of possibilities has its actual future. The actual future of a moment should be used to ascribe truth values to sentences about the future of this moment. To model the idea formally, the authors introduce a function—the so-called TRL-function—which assigns a history to every moment in the branching structure; this history is the actual future (and past) of that moment. Not every such function is appropriate. In the process of dispute with Belnap and his collaborators (recapitulated in section 5.3.4), the authors settle for the following constrains on TRL-function:

Deﬁnition 4.15 (TRL-function). Let hM,

1. ∀mm ∈ TRL(m);

0 0 0 2. ∀m,m0 (m < m ∧ m ∈ TRL(m)) → TRL(m) = TRL(m ) .

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The first conditions simply states that the actual history of moment m contains moment m and the second dictates that actuality “is stable,” which means that if moment m0 is in the actual future of moment m, then the actual history of m0 is the same as the actual history of m. We can use the notion of the TRL-function to indicate whether a moment is faithful to a history h. Definition 4.16 (Moment (un)faithful to a history). Let m ∈ h. We say that m is faithful to h iff TRL(m) = h, otherwise, m is unfaithful to h. At most one of the histories in the model contains uniquely faithful moments. Every other history has been “betrayed” by some of its moments. Some histories have been betrayed by its moments with many different histories. Worse still, there are models and TRL-functions which contain histories consisting of uniquely unfaithful moments.51 The definition of TRL-function guarantees that if we take a given moment m and a set of elementary possibilities open at that moment Πm (see def. 4.13, p. 100), we will find at most one history h in each elementary possibility such that all moments in h above m are faithful to h. Braüner et al.(2000) call these histories “counterfactual branches” (p. 204) and formally define them as follows: Definition 4.17 (Counterfactual branches). 0 0 C(m) = {h ∈ Hist|m ∈ h ∧ ∀m0>m(m ∈ h → TRL(m ) = h)}. TRL(m) is one among the counterfactual branches, while in other histories in C(m), moment m is their last unfaithful moment. Braüner et al.(2000) use set C(m) to offer a new semantics for the standard tempo-modal language. They begin with a branching structure enriched with a TRL-function hM, <, TRLi. Then, they define a valuation function as a map from the set of propositional variables to the power set of M, and then define truth of a sentence at a pair m/h, where h ∈ C(m). It means that in their semantics sentences are true with respect to pairs consisting of a moment and counterfactual branch passing through that moment. Then, Braüner et al.(2000) basically accept Ockhamist clauses for all the connectives, including the operator F:

CB 0 0 0 CB m/h|= Fφ iﬀ ∃m0 (m > m & m ∈ h & m /h|= φ) The only operator that exploits the notion of a counterfactual branch is the operator of historical possibility:

CB 0 CB m/h|= φ iﬀ ∀h0∈C(m)m/h |= φ It means that historical modalities are evaluated with respect to the counterfactual branches passing through the moment, rather than with respect to all histories passing through the moment, as is the case in Ockhamism. Thanks to this maneuver, the authors can overcome the the problem of previous TRL theories. In particular, the sentence φ → HFφ is valid in their semantics, which certainly is a desirable result (I discuss the semantics in more detail in section 5.3.4, p. 146).

51 To construct such a model, you can use the branching structure depicted on page 216. The history hω consists of uniquely unfaithful moments if we assume that ∀mTRL(m) = hn for some n ∈ N.

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I am unsatisﬁed, however, with the semantics of Braüner et al.(2000) My major worry is conceptual in nature. I think that the authors were carried away by the formal investigations and lost sight of the superior, theoretical aim. They wanted to provide a semantics which incorporates the idea of the “true future” of a moment (which they associate with the philosophy of William of Ockham). Peter Øhrstrøm(2009) describes this idea as follows:

A proper theory (. . . ) would have to include the idea of a proposition being true relatively to a moment of time (without any speciﬁcation of a chronicle and of a given selected history). Let us therefore investigate a truth- theory, which includes the idea of a true future in this sense. (Øhrstrøm, 2009, pp. 26).

Thus, a “true futurist” theory should incorporate the idea that a sentence is non- relatively true or false at a given moment. Nevertheless, the semantics of Braüner et al. (2000, repeated by Øhrstrøm, 2009) does not incorporate the idea of a sentence being true or false at a moment of time. The truth value of a sentence at a moment crucially depends on the choice of a counterfactual branch! The semantics of the future operator does not privilege any counterfactual branch as the default interpretation of F. Therefore, the theory of Braüner et al.(2000) turns out to be a history relativism of Belnap et al.(2001) with a slightly modified definition of historical possibility. 52 Thus, the authors abandoned “true futurism” to preserve the validity of p → HF p. If one decides to abandon the true futurism, however, one should go all the way to the intuitive Ockhamism, rather than stop half-way with a quasi-Ockhamism which uses counterfactual branches instead of histories. There are good formal reasons for not stopping half-way. Namely, even if it validates φ → HFφ, the semantics of Braüner et al.(2000) still generates counter-intuitive predictions. The authors themselves identify the formal difficulty connected to their semantics. Formally, it is similar to the problem of emerging possibilities, discussed in context of Alberto Zanardo’s theory. Remember that the definition of possibility operator takes into account only the counterfactual branches passing through m (i.e., those in C(m)). Thus, even if a given sentence φ is true at no such history and therefore ^φ is false at m, it might well happen that if you follow one of the counterfactual branches, let us call it h, to a later moment m0, the sentence φ might be true at some counterfactual branch at m0. Due to this feature, we can conclude that the sentence

F^F p → ^FF p is not valid under this semantics. Thus, we can sometimes truly say:

It will be possible that there will be a sea battle, but it is not possible that there will be a sea battle. 52The most surprising fact about this semantics is that the model contains information regarding what will actually follow the given moment (i.e., the TRL-function), but this piece of information is not used to interpret the future operator. The history TRL(m) is just as good as any other counterfactual branch as far as the interpretation of F is concerned.

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Øhrstrøm(2009) believes that no argument has been provided in favor of the validity of such a sentence and adds that “As long as no such arguments have been established, the true futurist position must be regarded as a possible answer to the problem of future contingency” (p. 30). I have already noted that this semantics hardly deserves that name of “the true futurist position,” but I also disagree with Øhrstrøm’s assessment of the logical status of the sentence F^F p → ^FF p. Doubtlessly, the question of validity is largely a matter of intuition (and my intuition speaks in favor of this sentence), but I think that there are good reasons to believe that falsifiability of this sentence is an unwelcome consequence. Our purpose, after all, is to model a temporal notion of possibility: what is possible given the state of the world at the moment. If we say that at moment m a sea battle cannot ensue, it means that the state of the world at m forecloses any future sea battle. Therefore, we cannot reasonably add that the state of the world at m admits a future state, say m0, which admits the sea battle. Let me introduce the dyadic relation between moments: R(m, m0) iff it is possible at m that m0 will follow. It is most natural to expect that such a relation is transitive—when m admits m0 and m0 admits m00, then m admits m00. And if the relation R is transitive, then F^F p → ^FF p is valid, which conflicts with the semantics of Braüner et al.(2000). Also, the semantics of Braüner et al.(2000) has, what I call, the problem of vanishing possibilities. Imagine that we study a radioactive particle which might (but does not have to) decay at any moment within a period of 2 seconds. Let me represent the situation as a tree with a single “no-decay” history h and a continuum of “decaying” histories branching off the original history at every single moment in the period [0,2]. The model is depicted in figure 4.4. I will denote by mx, x ∈ [0, 2] the moment in history h that which takes place at time x. Also, I will denote by hx the history that branches off h at moment mx. By φx, I will denote the sentence which says, “The particle decays at time x” (it might be encoded by Atx(p ∧ G¬p), where p stands for “The particle exists”). For example, h 1 is a history in which the particle decays after 3 one third of a second, at moment m 1 . Also, h 1 is the only history in which φ 1 is true. 3 3 3 Every moment mx in history h, in the period [0, 2], is an indeterministic event. At any such moment m, there are two elementary possibilities: either the particle decays at m or it continues to exist. Regarding the temporal topology of decay, I assume that for any x, mx is the last moment in hx at which the particle has not yet decayed (there might be no first moment at which the particle does not exist). Let us assume that TRL(m0) = h1, which means that the particle actually decays at time 1 (implying that ∀x∈[0,1]TRL(mx) = h1). Then, there are exactly two histories in C(m0), h0 and h1. These are the only two histories which satisfy the definition of counterfactual branches (see def. 4.17, p. 106). Given the semantics of , we need to conclude that m0 |= (φ0 ∨ φ1), i.e., it is necessary that the particle decays either right now or it will decay exactly 1s from now. Such a semantic conclusion is rather peculiar, given that the particle might well decay at any instant from the period [0, 2]. As far as the semantic theory is concerned, all the possibilities but two seem to have vanished. The authors could defend their theory by saying that the remaining possibilities have not yet emerged, but they will. This strategy would only by partially effective though. Indeed, for any x ∈ [0, 1], the sentence ^F^φx is true at m0 (because for any CB x ∈ [0, 1], mx/h1|= ^φx). The strategy will not work, however, for any x ∈ (1, 2]. For those values, the sentence G¬φx is true at m0 (because at m0 the operator G may

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h h1 h 1 h0 3

m 1 3

Figure 4.4: Possible decay.

follow only h0 or h1). We may try to express the later decays with other sentences, like ^F^F^φx. How- ever, the procedure does not need to work; it is eﬀective iﬀ h ∈ C(m1). However, there is no guarantee that h, along with h1, is the counterfactual branch at m1. Altogether different pattern of actualization might emerge. Assume for example that at the distance [0,1.5), the TRL function is guided by the following pattern:

TRL(mx) = hy, where y is the ﬁrst element greater than, or equal to, x in a 1 1 sequence (1 2 − n+1 )n∈N\0 1 This means that for x ∈ (0, 1], TRL(x) = TRL(m0) = h1. For x ∈ (1, 1 6 ], TRL(x) = 1 1 1 3 h 1 , for x ∈ (1 , 1 ], TRL(x) = h 1 , for x ∈ (1 , 1 ], TRL(x) = h 3 , etc. 1 6 6 4 1 4 4 10 1 10 If the TRL function is guided by the rule, one can express at m0 the idea that the particle decays at:

• x ∈ (0, 1] with a sentence ^F^φx;

1 • x ∈ (1, 1 6 ] with a sentence ^F^F^φx; 1 1 • x ∈ (1 6 , 1 4 ] with a sentence ^F^F^F^φx; 1 3 • x ∈ (1 4 , 1 10 ] with a sentence ^F^F^F^F^φx; • etc. Hence, we can somehow express the idea that the particle might decay before 1.5s. Nonetheless, the procedure of reiteration of F^ will never get us beyond this time. From the perspective of moment m0, we cannot “reach” the histories branching oﬀ h later than at 1.5s. The particle is sill as indeterministic and it is possible for it to decay at any moment between m0 and m2. However, the purely contingent matter, regarding which histories get actualized, prevents us from expressing the idea that the particle might decay after 1.5s. These possibilities somehow escaped our conceptual grip. The phenomena of

109 CHAPTER 4. SEMANTICS OF BRANCHING REALISM emerging and vanishing possibilities make me doubt whether I understand what notions are actually modeled by the semantics of Braüner et al.(2000) and how they are related to the ordinary notions of possibility, actuality, and time.

4.8.3 Continuations Another important relativist semantic was proposed by Tomasz Placek(2011). His project is an embodiment of Belnap et al.’s (2001) motto, “It would seem better to begin with a theory about more local incompatible possibilities” (Belnap et al., 2001, p. 197). He rephrases it as follows:

[L]ocal objects like continuations and events ﬁt the spirit of branching theories better than structures as large as histories. (Placek, 2011, p. 738)

To construct such “local” objects, the author begins with an ordinary branching structure hM, ≤i of moments ordered by the tempo-modal (Placek, 2011, p. 739). How- ever, his main purpose is to investigate the structure without the notion of a maximal course of events, i.e., a history. He substitutes the notion of a history with a new notion of a continuation of an event. A paradigmatic continuation is much less specific than a history; in particular, a single continuation is usually compatible with many mutually incompatible, further developments. To use the author’s example, the continuation of my opening the fridge might be extended to my taking a bottle of milk out of the fridge and it might extended to my taking a bottle of beer out of the fridge. Placek motivates his continuation-based theory in three ways. Firstly, the notion of a history is epistemically unattainable. It is not within our cognitive powers to grasp a course of evolution of the entire universe in all its minute details; in contrast, the notion of continuations can be measured with human yardsticks. Secondly, what follows from the previous point, the continuations seem to fit more naturally what we think about when we wonder what can happen. We routinely envisage possibilities which are very limited in time and space and which are highly under-specified in detail. Thirdly, the notion of a history (at least the one modeled by Belnap, 2003b), stands in a conflict with some models of general relativity, since the axioms of (Belnap, 2003b) require that every two events in a single spacetime have a common upper bound, while some models of general relativity violate this requirement (see Placek, 2011, p. 738). The semantics of possible continuations might be seen as an improvement of history relativism. In history relativism, as I have already stressed, the ascription of the truth status to a prediction of a contingent event makes no sense in a given context, unless a possible history is specified. Specification of a history seems to be too large a task, however, for an ordinary human being. Consequently, we would need to conclude that ascription of truth value to contingent predictions makes no sense for ordinary mor- tals. The semantics of continuations is relativist as well. In Placek’s view neither the meaning of a sentence, nor the context of its use specify the default continuation of use “for quite similar reasons as there is no history of use in the BT semantics—cf. (Belnap et al., 2001)” (Placek, 2011, p. 756). Therefore, also in this semantics, the ascription of truth value to a future contingent makes no sense, unless the relevant continuation

110 CHAPTER 4. SEMANTICS OF BRANCHING REALISM is “externally” provided. The only hope is that it is easier to supplement a continuation than a history, as it is easier to envisage a small, cognitively more manageable continuation, than a large, unimaginable history. Thus, Placek(2011) sets himself the goal of developing a formally precise theory of continuations. The theory becomes quite complicated, when we take into account the spatial dimension of branching structures. Fortunately, when we limit our investigation to interaction of modality with time, it is much easier to grasp the idea behind the theory. The author himself admits (p. 752) that our linguistic intuitions do not take into account the relativity introduced by Einstein’s theory, so it is safe to limit semantic investigations to a pre-relativistic model of branching. To formally capture the notion of a continuation, Placek(2011) uses the notion of an l-event. In our simplified case, an l-event is a non-empty subset of M, linearly ordered by <. On the level of semantics, continuations take over the role assigned to histories in Ockhamism. In particular, sentences are evaluated at pairs m/A, where m is a moment and A an l-event; the pair m/A is called an evaluation point. There is an additional proviso that m/A is an evaluation point only if {m} ∪ A is an l-event, i.e., m is compatible with A (it is an analog of the Ockhamist proviso that in an evaluation point m/h, m ∈ h). The definition of evaluation point is quite liberal; l-event A needs to be non-empty and compatible with m, but there are no other constraints. In most natural cases, a continuation of moment m slightly extends the moment into a future, but in principle a continuation of a moment might be endless, or on the other extreme, it might not extend the moment at all (in which case, it is slightly misleading to call l-event a continuation of m). Also, there might be “wholes” in l-events, or an l-event might be a singleton consisting of a moment of evaluation itself. Let us turn to the details of the semantic machinery. Importantly, we work with a language with metric temporal operators, let us call it Lmetr. It means that the language contains sentential variables, the usual truth-functional connectives, the modal operator of historical possibility ^, and a family of temporal operators Fx and Px, where x ranges over positive real numbers. I have already introduced the semantics of this operators in chapter3, but let me restate it to refresh your memory. The models are based on structures of the form hM, <, I, Ti, which are the usual branching structures with instants, supplemented with coordinalization function T which is an isomorphism between I and R. Thanks to the coordinalization function, we can represent instants numerically and we can define the distance (dist) between moments in the structure: dist(m1, m2) = x iff |T(im1 ) − T(im2 )| = x. A model for language Lmetr is a structure supplemented with a valuation function which maps a set of sentential variables Atom into a power set of M. Not only does the continuations replace histories in evaluation points, but they are also used in place of histories in interpretation of historical modalities. Strictly speaking, we need to use a fan of evaluation points:

Deﬁnition 4.18 (Instant-wise isomorphism; Fan of evaluation points). We say that two l-events A1 and A2 are instant-wise isomorphic iﬀ there is a bijection f : A1 7→ A2 such that ∀m∈A1 im = i f (m). Let m/A be an evaluation point. The fan Fm/A of evaluation points determined by m/A is the set of all evaluation points instant-wise isomorphic with m/A.

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0 0 If an evaluation point m/A enters the fan Fm/A, then l-event A needs to fulfill two conditions: firstly, A0 needs to be instant-wise isomorphic with A and, secondly, A0 needs to be consistent with m (otherwise m/A0 would not be an evaluation point). The name “fan” is meant to be evocative, if A goes above m into one of many futures, then Fm/A covers the l-events isomorphic to A in all the other futures extending m. If you draw possibilities in the form of a tree, the set Fm/A truly looks like an unfolded fan. We have all the necessary notions to define a notion of fulfillment (p≈C) of a sentence at an evaluation point. Definition 4.19. Let φ be a sentence of our language. Then: 1.m /Ap≈C p iff m ∈ V(p), for p ∈ var; 2. standard definition of truth-functional connectives;

C 0 ∗ 0 0 C 3.m /Ap≈ Fxφ iﬀ ∃m0∈W ∃m∗∈A(m < m ≤ m & dist(m, m ) = x & m /Ap≈ φ);

C 0 0 0 C 4.m /Ap≈ Pxφ iﬀ ∃m0∈W (m < m & dist(m, m ) = x & m /Ap≈ φ);

C 0 0 C 5.m /Ap≈ φ iff ∀m/A0 (m/A ∈ Fm/A ⇒ m/A p≈ φ). The historical possibility operator is a dual of historical necessity, ^ B ¬¬. The notion of fulfillment is very sensitive to the size of A, which is particularly vivid in the definition of future-tensed sentences. If an l-event A has an upper bound (as the ordinary continuations do), there is an x such that for every y, if y > x, then, for every C sentence φ, m/Ap6≈ Fyφ. This result reflects the idea that if a continuation does not extend enough into a future to reach the period that the sentence refers to, this continuation is unable to verify or falsify the sentence. This result affects other sentences which contain operator Fy such as ^Fyφ or Pz^Fyφ. These are also unfulfilled for sufficiently large y; the point of evaluation m/A just does not reach y units in a future.

x Definition 4.20 (m/A reaches x-units-above m, Rm(m/A)). Let m/A be an evaluation x point. We say that m/A reaches x-units-above m, Rm(m/A), iff ∃m1∈W ∃m2∈A(m < m1 ≤ m2 & dist(m, m1) = x). As we have already seen, unless an l-event is unbounded, it is “too small” to verify sentences which speak about a sufficiently remote future. To overcome the shortcom- ing, we need to be able to extend points of evaluation.

x Definition 4.21 (Extension of an evaluation point m/A, EXTm/A). Let m/A be an evaluation point. We say that m/A0 is an x-units-extension of m/A, in symbols m/A0 ∈ x 0 x 0 EXTm/A iff A ⊆ A and Rm(m/A ). Thanks to the notion of extension, we can shift from an auxiliary notion of fulfillment to the notion of definite truth. Definition 4.22 (Definite truth value). A sentence φ is definitely true at m/A, in sym- 0 0 C 0 x C bols, m/A|=φ iff ∃x∀m/A (m/A ∈ EXTm/A ⇒ m/A p≈ φ). A sentence φ is definitely false at M, m/A iff ¬φ is definitely true at m/A. A sentence φ is indefinite at m/A, in symbols m/A?=φ, iff φ is neither definitely true nor definitely false at m/A.

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A sentence is definitely true at a point of evaluation iff every “sufficiently long” extension of this point fulfills this sentence. I shall say that a sentence φ is valid in the semantics of continuations, in symbols |=Cφ, iff it is definitely true at every evaluation point of every model. The semantic definitions presented above might be slightly complicated, but on the conceptual level, they are very elegant. The author manages to systematically eliminate the concept of the history from the semantic analysis. Not only does history not feature as an element of a point of evaluation, but it does not show up at any other step of semantic analysis. In this feature, the project of Placek surpasses the alternative “localized” relativist semantics. They all get rid of a history as a parameter of truth, but they continue to use this notion in their theories. Zanardo(1998) uses them in his definition of recognized possibilities and in the definitions of semantic operators; Braüner et al.(2000), in the definitions of TRL-function and counterfactual branches; also Müller(2014) (whose theory I discuss in the following section) uses the notion to define transitions and to analyze the future tense operator. Let us now take a closer look at how the semantics functions in practice. It will be helpful to begin with a very simple model:

m4 m5

Greeks Persians win win

no sea battle m3 m2 sea battle

m1 Decision whether to ﬁght

Figure 4.5: Sea ﬁght.

Let us assume that dist(m1, m2) = dist(m1, m3) = 1 and let us study a few cases:

x 1. m1/{m1}?=F1(sea battle), because ∀x∃m /Am1/A ∈ Ext m3 ∈ A. Then, 1 m1/{m1} C C m1/Ap6≈ F1(sea battle) and therefore m1/{m1}6|= F1(sea battle). By analogy, x C ∀x∃m /Am1/A ∈ Ext m2 ∈ A. Then, m1/Ap6≈ ¬F1(sea battle) and therefore 1 m1/{m1} C m1/{m1}6|= ¬F1(sea battle).

2. m1/{m1, m2} |= F1(sea battle). This sentence is deﬁnitely true at this point, since x C there is an x (e.g., x = 1) such that ∀m /A(m1/A ∈ EXT ⇒ m1/Ap≈ 1 m1/{m1,m2} F1(sea battle)).

3. m1/{m1, m3} |= F1(¬sea battle) ∧ ^F1(sea battle). At every 1-unit-extension of m1/{m1, m3}, the sentence F1(sea battle) is satisﬁed. The sentence ^F1(sea battle) is satisﬁed as well, since in a fan of every 1-unit-extension of m1/{m1, m3}, there

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is an evaluation point m1/A such that m2 ∈ A (because A needs to be in instant- wise isomorphic with the extension of m1/{m1, m3}, so it needs to contain m2 which occupies the same instant as m3).

4. m1/{m1, m2}?=F2(Greeks win).

5. m1/{m1, m2} |= ^F2(Greeks win) ∧ ^F2(¬Greeks win). It is so, because every x-units-extension of m1/{m1, m2} sufficiently long to contain the result of the battle (for example m1/{m1, m2, m4}) generates a fan of evaluation points such that it contains an evaluation points at which F2(Greeks win) is satisfied and evaluation points at which F2(Greeks win) is not satisfied, and thus, at every C C point m1/A in the fan m1/Ap≈ ^F2(Greeks win) and m1/Ap≈ ^F2(¬Greeks win), so the conjunction of these two possibilities is definitely true.

6. m1/{m1, m2, m4} |= F2(Greeks win). Since continuations are local, we end up with a non-bivalent (post)semantics. In contrast with history relativism, some sentences might still be neither true nor false (see examples 1 and 4 above), even if we specify one of the continuations available at a context. Thanks to the liberal notion of the l-event, the postsemantics of continuations is very elastic. On the one extreme, we can simulate history relativism if we assume that in an evaluation point m/A, A needs to be equal to a whole history. On the other extreme, we can mimic supervaluationism if we assume that in an evaluation point m/A, A is always equal to {m}. In between, we can emulate assessment relativism if we 53 consider m1/{m1, m2} as the default format of an evaluation point. In his paper, Placek(2011) advertises his semantics with several case studies. In particular, he shows that: 1. Future is distinct from settled future. Placek is aware of the worrisome consequences resulting from semantic identification of future and settled future (we have discussed some of them in the context of Peirceanism). In the semantics of continuations, the notion of definite truth does involve quantification over all extensions, although the notion of definite truth cannot be identified with settled truth. We can express the thought with an observation that neither the semantic implication φ → φ, nor the postsemantic implication |=Cφ ⇒ |=Cφ, is valid. Example3 on page 113 attests to both these results—at m1/{m1, m3}, the sentence ¬F1(sea battle) is true, while the sentence ¬F1(sea battle) is false. It is a noteworthy feature that distinguishes postsemantics of continuations from postsemantics of supervaluations. In supervaluationism, the postsemantic implication from m||−S φ to m||−S φ holds at arbitrary context m. 2. Not all sentences in past tense are necessary. The guiding intuition of branching models is that past, contrary to the future, is settled. However, we should not jump to the conclusion that the implication (P) Pxφ → Pxφ is valid. For example, if φ B Fy p and y > x, it might well happen that PxFy p is definitely true while PxFy p is not. In fact, for y > x, the sentence PxFyφ is equivalent to

53In fact, we can cover only these cases of assessment relativism, when the context of assessment is below, equal, or above the context of use. The remaining cases, however, are not discussed even by MacFarlane.

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Fy−xφ. If we accepted (P), we could conclude that Fy−xφ → Fy−xφ is valid, but it has just been proved to be falsifiable in the previous point. 3. What was going to happen, might not have happened. Even though we are inclined to say that 100 years in the past, the second world war was in the future, we do not want to conclude that 100 years in the past, the second world war was inevitable. Thus, we do not want to accept validity of the implication from PxFy p to PxFy p. It is acceptable under the Peircean reading of future operator, but it is not valid in the semantics of continuations. As a counterexample, evaluate sentences P1F1(sea battle) and P1F1(sea battle) at point m2/{m2} in our toy-model 4.5 on page 113. All the examples investigated by Placek(2011) are in fact classic cases, studied already by Prior(1967), which distinguish Ockhamism from Peirceanism. In my opinion, they attest to the superiority of Ockhamist over Peircean semantics, so it is a good omen that the semantics (and postsemantics) of continuations behaves just as Ock- hamism in these cases. In fact, we can widely extend the results of Placek and prove that for sentences expressible in Lmetr, continuation-validity coincides with Ockhamist-validity. To prove this, it will be useful to first define the notion of the forward range of a sentence.

Deﬁnition 4.23 (Forward range). Let φ be a sentence of language Lmetr. The forward range of φ,FR(φ), is deﬁned by induction on complexity of φ as follows: 1. φ ∈ var ⇒ FR(φ) = 0; 2. φ = ψ ∗ χ ⇒ FR(φ) = max(FR(ψ), FR(χ)), for ∗ ∈ {∨, →};

3. φ = ∗ψ ⇒ FR(φ) = FR(ψ), for ∗ ∈ {¬, ^, Px|x∈R+ };

4. φ = Fx(φ) ⇒ FR(φ) = FR(ψ) + x. Observe that the forward range of a sentence is distinct from its “degree of futurity.” Degree of futurity informs us which time(s) in the future the sentence is “really about,” while the forward range says how far into the future we need to reach in the process of evaluation of the sentence. These two notions might come apart. for example the forward range of the sentence F100P99 p equals 100, while its degree of futurity equals 1. It might be useful to remind that our metric temporal operators are so deﬁned that the subscripts range uniquely over positive reals. Our theorem is a consequence of the following lemma:

Lemma 4.1. Let M B hM, ≤, I, X, Vi, and φ a sentence of language Lmetr. Then, for FR(φ) arbitrary A ⊆ h such that Rm (m/A), M, m/h |= φ ⇔ M, m/Ap≈Cφ

Proof. The right to left direction is straightforward. If m/Ap≈Cφ for arbitrary large A ⊆ h, then m/Ap≈Cφ, for A = h. For A = h, the deﬁnitions of |= and p≈C coincide, therefore, M, m/h |= φ. The left to right direction can be proved by induction on complexity of φ:

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1. For p ∈ var: If M, m/h |= p, then m ∈ V(p), then M, m/Ap≈Cφ, for arbitrary A ⊆ h. 2. For φ ∈ {ψ ∨ χ, ψ → χ, ¬ψ}, the induction is straightforward.

0 0 3. Let φ B ^ψ. If M, m/h |= ^ψ, then ∃h0 (m ∈ h & m/h |= ψ). Let us now take FR(φ) a fan of evaluation points Fm/A for arbitrary A ⊆ h and such that Rm (m/A). Since all evaluation points in Fm/A are instant-wise isomorphic, all of them reach FR(φ)-units-above-m. And as FR(^ψ) = FR(ψ), all of them reach FR(ψ)-units- 0 0 0 0 above-m. Of course, ∃m/A ∈Fm/A A ⊆ h , where h is our chosen history at which 0 0 FR(ψ) 0 0 ψ is true. Since A ⊆ h and Rm (m/A ) and m/h |= ψ, we use the inductive 0 C 0 hypothesis and conclude that m/A p≈ ψ. Since m/A ∈ Fm/A, then, by fulﬁllment deﬁnition for ^ (def. 4.19), M, m/Ap≈C^ψ.

0 0 4. Let φ B Pyψ. If M, m/h |= Pyψ, then ∃m0 (m < m and dist(m , m) = y and M, m0/h |= ψ). We can conclude, by inductive assumption, that M, m0/Ap≈Cψ for FR(ψ) 0 0 arbitrary A ⊆ h such that Rm0 (m /A). Since m < m and FR(φ) = FR(ψ), FR(φ) FR(ψ) 0 then if Rm (m/A), then Rm0 (m /A). Thus, for arbitrary A ⊆ h such that FR(φ) 0 0 0 C Rm (m/A), there is m < m such that dist(m , m) = y and m /Ap≈ ψ. And C FR(Pyψ) consequently, M, m/Ap≈ Pyψ for arbitrary A ⊆ h such that Rm (m/A).

0 0 5. For φ B Fyψ. Assume that m/h |= Fyψ, then ∃m0 (m < m and dist(m, m ) = y and m0/h |= ψ), then, by inductive assumption, for arbitrary A ⊆ h such that FR(ψ) 0 0 C 0 Rm0 (m /A), M, m /Ap≈ ψ. Observe that since A ⊆ h and dist(m, m ) = y and 0 FR(ψ) 0 FR(ψ)+y FR(ψ)+y m < m , then (∗) Rm0 (m /A) iﬀ Rm (m/A). Thus Rm (m/A), and there-

fore ∃m1∈M∃m2∈A(m < m1 ≤ m2 & dist(m, m1) = FR(ψ) + y). Since FR(ψ) ≥ 0 0 0 0 0 and dist(m, m ) = y, so ∃m2∈Am ≤ m2. Therefore, ∃m >m∃m2∈A(m < m ≤ m2 0 0 C and dist(m, m ) = y and M, m /Ap≈ ψ). By truth clause of Fy, it means that C FR(ψ) 0 M, m/Ap≈ Fyφ. It holds for arbitrary A such that A ⊆ h and Rm0 (m /A). Re- FR(ψ) 0 FR(ψ)+y member that by (∗) Rm0 (m /A) iﬀ Rm (m/A) and observe that FR(ψ) + y = FR(Fyψ) FR(Fyψ). Thus, we can conclude that for arbitrary A ⊆ h such that Rm (m/A), C M, m/Ap≈ Fyψ.

We say that the sentence φ of language Lmetr is valid in the semantics of continuations, |=Cφ, iﬀ at an arbitrary point m/A in an arbitrary branching model with instances M, M, m/A|=Cφ. Thanks to lemma 4.1, we can easily demonstrate the general result: Theorem 1. |= φ iﬀ |=Cφ

Proof. For left to right direction, assume that |= φ and, for reductio, that 6|=Cφ. Since C C 0 x 6|= φ, then ∃M∃m/A∈MM, m/A6|= φ. Thus, for every x, there is m/A ∈ EXTm/A such 0 C 0 FR(φ) that M, m/A p6≈ φ. In particular, for x = FR(φ), there is m/A ∈ EXTm/A such that m/A0p6≈Cφ. Since m/A0 is an FR(φ)-units-extension of m/A, then by deﬁnition 4.21, FR(φ) 0 Rm (m/A ). Since every linearly ordered set in M is a subset of some history, then, for some h, A0 ⊆ h. By lemma 4.1, we can conclude that M, m/h 6|= φ, which contradicts our assumption that |= φ.

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For right to left direction, assume that |=Cφ and, for reductio, that 6|= φ. Since 6|= φ, ∃M∃m/hM, m/h 6|= φ. By lemma 4.1 again, we can conclude that for arbitrary A ⊆ h such FR(φ) 0 0 C 0 x C that Rm (m/A), M, m/Ap6≈ φ. It means that ¬∃x∀m/A (m/A ∈ Extm/A ⇒ m/A p≈ φ). Consequently, m/A6|=Cφ, which contradict the assumption that |=Cφ. Therefore, the set of Ockhamist validities coincide with the set of sentences valid in the semantics of continuations. As a consequence, the semantics of continuations is not an easy prey for objections based on purely semantic considerations. The semantics can safely rest on the intuitive truths of Ockhamism. Nevertheless, the results presented above are not as comforting for a continuation theorist as they might seem. The problem lies in the fact that Placek limited his investigations to sentences of a language with metric operators. However, when we supplement the language with non-metric tense operators, the results generated by the semantics of continuations are more problematic. Placek(2011, p. 753) motivates his limitation of the language as follows: “Since l-events can be small, it is preferable to work with metric tenses.” His motivation is not convincing for three reasons: firstly, metric tenses can reach very far into the future, so they do not seem to be better suited for the “small” l-events; secondly, in the process of evaluation of the definite truths, we are entitled to extend small l-events to very large l-events; thirdly, even in the framework of continuations, it would be very useful to have a simple, future-tense operators; for example, it is reasonable to assume that the sentence, “This chapter will be finished,” is definitely true in the possible continuations in which you are reading these words. To understand this thought, we are not required to specify at which exact future moment the chapter will have been accomplished. Thus, there are good reasons to extend the language with non-metric tenses. For- mally, it is easy enough to modify the notion of satisfaction to incorporate non-metric tenses: Definition 4.24 (Fullfillment of temporal operators). Let M be a model and φ a sentence of Lmetr ∪ {F, P}. Then:

C 0 ∗ 0 C 1. M, m/Ap≈ Fφ iﬀ ∃m0∈W ∃m∗∈A(m < m ≤ m ∧ M, m /Ap≈ φ);

C 0 0 C 2. M, m/Ap≈ Pφ iff ∃m0∈W (m < m ∧ M, m /Ap≈ φ). We can retain the natural duality of F and P with G and H, respectively. Intuitively speaking, the sentence Fφ is fulfilled at a moment, in a continuation, iff the continuation reaches far enough to find another moment where φ is fulfilled. The sentence Gφ is fulfilled by a moment/continuation pair iff every future moment within a range of the continuation fulfills φ. The smaller the continuation is, the less F-sentences (i.e., sentences with F as their primary operator) are going to be true and the more G-sentences are going to be true. In the extreme case, where no moment in A is later than m, no F-sentence is fulfilled and every G sentences is. This is intuitively correct; if the continuation does not reach far into the future, not many sentences about the future are going to be satisfied by this continuation. We do not change the definition of definite truth, a sentence is definitely true at m/A iff for some real number x, every extension of m/A reaching at least x units into the future of m fulfills the sentence. The introduction of non-metric temporal operators

117 CHAPTER 4. SEMANTICS OF BRANCHING REALISM introduces a new aspect of indeterminacy. So far, a sentence could be neither definitely true, nor definitely false only due to indeterminism. Now, a sentence can be indeterminate due to its infinite range. Consider for example a model MINF-p based on structure F B hM, ≤, I, Xi, where ≤ is a linear order, and assume that the propositional constant V(p) = {m| X(im) ∈ N} (i.e, p is true at all and only moments which correspond to natural numbers). As an example, take sentence p to stand for “A bugle call is sounded from the tower of St Mary’s Church in Kraków,” and a unit to represent an interval of an hour. We need to slightly extend the seven centuries tradition of the call and imagine a scenario in which the call is sounded every hour for eternity. Now consider the sentence, “It is always going to be the case that another bugle call will sound,” (GF p) evaluated at a moment/continuation pair m0/{m0}. The sentence p is true at arbitrary large natural number, so, intuitively, the sentence GF p should be definitely true. But in the theory of continuation it is not. There is no number x such that every x C x m0/A ∈ EXT , m0/Ap≈ GF p, because for arbitrary x, there is an m0/A ∈ EXT , {m0} m0 C C where A has the maximal element mmax. At m0/Ap6≈ GFφ, since at mmax/Ap6≈ F p. So, GF p is not definitely true at m0/{m0}, but it is is not definitely false either, because for x m /A ∈ EXT x m /Ap≈CGF p every , there is an extension 0 m0 such that 0 (any extension which does not have the upper bound in M is a good candidate for A). Therefore, at m0/{m0} in our model, it is indeterminate whether there is always going to be a moment which has a bugle call in its future. The indeterminacy results from the fact that the sentence talks about arbitrary remote future, not from indeterminism of any sort. The observation reveals an interesting aspect of the consequence relation for definite truth. Contrary to Ockhamist truth, it is not closed under the rule of generalization for operator G. In Ockhamism, we have that if a sentence φ is true at every moment/history pair in a model, then the sentence Gφ is also true at every moment/history pair. However, the previous example can be used to show that it is not so in the semantics of continuations. After all, the sentence F p is definitely true at every mo- 0 INF-p 0 x ment/continuation pair m/A in model M , that is ∀m∀A∃x∀m/A (m/A ∈ EXTm/A ⇒ m/A0|=C F p) (it is sufficient that x is long enough to reach to the first natural number above X(im)). We have just seen, however, that the sentence GF p is not definitely true at some moment/continuations pairs in MINF-p. So, the generalization rule does not hold. Generally, if a sentence F(p∨¬p) is valid in Ockhamism, then GF(p∨¬p) is valid as well. It does not holds in the semantics of continuations. Since we limit our attention to structures with no maximal elements, |=C F(p ∨ ¬p). Nonetheless, 6|=CGF(p ∨ ¬p). The sentence is indeterminate at every m/A, where A has an upper bound in M. To 0 0 0 x C see that, it is enough to notice that ∀x∃m/A (m/A ∈ EXTm/A & m/A p6≈ GF(p ∨ ¬p)). 0 0 We just need to choose any m/A such that the upper bound mu of A is an element of A0 (for example, m/{m}, for x = 0). In such case, we have that m/A0p6≈CGF(p ∨ ¬p), 0 C 0 because mu/A p6≈ F(p ∨ ¬p), as there are no moments later than mu in A . So, the sentence GF(p ∨ ¬p), contrary to the sentence F(p ∨ ¬p), is not valid in the semantics of continuations. The particular problem just presented could be overcome with an ad hoc require- 0 0 0 0 0 x A A ment that ∀x∀m/A∀m/A (m/A ∈ EXTm/A ⇒ mu < A ), where mu is the upper bound of 0 A . Nevertheless, the general problem does not disappear. Notice that Fy(p ∨ ¬p) and

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G(p ∨ ¬p) are both valid in every structure with no maximal elements (i.e., definitely true, at every point of evaluation in every model based on such structure). However, the sentence GFy(p ∨ ¬p) is not valid in such structures. It is due to the fact that no 0 x matter which x you choose, some m/A ∈ EXTm/A are not going to fulfill GFy(p ∨ ¬p). In particular, if A0 is upper bounded by m0, it is sufficient to travel with operator G ∗ ∗ 0 ∗ 0 C to a moment m ∈ A such that dist(m , m ) < y. Then m /A p6≈ Fy(p ∨ ¬p) and thus C m/Ap6≈ GFy(p ∨ ¬p). Since x is arbitrary, we have that GFy(p ∨ ¬p) is not definitely true at m/A (it is not definitely false either, since for every upper unbounded A and any C y, m/Ap≈ GFy(p ∨ ¬p)). To sum up, Placek(2011) presents a compelling modification of relativism which is completely free of the notion of history. It is still a version of relativism, i.e., it is meaningless to call a future contingent true/false in a context, unless a possible continuation is specified. Nonetheless, it might be seen as an improvement over the history relativism, since it seems to be easier to specify a possible continuation than a whole history (however, I will argue in section 4.8.5 that it is not much easier). On the strictly logical ground, the semantics turns out to be equivalent to Ockhamism, if the language does not contain non-metric temporal operators. However, when it is enriched with such operators, it generates some problematic results.

4.8.4 Sets of transitions Another modification of the relativist (post)semantics has recently been advocated by Thomas Müller(2014). He shares Placek’s concerns that: [E]ven though histories are not possible worlds, they are still large structures with a global ring to them. (. . . ) [A]nd epistemic access to whole histories of our world is impossible. (Müller, 2014, p. 344) Epistemic inaccessibility of histories is a major worry for history relativists, since they postulate that the process of semantic evaluation can begin only when a unique history is specified. The problem with such a theory is well-summarized by Müller himself: If we want to evaluate a sentence about the future, we normally need to specify which of the equally possible futures we mean to refer to, for otherwise no assessment may be possible. But do we really need to specify a full history, a full course of events from the beginning till the end of time? That seems a bit too much, really, and it can’t be what is going on when we assess sentences containing the future tense: we have to make do with much more limited information. (Müller, 2014, p. 354). Indeed, if one insists that it makes no sense to ascribe a truth value to a sentence at a context, unless a particular history is specified, it might help to have the slightest idea what it means to specify a history. One way to go around this problem is by replacement of big, epistemically unavailable histories with smaller, more approachable objects, which are easier to specify.54

54Otherwise, one can side with Belnap et al.(2001) and insist that, for all practical purposes, there is no need to specify a history or to ascribe a truth value to a sentence in a context.

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h1 h2 h3

Is president Is not president

Election day m3 m2 Election day

does not run runs

m1 Decision whether to run for president

Figure 4.6: The election.

A cost of such a limitation is that much of the information inscribed in histories needs to be abandoned and cannot be later used for semantic purposes. Müller argues, however, that the loss is not that grave, since we rarely need that much detail regarding possible future developments. “If I deliberate where to go next weekend, for example, I will map out individual options separately (to a very limited degree of detail of course), and there is no need for me to ‘carve up’ any given larger structure containing all these possibilities into individual, consistent scenarios” (Müller, 2014, p. 351). There is another source of motivation behind Müller’s project (also shared with Placek, 2011). Namely, the specific definition of a history employed by the branching spacetimes theory of Belnap(2003b) does not cohere well with some spacetime models of general relativity: “for non-time-orientable space-times, the BST approach to modal consistency is inappropriate” (Müller, 2014, p. 352). In fact, there is one more reason that pushes Müller into the direction of a “local” theory of possibilities. He mentions it only in passing in (Müller, 2014), but dwells on it in (Müller, 2010). He believes that the localized and limited notion of (in)determinism is helpful, while we try to develop a worldview which incorporates both the “manifest image” (with concepts such as freedom, agency, responsibility etc.) and the “scientific image” (with concepts such as space, time, mass, force, etc.). The theory becomes quite complicated if one wants to incorporate the spatial dimension present in the BST theory that Müller focuses on. I will simplify Müller’s theory for our purposes and limit the exposition to the temporal aspect of branching. Fortunately, as far as the semantic issues are concerned, the author himself limits his investigation to the non-spatial models of branching and I shall follow. There is a natural way to spell out an immediate possibility open at a moment. Such a possibility is a set of histories forming a “uniform group.” It means that they do not split at this moment. The set of such histories is called an elementary possibility and the set of such possibilities available at m is denoted by Πm (see def. 4.13 on p. 100). A simple example depicted in figure 4.6 might help. Let us consider a person who decides whether to run for president. She is a serious candidate, so if she decided to run,

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she might win. There are three histories passing through moment m1. Nevertheless, there are only two elements of partition Πm1 = {{h1}, {h2, h3}} (since h2 and h2 are undivided at m1, they end up in one set, see def. 4.13). Thus, although there are three possible histories passing through m1, there are only two possibilities open at m1. In a sense, it might even be better to call elements of Πm “immediate possibilities of m.” Becoming a president is an open option for the candidate at m1, but this option is not available immediately. At m1, there are only two immediate possibilities: to decide to run and to decide not to run. The notion of elementary possibility allows to introduce the concept of a transition:

Deﬁnition 4.25 (Basic transition). A basic transition from m to A, m A, is an ordered pair hm, Ai, where A ∈ Πm. Each basic transition consists of a moment and one of its elementary possibilities. One can think of transitions as immediately possible ways to settle the indeterminacy residing at m. Observe that if moment m is deterministic, i.e., if no histories split at moment m, there is only one elementary possibility following m and the transition can be said to be trivial. We can combine basic transitions into larger sets. Some such sets are “consistent,” others are not. At the picture above, the set of transitions T : {m1 {h2, h3}, m2 {h3}} is consistent, this two transitions might “happen together” (the candidate who 0 runs for presidency can loose). On the other hand, the set of transitions T B {m1 {h1}, m2 {h2}} is not consistent since the move from m1 to h1 precludes the move from m2 to h2 (if someone does not run for president, she cannot win). A consistent set of transitions recounts a coherent (although typically incomplete) story about how the world can develop. Formally speaking:

Definition 4.26 (Consistent set of transitions). An indexed set of transitions TI B T {mi Ai}i∈I is consistent iff i∈I Ai , ∅. It means that a set of transitions is consistent iff there is at least one possible history which attests to the coherence of the “story” described by the set. Let us use the symbol HT to designate the set of all histories “preserved” by the set of transitions TI, i.e., I T HTI B i∈I Ai (H∅ B Hist). We say that the set of transitions TI admits a moment m iff (HTI ∩ Hm) , ∅. To simplify the notation, I will henceforth skip the index I, when I talk about the sets of transitions. Finally, we have all the necessary notions to introduce transition semantics. The crucial innovation is that sentences are evaluated at points m/T, where T is a consistent set of transitions and T admits m.55 Now, let us consider the standard temporal language L B {Atoms ∪ {¬, ∨, F, P}} (the author does not include an operator of historical possibility) and a typical branching model M B hM, ≤, Vi, where V : Atoms 7→ P(M). The truth conditions for sentences of the language are defined as follows: Definition 4.27 (Truth in the semantics of transitions).

55It is yet another incarnation of the basic requirement of Ockhamism. Remember that in Ockhamism, we require that every evaluation point m/h is “compatible,” i.e., that m ∈ h. In the semantics of continuations, we also require this kind of compatibility, i.e., that {m} ∪ A is linearly ordered by < in any evaluation point m/A.

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1. M, m/T|=T p iﬀ m ∈ V(p); 2. standard deﬁnition of truth-functional connectives;

0 0 T 0 T 3. M, m/T|=Pφ iff ∀h∈Hm∩HT ∃m ∈h(m < m and M, m /T|=φ); 0 0 T 0 T 4. M, m/T|=Fφ iff ∀h∈Hm∩HT ∃m ∈h(m < m and M, m /T|=φ). The definition of the temporal operator F partially resembles Zanardo’s definition. It also renders Müller’s semantics a particular mix of Ockhamism and Peirceanism. It preserves the Ockhamist idea that a truth value of a sentence is relative to a modal parameter. Nonetheless, his modal parameter T is under-specified and it admits many different specific developments, i.e., histories. At this juncture, the Peircean idea is incorporated and a sentence is true at m/T iff it is true at some moment in every specific history admitted by set T. One more feature of the semantic definition deserves a comment. After all, the whole project begins with a criticism of “large” and “global” histories, but the concept of a history is explicitly used in the definition of temporal operators. In this respect Müller(2014) is less radical than Placek(2011), who carries out his whole semantic project without resorting to the concept of a history. However, it is not an inconsistency on behalf of Müller, as his major problem with histories is not that they are simply too large, but that they are too large to specify; and a history needs to be specified only when it features as an element of a point of evaluation (to address the initialization failure). Müller gets rid of a history as an element of a points of evaluation and replaces it with a set of transitions. The specification is nevertheless necessary and, according to Müller, the context will not do the job: “a context of utterance cannot supply a ‘true future of the utterance’ ” (p. 347, see also p. 350). Therefore, the specification needs to be done “manually.” The only hope is that the task will be easier, because a set of transitions typically encodes much less information than a history. I am not entirely convinced that specification of a set of transitions is so much easier than specification of a history, but before I discus the point, let us better understand the mechanism of Müller’s theory. Let us experiment with the model depicted in figure 4.6 on page 120. Let Hr = {h2, h3} and Hp = {h2}. Remember that H∅ = Hist. Then, let us look at a few interesting sentences and their truth conditions: T T 1. m1/∅|=F(runs) ∨ ¬F(runs) 5. m1/Hr|=F(runs)

T T 2. m1/∅6|=F(runs) ∨ F¬(runs) 6. m1/Hr|=¬F(wins)

T T 3. m1/∅6|=F(runs) 7. m1/Hr6|=F(wins)

T T 4. m1/∅6|=F¬(runs) 8. m1/Hp|=F(wins) Indeed, the semantics is located right in the middle between Peirceanism and Ock- hamism. One problem of Peirceanism was that for any contingent F1φ, the sentence F1φ∨F1¬φ was false. This feature distinguishes Peirceanism from Ockhamism, where the disjunction is always true. The behavior of the transitions semantics is more nu- anced, generally speaking: the more extensive the set of transitions is, the more sentences of the form Fnφ ∨ Fn¬φ are true. For large sets T, the semantics behaves more like Ockhamism, while for small sets, it resembles Peirceanism.

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In fact, Müller observes that in the extreme case, when HT is a singleton, the clause of Fφ coincides with the Ockhamist deﬁnition. In the other extreme, when HT = Hist (e.g., when T = ∅), the truth clause of F turns into the Peircean F. In the in-between cases, the semantic behaves like Ockhamism for the portion of the future which is speciﬁed by the set of transitions, and like Peirceanism later on. It will be useful to state the relation between Peirceanism and the semantics of transitions explicitly: Lemma 4.2. Let M = hM, <, Vi be a branching model, m a moment in M, and φ a sentence of the temporal language, then:

M, m|=Pφ iﬀ M, m/∅|=T φ

Proof. The lemma can be easily proven by induction on complexity of φ. Let us take a look at one of the temporal operators only: Let φ = Pψ. M, m/∅|=T Pψ iff 0 0 0 T ∀h∈Hm∩H∅ ∃m ∈h(m < m and M, m /∅|=ψ). Remember that H∅ = Hist, so Hm ∩ H∅ = Hm. 0 0 T 0 T Hence, M, m/∅|=Pψ iff ∀h∈Hm ∃m ∈h(m < m and M, m /∅|=ψ) iff (by inductive assump- 0 0 0 0 P 0 tion) ∀h∈Hm ∃m ∈h(m < m and M, m |=ψ) iff (by backward linearity of <) ∃m

Deﬁnition 4.28 (Restriction of a model to a set of transitions). Let M B hM,

Lemma 4.3. Let M = hM,

N M Hist = HT M Proof. The inclusion from right to left is straightforward. If h is a history in HT , then all moments in h end up, by deﬁnition, in N. Since we add no elements to N which were absent from M, and since we preserve the original ordering, then, as h is a maximal chain in M, it is also a maximal chain in N, and so, a history in N. The left to right inclusion requires a little more work. Let us assume that h∗ ∈ N ∗ M ∗ ∗ Hist , and, for reductio, that h < HT . As h ⊆ N and h is linearly ordered, the fact ∗ M ∗ M ∗ M that h < HT must mean one of two things: (a) h < Hist or (b) h ∈ Hist , but ∗ M h < HT . Let us prove that each of these two assumption leads to a contradiction. First, let us assume that (a) h∗ < HistM. Then, h∗ is not a maximal linearly ordered ∗ subset of M. Consequently, ∃m0∈M\N ∀m∈h (m0

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M ∃m∈h∗ m0 M mx. Since h0 < Hx, then, h0 must have split with hx at mx or earlier. Therefore, ∗ mw < h0. We shall see that (*) ∀m(m ∈ h → m mw → m < h ). Thus, h must branch from hw at mw or earlier. The same is true for any other h ∈ Hw. Therefore, ∗ there is a “counterexample” moment mc such that mc ∈ h and ∀h∈Hw mc < hw. If M ∀h∈H mc hw, then ∀ M mc h (because H ⊆ Hw). Remember, however, that w < h∈HT < T S M N = { h|h ∈ HT }. So, mc < N. But mc ∈ h∗ ⊆ N, so mc ∈ N. Thus, point (b) N M also leads to a contradiction. Therefore, we have established that Hist ⊆ HT , which concludes our proof. Observe that in our proof we have not used the so-called prior choice principle (PCP) in our proof. It is an axiom which states that if h1 , h2, then there is a maximal element in h1 ∩h2. Thomas Müller(2014) assumes the principle when he introduces his models for transition semantics, but the assumption is not obligatory for the deﬁnition of operators, so the equivalence of Peircean truth and transition truth holds independently of PCP.

M M N N Corollary 4.1. (Hm ∩ HT ) = (Hm ∩ H∅ )

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M M M M N Proof. If h ∈ (Hm ∩ HT ), then h ∈ HT . By lemma 4.3, HT = Hist and, by deﬁnition, N N N M N N N Hist = H∅ . So h ∈ H∅ . Since h ∈ Hm , m ∈ h, so h ∈ Hm. Hence, h ∈ (Hm ∩ H∅ ). N N N N N Conversely, if h ∈ (Hm ∩ H∅ ), then h ∈ H∅ . As H∅ = Hist and, by lemma N M M N M 4.3, Hist = HT , then h ∈ HT . Since h ∈ Hm, m ∈ h, thus h ∈ Hm . Therefore, M M m ∈ (Hm ∩ HT ). To prove the main theorem, we need one more, simple lemma: Lemma 4.4. Let M be any branching model and T a consistent set of transitions in M. Let N B MT . Then, for any m ∈ N and any sentence φ of the temporal language: M, m/T|=T φ iﬀ N, m/∅|=T φ

Proof. It is a straightforward proof by induction on complexity of φ. Let us study the case when φ = Fψ: T 0 0 T M M, m/T|=Fψ iff ∀ M M ∃m0∈h(m < m and M, m /T|=ψ). By corollary 4.1,(H ∩ h∈Hm ∩HT m M N N 0 0 T H ) = (H ∩ H ). So, ∀ N N ∃m0∈h(m < m and M, m /T|=ψ). By inductive as- T m ∅ h∈Hm ∩H∅ 0 0 T T sumption, ∀ N N ∃m0∈h(m < m and N, m /∅|=ψ), which means that N, m/∅|=Fψ. h∈Hm ∩H∅ We can finally turn to our theorem: Theorem 2. Let φ be a sentence of the temporal language, then |=Pφ iff |=T φ Proof. The right to left direction is evident. If |=T φ, then, in particular, for every M and every m, M, m/∅|=T φ. However, by lemma 4.2, M, m/∅|=T φ iff M, m|=Pφ. Thus, we can conclude that for every M and m, M, m|=Pφ, which means that |=Pφ. The left to right direction is by the point as easy. Assume that for some sentence φ, P T T T |=φ and 6|=φ. It means that for some M, m/T, M, m/T6|=φ. By lemma 4.4, MT , m/∅6|=φ P P and by lemma 4.2, MT , m6|= φ. Therefore, 6|= φ, which contradicts our assumption that |=Pφ. A close analog of this proof can be used to establish the equivalence of consequence relations of both semantics: Corollary 4.2. Let Γ be a set of sentences of the temporal language and φ a sentence of the language, then:

Γ|=Pφ iﬀ Γ|=T φ Thus, we have proven that the logic of Peirceanism perfectly coincides with the logic of transitions. Since Peircean semantics has been proven by Burgess(1980) to be strongly complete, we can adopt his result to show that the semantics of transitions is strongly complete with respect to the same formal system. It is worth knowing that we can concisely axiomatize the semantics of transitions, but it also means that all the intuitive problems of Peircean semantics are preserved by the semantics of transitions. Thomas Müller(2014) is willing to accept the consequence: “Peircean behavior in case (b) is the price to be paid for the fact that the given clause does not leave any cases undecided and does not have to rely on supervaluations or similar,” but, in my view, it is a high price to pay.

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Regardless of logical similarities, the notions modeled by Peircean semantics and the semantics of transitions are signiﬁcantly diﬀerent. When a Peircean says that φ will happen (as of a moment m), they should be understood as saying that φ is settled to happen (as of a moment m). On the other hand, when a transition semanticist says that φ will happen (as of a moment m and a set of basic transitions T), it is most natural to understand her as saying that φ is settled to happen (as of moment m) given the scenario encoded by T does unfold. Thus, the truth of a sentence at a set of transition might be understood as a conditional temporal necessity (φ is historically necessary at moment m under the condition that T).

4.8.5 A problem with local relativism Remember that history relativists claim that the context initializes a moment of evaluation, but it does not initialize a history of evaluation. They conclude that the assignment of a truth value to a sentence (or, generally, denotation to an expression) is not possible, unless a history is explicitly specified. Tomasz Placek and Thomas Müller have diagnosed a problem with such an attitude. Namely, it is hard to “specify” a suitable value of the history parameter. It seems that to specify a history we need to describe it in all its details. However, it is to ask the impossible. Thus, we seems to be obliged to conclude that it simply makes no sense to say that a non-modal sentence has a truth value in a context and to say that a non-modal term denotes in a context. This is a rather radical consequence. Many have the strong conviction that it does make sense to say that a sentence about the future possesses a truth status (it is true, false, or neither). For those, modal relativism is a less appealing option. Placek(2011) and Müller(2014) do not want to abandon relativism altogether, but only alleviate its radical consequence. They try to avoid the final conclusion that the truth ascription to sentences at contexts makes no sense by replacing histories with smaller objects. Thus, when we want to apply their semantics to analyze a particular sentence used in a particular context, we can do it without describing a history in all its infinite detail. We just need to specify a smaller object such as a continuation or a set of transitions, and then employ a semantics which uses those as parameters of truth. I find both these projects very compelling, formal alternatives to Ockhamism. Nonetheless, when it comes to philosophical motivation, I am still concerned if they can truly deliver what they promise. Specifically, I am not sure if we are in a much better position to specify a smaller object like a continuation or a set of transitions than a history. To see how difficult a task it is, let us work with Placek’s example: After opening the fridge, I can take out either beer or milk. (Placek, 2011, p. 738)

I claim that the description of the two scenarios presented above is much too coarse- grained to specify a continuation of Placek or a set of transitions of Müller. Observe that I can open the door of the fridge smoothly, but I can also rapidly yank the handle. These two options are not jointly possible, so they cannot belong to a single possible continuation which extends all the way to the fridge being open. However, when I want to apply the semantics of Placek(2011) or Müller(2014), I need to specify exactly one

126 CHAPTER 4. SEMANTICS OF BRANCHING REALISM of these possible ways. Their semantic parameters allow for many incompatible continuations which follow the opening of the fridge, but up to that point, exactly one modally consistent scenario needs to be specified. Therefore, I need to specify how I open the door of the fridge: slowly or quickly (and if I do it quickly, how quickly exactly, and at which angle do I hold the handle). Further, when I open the door I can smile or frown. These two are not jointly possible. Therefore, I need to specify my facial expression. Then, when the door is open I might blink or not, and this detail needs to be specified as well, since the two are not jointly possible, etc. It is evident that the list of such details is extremely extensive. In a nutshell, there are plenty, mutually incompatible ways to enter a kitchen, open a fridge, and grab a bottle of milk or beer. So, specifying a unique possible continuation, even if it concerns a relatively short period, is an extremely laborious task, arguably, an impossible one. After all, who could describe all the points of indeterminacy between my entering the kitchen and grabbing a bottle? Thus, in another form, the problem of history relativism returns in continuation relativism. To ascribe a truth value (or a lack thereof) to a sentence, we need to specify a single continuation, but it is not within human cognitive powers to specify one. Therefore, the truth ascription to non-modal sentences is unintelligible. Consequently, the semantics of continuations suffers from the very same kind of objection that it was designed to repudiate.56 Müller(2014, p. 351), also argues that smaller objects like continuations are better suited to model our thinking about possible scenarios, which is required, among other things, for planning. In general, I agree that thinking about future possibilities is helpful for planing. For example, if Tomasz plans to offer white coffee to his guests, it is helpful for him to realize that it is possible to get the milk out of his fridge. How- ever, I have already argued that this description is far too coarse-grained to specify the continuation or the set of transitions. Moreover, I claim that the ordinary Ockhamist semantics, which uses histories rather than continuations, is perfectly well suited to grasp the representation required for planning. Observe that when Tomasz plans to offer white coffee to his guests, all he needs to think about is whether it is possible to (somehow) open a fridge and (somehow) take the bottle out and that it is then possible to (somehow) use the milk in the bottle to prepare white coffee. Notice that this thought can be perfectly well expressed in standard Ockhamism. Let p stand for opening the fridge, q for taking the bottle out of the fridge and r for preparing a cup of white coffee. For strategic planning it is useful to know that: • ^(F(p ∧ Fq)), • (F(q → ^Fr)). Notice that to assess these two statements as true of false, we do not need to specify a unique history. All that is required, given the standard Ockhamist semantics, is that there exists some possible history which contains p followed by q, and that in every history, q might be followed by r. There might be plenty, slightly different histories in which Tomasz opens a fridge and grabs a bottle of milk, but we do not need to take special care to select exactly one of them; any history will do. Moreover, we do

56The semantics have other purposes, e.g., compatibility with general relativity. My argument does not undermine these achievements.

127 CHAPTER 4. SEMANTICS OF BRANCHING REALISM not need to cognitively represent any single history in all its details. All we need to represent are opening the fridge, taking the milk out, and preparing white coﬀee, and the appropriate tempo-modal relations between these three. And to do this is a peace of cake for such an exquisite planner as Tomasz. Thus, I think that the semantics of continuations is not inherently better suited to represent activities such as planning. The standard Ockhamism does it well enough. The semantic of continuations has mostly a post-semantic utility. The move from histories to continuations might be a useful step for a postsemantic relativist who thinks (reasonably) that we cannot specify a full possible history, but who does not want to conclude that it simply makes no sense to say that a contingent sentence is true or false. However, they still need to have the means to specify a single possible continuation, which, as we have seen, is not an easy task. Therefore, it seems that the relativist need to learn to live with the idea that the question whether the sentence like “There will be a sea battle tomorrow” is true in a context simply makes no sense. The view is not going to be satisfactory for those who think that the question does make sense. They need to give up the relativist postsemantics. If they additionally believe that the sentence, “There will be a sea battle,” can be true and that it is true iﬀ a sea battle will actually follow the event of utterance, then they need to abandon anti-futurism altogether and accept some futuristic alternative. I describe some of the alternatives in the following chapters.

128 Chapter 5

Thin Red Line

A variety of philosophers have found branching representation of reality somewhat incomplete. Granting that if indeterminism is true, there are many alternative courses of events, they insist that it is not the end of the story. They think that the possible courses of events depicted by the branching structure should be supplemented with the actual course of events. They are guided by what I call an actualist insight, which was well expressed by a cosmologist, George Ellis:

Things could have been diﬀerent, but second by second, one speciﬁc evo- lutionary history out of all the possibilities is chosen, takes place, and gets cast in stone. (Ellis, 2006, pp. 1812–13).

The actualists tend to think that branching structure is well equipped to capture that “things could have been different,” but does not have sufficient resources to account for the fact that “one specific history takes place.” Therefore, they argue that branching needs to be in some way supplemented. One way to merge the branching model with actuality is called the Thin Red Line theory. I will first outline the theory and then discuss numerous objections it provoked. I conclude that it is not a viable theory. In the following chapter, I propose an alternative strategy to account for the actualist insight.

5.1 Metaphysics of the Thin Red Line

Down-to-earth and quite suggestive phrasing of the actualist insight have been proposed by Belnap et al.(2001): Sure, there are many things that might happen, but only one of them is what really will happen. (Belnap et al., 2001, p. 160). Prima facie, there are two aspects of reality involved in the fragment above. On the one hand, there are the possible ways in which the world can develop, and on the other,

129 CHAPTER 5. THIN RED LINE the actual way in which it does develop. The authors rightly observe that “[i]f one has objective leanings at all, it is easy to feel a need to treat both conjuncts with equal objectivity” (Belnap et al., 2001, p. 161). To introduce the objective actuality on the scene, they propose the following metaphysical picture:

It seems good to try to combine objective indeterminism with an objective actual future. One is thereby tempted to continue to represent objective indeterminism by postulating that our world (up to an idealization) is treelike, but to hold in addition that there is a distinguished history, the Thin Red Line, which we abbreviate as TRL.(Belnap et al., 2001, p. 161)

To give a proper due to the modal aspect of reality, they accept modal neutrality inherent in Genuinely Realistic account of branching. Additionally, to incorporate the actualist insight, they distinguish the history which is being realized—the Thin Red Line. The metaphysical picture was entertained already by Belnap and Green(1994). They described the idea that just one history in the whole branching model is actualized as the Absolute Thin Red Line.

Our World The actual course(s) of events

I ﬁnd such an amalgam of ideas slightly confusing, since it implies that the actual course of events is only a part of the world, with a plethora of non-actual events that occupy other parts of the world! In fact, what actually happens is only a tiny fraction of everything that happens in the world.1 Such an unusual vision results from the attempt to satisfy two conﬂicting desires. On the one hand, to uphold modal neutrality and not to single out any history as the accurate perspective on reality. On the other hand, to distinguish one of the histories from all the others, i.e., the history that is actualized. We will see that this combination of ideas creates a substantial tension. Belnap et al.

1Those familiar with discussions in the philosophy of time will recognize that this metaphysical picture is a modal analogue of a position which is sometimes described as a “moving spotlight.” According to this view in philosophy of time, the world contains all the events which happen at all times. Nonetheless, a portion of these events are objectively distinguished as “the present.” Thus, there are events happening in the word which are objectively present and there are many more events which are not. We shall see that some of the traditional problems of the moving spotlight also aﬀect its modal analog—the Thin Red Line.

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(2001) exploit it to construct a number of specific objections against the notion of a distinguished, actual possibility within our branching world. Having diagnosed a conflict between the branching world and the non-branching actuality, Belnap et al.(2001) eventually decide to take the side of the branching world. Later, when they once again reflect on their actualist slogan—that many things might happen, but only one will—they decide that only one of its part can be right. They conclude that “might” does depict an objective feature of reality, while “will” does not. To dispel the actualist tendencies they write that One ought not be taken in by a definite description such as “the future” or “what will happen.” (Belnap et al., 2001, p. 168) Since the particular version of actualism described as the Thin Red Line looks rather unpromising from the outset, one might suspect that Belnap and Green had invented a straw man that was later annihilated by Belnap et al.(2001) It would not be a fair of assessment of their project though. Historical research reveals that many proponents of actuality within the context of branching might easily leave the reader with an impression that they endorse or at least seriously consider a sort of metaphysical picture which Belnap and Green have named the Thin Red Line.2 The first mention of the TRL-like theory is, as far as I know, due to Arthur Prior. In a paper from 1966, he discusses a number of systems of temporal logic, including Occamism: In the metatheory of the Occamist systems, we may define an O or an O0 model as a line (without beginning or end) which breaks up into branches as it moves from left to right (this being interpretable as a movement from past to future), so that from any point on it there is only one route to the left (backward into the past) but a number of alternative routes to the right (forward into the future). In each O or O0 model there is a single designated route from left to right, taking one direction only at each fork. This represents the actual course of events. (Prior, 1966, p. 157) The “Occamist” system of Prior(1966) is crucially di fferent from “Ockhamist” system of Prior(1967). Namely, every Occamist model is equipped with a “designated route,”3 while there is no such route in Ockhamist models. Due to incorporation of “the actual course of events,” Prior(1966) was able to distinguish “actual” assignments of truth values from “prima facie” assignments. In Ockhamism, we are left with prima facie assignments only. It is not entirely clear to me whether the Occamist model is a good example of the TRL model. Nonetheless, a branching world with an actual part can be suggested by the

2To be fair to the alleged Thin Red Liners as well, I should note they were mostly concerned with the semantic and logical aspect of the theory. They were rarely sufficiently explicit regarding their metaphysical views to firmly categorize them as the Thin Red Liners in Belnap’s sense of the term. For example the papers of Peter Øhrstrøm might sometimes suggest a metaphysical picture similar to Belnap’s Thin Red Line. He even wrote a paper “In defense of the Thin Red Line” (2009). However, in personal communication, he explicitly rejected his commitment to the version of actualism just presented. Nonetheless, he admitted that there is need for more clarity regarding the exact metaphysical import of the notion of the Thin Red Line. 3The difference between O and O0 is irrelevant for our purposes.

131 CHAPTER 5. THIN RED LINE description above. This suggestion is taken forward by Rescher and Urquhart in their Temporal Logic (1971). They discuss three views—the Leibnizian, the Stoic, and the Epicurean—on the relation between possibility and actuality (Rescher and Urquhart, 1971, pp. 200–202). To compare and contrast the three views, they introduce the following picture: Let us represent the world as an infinite tree branching toward the future, whose branch points (nodes) represent junctures (events of certain sort, viz. C-events) at which one among different alternatives come to be realized. The actual course of history will be one among the branches of such a tree.(Rescher and Urquhart, 1971, p. 201, emphasis mine, C-events stand for chancy events). This fragment might be seen as an archetype of the Thin Red Line theory. On the one hand, the authors admit that the world is accurately represented as a branching tree—it consists of all the possibilities. On the other hand, the authors do not want to give up the actualist intuition, even within such conceived, modally extensive world. So, they claim that only one of the branches of the tree is the actual course of history. They to try to to have a cake of modally neutral world, eat a cake and claim that, after all, only one of the possibilities is actual. The tension between the Genuinely Realistic account of possibilities and the actualist insight is detectable in other parts of their book as well. For example, Rescher and Urquhart claim that [T]here is but a single “course of time” itself, although there may be a multiplicity of possible future courses of events within this unique time. (Rescher and Urquhart, 1971, p. 72). We are confronted here with an idea that numerous possible courses of events take place within a single, linear course of time. It is a rather non-standard idea, one would naturally assume that what takes place in the linear course of time is the linear course of actual events rather than the multitude of possible courses of events. If all the possible futures take place in unique time, then it seems that we cannot reasonably talk about “what will actually turn out” and we need to give up the bivalence of future contingents. The actualist tendencies of Rescher and Urquhart do not allow bite this bullet and they insist that On our view, (. . . ) it is plausible to characterize a future contingency as true or false (albeit one cannot say which). The (temporally definite) future-contingent proposition p is, even as of n [i.e., as of now], either true or false (whatever ultimately turns out), although it does not yet (i.e., at n) possess this truth value in a determinate way. (Rescher and Urquhart, 1971, p. 73) Rescher and Urquhart are not the only ones who could have suggested the TRL-like view to Belnap and Green. Let us take the excellent work by McKim and Davis(1976) as an another example. The authors aim is to semantically distinguish the modal and the factual future tense. In the process they write, very much along the lines of Rescher and Urquhart:

132 CHAPTER 5. THIN RED LINE

In a branching time world, commitment to a real future is essentially commitment to the claim that there is some unique branch, i.e., some connected sequence of states, which will be actualized as time passes. (McKim and Davis, 1976, p. 234). Again, it is suggested that within the world, which is branching, there is a region which is somehow special, since it is the only actualized part of the world. A similar idea might be extracted from the writings of Peter Øhrstrøm:

In [the last system I will discuss] it is assumed that there is a branch, f , in the branch structure which represents the factual course of events. (. . . ) In this way the times in the structure are divided into two groups, the factual and the counterfactual. (Øhrstrøm, 1981, pp. 88–89).

Once again courses of events are divided into two categories. The course of events that factually takes place, and the courses of events that do happen in time, but they do not factually take place. It is not surprising then, that when Belnap and Green ﬁrst reﬂected on how to combine branching notion of possibility with the idea of a single actual course of events, they summarize the attempts of their predecessors in the following word:

TRL represents the actual history, the one and only actual history in all of Our World. If you metaphorically stand outside Our World, you will see it clearly marked. (Belnap and Green, 1994, p. 379)

5.2 Semantic impact

We shall see that such conceived idea of the Thin Red Line has a number of substantial drawbacks. Nonetheless, it is important to realize that it also has at least one important virtue—it can be used to address the initialization failure. Thanks to the distinguished actuality, we can clearly differentiate between the plain future “will” and the modalized future “might” or “must.” In fact, the main motivation behind the theories of Prior (1966), Rescher and Urquhart(1971), McKim and Davis(1976), or Øhrstrøm(1981) was to give a clear account of non-modalized future tense within indeterministic setting. The metaphysical background was just briefly sketched to justify the semantic decisions. Partly due to the semantic orientation, the precise metaphysical commit- ments of actualism within the context of branching have not been spelled out carefully enough. The TRL response to initialization failure is pretty much straightforward. At any moment, there are many different future continuations. However, to assign a truth value to a sentence about the future, one should appeal to the single history which will be actualized—the TRL. The other available histories can be used to analyze the sentences about what might and must happen, but the truth value of the sentences which say what will happen should depend on what will in fact happen. However, this simple insight has proven to be notoriously difficult to capture formally. The simplest possible idea would be to enrich the notion of the model, so it

133 CHAPTER 5. THIN RED LINE becomes a quadruple M B hM, ≤, TRL, Vi, where TRL represents the actual history. Then, we can use the absolute TRL (a-trl) to interpret the future tense operator:

a-trl 0 0 0 a-trl Definition 5.1. M, m|= Fφ iff ∃m0 (m > m & m ∈ TRL & M, m |= φ). On the first sight, it is exactly what we wanted to achieve. The truth value of a sentence in future tense depends on what will actually happen. Also, this simple maneuver helps to easily solve the initialization failure. Thanks to the existence of the Thin Red Line, we do not need to semantically relativize the truth value of a sentence to a history (there is no h on the left-hand side of |=a-trl). So, we do not need to initialize the history parameter and the initialization failure does not arise. Then, we can use a simple, “deflationary” postsemantics:

Definition 5.2. m||−a-trlφ iff m|=a-trlφ. This theory is very clean and simple on the postsemantic level, but it is faulty on the level of semantic proper. This difficulty has been realized already by Richmond Thomason(1970) who briefly considered (pp. 270–271) a version of TRL semantics and makes the following comment:

β Suppose that β is in the real future of α; then what of the point γ? It isn’t in real time, and yet in order to evaluate tensed formulas at γ α we must provide it a real future. (Thomason, 1970, p. 271) γ δ

The problem that Thomason alludes to is that the interpretation of the operator F requires the “real future,” but at some moments of the tree, the real future is missing. So, it is hard to analyze the operator F at such moments. At ﬁrst, it might seem like an artiﬁcial problem. After all, why would we ever need to evaluate tensed formulas at points which are not in the real time? Well, there is an important reason why we do need to be able to do it. If we want the semantics to be compositional, we need some way to interpret the future operator outside of the TRL. After all, the future operator might be embedded in scope of modal and temporal operators which shift the point of evaluation outside of the Thin Red Line, as is well illustrated by the example studied by Belnap and Green:

The coin will come up heads. It is possible, though, that it will come up tails, and then later (*) it will come up tails again (though at that moment it could come up heads), and then, inevitably, still later it will come up tails yet again. The trouble is that at (*) the example says that tails will happen, not merely that it might, whereas the explanation of the future tense given above presupposed that the moment of evaluation was in the TRL. (Belnap and Green, 1994, p. 379)

134 CHAPTER 5. THIN RED LINE

The proponents of the TRL took seriously Thomason’s imperative that “we must provide γ a real future” and in order to evaluate future-tensed formulas outside of the TRL, they decided to provide real/actual futures to the moments outside of the actual course of events.4 This decision marks an important shift in the approach to the TRL. In face of the semantic diﬃculties, the TRLers abandon the more natural idea of the Absolute Thin Red Line which says that exactly one of the possibilities is being actualized in the course of time and move to a metaphysically less transparent notion of a Functional Thin Red Line which implies that even if some moment will not be actualized, it has an actual future of its own. Formally, the technique is very simple, we just replace a single TRL with a TRL-function trl f cn : M 7→ Hist such that ∀mm ∈ TRL(m). Intuitively, trl f cn(m) is meant to designate the history actual at moment m. With the concept of the trl f cn, we can interpret the future tense at any moment of the tree:

f-trl 0 0 0 f-trl Definition 5.3. M, m|= Fφ iff ∃m0 (m > m & m ∈ trl f cn(m)& M, m |= φ). Thanks to this definition, the future tense is defined at every moment in the treelike world. The semantics does not require relativization to histories, so the initialization failure is averted. Consequently, we can stick to the simple, deflationary postsemantics. The replacement of TRL with trl f cn results with a semantically more appealing system (we shall see in sec. 5.3.4 however, that it is not free of difficulties), but it forces the TRLers to sacrifice part of their conceptual chastity. When they initially incorporated the notion of the actual course of events into the context of branching they explicitly argued that there is just one possible course of events which gets actualized as the time goes by. The actualist intuition supports this verdict. However, in face of the formal problem explicated by Belnap and Green, they need to give up this assumption and agree that each moment in the branching world has its own actual future. The hint towards such relativization is noticeable already in (Rescher and Urquhart, 1971). Initially thy claim that “The actual course of history will be one among the branches of such a tree” (p. 201), but they later add that the “branch points represent junctures at which one among different alternatives come to be realized” (p. 201). How- ever, if two branch points are not a part of a single history and some alternative comes to be realized at each of them, then there cannot be just one actual course of history in the whole tree. Thus, it seems that we require an actual history for every moment in the tree. Similarly McKim and Davis(1976), when they first motivate their formal system, claim that “In a branching world (. . . ) some unique branch (. . . ) will be actualized as time passes” (p. 234). However, when they construct the formal system, they introduce a function f which “assigns each moment its actual future” (p. 235). The same kind of function is being used by Barcellan and Zanardo(1999) or Øhrstrøm(2009). We have moved from Thin Red Line, to Thin Red Lines that can be pictorially represented on the following tree:

4Importantly, Prior(1966) did not make this assumption. Perhaps he later realized that it might be necessary, while being very problematic, and that is why he discarded of the actual assignments altogether in (Prior, 1967).

135 CHAPTER 5. THIN RED LINE

Our World The actual course(s) of events

We entertain the idea that even if there is just one possibility which is being actualized in the course of time, the TRL, the possible events which do not get actualized have their own “quasi-actual” possibilities which are actualized from “their” point of view. At most one of the possible futures is the “real” futures, but the moments not in the real future have their own futures which are not real, but “real-for-them.” So, to interpret a future tense, we use the real future and if it is missing, we use the real futures of a “second grade.” They might not be “really” real, but they are more real than other available candidates. I sympathize with Thomason’ comments on this idea: At this point we begin to lose track of what a “real future” is, and plainly it would be better to just return to a linear conception of time. (Thomason, 1970, p. 271) I think that the semantic difficulty which the Thin Red Liners have encountered is grounded in the problematic metaphysical vision they initially endorsed. They maintain that the actual course of events is linearly ordered, but the world is not. They would like to attach the interpretation of temporal connectives to the actual course of events, but the actual course of events is not “present” at some parts of the world. How to interpret the temporal connectives at these non-actual moments? What are the non-actual people are talking about, when they talk about “the future”? It seems to me that a metaphysical vision which distinguishes actual and non-actual parts of the world is not easy to maintain. If one accepts modal neutrality and admits that the world has a treelike structure of events, it is hard to find any ground to absolutely distinguish a particular part of the world which we occupy, as the actual part. However, I am not convinced that Belnap et al.(2001) draw the right conclusion this observation. They claim that in face of the problems with the TRL, one should stick to the treelike world and simply abandon the notion of the actual course of events. I will argue that one can do the reverse: to abandon the idea of a treelike world and continue to believe that only one of the (objectively) possible courses of events is actualized by the (indeterministic) procession of events in (non-branching) time. Let me postpone this discussion until the next chapter and focus on the specific problems with the TRL first.

136 CHAPTER 5. THIN RED LINE

5.3 Objections to the Thin Red Line

Nuel Belnap and his collaborators (Belnap and Green, 1994; Belnap et al., 2001) have exposed many points of weakness of the Thin Red Line theory. They all stem, in one way or another, from one single peculiar feature of this view: that within the branching world, one can distinguish actual and non-actual parts. These arguments establish that a world is a very peculiar place to live in, especially for those who inhabit its non-actual parts. I will ﬁrst present problems with the Thin Red Line grounded in metaphysics and epistemology and then focus on the semantic oddities. Finally, I will mention the postsemantic diﬃculties that result from such a metaphysical position. Sections 5.3.1– 5.3.4 draw heavily on the material I included in (Wawer, 2014).

5.3.1 Metaphysics Let me begin with a metaphysical problem with the TRL diagnosed by Belnap et al. (2001). The authors ask:

What in the structure of our world could determine a single possibility from among all the others to be “actual”? As far as we know, there is nothing in any science that would help. (Belnap et al., 2001, p. 162)

In the Genuinely Realistic account of possibility, “being actual” is indeed a problematic property. The authors try to figure out what “makes” one of the possibilities the actual possibility. What distinguishes “actual” parts of our world (actual possibilities) from “non-actual” parts of our world (non-actual possibilities). After all, all the possibilities are “made of the same stuff,” they are occupied by objects with are massive, resilient, colorful, electrically charged, etc. Thus, no “ordinary” property can make a difference and distinguish one of the possibilities as “the actual one.” There- fore, actuality needs to be some kind of mysterious property which is not detectable by ordinary measuring devices. One has every right to be skeptical about any theory which postulates this kind of of peculiar properties.

5.3.2 Epistemology The mysterious property of “being actual” generates epistemological worries. Let TRLabs designate the unique actual part of the branching world. Then, the authors ask:

[H]ow we could know whether we are on TRLabs. How could we ﬁnd out? (Belnap et al., 2001, p. 163)

It is a perfectly reasonable question in this context. After all, the world is branching and every part of it is equally real, tangible, and vivid. Then, if we are told that only some of these parts have an extra quality of being actual, the quality undetectable by senses, then we have every right to be concerned whether we are actual or not. How can we ever ﬁnd out if the part of the world which we happen to occupy possesses this extra quality? In fact, it is immensely unlikely, that we are actual! At every indeterministic

137 CHAPTER 5. THIN RED LINE juncture in the past, the actuality might have “followed” an alternative scenario. It is perfectly possible that in actuality the dinosaurs still rumble around while humans have never evolved. And we, and all our concerns are, unbeknown to us, just a sheer, non-actual possibility. The mere fact that we can reasonably entertain this idea in the world with a Thin Red Line shows that there is something wrong with such a world. Belnap et al.(2001) diagnose, that the faulty element is an unnecessary addition of the actual possibility. A similar argument convinced David Lewis to reject the absolute and accept the indexical notion of actuality (Lewis, 1986, pp. 92–6).

5.3.3 Actuality Another problematic feature of the TRL theory that Belnap et al.(2001) point out is, surprisingly, its treatment of actuality:

The TRL theory also has troubles with actuality. (. . . ) As Lewis has argued (Lewis, 1970a), this world’s being the actual world does not favor it over any others, but is just a reﬂection of the fact that this is the world at which we are conversing. To suppose that there is one from among the histories in Our World that is the absolutely actual history is rather like purporting to stand outside Lewis’s realm of concrete possibilia and pointing to the one that is actual. But this is wrong in both cases. (Belnap et al., 2001, p. 163)

I understand this objection along the following lines: David Lewis has argued, convincingly, that as soon as one accepts a plurality of concrete possible worlds, one should not distinguish one of these worlds as possessing a special property of being actual (see Lewis, 1970a, p. 187). Instead, in the setting of Genuine Modal Realism, it is most reasonable to take actuality to be a perspective dependent feature—each possible world is actual from its own perspective:

The actual world is not special in itself, but only in the special relation it bears to the ontological arguer. Other worlds bear the same relation to other ontological arguers. (Lewis, 1970a, p. 187).

Then, if Lewis is right, it is natural to expect that the same kind of mechanism should hold for Genuine Branching Realism. If one accepts that all overlapping possibilities are equally real, concrete courses of events (as the Thin Red Line theorists seem to accept), then it is natural to expect that the best treatment of actuality is the indexical treatment of actuality. All possible circumstances are actual, from their own perspectives (the details need to be more intricate in case of branching since possible histories/worlds overlap and Lewis’ worlds do not). However, the Thin Red Liners go against Lewis’ advice and despite commitment to Genuine Modal Realism, they treat actuality as something more than just an indexical expression. In their opinion, the actual course of events is absolutely, and not merely relatively, distinct from possible courses of events. Thus, one part of the TRL theory strongly suggests a purely indexical account of actuality, but another part of the theory openly goes against this

138 CHAPTER 5. THIN RED LINE perspectival account. We end up with a peculiar theory within which the ontological arguers who do not inhabit the TRL think that they are actual, but they are mistaken.

5.3.4 Semantics Finally, I can turn to the most elaborated family of objections regarding different versions of TRL theories. They can be loosely described as semantic or logical objections. The concept of the Thin Red Line has a military connotation. It refers to a certain defensive complex that consists of small in number, but strategically deployed and well-equipped military units. I will use the military overtone to explicate the formal problems which different version of TRL theories encounter. We shall see that the semantic Thin Red Line was severely bombarded, especially in 1994 and 2001. However, the defenders have never surrendered and often returned fire. I am going to present the dialectic of the development of the concept of the TRL in a series of such “attacks” and “defenses.” I organize these military maneuvers in logical, rather than chronological fashion, which means that I describe various versions of TRL semantics from the simplest to the more complicated as responses to increasingly challenging arguments presented by its critics.

Building the ﬁrst trenches: TRL1 The basic TRL semantics, found in (Øhrstrøm, 1981), is based on an idea that we need to intimately bind the interpretation of the F operator with the TRLh. It will be then useful to incorporate this information to the deﬁnition of the model:

Definition 5.4 (TRL1-model). ATRL1-model M is a quadruple hM, ≤, TRLh, Vi where hM, ≤, Vi is a branching model and TRLh ∈ Hist. The sentence “There will be a sea battle” is true iff there will be a sea battle in the actual future. If we use TRLh to interpret F, we can get rid of the history parameter of evaluation altogether. The following definition naturally gets across this idea.

Definition 5.5 (Sentence φ is true in TRL1-model M at m). • M, m|=a-trl p iff m ∈ V(p) where p ∈ Atom; • standard definitions for truth-functional connectives; • M, m|=a-trlPφ iff ∃m0(m0 < m ∧ M, m0|=a-trlφ);

a-trl 0 0 0 0 a-trl • M, m|= Fφ iﬀ ∃m (m < m ∧ m ∈ TRLh ∧ M, m |= φ);

a-trl 0 0 0 a-trl • M, m|= F φ iﬀ ∃m (m < m ∧ M, m |= φ);

a-trl 0 0 0 0 a-trl • M, m|= F φ iff ∀h(m ∈ h → ∃m (m ∈ h ∧ m < m ∧ M, m |= φ)). Thanks to the removal of the history parameter from our semantics, we can bind interpretation of F with TRLh avoid the initialization failure, but we can no longer use it to define modal connectives. Notice that the definitions above render F and F essentially tempo-modal operators. To evaluate a formula containing F or F we need

139 CHAPTER 5. THIN RED LINE to take into account both a temporal factor (a future moment m) and modal factor (the history in which the moment is situated). The intended meaning of F is ‘possibly in the future’ or simply “it might be that” and F can be read as “necessarily in the future” or “it is inevitable that.” Simply put: modal operators look “sideways” at other histories but also “forwards” into the future. The future tense operator looks only forwards, and only into the actual future. The deﬁnitions proposed above mimic the idea of the Peircean sense of operators f and F. We need to deﬁne F and F separately since f and F are not dual in Peircean semantics (for more details, cf. section 4.3, and also Prior, 1967, Ch. 7; Barcellan and Zanardo, 1999, p. 3; Belnap et al., 2001, p. 161).

The ﬁrst shots: Truth values outside the TRL

I have already introduced the the crucial logical objection against the TRL1. It comes down to the question of interpretation of operator F at moments not in TRLh. Belnap and Green(1994, p. 379) write that “Branching +TRL has the defect that it gives no account of the future tense relative to moments that do not lie in the TRLh.” In fact, that is not quite right since according to the TRL1 all the future tensed sentences evaluated outside the TRL are simply false. Anyway, it is still a very serious objection. It seems that in TRL1 operator F cannot be embedded in scope of modal connective like F or F . This objection was recognized as a fatal one even by the most persistent defenders of the TRL (see e.g., Braüner et al., 1998). As a result, they decided to reconsider the notion of the TRL in a way that accounts for tensed statements at arbitrary evaluation points of a model.

Battle lines re-drawn: TRL functions In response to the attack, most TRLers decided to regroup. The generally accepted strategy was to posit a TRL at every point of the model using the trl f cn (see McKim and Davis, 1976; Braüner et al., 1998, 2000; Barcellan and Zanardo, 1999; Øhrstrøm, 2009).5 The function, intuitively speaking, picks for each moment in a model its actual future (in general, existence of such function depends on the axiom of choice). Obvi- ously, not every function f : M 7→ Hist will do. Some constraints must be imposed for it to represent the intended idea. First of all, since the function is about to pick the actual history for a moment, the moment had better be a part of this history. So, the minimal constraint is the following:

Condition 1: ∀m∈M m ∈ trl f cn(m). (McKim and Davis, 1976, p. 235) Let us try to deﬁne a new notion of a model:

Definition 5.6 (TRL2-model). ATRL2-model M is a quadruple hM, ≤, trl f cn, Vi where hM, ≤, Vi is a branching model and trl f cn satisfies Condition 1. We can now redefine the truth clause for the future operator (the rest of the connectives are interpreted as stated in definition 5.5).

Deﬁnition 5.7 (Fφ is true in TRL2 model M at m).

5There were some exceptions to this strategy like (Malpass and Wawer, 2012) or (Wawer, 2014).

140 CHAPTER 5. THIN RED LINE

f-trl 0 0 0 0 f-trl • M, m|= Fφ iﬀ ∃m (m < m ∧ m ∈ trl f cn(m) ∧ M, m |= φ). This maneuver allows for a compositional embedding of F. We know how to interpret the sentence like P F (q ∧ Fq). So one attack was parried, but the ﬁeld was far from peaceful.

Gap in the lines: F p → FF p fails

It turns out that this general definition of trl f cn is not completely satisfactory. One of the most striking deficiencies is the fact that under TRL2 semantics neither of these two very intuitive sentences: FFφ → Fφ and Fφ → FFφ is valid. In the usual temporal logic, they define, respectively, transitivity and density of the accessibility relation. However, under TRL2 semantics they change their usual meaning and fail for different reasons. To see that none of the two sentences is valid, consider the following simple TRL2 model M B hM, ≤, trl , Vi such that: h1 h2 f cn m3 ¬p m4 p

• M = {m1, m2, m3, m4} m ¬p • {m1, m2, m3} = h1, {m1, m2, m4} = h2 2

• m1 < m2 & m2 < m3 & m2 < m4 & m4 m3 & m3 m4 m1 ¬p • trl f cn(m1) = h1, trl f cn(m2) = h2

• m1, m2, m3 < V(p), m4 ∈ V(p) f-trl f-trl f-trl Observe that in this model: M, m1|= FF p since M, m2|= F p but M, m12 F p. f-trl f-trl Therefore, FF p → F p is not valid. Similarly M, m1|= F¬p and M, m12 FF¬p so the converse implication is not valid either. This troublesome consequence is due to the fact that moments in the tree might not “accord” with respect to their TRLs. As visible in the foregoing example, the TRLs of m1 and m2 are diﬀerent even though m2 is in the TRL of m1.

Overzealous defense: trl f cn excludes branching One of the ways to cure the aforementioned ﬂaw is to impose an additional constraint on trl f cn. In order to avoid “disagreement” between moments in the tree, Belnap and Green(1994) suggested the following move:

Condition 2: ∀m1,m2 m1 ≤ m2 ⇒ trl f cn(m1) = trl f cn(m2). (Belnap and Green, 1994, p. 380)

Definition 5.8 (TRL3-model). ATRL3-model M is a quadruple hM, ≤, trl f cn, Vi where hM, ≤, Vi is a branching model and trl f cn satisfies Conditions 1 and 2. The problems from the previous section disappear: both FFφ → Fφ and Fφ → FFφ are valid (in any densely ordered frame). Nevertheless, the price is very high. As a result of Condition 2, we exclude any branching TRL3-models. A very easy proof is sufficient to establish it:

141 CHAPTER 5. THIN RED LINE

1. Assume that there is a branching TRL3-model, that is, there are m0, m1, m2 ∈ M such that:

(a) m1 ≮ m2 and m2 ≮ m1, and m1 , m2;

(b) m0 ≤ m1 and m0 ≤ m2.

2. Since m0 ≤ m1 and m0 ≤ m2, then, by Condition 2, trl f cn(m0) = trl f cn(m1) = trl f cn(m2). Let trl f cn(m0) be a history h.

3. By Condition 1, m0, m1, m2 ∈ h.

4. By deﬁnition of a history as a linearly ordered set we have that m1 < m2 or m2 < m1 or m1 = m2 which contradicts 1(a).

A very similar proof was used by Belnap and Green(1994, p. 380) to establish the fundamental discrepancy between the idea of the actual history and the branching representation of ontic indeterminism. However, Belnap and Green’s response to the problems is too hasty and one should (and the TRLers did) take more moderate steps in the campaign.

Cautious defense: trl f cn allows branching

The philosophers and logicians arguing in favor of the TRL acknowledged that trl f cn should be somehow constrained. Nevertheless, Condition 2 proposed by Belnap and Green(1994) is evidently too strong. The middle way was ﬁrst noticed by McKim and Davis(1976). They proposed a weaker cousin of Condition 2 which did not force the deterministic conclusion.6 Condition 3: ∀m1,m2 (m1 < m2&m2 ∈ trl f cn(m1)) ⇒ trl f cn(m1) = trl f cn(m2) .(McKim and Davis, 1976, p. 235)

The condition is thought as follows: if a history h is picked as the future of a given moment m1, then every m2 > m1 which is in h must conﬁrm m1’s “choice.” Nonethe- less, the moments above m1 that are not in h are free to choose otherwise (unless their antecedents above m1 enforce some choice upon them). Having introduced the new condition we can slightly reconstruct the TRL-model.

Deﬁnition 5.9 (TRL4-model). ATRL4-model M is a quadruple hM, ≤, trl f cn, Vi where where hM, ≤, Vi is a branching model and trl f cn satisﬁes Conditions 1 and 3. Condition 3 does not exclude branching models and ensure validity of FFφ → Fφ and Fφ → FFφ (the second, only in densely ordered frames). Some authors (e.g., Barcellan and Zanardo, 1999; Braüner et al., 1998) considered one additional and quite natural condition on trl f cn, namely:

0 0 Condition 4: ∃m∈M∀m0 m < m ⇒ trl f cn(m ) = TRL(m)

6A very similar condition was proposed, in a slightly diﬀerent context, by Thomason and Gupta(1980).

142 CHAPTER 5. THIN RED LINE

Condition 4, together with Conditions 1 and 3, guarantee that one (and only one) of the histories in the tree is special in a sense that it is picked as the actual history by ∗ ∗ each its member. It means that ∃h∗ ∀m∈h∗ trl f cn(m) = h . The history h is called “real” by Barcellan and Zanardo(1999) and “normal” by Braüner et al.(1998). As we shall soon see, it is not the end of the war. Belnap et al.(2001) have found their way across the TRL4 trenches and attacked again.

Another storm: φ → HFφ fails

Belnap et al.(2001) pointed out that TRL4 is not a foolproof tactic either. The most important disadvantage is that it fails to validate φ → HFφ, while this sentence and its counterpart, φ → GPφ, guarantee a certain minimal symmetry between past and future: a moment m1 is in the past of a moment m2 if and only if m2 is in the future of m1. This very feature fails under the TRL4 semantics. To see this, let us examine the following TRL4-model:

h1 h2 • M = {m1, m2, m3} m2 ¬p m3 p

• {m1, m2} = h1, {m1, m3} = h2

m1 ¬p • m1 < m2 & m1 < m3 & m2 m3 & m3 m2

• trl f cn(m1) = h1, trl f cn(m2) = h1, trl f cn(m3) = h2

• m1, m2 < V(p), m3 ∈ V(p) f-trl f-trl Evidently M, m3|= p but M, m32 HF p because ∃m m < m3 (namely m1) such f-trl f-trl that m2 F p. Therefore, M, m32 p → HF p. The bizarre nature of this consequence is well illustrated by Belnap et al.’s (2001) example (2001, p. 166), slightly adjusted to the notation of our case. Let m1 happen at 1:00 P.M., m2 and m3 both happen at 2:00 P.M., and let p mean “The coin lands tails”:

Now picture Jack at the moment of use, m3, where the coin landed tails at 2:00 P.M. It would seem that in order to speak truly at m3, Jack would be obliged to say “The coin has landed tails, but this is not what was going to happen at 1:00 P.M. At 1:00 P.M. the coin was going to land heads. It’s just that it didn’t.” (Belnap et al., 2001, p. 166)

A related problematic sentence is ^F(p ∧ PG¬p). It can be true in TRL4 semantics while it basically says that it is possible that something will happen even though it was not going to happen.7 Another troublesome example pointed out by Belnap et al.(2001) is that the sentence Fφ → FPFφ is valid in TRL4. It is translated by Belnap et al.(2001) into “That something will happen does indeed imply that it is inevitable that it will be true that it was going to happen” (Belnap et al., 2001, p. 167). Which is dangerously close to the deterministic Fφ → Fφ saying that whatever will happen, will happen out of necessity.

7This claim seems unacceptable to me, but Patrick Todd(2015b) have recently tried to defend its plausibility.

143 CHAPTER 5. THIN RED LINE

How to ﬁght?

Do Not budge an inch The first of the tactics is to stick to the TRL4 solution and somehow explain away the counter-intuitive consequences. For example Barcellan and Zanardo(1999) appeal to the research in Artificial Intelligence and claim that at any moment m, trl f cn(m) picks a history which “best fits suitable criteria like minimal change principles, probability, typicality and others” (Barcellan and Zanardo, 1999, p. 7). In models whose trl f cn satisfy Condition 4, the sentence φ → HFφ might serve as a test of “proper development” of a course of events since we have that φ → HFφ is valid at m iff m ∈ h∗. It means that the “real” history h∗ develops in the best possible agreement with the criteria.8 The authors use trl f cn to interpret operator fA

Deﬁnition 5.10 ( fA is true in TRL4 model M at m).

f-trl 0 0 0 0 f-trl • M, m|= fAφ iﬀ ∃m (m < m ∧ m ∈ trl f cn(m) ∧ M, m |= φ). Then the authors add the following comment:

According to this point of view, the satisﬁability of the negation of p → H fA p has a quite reasonable reading: if this formula can be falsiﬁed at a given moment t, then, somewhere in the past of t, the world did not develop according to our “preferred developments” criteria. (Barcellan and Zanardo, 1999, p. 7)

I agree with Barcellan and Zanardo that if the role of trl f cn is to pick the most probable or most typical history, then we should not be surprised with an occasional failure of p → H fA p. After all, improbable and untypical things happen every now and then. However, if we agree with this interpretation of the function, it can no longer serve as a basis for interpretation of the future tense. We should understand fAφ as “according to our preferred criteria, φ should happen.” But the interpretation of sentences in the future tense does not depend on what should happen, it depends on what will happen. Therefore, it is misleading on the part of Barcellan and Zanardo to call the TRL-function “an actualizing function” (Barcellan and Zanardo, 1999, p. 5, def. 2.1) and to call the values of trl f cn actual futures. Also, it is misleading bind the interpretation of the future tense with the values of the function trl f cn. The authors can explain why φ → H fAφ fails sometimes, but they can do that only if they assume that fA does not mean “in the future,” but something like “in the most likely future.” Another rationalization of the failure of φ → HFφ is proposed by Braüner et al. (2000). Their argument goes as follows: The counter-factual assumption of q does not invalidate the truth of the past prediction PF¬q. If I am awake now, it certainly was true yesterday that I was going to be awake after one day. The prediction was true (but

8This view is a distant echo of the Leibniz’s ideas. He also proposed a set of criteria—simplicity of laws, richness in phenomena, happiness of minds, maximality of perspectives—and then his “real” history is simply the best of possible worlds which God chooses to actualize.

144 CHAPTER 5. THIN RED LINE

of course not necessary) even if I now—while being awake—imagine that I were asleep. For this reason one might say, that the truth of q ∧ PF¬q, where q stands for “I am asleep,” is in fact conceivable. (Braüner et al., 2000, p. 203) The authors try to explain the failure of q ∧ PF¬q.9 To explain this failure, they use a rather controversial argument. They begin with a simple observation that Peter is awake (let q stand for the sentence, “Peter is awake”). If it is so, then it was true a day before that Peter would be awake on the next day (P1F1q is true). One would expect that q is true as well, after all, Peter is awake, but here comes a surprise— q is false. So, even though Peter is awake, the sentence “Peter is awake” is false because Peter imagines that he is asleep. Thus, the truth of P1F1q is grounded in the actual state of aﬀairs, while the truth of ¬q is grounded in an imaginary state of aﬀairs. Therefore, the modal status of the sentence q ∧ P1F1q changes from one conjunct to another, while we naturally assume that both conjuncts should be evaluated in the same circumstances (either real, or imaginary) unless there is a modal operator which signals the shift of circumstances. But there is no operator in the logical form of the sentence they consider. Therefore, we should either focus on the actual situation in which PnFn¬q is true, but q is false, or consider the imaginary situation in which q is true, but PnFn¬q false (in any case q ∧ P1F1¬q is false). Anyhow, the authors themselves admit that their “piece of argumentation is somewhat strained” (Braüner et al., 2000, p. 203) and carry on to present another proposal.

Counter-attack: Counter-factual Thin Red Lines Braüner et al.(2000) were dis- satisfied with TRL4 for a couple of reasons. First of all, it did not validate φ → HFφ and secondly, it did not provide the straightforward interpretation of modal operators (presumably the authors would not be content with the account of F and F proposed in definition 5.5). Therefore, they decided to devise a new TRL semantics which would deal with these problems. I have already described their approach in the previous chapter (see section 4.8.2), so let me just briefly restate the major points. In their semantic project, Braüner et al. utilized the concept of trl f cn to define the set of “counterfactual branches” of moment m:

Deﬁnition 5.11 (Counterfactual branches). Let trl f cn : M 7→ Hist be a function satisfying Conditions 1 and 3. The set C(m) of counterfactual branches of a moment m is: 0 0 0 C(m): = {h ∈ Hist|m ∈ h ∧ ∀m > m(m ∈ h → trl f cn(m ) = h)}.

Due to Condition 3, trl f cn(m) ∈ C(m) but the set might be larger. If there are many elementary possibilities open at m, then C(m) might contain a single representative from each elementary possibility. With the notion of counterfactual branches (cb) at our disposal, we can deﬁne new truth clauses for the connectives in the TRL4-model.

9In fact, they are not suﬃciently precise, as the truly surprising case that cries for explanation is the failure of q ∧ HF¬q, or a metric q ∧ PnFn¬q; they correct their mistake in (Øhrstrøm and Hasle, 2011).

145 CHAPTER 5. THIN RED LINE

Deﬁnition 5.12 (Sentence φ is true in TRL4 model M at m/h pair). Let M be a TRL4 model. Assume that for every pair m/h, m ∈ h ∈ C(m), then:

• M, m/h|=cb p iﬀ m ∈ V(p) where p ∈ Atom; • standard deﬁnitions for truth-functional connectives;

• M, m/h|=cbPφ iﬀ ∃m0(m0 < m ∧ M, m0/h|=cbφ); • M, m/h|=cbFφ iﬀ ∃m0(m < m0 ∧ m0 ∈ h ∧ M, m0/h|=cbφ);

• M, m/h|=cb^φ iﬀ ∃h0(h0 ∈ C(m) ∧ M, m/h0|=cbφ).

Operators G, H, and are duals of F, P, ^ respectively.10 It is easy to see that, given these new clauses, sentences FFφ → Fφ and φ → HFφ are valid while sentences ^F(p ∧ PG¬p) and Fφ → FPFφ are not. Additionally, we have a clear interpretation of modal operator ^ analogous to its Ockhamist interpretation. Hence Braüner et al.(2000) achieved the aims they stated but this solution is not without objections. The first of these is formulated by Braüner et al. themselves. They observe that their new semantics invalidates a sentence F^φ → ^Fφ. To see that it is problematic consider the following two examples: (1) “Tomorrow, I might have dinner out” and (2) “Necessarily, tomorrow I will stay home” According to our new semantics, (1) and (2) might well be true at the same moment m. Nonetheless, I believe this approach faces a more important, conceptual difficulty. It seems to betray the fundamental principles motivating the introduction of the concept of the TRL. Introduction of a history parameter to an index re-opens the gates for the initialization failure, since the sentences do not have non-relative, history-independent truth values! Remember that securing such truth values was one of the main motiva- tions for constructing the TRL semantics in the first place. The only difference between “pure” Ockhamism and this version of the TRL is that we introduce a slightly modified notion of possibility. I think that this was not what all the fuss with the TRL was about. Consequently, I do not consider this attempt by Braüner et al. to be a promising line of defense of the notion of the TRL. To sum up, the Thin Red Line theory turns out to be problematic on the semantic ground. In particular, it has substantial problems securing the validity of a basic principle of temporal logic: φ → HFφ. I will turn now to one more problem of the Thin Red Line theory, postsemantic rather than semantic in nature.

5.3.5 Postsemantics The criticism of the “logical” aspect of the TRL has undergone a characteristic shift from (Belnap and Green, 1994) to (Belnap et al., 2001). When Belnap and Green

10In 1998, Braüner et al. had proposed yet another version of the TRL semantics. It is technically more so- phisticated since they evaluate sentences at pairs m/trl f cn rather than m/h. However, in their sentences, trl f cn plays the exact formal role that the history parameter does in the just described semantics. Consequently, the discussion below applies, mutatis mutandis, to this earlier theory of Braüner et al.

146 CHAPTER 5. THIN RED LINE discuss the so-called absolute TRL models with a single actual history, they focus on the purely semantic problem. They are worried that it is diﬃcult to deﬁne the future operator at moments outside of the distinguished history in such models. By contrast, Belnap et al.(2001) criticize the absolute TRL from another angle:

The TRL theory sounds all right, but it is not. It has the “logical” defect that it gives no account whatsoever of predictive speech acts occurring at moments of use that lie oﬀ the TRL and is by so much useless. (Belnap et al., 2001, p. 162)

The authors are no longer interested, at least not directly, in the analysis of the embedded future tense connective, but in analysis of speech acts11 which take place in the contexts that lie outside of the actual course of events. Thus, they ask a paradigmatic postsemantic question in the sense of (MacFarlane, 2003)—how to use a semantic machinery (for example, TRL1 or Ockhamism), to analyze speech acts that happen not to take place on the TRL? Another fragment conﬁrms this interpretation of Belnap et al.’s (2001) challenge:

We have no trouble with predictions that will be or have been made, but we have no way of understanding predictions that might have been made. We have no way of getting a grip on “Had things gone otherwise, Jack would have asserted the following: ‘It will (eventually) rain.’” Given the context of Jack’s assertion, the TRL is no longer able to guide us in understanding his reference to his future. (Belnap et al., 2001, p. 162)

The authors clearly expect that the TRLer should be able not only answer how to evaluate the sentence, “Had things gone otherwise, it would rain,” used in an actual context, but also how to evaluate the sentence “It will rain” used in a context in which thing had gone otherwise. To illustrate the diﬀerence, let us have a look at the model depicted below and let us consider a sentence:

(S) It might have been the case that the coin would land heads. .

I shall use a simpliﬁed deﬁnition of Might-have-been operator introduced by Bel- nap et al.(2001, p. 245) and encode it as P^F. So, sentence (S) has the logical form P^FF(heads).

11“Predictive speech act” might refer to any speech act that uses the future tensed sentence as its vehicle. Belnap et al.(2001) focus on the speech act of assertion, but Belnap extends it to bets, promises, orders, and others.

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P^FF(heads) F(heads)

Our World The actual course of events

The properly semantic problem regards interpretation of the sentence P^FF(heads) used in an actual context. In particular, the underlined occurrence of operator F. The worry is that when the first three operators P^F shift the semantic parameter to a non- actual moment and a non-actual history, then the embedded sentence F(heads) will end up false or meaningless at the new index. The postsemantic question concerns the interpretation of the sentence F(heads) used at a non-actual context (just before the non-actual toss). It is not one and the same question. The picture above indicates it quite clearly. On the one hand, he sentence P^FF(heads) is evidently true used at the actual context (just as the sentence P^FF(tails) is). On the other hand, it is entirely unclear if we should call the sentence F(heads) true, while used in the non-actual context. To put things differently, it is a different thing to say if it is true that φ is possible than to say if φ is true used in a possible context. I am by no means the first one to raise this distinction, the locus classicus is (Evans, 1985). In his paper, Evans contrasts a certain semantic theory of temporal logic, T3, with a semantics of modal logic: [T]he semantic value a complex tensed sentence possesses in a context is, according to T3, a function of the semantic value which the embedded sentence would possess in another context; this is not true of the semantic values of complex statements of a modal logic, or indeed of any other known logic. (Evans, 1985, p. 361) I agree with Evans that it is possible to provide a compositional semantics which does not presuppose that we need be able to evaluate the embedded sentences in other context (in particular, in non-actual contexts). In fact, I proposed a version of such semantics in (Wawer, 2014). The message for the current discussion is that it is possible to answer Belnap and Green’s semantic challenge without answering Belnap et al.’s (2001) postsemantic challenge. To answer the postsemantic challenge, one needs to construct a theory which would be able to do something like what Evan’s theory T3 was supposed to do. Namely, to compute the semantic value of a sentence in a context in terms of the semantic value that the embedded sentences posses in other contexts. As Evans have noticed, it is a

148 CHAPTER 5. THIN RED LINE rather unusual requirement, rarely assumed for semantic theories. In some cases, this requirement simply cannot be met. As Kaplan(1989) makes clear, we cannot establish the truth value of a sentence “I could not exist” in terms of the semantic value which the sentence “I do not exist” possesses in another context (there is no context in which I truly say “I do not exist”). Moreover, I agree with Evans, that such requirement is too demanding, especially in case of modal logic. In fact, to satisfy this requirement, one would need to presuppose Genuine Branching Realism and give up the actualist ideals altogether. Nonetheless, some authors have tried to answer this problem. I will ﬁrst discuss the attempt of MacFarlane(2003, 2014). He argues that there are su ﬃcient resources in the TRL theory to answer Belnap et al.’s (2001) postsemantic objection, but he rejected this postsemantics for independent reasons. Then, I will recount the attempt of Malpass and myself (2012) and argue that it was also mistaken. In the end, I explain my current attitude towards the postsemantic objection.

5.3.5.1 TRL-functions The postsemantic challenge has been ﬁrst addressed by John MacFarlane(2003). In fact, he was the ﬁrst to clearly distinguish the semantic and postsemantic dimension of Belnap et al.’s (2001) objection. He agreed with Belnap et al.(2001) that the absolute TRL is not a promising outset for the accurate postsemantic theory since it postulates just one actual history and this history is simply not available at non-actual context. However, he noticed that one can use trl f cn in a non-standard manner and instead of using it to semantically interpret future operator, he used it to initialize the history of evaluation at a given context (and thus, to prevent the initialization failure and answer the postsemantic challenge).

f-trl Postsemantic def. 5.13 (trl f cn postsemantics). m||− φ iﬀ M, m/trl f cn(m) |= φ. Thanks to this deﬁnition, we are able to say if a sentence is true at any context on the tree, so the postsemantic problem of Belnap et al.(2001) is averted. Moreover, all the properly semantic problems characteristic to TRL theories disappear:

This proposal is not touched by Belnap and Green’s semantic arguments against the use of a thin red line (Facing the Future, pp. 160–170). It uses the very same semantics proper as Belnap and Green endorse, and appeals to the thin red line only in the postsemantics. (MacFarlane, 2003, p. 330, n. 10)

So, we do not need to modify the standard Ockhamist semantics to subscribe to a TRL theory. In particular, the semantics of future operator is history dependent, but thanks to trl f cn postsemantics, it does not generate the initialization failure. When the sentence is used at context m, we begin the process of evaluation at trl f cn(m). Then, we use the standard Ockhamist deﬁnitions in the process of evaluation of the sentence used. Due to this simple maneuver, we can leave behind, at a single stroke, all the semantic problems recapitulated in the previous section. Since we accept the Ockhamist semantics, we inherit all the Ockhamist validities, in particular, φ → HFφ, FFφ → Fφ,

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and F^φ → ^Fφ. We avoid logical complications because in trl f cn postsemantics the role of the trl f cn is not to semantically interpret operator F, but to initialize the history parameter. So, thanks to the trl f cn, the context initializes both semantic parameters. It means that when a sentence is used at m, he temporal operators shift the moment of evaluation up and down the history trl f cn(m). If we consider a sentence like ^Fq, however, then the modal operator ﬁrst shifts the history of evaluation from trl f cn(m) to some other h ∈ Hm and then, the temporal parameter travels along h rather than along trl f cn(m). Thanks to this technique, the problem of interpretation of operator F at moments outside the TRL does not arise. Thanks to the postsemantic application of trl f cn we kill two birds with one stone. Firstly, we answer the postsemantic challenge and account for prediction made at arbitrary contexts. Secondly, we secure all the reasonable Ockhamist validities on the semantic level. Of course, the shift from semantic to postsemantic level of analysis does not by itself answer the metaphysical or epistemological problems described in sections 5.3.1, 5.3.2, and 5.3.3. However, it doubtlessly is a signiﬁcant improvement. MacFarlane’s postsemantic proposal has been appreciated by Øhrstrøm and Hasle, the major champions of the semantic version of the Thin Red Line (see Øhrstrøm and Hasle, 2011, sec. 5.3). Nonetheless, MacFarlane himself rejected it since he believed, convinced by Belnap and Green(1994), that metaphysics of the Thin Red Line is not compatible with indeterminism (I dwell on his arguments in the next chapter). In his later work, he gives up this conviction and he admits that the assumption of indeterminism does not (at least not straightforwardly) contradicts the idea of the actual future (see 12 MacFarlane, 2014, p. 209). Nonetheless, MacFarlane rejects trl f cn postsemantics

There is good reason to reject this picture. The reason is not metaphysical, but semantic—or, rather, postsemantic. The Thin Red Line view yields bizarre predictions about merely counterfactual retrospective assessments of future contingent claims. (MacFarlane, 2014, p. 209)13

To see what the problem consists in, let us consider the following TRL model:

12I like to think, immodestly, that this concession is partly due to the conversations we had in 2012, during my research visit in Berkeley which coincided with Professor MacFarlane’s works on his book. 13It is a very interesting note given my argument that it is MacFarlane’s postsemantic relativism that generates “bizarre predictions about merely counterfactual retrospective assessments of future contingent claims” (see section 7.7).

150 CHAPTER 5. THIN RED LINE

Yesterday:Accurate:“Tomorrow:Sunny”

h1 h2

m1: m2:!

m0 Tomorrow:Sunny

The red arrow indicates that h1 is the actual future of m0. At m0, Jake makes an assertion using a sentence (S) “It will be sunny tomorrow.” Now, our task is to assess the accuracy of this assertion. Remember that MacFarlane is assuming the truth norm of assertion, i.e.,: If an act of assertion is accurate, then the sentence asserted is true in the context in which the act takes place.

Things look reasonably well at m1.

Now imagine someone at m1 looking back at Jake’s assertion and wondering about its accuracy. This assessor will take the accuracy of Jake’s assertion to depend on whether the sentence he asserted, (S), is true at the context in which he asserted it, m0. Since, according to the Thin Red Line view, (S) is true at m0, the assessor should take Jake to have made an accurate assertion, not one he needs to retract. And this seems right; after all, the assessor has only to feel the sun on her skin to know that Jake’s assertion was accurate. (MacFarlane, 2014, p. 210, notation modified) Actually, the inference which MacFarlane makes in the third sentence does not hold. The mere fact that the sentence which Jake asserts at m0 is true at m0 does not imply that “the assessor should take Jake to have made an accurate assertion.” After all, according to MacFarlane himself, the truth in the context is a necessary condition of accuracy of an act of assertion, but it might well not be a sufficient one.14 However, assuming that other reasonable conditions are satisfied (for example, Jake believes that it will be sunny, he is justified to believe this, it is probable that it will be sunny, etc.) and that these conditions are jointly sufficient to grant accuracy, the assessor should take Jake to have made an accurate (act of) assertion indeed. “And that seems right.”

14This fallacy reinforces the claim I made in the previous chapter, that MacFarlane sometimes means the proposition asserted rather than the act of assertion when he writes about the “assertion.” In those cases, “to make an accurate assertion” simply means “to assert a true proposition.” Then, an assertion is accurate iﬀ it is true.

151 CHAPTER 5. THIN RED LINE

Things don’t work so well, though, if we imagine someone at m2 looking back and assessing Jake’s assertion at m0. As before, the assessor should take Jake to have spoken accurately just in case (S) is true at m0. Since, according to the Thin Red Line view, (S) is true at m0, the assessor should take Jake to have spoken accurately. But that seems wrong; the assessor has only to feel the rain on her skin to know that Jake’s assertion was inaccurate. (MacFarlane, 2014, p. 210, notation modiﬁed).

It is the essence of MacFarlane’s postsemantic problem with the TRL—the problem of possible retrospective accuracy assessments. In my opinion, this diﬃculty stems from the problematic TRL metaphysics that Belnap et al.(2001) have described. Only if we assume that the modal standpoint of every history is equally legitimate and at the same time assume that only one of the histories is “the actual,” we can construct MacFarlane’s example. After all, he asks us to imagine a person—the assessor—who feels the rain on her skin at m2 and assess an utterance that has taken place in what she thinks was her actual past, but the assessor does not occupy the really actual future of her past. Even if she believes the contrary, her feeling the rain and the the rain itself are a merely possible, non-actualized future of her own past. It is a highly unusual idea and it can take place only in the TRL-like world that Belnap et al.(2001) have depicted. This metaphysical conception is highly extraordinary and it is no surprise that it generates very peculiar consequences also on the postsemantic level. I take MacFarlane’s postsemantic argument as a still another nail in the coﬃn of this metaphysics.

5.3.5.2 Supervaluational Thin Red Line I would like to mention one more attempt to solve the postsemantic problem of possible predictions due to Alex Malpass and myself (Malpass and Wawer, 2012). In the first place, I would like to stress that I present my own opinion about our joint endeavor. Alex Malpass might not agree with my retrospective assessment of our joint work. Our project stems from a firm conviction that introduction of “alternative actual futures,” i.e., actual futures of merely possible moments, is a mistake. We argued that there should be one and only one actual course of events in the whole branching model—the unique possibility which gets actualized in the course of time (I still think that it is a good guiding principle, if one wants to incorporate actuality into the realm of branching possibilities). Our reasons were twofold, firstly we shared Richmond Thomason’s intuition that as soon as we introduce multiplicity of actual futures into the branching model, “we begin to lose track of what a ‘real future’ is” (Thomason, 1970, p. 271). We much preferred the simple-minded idea of a single actual course of events actualized in the procession of time. Secondly, we were concerned that if one introduces an actual future for every possible moment, then, if one also wants to preserve reasonable validities on the semantic ground, then one needs to concede that each history in the model is “actual-for-itself.” (see Malpass and Wawer, 2012, p. 128). But then, the Ockhamist initialization problem would resurface in a still another form. All the theoretical benefits from introducing of the TRL would be lost. Therefore, we did everything to stick to a natural idea that one and only one of

152 CHAPTER 5. THIN RED LINE the possibilities depicted in the branching model is the possibility that gets actualized in the procession of time. However, we were aware that Belnap et al.(2001) have presented their postsemantic objection against such simplistic idea of the TRL and the main purpose of our paper was to respond to this objection. That is, we wanted, as requested by Belnap et al.(2001), to account of the predictive speech acts that take place outside of the actual possibility. In some sense, we anticipated that there is something slightly suspicious about the idea of ascribing truth values to sentences used at non-actual moments. We even say that Non-actual moments do not have actual futures, therefore (. . . ) the complaint of Belnap et al.(2001) asks for what cannot be done. (p. 126) But we eventually give in and declare that: [W]e will account for predictions situated at moments not in the TRL. (p. 127) Since we did not want to embrace the idea of multiple actualities in the model, we realized that “we need to come up with a way of thinking of non-actual predictions, while keeping the TRL fixed” (p. 128). We have eventually come up with is a kind of patchwork theory of predictions. We carefully distinguished actual predictions from non-actual predictions. In the simplest terms, our idea was that: • A prediction that is actually made is true iff what it says will actually happen. • A prediction that could have been made is true iff it would be true, if made. Later, we argued that the question about what will happen is entirely different from the question about what would happen. In case of the former, there is a reliable and solid aspect of reality that settles this issue, namely the flow of time. In case of the latter there is no comparable feature. We express our conviction as follows:

[I]t is the passage of time that resolves future contingents one way or the other. At the same time, the passage of time, no matter how long-lasting, will never resolve a non-actual future contingent in a similar manner. To the philosophical logician who holds the view, there is therefore a requirement to treat actual future contingents and merely possible future contingents diﬀerently. (p. 129)

The distinction on the metaphysical level is paired with a diﬀerence in epistemic access. In case of sentences about the future, there is a relatively straightforward way to learn if there are true, namely—wait and see. In case of sentences about what would happen, the epistemic access is much more limited. All we can do to learn what would be the case is to infer what would be settled to happen given how the world actually is. For example, if I have a rigged, double-headed coin in my pocket, then I do know that it would have landed heads, if I had tossed it. I can learn this solely on the basis of properties of the actual coin and the tossing set-up. In many cases, though, there is

153 CHAPTER 5. THIN RED LINE absolutely no way of knowing what would happen. If the coin in my pocket is a fair, indeterministic coin, then there is absolutely no way to figure out how it would have landed. Due to the difference in epistemic access, we can reasonably make a guess that that the coin will land heads or bet that it will, but it makes little sense to guess or bet that the coin would have landed heads. From metaphysical and epistemic difference between the actual and the possible, we derive a semantic conclusion. The case of actual predictions is reasonably clear, their truth value depends on what will in fact happen. The case of non-actual predictions is different.

Imagine I hold in my hand a fair coin. I don’t ﬂip the coin but I could have done so. Moreover, I could have said, just before the possible toss, that the coin would show heads. Belnap asks whether this possible statement is true of false. To us, because it is a fair coin and it wasn’t ﬂipped, it seems that this assertion cannot be counted as true. Neither would it be true if we substitute “tails” for “heads.” (p. 129)

Therefore, we aimed at the semantic theory which would render actual predictions of contingent events true or false while at the same time it would render non-actual predictions of contingent events neither true nor false. We developed a series of theories that meant to grasp the distinction between actual and non-actual predictions. Initially, we tried to solve the Belnap et al.’s (2001) problem of non-actual predictions with properly semantic means. That is, we proposed non-standard ways to interpret the future operator at non-actual moments (we first replaced operator “will” with “would,” and then offered a non-standard reading of “will” at non-actual moment). However, all these attempts fell pray to semantic objection. As we have witnessed so many times already, as soon as one starts to mingle with the standard Ockhamist definitions, one is going to generate some fairly non-intuitive consequences on the semantic level. I refer an interested reader to appendix 7.8 for description of our initial failures. The lesson which we have learned a hard way is that one should not question the standard understanding of the temporal and modal connectives. We have finally come to our senses and decided to preserve the standard, Ockhamist semantics for temporal and modal operators. To express the difference between actual and non-actual predictions, we designed a theory which we have called a Supervalu- ational Thin Red Line (STRL for short). It is Supervaluational, since we incorporate elements of supervaluationism at non-actual moments and it is Thin Red Line, since it continue to stress the special status of the actual predictions. We wanted to achieve the effect that the truth value of a prediction at a moment is “sensitive” to the existence of the TRL. If a prediction is made at an actual moment, its truth value should depend on what is the case in the actual course of events. If it is made at a non-actual moment, it should depend on what is possible and necessary at this non-actual moment. In our original text, it is not entirely clear what we meant when we write about a truth value of a sentence at a moment. In particular, it is not clear if we talked about the truth value of a sentence at a semantic index or the truth value of a sentence at a context, we just distinguish two levels of truth. When I now reflect on our efforts, however, I realize

154 CHAPTER 5. THIN RED LINE that our primary aim was to address Belnap et al.’s (2001) postsemantic challenge—to give an account of predictive speech acts occurring at non-actual moments. Thus, it is most reasonable to assume that our theory was meant to explicate the postsemantic notion of truth of a sentence at a context. Thus, we can phrase the idea of Malpass and myself in form of the following postsemantic deﬁnition: Deﬁnition 5.14 (STRL postsemantics).

m||−strlφ iﬀ ∀h(m ∈ h ⇒ M, m/h |= φ) or M, m/TRL |= φ.

We say that formula φ is false at a context iff its negation is true at the context. Our definition presupposes that one, and only one, of the histories in the branching model is distinguished as the actual history, so an STRL-model is a TRL1 model, i.e., a quadruple hM, ≤, TRLh, Vi where TRLh ∈ Hist. Thanks to the disjunctive character of our postsemantic definition, we can clearly distinguish between actual and non-actual predictions. On the one hand, an actual prediction, “The coin will land heads,” is either true or false (depending on what the result will be). On the other hand, the same prediction made at a non-actual moment is neither true nor false (it reflects the idea that there is nothing in the world that could resolve how the possible coin lands). At the same time, our postsemantics allows for some non-actual predictions to be true, namely, the ones which are non-contingent. For example, if a non-actual toss is made with a rigged, double-headed coin then the non-actual prediction, “The coin will land heads,” is true (it reflects the idea that given how the world actually is, there is something to resolve the result of this possible toss). Some non-actual claims concerning a fair coin might have truth values. For example, “The coin might land heads,” is true and, “The coin must land heads,” is false. We thus arrived at a formal definition which gives due to philosophical, and linguistic motivation which guided our project. Also, we answered Belnap’s worry that the “TRL theory (. . . ) gives no account whatsoever of predictive speech acts occurring at moments of use that lie off the TRL” (Belnap et al., 2001, p. 162). On top of that, we have established that such defined notion of truth has quite desirable formal properties.

Some properties of STRL theory Thanks to the postsemantic character of our theory, we can easily dispel, just at MacFarlane(2003) did, the worries that TRL theory is logically unsound. In particular, sentences which were problematic for some earlier proposals: Fφ ∨ F¬φ, φ → HFφ, FFφ → Fφ, and F^φ → ^Fφ are all true in every context under STRL. In fact, we established that the notion of STRL-validity coincides with Ockhamist truth. However, the issue is quite subtle since it turns out that on the level of validity-in-a-structure, STRL-validity and Ockhamist validity might diverge. Nonetheless, they converge on the more general levels of validity. For precise deﬁni- tions, proofs and examples, consult appendix 7.9. Regarding the notion of semantic consequence, we can just as easily dismiss any potential worries if we assume that the notion of semantic consequence should concern preservation of truth-at-index rather than truth-at-context. Then STRL consequence just is the Ockhamist consequence. However, even if we focus on the preservation of truth-at-context, the things look surprisingly well for STRL. Even though it has a

155 CHAPTER 5. THIN RED LINE supervaluational component, it avoids the problems of supervaluationist consequence relation raised by Williamson(1994) and Tweedale(2004) (see section 4.5). It turns out that, thanks to the existence of the TRL, STRL consequence behaves considerably better than its supervaluational cousin. Things are rather intricate though since there is a limited notion of consequence—consequence-at-a-ﬁxed-context-in-a-ﬁxed- structure—which can be used to restate the arguments of Williamson and Tweedale.I recapitulate in appendix 7.10 the results regarding the semantic consequence relation that Malpass and I have presented in our paper. Finally, we investigated two distinct ways in which STRL semantics could be aug- mented with a truth operator. I present the details of our studies in appendix 7.11.

Problems with STRL Andrea Iacona(2014) have recently published a paper where he pointed out some difficulties of our account. In my opinion, his problems result mainly from misunderstanding of our technical apparatus (partially induced by our insufficient precision) and I do not think they are fatal for our theory. They revolve, however, around a particular point of our view which I myself find highly controversial today. I will first respond to the specific worries raises by Iacona and then articulate the most problematic feature of our theory. I will end the chapter with reassessment of the postsemantic challenge of Belnap et al.(2001). According to Iacona, the most problematic feature of STRL theory is that it presupposes a notion of a model where one and only one of the histories is distinguished as the actual history (TRL). It is doubtlessly true, we were fairly explicit about this assumption:

A TRL structure T is a pair hF, TRLi, where F is a branching time structure and TRL is a distinguished history of the model–—the history which represents the actual course of events through time. (p. 124)15

Thus, we have built into our semantic model, the information that a particular history—TRL—represents the actual course of events. Iacona assesses this idea in rather sever terms “The very idea that a special formal apparatus in which actuality is represented should be tailored to Ockhamism is wrongheaded” (Iacona, 2014, p. 2651). His most general objection is that we were too rigid in our theorizing, to the point that we run ourselves into serious trouble:

If the actual history is ﬁxed once and for all in the model, it turns out that sentences lose some basic semantic properties when they are evaluated at non-actual moments. However, there is a sense in which one may expect that all moments are alike with respect to those properties, namely, the sense in which every moment is actual from its own point of view. The inability to account for this sense may be called the rigidity problem.

Iacona argues that we failed to notice that even the non-actual moments are (in a sense) actual, and due to this failure, our theory suﬀers major diﬃculties. In general,

15In fact, not only did we assume that only one of the possibilities is the actual possibility, but we further postulated that no operator in the object language should be able to shift the actual history from one value to another (p. 129).

156 CHAPTER 5. THIN RED LINE

I believe that Iacona’s objections rely on certain misunderstanding of our position. Namely, in his objections, Iacona stresses that

• even the non-actual moments are actual for themselves, while we insist in our theory that

• only the actual moments are actual. The confusion of these two perspectives might easily make our theory look more problematic than it actually is. This misunderstanding is closely related to a specific feature of out theory that Iacona neglects. Namely, we use two notions of truth to explicate our view: truth-at-moment and truth-at-moment-history-pair. I have already argued that given the purpose of our theory, it is most reasonable to assume that the former corresponds to truth-at-context, while the latter relates to truth-at-index. As far as truth at index is concerned, it is reasonable to expect that every index has “equal rights.” Iacona likes to express the “democracy” among indexes with the phrase that each index is actual for itself. I am not certain if it is the most fortunate façon de parler, but I generally agree with the author’s idea that as far as the semantics proper is concerned, all indexes are equally good. Otherwise, we would likely generate some logical oddities. Importantly, in our STRL theory all the indexes are on a par. We do not postulate that semantics is different in actual and non-actual circumstances. We only stress that we should treat predictions made in actual and in non-actual contexts differently. In the process of his criticism, however, Iacona persistently understands our notion of truth-at-moment as if it referred to truth-at-index rather than truth-at-context. Then, it seems as if semantic properties of expressions changed from one index to another which makes our theory sounds more controversial than it should.16 The author derives a number of more specific objections out of his general critical observation. I divide those difficulties into four categories, (a) the problem with truthmakers, (b) the semantics of operator “actually,” (c) the problem with counterfactuals, (d) the meta-linguistic actuality shift. Let me begin with the problem of truth makers.

[W]e get that sometimes—at actual moments—the sentence is true because it is true in the actual history (in virtue of the second disjunct), while at other times—at non-actual moments—it is true because it is true in all histories (in virtue of the ﬁrst disjunct). (Iacona, 2014, p. 2639)

We were aware of this feature of our theory, of course, and we were happy to embrace it. Remember that we had a speciﬁc task in mind: “to come up with a way of thinking of non-actual predictions, while keeping the TRL ﬁxed” (p. 128). Thus, whenever we evaluate a sentence used at a particular moment, we hold in the back of

16To be honest, we have given a reader a fair chance to incorrectly understand our project (mainly because we have not understood it precisely enough ourselves, at least I did not). For example, we introduced two notions of truth but we did not elaborate on their signiﬁcance and the relations between them. Also, when we talked about “points of evaluation” we sometimes referred to moments and sometimes moment/history pairs. Thus, it was fair on behalf of Iacona to point this ambiguity.

157 CHAPTER 5. THIN RED LINE our heads the implicit assumption that TRL is the history which accurately represents the world’s temporal evolution. Thus, whenever we reason about a truth value of a sentence at moment m, our reasoning has a following format: given that TRL is the actual course of history, what is the truth value of a sentence used at a moment m. This attitude of ours is clearly revealed when we compare predictions made at actual and non-actual moments:

[T]o assess the truth value of an actual sentence about the future one just needs to (wait and) see what the actual future is like, while the assessment of the truth value of the non-actual prediction demands something diﬀerent—namely reasoning about what would be possible and what would be necessary at this non-actual moment. (p. 131)

Thus, the truthmakers of actual and non-actual predictions are radically different. A sentence used at an actual context is true or false depending on what is actually the case, while the truth value of a non-actual sentence, used at a non-actual moment m depends on what the possibility m contains (this idea is encoded by definition 5.14, 155). When we assess the truth value of a sentence used at an unrealized possibility, we check how much information can be retrieved from this possibility. Iacona argues that this way of thinking about actual and non-actual predictions is unacceptable since it generates serious semantic complications. In particular, he observes that in our theory ∨ is truth-functional at actual moments and non-truth-functional at non- actual moments. (Iacona, 2014, p. 2640) From this, he concludes that that “the meaning of a logical constant varies with the moment of evaluation” (Iacona, 2014, p. 2640). I admit that this view would be highly problematic, but I do not think that one needs to make this conclusion in our case. Remember that Malpass and I distinguish two notions of truth: ||−strl and |=. The first is applied to sentences at contexts while the second to sentences at indexes. As far as truth at index is concerned, our notion of disjunction is fully truth-conditional (we simply accept, the classical definition of ∨). Only in case of truth-at-context, as Iacona noticed, the truth value of a disjunction at a non-actual context does not depend functionally on the truth value of the disjuncts at this context. Now, we need to ask whether we should require from the logical constants to be functions of truth values at indexes or function of truth values at contexts. In my opinion, Lewis(1970b) and Kaplan(1989) convincingly argued that we should assume the former. For example Kaplan’s “content” is a function from possible circumstances to truth values rather than from context to truth values. Moreover, Kaplan shows that it needs to be so, if we want to arrive at an accurate interpretation of indexical expressions. Malpass and I use Ockhamism as our semantics proper, so the meaning of logical connectives in our theory parasites on their (truth-functional) meaning in Ockhamism. Our principal goal is not to specify intension of linguistic expressions, but to explain how to move from intension to extension in a particular context. I admit that the distinction between truth at context and truth at index has not been so explicit in our original paper. However, given the use we make of our theory, it is safe

158 CHAPTER 5. THIN RED LINE to assume that STRL-truth should be identified with truth at a context and when it is, then the objection simply does not apply to our theory. Moreover, at some points of our paper it is quite explicit that we construe meaning in terms of truth at moment/history pairs (indexes) rather than in terms of truth at moments (contexts). For example, in a more technical part of our paper we discuss a semantic definition of a truth operator which behaves differently at moments that belong to TRL and differently on those which do not (see def. 7.27, p. 248). Then, we make a telling comment about this idea:

What is even more unsettling is that the meaning of the language seems to change from one point to another. In particular, the meaning of the operator “It is true that” is quite diﬀerent depending on whether it operates in actual or non-actual circumstances. As a result, it seems for example that T-schema (φ ↔ Trφ) or bivalence (Trφ ∨ Flφ) are only contingently true; they hold in the actual world only. This is an idea which we feel reluctant to accept. What distinguishes actual from non-actual moments is the manner in which we can asses truth value of formulae and not the meaning of the language. (p. 138)

Thus, we assumed that what constitutes the meaning of an expression is the pattern of truth values at moment/history pairs, rather than at moments, as Iacona assumes. Another objection that can be distilled out of Iacona’s paper concerns the behavior of operator “actually” within STRL. It is related to the previous one since it also presupposes that meaning of a semantic operator depends on the pattern of truth values at contexts rather than at indexes. Iacona focuses on the so-called “secondary” sense of the word “actually,” discussed by Lewis(1970a, p. 185). In this sense of the word, “actually” does not refer to the unique, actual world, but it is supposed to be a redundant operator which inherits its reference from the modal operators in scope of which it is embedded. An example where “actually” is used in the secondary meaning given by the author is:

If this city were bigger, it would actually have more buildings.

In fact, I was convinced by Max Cresswell(1990) that even in such cases, “actually” preserves its indexical nature.17 Let us not bother with these subtleties, however, and focus on the semantically redundant sense of “actually.” The author believes that “there is no way to make sense of this reading if it is assumed that the only history to which actuality can be ascribed is TRL”(Iacona, 2014, p. 2641). The objection is a bit far-fetched, since we provide no treatment of any kind of “actually” operator, index-

17Consider for example the sentence If this city were bigger, it might still be believed to be smaller than it actually would be. In this case, operator “actually,” embedded in the modal operator(s) “it might be believed that” does not refer to the possible world in which the city is smaller, as the “secondary” reading of “Actually” would require. Also, it does not refer to the actual world, in which the city is not bigger. To properly interpret this sentence, “actually” needs to refer to the possible world, in which the city is bigger. Thus, the mechanism is still indexical, just the point of reference has been “deferred” to another, “quasi-actual” world.

159 CHAPTER 5. THIN RED LINE ical or not, in our paper. We also explicitly admit that we do not (p. 139).18 In fact, Iacona softens his claim on the following page, and he says that “Nothing prevents us from thinking that a TRL semantics with fixed actual history can leave room for some understanding of ‘actual’ that mimics the indexical [i.e., secondary—J.W.] account of actuality” (p. 2642). Nonetheless, he still claims that under the “secondary” reading of “actually” our semantics identifies “actually” with “necessarily” at non-actual moments, and he finds it an undesirable result (Iacona, 2014, p. 2642). Regarding this last objection, Iacona is plainly mistaken. The deflationary definition of @ that Iacona uses is very simple: Definition 5.15. M, m/h |= @φ iff M, m/h |= φ. It is important to notice that it is a semantic, rather than postsemantic definition. Thus, it is reasonable to assume that it indeed provides the meaning of the connective @. Then, Iacona claims that “@ expresses necessity at m, that is, truth in all courses of events that are possible at m” (Iacona, 2014, p. 2642) while it is simply not true! The pattern of truth conditions of @φ and φ are different both at actual and non-actual moments. Consider any non-actual m in a model M such that h1 and h2 “pass through” m (i.e., m ∈ h1 and m ∈ h2). Assume also that sentence p is true at some m1 ∈ h1 such that m1 > m and that for every m2 > m in h2, p is false at m2. In such a case, we have that M, m/h1 |= @F p and M, m/h1 6|= F p. Since we identify the semantic value of a sentence with a set of indexes at which the sentence is true (as is usual), we can conclude that this example is sufficient to show that the semantic values of @F p and F p are different. Thus, against Iacona’s claims, they do differ in meaning. There is indeed a connection between @F p and F p at non-actual moments that can be captured in terms of truth at non-actual context. Observe that given STRL postsemantics, for any non-actual context m, we have that: m||−strl@φ iff m||−strlφ That is, @φ and φ are true at the same non-actual contexts. In our account, it means that the conditions which are sufficient to ground the truth of a non-actual prediction @φ, need to be strong enough to ground the truth of φ as well. That is, there needs to be something in the actual world which necessitates the truth of φ. However, even on the postsemantic ground, we have very clear ways to distinguish secondary actuality from necessity. In particular, at a non-actual context m at which F p is contingent, we have that m||−/strl¬@F p while m||−strl¬F p which means that @F p and F p are false at different non-actual contexts. It also implies that it is much easier to reject a non-actual sentence F p (it is sufficient that F p might be false in the possible circumstances), then it is to reject a non-actual sentence @F p (in this case, F p must be false in the possible circumstances). For other sort of examples, notice that if F p is contingent at m (no matter actual or not), then:

18To be fair, we had also promised that we would deliver a paper focused on this precise issue and we nerved did. However, consult (Wawer, 2014, pp. 394–398) and section 6.2 for an extended discussion of “primary” reading of “actually.”

160 CHAPTER 5. THIN RED LINE

• m||−strl@F p ∨ @¬F p, while m||−/strlF p ∨ ¬F p, • m||−/strl@F p → F p,

• m||−strl^@F p ∧ ^@¬F p, while m||−/strl^F p ∧ ^¬F p. All these show that there are plenty of ways to distinguish (deﬂationary) actuality from necessity, both on semantic and on postsemantic level which answers Iacona’s objection that our account confuses the two. The next worry raised by Iacona concerns our account of counterfactuals. Remem- ber that we construe non-actual predictions in terms of a counterfactual construction: a possible prediction is true iﬀ it would be true, if made. Then, when we consider the possible prediction, “The coin will land heads,” we argue that given that the toss does not take place, there is no telling how the fair coin would have landed, and thus, we should not ascribe the truth value to this non-actual prediction. Iacona admits that each result of a toss is an unactualized possibility, but he carries on:

Yet this does not entail that there is no answer to the question of what would have happened if the coin had been tossed. For what really matters to that question is not which possibility is actual, but which possibility would be actual. (Iacona, 2014, p. 2640).

However, even if we rephrase our counterfactual along the lines proposed by Ia- cona, it does not aﬀect my intuitions. Just as I thought that there is no telling regarding how the coin would have landed, I think that there is no telling regarding how the coin would have actually landed. In this case, the addition of “actually” is entirely super- ﬂuous and does not help a bit to solve the mystery of the possible result. I stick to my guns and I claim that no investigation, no matter how long and thorough can ever establish the answer to this question. Therefore, I still think that no truth value should be ascribed to this non-actual predictions. In the meanwhile, I realized that Malpass and I have unknowingly used a condition which Michael Dummett(1976, p. 52) calls principle C:

(C) If a statement is true, there must be something in virtue of which it is true.

Since we think that there is nothing in virtue of which the would-be statement is true, then it is not true. We share this general diagnosis with Dummett:

[T]he principle C may at ﬁrst strike one as empty. We feel its force only when we consider something which appears a violation of it. The most obvious such violation is provided by a counterfactual conditional alleged to be true even though there is nothing which, if we knew of it, we should accept as a ground for its truth: for instance, those counterfactuals asserted by one school of theologians to be the objects of God’s scientia media, re- lating to the behaviour, had they been created, of beings endowed with free will whom, on the basis of such knowledge, God decided not to create. Most people naturally feel a strong objection to such a conception, precisely on the ground that, in such a case, there would be nothing to

161 CHAPTER 5. THIN RED LINE

make the counterfactual true. This objection is based upon the thesis that a counterfactual cannot be, as I shall say, barely true.(Dummett, 1976, pp. 52–53).

At this issue, Dummett and us (and “most people” if Dummett is to be trusted) clearly disagree with Iacona who believes that

[I]t is plausible that, just as the truth of an actual prediction depends on what will happen, the truth of a counterfactual prediction depends on what would happen if certain conditions were to obtain. (Iacona, 2014, p. 2640).

In contrast, Malpass and I believe that the truth value of an actual prediction can be grounded in what will happen, while the the truth value of a would-be prediction cannot be grounded in what would happen. It might be partially explained by our conviction that the (linear) passage of time is a sufficiently solid feature of reality to ground truth values of sentences about the future, while there is nothing comparably solid to ground the truths about the counterfactual future. Against Iacona, I do not think that our theory is self-defeating or unintelligible, but I no longer believe that we chose the most fortunate format to convey our idea. Now, I think that it is methodologically more healthy to clearly distinguish the two insights we packed into one project. In particular, I think that the question of the truth value of a sentence in a context should be strongly distinguished from the question of appropriate treatment of counterfactuals. Our idea is better grasped by conjunction of two independent beliefs (i) that sentences about the future, including the future contingents, are either true or false at all contexts and (ii) that “contingent” counterfactuals are neither true nor false at all contexts. I tried to defend the first of these claims in (Wawer, 2014), the theory which preserves the second insight is drafted in (Wawer and Wronski´ , 2015). Let me now turn to the last of the objections. It concerns a meta-linguistic actuality shift. I think that it really constitutes the essence of Iacona’s problem and nicely pictures a different approach that Iacona and us take towards our theory. The problem can be very briefly summarized by one of his comments about our project:

[S]uch a semantics seems incapable of making sense of the counterfactual hypothesis that one of the courses of events that include a given non-actual moment is the actual course of events. (Iacona, 2014, p. 2642).

One might wonder why Iacona wants to make a hypothesis that a course of events which contain a non-actual moment is an actual course of events. He thinks that it is necessary for interpretation of some sentences of the object language:

The point is rather that certain results concerning simple formulas of the object language seem to conﬂict with certain thoughts that we can express in the metalanguage by means of constructions in subjunctive form. (Ia- cona, 2014, p. 2641).

Let me try to understand this objection with a speciﬁc example. Let us use a model discussed by Iacona in his paper (p. 2639):

162 CHAPTER 5. THIN RED LINE

h1 h3 h2 m1 p m4 m2

Let us now consider a truth value of the sentence P^FF p evaluated at pair m2/h2. The standard Ockhamist rules dictate that the sentence is true at moment m2 and history h2 (m2/h2 |= P^FF p) because at moment m3 and history h1 it is true that F p (m3/h1 |= F p). Iacona prefers to interpret m3/h1 |= F p with a counterfactual construction “If h1 were the actual history, m1 would obtain, hence F p would be true at m3”(Iacona, 2014, pp. 2640–2641). The key insight here is that to interpret a truth value of a sentence like P^FF p, when if it is used at an actual moment in the actual history, we need to check the truth value of the sentence F p at some other moment and some other history. At this point, we could not agree more! Iacona must have not analyzed our definition carefully enough since when you consult our postsemantics, you will clearly see that strl m2||− P^FF p iff m2/h2 |= P^FF p iff there is an earlier moment (like m0) such that there is a history passing through that moment (like h1), such that there is a later moment in this history (like m3) at which F p is true with respect to this history. If this is what it means to make sense of “the counterfactual hypothesis that one of the courses of events that include a given non-actual moment is the actual course of events,” then, against Iacona’s complaint, we can make perfect sense of this hypothesis. In fact, whenever we assess a sentence like P^Fq used at an actual context we do consider the truth value of q at a non-actual moment in a non-actual history. We just prefer not to read m/h |= φ as “had m and h been actual, φ would be the case,” but it is a purely terminological matter. Even in our model with a single actual history, we can adopt Iacona’s meta-semantic jargon and describe the truth at moment/history pair along these counterfactual lines. Importantly, it does not force us to shift the actual history from one place to another in the model. It just requires us to shift the value of history parameter in the semantic index and we are happy to do this whenever necessary. When we consider a non-actual moment as a context of use of a sentence, however, then we do not want to assume that it is actual: Note that the intended interpretation of M, m||−strlφ, where m is off the TRL, is not “had m been actual, then φ would have been true” (which is a subjunctive conditional), but something like “m is the actual world’s potential in which φ happens.”19 (p. 133) Nonetheless, it is a different issue than the one that Iacona alludes to. In particular, one does not need to consider non-actual context as actual to understand the “simple formu-

19Perhaps, it would be more fortunate to say, “actual world’s potential m grounds the truth of φ.”

163 CHAPTER 5. THIN RED LINE las of the object language.” Therefore, the simple semantic consideration mentioned by Iacona do not force us to think along these lines. Importantly, our decision to not to interpret non-actual context as if they were actual is not just our whim. Remember that the whole point of our project was to answer Belnap et al.’s (2001) postsemantic challenge: The TRL theory sounds all right, but it is not. It has the “logical” defect that it gives no account whatsoever of predictive speech acts occurring at moments of use that lie off the TRL and is by so much useless. (Belnap et al., 2001, p. 162, emphasis mine) Thus Belnap et al.(2001) explicitly require a proponent of a TRL theory to account for speech acts made at contexts off the TRL. Remember that TRL is the actual course of events. Consequently, the postsemantic challenge explicitly calls for a treatment of sentences used at moments which are not actual. Thus, if we want to address the postsemantic challenge, we cannot assume that non-actual context are actual. To answer the postsemantic challenge, we had to to explain how to assess sentences used at non-actual context. It is exactly what we did in our postsemantic definition 5.14 (p. 155) and it is a feature of our theory that I now find highly controversial. After all, when one considers a sentence made at a particular context m, it is nothing but most natural to assume that moment m is an actual moment which is a part of the actual history. It is not even entirely clear to me what it means to use a sentence at a context which is not actual. And this brings us to the heart of the matter. I think that we, just as many TRL theorists before us, got involved into a game we should not have played. Our original sin was to accept Belnap’s challenge an try give some “account whatsoever of predictive speech acts occurring at moments of use that lie off the TRL.” What should have immediately raised our eyebrows, but did not, is the presupposition of the challenge. Namely, Belnap et al.(2001) assume that there are predictive speech acts occurring in non-actual circumstances. But it should have been clear for the actualists like us that the only acts that occur, are the act which actually occur. Thus, there are no speech acts that “lie off the TRL”! Therefore, there is nothing to give an account of. The challenge never gets off the ground. There is a non-controversial sense in which we can account for possible predictions. After all, a possible prediction is a prediction that could have been made. And it is very easy to account such prediction, namely, the sentence that could have been used, could have been true or could have been false. The problem arises only when we try to answer if a prediction that could have been made is true. Then, there is a natural tendency to introduce an actual future for every possible moment, so we can finally say if a possible prediction is true. Malpass and I fell into the trap. We considered a prediction that could have been made and we asked “whether this possible statement is true of false” (p. 129). I think now that we should not have gotten engaged into this project. We should have firmly insisted from the outset that there are no non-actual contexts, so there is no need to account of sentences used at non-actual contexts (this is what I did in Wawer, 2014, p. 387). As soon as we begin to consider non-actual contexts, we give up our actualist credentials. It has been noticed already by Evans that when we admit sentences made at non-actual context, then

164 CHAPTER 5. THIN RED LINE

On this view, we must regard ordinary statements as containing a reference to the world of utterance, in the way tensed statements contain a reference to the time of utterance. To think in this way about possible worlds seems to commit one to an unacceptable form of modal realism—the doctrine that other possible worlds exist in exactly the same sense in which the actual world exists, and diﬀer from it merely in not being the ones we happen to inhabit. (Evans, 1985, p. 363).

When we accepted Belnap et al.’s (2001) challenge, we implicitly accepted the Genuinely Realistic way of looking on the tree of possibilities. We allowed into the world plenty of people who leave in non-actual possibilities and who make predictions of their own. When we constructed our theory, we implicitly admitted that these people are real and promised to tell them if what they say is true. Therefore, when Belnap et al.(2001, p. 162) complain that “the TRL (. . . ) gives no account whatsoever of predictive speech acts occurring at moments of use that lie oﬀ the TRL” one should not be much concerned. It is not a “logical defect,” as Belnap et al. (2001) describe it, but it is the most reasonable consequence of the actualist worldview. When we try to answer if a sentence is true when used in a non-actual context, then

this is just what commits us to a particular way of looking at the matter. (. . . ) The decisive movement in the conjuring trick has been made, and it was the very one that we thought quite innocent. (Wittgenstein, Philo- sophical Investigations, §308.)

If we do not want to get tricked, then the best way to address Belnap et al.’s (2001) postsemantic worry, is not to worry about it at all. A philosophical problem has once again been created by an ill-posed question . In the next chapter, I oﬀer an elaborated therapy which helps to get actualism straight.

165 Chapter 6

Branching Actualism

Let me quickly recapitulate what brought us here. Firstly, I argued that it is misleading to think about the branching structure as representing time. The considerations that underlie the branching representation are not suﬃcient to justify the claim that the earlier-later relation is non-linear. Then, I discussed a theory, which I called Genuine Branching Realism. According to this theory a course of events that we participate in is equally real as any other alternative scenario. No modal point of view is privileged and what happens from our modal point of view is just as much a part of reality as what happens from the points of view of alternative histories. Or, as Belnap et al. (2001) put it, Our World is branching. I have then described multiple strategies of semantic and postsemantic assessment of sentences used in the branching world. Later, I have introduced what I called the actualist insight that stresses the absolute diﬀerence between actuality and possibility. We have considered one way to capture the actualist insight—the co-called Thin Red Line. I have recounted numerous arguments, both metaphysical and semantic, against such combination of ideas. The core of the criticism is that it is very hard to hold at the same time (a) that no modal point of view on reality is privileged (the world is branching) and (b) that one of the possibilities is absolutely actual.

6.1 Metaphysical background

In this chapter, I am going to defend the actualist insight. However, I do not intend to formally or conceptually develop another version of the Thin Red Line theory. I think that if one accepts modal neutrality and grants that the world has a treelike structure, then there is no place for the absolute distinction between the actual and the possible. In this context, actuality can be conceived, at most, as a relative feature of reality. Faced with a conflict between the branching world and the non-branching actuality, the Branching Realists firmly stand on the side of the branching world. They are willing to sacrifice the absolute distinction between what is actual and what is possible to preserve the modal neutrality. I will argue that in face of this conflict, it is equally reasonable to side with non-

166 CHAPTER 6. BRANCHING ACTUALISM branching actuality and abandon the idea that the world is branching. Thus, I agree with Belnap et al.(2001) that the arguments against TRL constitute a reductio of this theory, but I disagree with respect to which of the premises should be rejected. In this chapter, I reject modal neutrality and assume that our point of view is special. It means that “our” mooded facts regarding what is actual and what is possible are the absolute mooded facts. I assume that, if determinism is false, then possible courses of events do branch but the actual course of events does not. Only within such conceived reality, we can really appreciate both sides of Belnap et al.’s (2001) slogan:

Sure, there are many things that might happen, but only one of them is what really will happen. (Belnap et al., 2001, p. 160).

The actualists commonly believe that the semantic diﬀerence between might and will (and generally, between possibly and actually) has metaphysical underpinning. A famous fragment by Kripke makes the distinction quite clearly:

Hence there are thirty-six possible states of the pair of dice, as far as the numbers shown face-up are concerned, though only one of these states corresponds to the way the dice actually will come out. (. . . ) Now the “actual world” in this case is the state of the dice that is actually realized. Another entity, more “concrete” than this state, is the Lesniewskian-Goodmanian physical entity which is the “sum” of the two dice. (Kripke, 1980, pp. 16–17).

When we transfer this idea to the branching setting, we can say that there are many possible ways in which the world may develop (the branching tree of possibilities). However, only one of these ways correspond to how the world actually develops. It can be called the actualized possible course of history and it should be distinguished from the world itself—the concrete reality which surrounds us. The diﬀerence between what is possible and what is actual has been explicated in more than one way among the modal actualists. Some phrase it in terms of existence: what is actual exists while what is possible does not. Others prefer to appeal to concreteness: what is actual is concrete while what is possible is abstract. Others still, like Aristotle, take this distinction to be conceptually primitive. I do not want to decide this issue, but I will sometimes describe the actual world as concrete and possibilities as abstract to stress the diﬀerence between the actual and the possible. We should be careful when we use the phrase like “the actual world” or “what actually happens” since it might be ambiguous in the actualist setting. It might refer either to the possible course of events which is realized or to the concrete entity that actualizes one of the possibilities. To stress the distinction, I will use the adjective actualized to describe the possibility and the term world to refer to the concrete entity. When I use the phrase like “the actual world,” “the actuality” or “what is actual” it always refers to the world rather than to the actualized possibility. I am going to call Branching Actualism the position that separates the sphere of branching possibilities from the concrete actual reality that we are surrounded by. I will contunue to call the elements of the branching strucutre momentary possibilities or possible moments,

167 CHAPTER 6. BRANCHING ACTUALISM while, when I mean to refer to the instantaneous slices of the actual world, I will call them events. Let us compare Branching Realism and Actualism with a particular example. Ac- cording to Branching Realism, from our modal viewpoint, there was no third world war in the 20th century, but from a diﬀerent viewpoint on the world, there was one. None of these two viewpoints is somehow privileged or special, so whether there was a third world war in the 20th century is a relative matter. In contrast, for Branching Actualism, it is an absolute fact about our world that there was no third world war in the 20th century, and it does not require relativization to a particular modal perspective. We cannot abstract from our own modal vista, since our perspective on the world is the only perspective. The realist position naturally corresponds with the “democracy” inherent to the semantics of modal logic. Observe that when I introduced Ockhamist semantics, I did not distinguish any moment/history pair as the actual moment/history pair. All of them had the same semantic status and each of them can be provisionally posited to be the actual moment/history pair. Robert Stalnaker have argued, however, that the modal realists unjustiﬁably draw a metaphysical lesson from the democracy inherent in the semantic theory:

Thesis three [i.e., the indexical nature of the adjective “actual”] seems to imply that the actuality of the actual world—the attribute in virtue of which it is actual—is a world-relative attribute. (. . . ) But if there is no absolute property of actuality, does this not mean that, looking at things from an objective point of view, merely possible people and their surroundings are just as real as we and ours? The mistake of this reasoning, I think, is in the assumption that the absolute standpoint is a neutral one, distinct from the view from within any possible world. The problem is avoided when one recognizes that the standpoint of the actual world is the absolute standpoint, and that it is part of the concept of actuality that this should be so. (Stalnaker, 1976, p. 69)

Thus, the actualist argue that the neutrality inherent to the semantics of modal logic does not transfer to the metaphysical neutrality. We can claim that no possible circumstances are privileged, as a semantic parameter, but also insist that only one of these possible circumstances correspond to how things really are. The remaining possible circumstances represent how the things could be. In other words, we should not get carried away and even it turns out that the treelike model is the most useful model of a tempo-modal language, we should not jump to the conclusion that the world is a treelike structure.1 Let us further explore the actualist alternative. It might help, if we ﬁrst draw the treelike structure that represents all the possible scenarios.

1A similar point has been raised by Saul Kripke: The apparatus of possible words has (I hope) been very useful as far as the set-theoretic model-theory of quantiﬁed modal logic is concerned, but has encouraged philosophical pseudo-problems and misleading pictures. (Kripke, 1980, p. 48, n. 15)

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Possibilities

According to actualists, it is an incomplete model of reality. It captures the possibilities but it leaves out the actuality. Let us ﬁll in this gap, keeping in mind that at each instant, one of the possibilities is actualized.

actualizes Possible courses of events World at t0 t1

actualizes World at t1 t

Possible courses of events t actualizes 2 World at t2

Possible courses of events There is a reciprocal inﬂuence between how the world does develop and how it can develop. On the one hand, the world’s development is limited by what is really possible. If no possibility allows for a travel faster than a speed of light, then nothing in the world ever travels faster than the speed of light. On the other hand, the world’s actual development restricts the available possibilities. Given how the world is like at

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t1 at the diagram above, there are only two ways in which it can develop, even though at earlier time t0, there are six of them. Thus, what actually happens in the world at a time limits what can happen later on. As a result, there is a mutual limitation of the possible and the actual.2 It is important to stress that the world’s development is an indeterministic process. At the diagram above, each of the six scenarios is available at t0—the world really can go each way. The mere fact that the world will be in one speciﬁc state at t1 does not limit the number of possibilities available at t0. More generally, the fact that things go in one speciﬁc way does not imply that they have to go the way they do. The metaphysical idea described above resembles a view which MacFarlane(2014, p. 212) called a moving dot:

[E]ven if I’m now located in many worlds that overlap in the present but diverge in the future, there’s a fact of the matter as to which one is the actual future.3 To ﬁnd out, I just have to wait and see what happens as I travel forward in time. (MacFarlane, 2014, p. 212). The name “moving dot” is quite accurate. After all, if you look at the diagram above, there is a red dot moving up the tree of possibilities as the time goes by. It corresponds to the fact that at each instant of time, the world actualizes exactly one of the instantaneous possibilities available at that time. MacFarlane is rather distrustful about this idea

But this “moving dot” picture embodies a fundamental confusion. We’ve already represented time as one of the spatial dimensions of our tree. What could possibly be represented by the motion of a point along this dimension? Certainly not a process that takes place in time, since all such processes are already represented spatially on the tree. (. . . ) If worlds branch, then we branch too.(MacFarlane, 2014, p. 212).

How can I then propose the idea based on such a fundamental confusion? Well, I think that there are subtle but important diﬀerences between MacFarlane’s model of the moving dot and mine. First of all, he presupposes that “I’m located in many worlds that overlap in the present.” But if you consult the diagram above, you will clearly see that I am not. I am located in one world only (the blue dot on the right). The possibility presently actualized by the world might be located in many possible scenarios, but it does not imply that the world-at-present is a part of more than one course of events. Thus, in his criticism of a “moving dot,” MacFarlane presupposes a realistic account of branching possibilities—he assumes that they are composed of concrete objects and

2Arthur Prior(1968, ch. VI) has argued that this kind of limitation is best justiﬁed within Aristotelian metaphysics. In this framework, the actual substances and their modal features like essences or dispositions naturally limit the amount of available possibilities. I shall brieﬂy discuss this idea in due course. 3This sentence might be slightly misleading since it suggests that there is now the fact of the matter regarding which of the possible futures will happen. That is, it suggests that the actual future is somehow present among the things which exist now. However, the actual future is what will happen and what will happen, will happen, when it will happen, not now. Thus, it would be more fortunate to say that there will be a fact of the matter regarding which of the possible futures will be actualized.

170 CHAPTER 6. BRANCHING ACTUALISM processes.4 Within such realistic account of branching, it is indeed the case, as MacFar- lane insists, that all processes that take place in time are represented on the tree. After all, the branches consist of causally connected, concrete events. However, within the actualist account I advocate, no single physical process is represented on the tree. The tree represents the possible paths of evolution, but even if we represent all the possible paths of evolution, the description of reality is incomplete. We still need to represent all the actual temporal processes taking place in the world. This last part is represented be the “moving blue dot” which induces the red dot moving up the tree of possibilities. Thus, I disagree with MacFarlane that the idea that only one of the possible states is being actualized at every instant of time involves a “fundamental confusion.” The fact that on the diagram above, there literally is a red dot changing its location from one cell to another is just a matter of representation. To save a bit of paper and simplify the drawing, we can superimpose the cells and obtain the following pictorial representation:

Possible courses of events The course of events

Within this picture, the Kripke’s distinction between “actual world” and “another entity, more ‘concrete’ ” is clearly depicted. The blue line on the right represents how the “concrete entity” evolves in time, while the red line on the tree represent the possible course of events actualized by the procession of events in time. The picture above is meant to convey the exact same information as an inﬁnite series of separate cells. In particular, it is not meant to suggest that the whole history of the world is somehow “ready” or “present” from the dawn of time. The history of the world is constituted by what happens in the world in the successive moments of time, so it cannot be present at any single moment of time. Let me point out at this instant, that I do not intend to side with any particular position in philosophy of time in my investigations. I side with actualism and I reject modal neutrality but with respect to the analogous temporal issue I intend to remain neutral. I do not want to prejudge whether there is absolute diﬀerence between what is present and what is past or future.5

4Incidentally, this fragment calls into question MacFarlane’s earlier reassurance that “We do not assume that our worlds are real, concrete wholes, as in Lewis (1986). Nor do we assume that they are ersatz representations” (MacFarlane, 2014, p. 203). 5As might be clear by now, I tend to think that there is certain dis-analogy between modal and temporal case. It would require at least another dissertation to analyze this issue in details, so let me just say that on the level of “gut feeling” I agree with the following statement of Kit Fine: The case of time is perplexing in a way that these other cases are not. On the one hand, (. . . )[w]hat goes on in the present and at other times is somehow part of the same all-

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The world should not be identified with the possibility actualized by the world (the “red” possibility) for plurality of reasons. Fist, the world is concrete while possibilities are abstract. Second, the “red” possibility is temporally extensive while the world itself might be a temporally “thin,” presentist world. Last, but not least, the information contained in the “red” possibility is somewhat limited. It encodes which scenario is actualized but does not indicate whether alternative scenarios are available. Only when both aspects of are covered, however, we give the full account of the reality. Thus, the tree of possibilities is an important aspect of the world, while it is completely irrelevant for the red possibility. The red history would be exactly the same if there were no other possibilities, but the world would be substantially different if there were no other possibilities. We can call the possible course of events actualized by the world the Thin Red Line. However, this perspective on the TRL is very different from that of Belnap et al.’s (2001). On the one hand, Belnap et al.(2001) think that the world is a branching object with a “red,” actual part. On the other hand, in the view I present, the world does not branch in time. Branching takes place among possible scenarios and the “red” possibility is one of them, the one that is (indeterministically) actualized by what happens in the world at successive instants of time. Therefore, I disagree with Belnap et al.(2001) that

To the extent that common sense asks for (. . . ) a unique naturally given “actual history” to which a given utterance-event belongs, to that extent, common sense is asking for something it cannot have. (Belnap et al., 2001, p. 206)

Let me brieﬂy mention one particular controversy (I develop and discuss this objection in section 6.4.1). The Branching Realists often criticize actualism with the following objection: if only one of the possibilities is actualized—the single “red” possibility—then, in an important sense, the other possibilities are not real. The non- actualized possibilities might represent our limited knowledge, but they do not represent how the world can really be like. Thus, if only one of the histories is marked red then, in an important sense, the world cannot develop any other way than how the red line dictates. The world, in the process of its development, can do nothing but follow the invisible redness which binds its steps as it marches through the maze of (quasi) possibilities. This mode of presentation completely distorts the actualist insight, however. There is indeed a necessary connection between how the world is like and which possibility is actualized, but the critics mistake the direction of dependence. It is not that the world develops as it does because it somehow has to follow what the actualized possibility dictates. It is the other way around: the actualized possibility is as it is because it has to “follow” the world’s development. As the time goes by, the world realizes exactly one of the available options at every indeterministic juncture. Depending on how the

encompassing reality in a way in which what goes on in the actual world and in other possible worlds is not. On the other hand, (. . . ) [w]hat goes on in the future, or in the past, does not seem real to the same extent or in the same way as what goes on in the present. Thus, the past and future appear to have some kind of intermediate status. (Fine, 2005, pp. 285–286)

172 CHAPTER 6. BRANCHING ACTUALISM events in the world proceed, exactly one of the available possibilities gets actualized. But there should be no feeling of determinism about it. Even though it is necessarily that one of the possibilities is actualized, there is no single possibility which is realized out of necessity. One more proviso is due. The idea I have sketched appeals to the concept of “the world at a time.” However, relativity theory strongly suggests that this concept is not fundamentally adequate. Whether two events happen at the same time is a relative matter—it depends on an observer who is abstractly represented as a frame of reference. Even if two events happen at the same time in one frame of reference, they may happen at different times in another frame of reference. The idea I present suits well enough the everyday idea of time, but it is not sufficiently accurate if we want to take into account the scientific worldview. If we try to generalize the picture, however, the complexity level quickly increases. It is much easier to introduce branching among possible temporal scenarios than branching among possible spatiotemporal scenarios. The considerations outlined in (Belnap, 2003b) suggest that the best strategy is to spatially “localize” possibilities and to replace possibility at a time with a possibility at a spacetime point.6 We can then try to organize the localized possibilities into a large, branching-like structure. To incorporate the actualist insight into this setting, one needs to insist that every “solid” event which actually takes place at a specific spatiotemporal location actualizes exactly one of the spatiotemporally local possibilities available for this location. As actual events change, along time-like curves, they always actualize one of available possibilities. Thus, what happens after a given event is constrained by what is possible at this event. At the same time, the possible continuations of a spatiotemporally local event are limited by what has happened in the causal past of this event.7 Thus, the reciprocal relations between possibility and actuality is analogous in temporal and in spatiotemporal case. A lot more should be said on this subject, but is is my conviction that the actualist insight is not intrinsically connected to the pre-relativist account of time.

6.2 Semantic impact

Let us investigate how the assumption of modal actualism inﬂuences the treatment of future contingents. First, let us recall the initialization problem. Within a language that contains both temporal and modal operators, we need two parameters to assign truths value to sentences: a temporal parameter (a time or a moment) and a modal parameter (a world or a history). However, sentences of our language often do not specify the

6Interestingly, Fine(2005, sec. 10) and Pooley(2013, sec. VII and IX) argued that such “local” account of possible states is at odds with perspectival metaphysics. They believe that even if we agree that the facts essentially depend on a perspective (viewpoint, perspective, etc.), then it is unlikely, that a spacetime point is a reasonable candidate for a perspective. They are concerned that if we took a such a local notion of perspective, then we are forced to metaphysically privilege what is here over what is elsewhere (i.e., in space-like related regions). They both ﬁnd this consequence highly unwelcome. 7Unless the modal funny business is allowed in which case the possible continuations available for one event might be, in a sense, limited by what happens at a diﬀerent event. The modal funny business is a one way to understand the EPR-like phenomena in quantum mechanics.

173 CHAPTER 6. BRANCHING ACTUALISM relevant time and they hardly ever specify the relevant possible world. A natural reaction is to appeal to the time and world at which the utterance take place. The branching theorists, however, almost unanimously rejected this option. I believe that they were reluctant to endorse this idea because they implicitly endorsed the Genuine Realist account of possibilities. They thought of possibilities along the Lewisian lines: as concrete, partially overlapping physical entities. Then, they cannot appeal to the world at which the utterance takes place, because the concrete act of utterance takes place in many worlds! Consequently, one essential semantic parameter is left unspeciﬁed by the context. This diﬃculty of modal realists is recognized by John MacFarlane:

David Lewis sees this point very clearly. He acknowledges that if an utterance of a future contingent belongs to more than one possible future history, we cannot appeal to “the actual future” to secure it a determinate truth value. (MacFarlane, 2003, p. 326).

So, in the realistic perspective, we are “tree dwellers” and utterances we make take place in many overlapping worlds, as is depicted in figure 6.1. Things appear differently from the The coin will actualist perspective. Remember that land heads. actualism stresses the fundamental distinction between the actual world and the sphere of possibilities. An utterance is a physical act which takes place in the actual world and there is only one such world. This world neither begins Figure 6.1: Utterance in many worlds. nor ends with the act of Jack’s utterance. Some events has preceded and some will follow the act. Among the events that will follow, will be the coin’s landing on one of its sides. We can represent the situation pictorially, as on figure 6.2.

actualizes The coin will The coin will t land heads. land heads.

Possible courses of events The course of events

Figure 6.2: Utterance in one world.

The concrete, physical act utterance takes place on the right-hand side of the ﬁgure, in the actual world. The event at which it takes place is distinguished with a dot. The continuous line represents the events that have happened before the act of utterance

174 CHAPTER 6. BRANCHING ACTUALISM and will happen after the act. Since the act of utterance actually takes place, it certainly is possible. The possibility of the act occurring is depicted on the tree (it is indicated by a red dot). By uttering the words, “The coin will land heads,” Jack actualizes this possibility. In general, as the world develops in time, it actualizes one of the available possibilities. The unique possibility that has been and will be actualized is indicated in red on the tree of possibilities. It does not need to be the one that I have indicated, it just has to be one particular possibility, and for purposes of exposition, I assumed that the “heads”-possibility will be actualized. With this metaphysical picture in the back of our heads, let us return to the semantics of future contingents. We can begin with a simplified case of the language that contains solely temporal operators. Let us consider the sentence “The coin will land heads” (F p) used at a particular context. Within the branching realist picture (as depicted in figure 6.1), it is entirely unclear how to evaluate this sentence. There are many distinct courses of events that contain the utterance and it is uncertain which one to use for semantic purposes. Within the actualist model, though, the issue seems different. In this model, the course of events that contains the utterance is linear. One might want to exploit this feature of reality for semantic purposes. A very simple procedure recommends itself: To analyze sentences in future and past tense used in the actual world, we just need to “travel” up-and-down the linear time. The truth value of a sentence in past tense depends on what was actually the case at earlier times and the truth value of a sentence in future tense depends on what will actually be the case at later times. So, the fact that possibilities are branching does not affects the interpretation of sentences in past and future tense. Given that the world develops linearly, it is clear how to interpret the temporal operators. It is not to say that the branching possibilities serve no semantic purpose. Quite the contrary, even if we live in the world that does not branch in time, we may want to talk about what was, is, and will be really possible. Then, the linear structure is insufficient, as it merely depicts what did, does, and will in fact happen. This point was clearly articulated by McKim and Davis:

To semantically interpret historical modalities, we need to resort to the branching structure of possibilities and interpret sentences on the tree, at moment/history pairs. But then, the initialization problem returns and we need to specify the possible moment and the possible history relevant for semantic evaluation. Let us see if the actualist context can help to decide the relevant semantic parameters. One part of the task is relatively straightforward. In actualism, there is an intimate relation between the world and its possibilities: at any time, the physical world realizes just one of the momentary possibilities available at the time. Thus, the world, as it is at the time of utterance, also actualizes a single momentary possibility. Quite evidently, this very momentary possibility is relevant for semantic purposes. The possible moment

175 CHAPTER 6. BRANCHING ACTUALISM actualized by the event at which the utterance takes place, is the possible moment initialized by the context. What about the second parameter—the possible history? An analogous procedure recommends itself: The possible history actualized by the world in which the utterance takes place, is the possible history initialized by the context. In actualist setting, the utterance takes place in one world only. This world, throughout the period of its existence has and will actualize a single possible scenario and it is the scenario that should be used for semantic purposes. Thus, within actualist setting, there is a quite natural way to accept Twardowski’s dictum: “Circumstances accompanying speaker’s words complement what the words do not express” (Twardowski, 1900, p. 6). The “complemented” parameters are the possible moment actualized at the time of utterance (the moment of the context) and the possible history actualized by the world of utterance (the history of the context). The initialization problem disappears and we have a straightforward way of using the Ockhamist semantics for evaluation of sentences used at contexts. We can readily translate this idea into a postsemantic theory: f Definition 6.1 (Futurism). c||−φ iff mc/hc |= φ. A sentence is true at context c if and only if the sentence is true at the possible moment mc initialized by the context and the possible history hc initialized by the context. Since the history of the context is identical to the actualized history, I shall sometimes refer to it at h@. The idea behind the postsemantic futurism is very simple. The truth value of the sentences used in present tense depends on how the world is like at the time of utterance. The truth value of the sentence in past tense depends on how the world was like prior to the time of utterance. The truth value of the sentence in future tense depends on how the world will be like after the time of utterance. The truth value of a modal sentence depends on what possibilities are available, given the state of the world at the time of utterance. If we limit ourselves to non-modal sentences used in our world, we simply “travel” up and down the linear time and the semantics of this fragment of our language is just a semantics of linear time. In this limited case, the structure of branching possibilities makes no impact on the semantic process. In those cases, the only relevant thing is what has and will actually happen. I should stress at this point that this idea is not universally accepted among actualists. Many among them insist that even though the world will be in a unique actual state tomorrow and even though the sentence talks about what will happen tomorrow, its truth value at present depends on what happens at present (in particular, on what is possible and what is necessary at present). This branch of actualists usually make substantial use of the whole structure of possibilities when they evaluate sentences about the future. I once thought that this split has metaphysical underpinning (I expressed this view in Wawer, 2014), but I am now more inclined to say that the quarrel is motivated by semantic considerations. The clash between these two lines of actualism is nicely exemplified by an early 20th century debate between Kotarbinski´ (1913) and Le sniewski´ (1913). I discuss these two contrasting attitudes in section 6.5. Futurism exemplifies the brand of actualism that makes the truth value of a sentence uttered at one time conditional on what happens at another time. Strictly speaking,

176 CHAPTER 6. BRANCHING ACTUALISM temporal parameters of our semantics do not travel up and down the actual course of events. They travel along the possible course of events that has and will be actualized. In a sense, the actualized course of events (the “red” possibility) stands “proxy” for the actual world in the realm of possibilities. The shift from the actual to the actualized is safe though. It is not going to distort any semantic values since the actualized possibility corresponds to what has and what will happen in the actual world. The use of the proxy is necessary since we want to preserve the uniform (and compositional) treatment of temporal and modal operators that is offered by Ockhamism. If we required that the temporal operators (P and F) were always interpreted in our world, then it would be difficult to interpret the sentence like “The coin might land tails” (^Fq) uttered at an actual context. For the proper interpretation of this sentence, it is crucial that the operator F does not travel up the actual world, but up the possible scenario where the coin lands tails. Therefore, it is useful to interpret sentences at the actualized possibility (hc) rather than in the actual world since it is then clear how to interpret modal connectives. The futurist postsemantics does not limit the interpretation of future and past operators to the actualized course of events. It just requires that the interpretation of any sentence begins at the actualized course of events. Then, if need be, hc can be replaced with another history. For example, if we consider the branching model depicted at figure 6.2, but change the sentence uttered by Jack to “The coin might land tails” (^F(tails)) then, the futurist postsemantics, together with Ockhamist semantics, dictate that:

1. c ^F(tails) iﬀ

2. mc/hc |= ^F(tails) iﬀ

3. ∃h(mc ∈ h & mc/h |= F(tails)) iﬀ

4. ∃h∃m(mc ∈ h & m ∈ h & m > mc & m/h |= tails).

The model indicates that there is a possible moment/history pair satisfying this condition, so the sentence “The coin might land tails,” actually used by Jack, is true. Importantly, what happens in hc after mc is irrelevant for the truth of this sentence. It can be contrasted with the sentence “The coin will land heads”:

1. c F(heads) iﬀ

2. mc/hc |= F(heads) iﬀ

3. ∃m(m > mc & m ∈ hc & m/hc |= heads).

In this case, history hc is crucial for evaluation of the sentence (and it should, since it “follows” what happens in the world and the truth value of sentences about the future should depend on what will happen). In our case, the coin actually lands heads after Jack’s utterance, so the sentence “The coin lands heads” is true later in the “red” possibility hc, so Jack utters a true sentence. The general postsemantic procedure is applicable to more complex cases like “The coin might not have been tossed” (P^F¬(toss)). Whenever we interpret a sentence

177 CHAPTER 6. BRANCHING ACTUALISM used in a context we “jump” to the appropriate, context-designated parameters and then apply the reliable Ockhamist semantic clauses (I discuss more complex cases in section 6.3.4). Therefore, thanks to the futurist postsemantics, we can avoid the initialization problem, but we can use all the desirable features of Ockhamist semantics. In my account all the interesting stuff happens when we shift from a sentence at a context to its representation in the Ockhamist model. This attitude of mine should be strongly contrasted with the mainstream approach. The most common procedure among the logicians who wanted to interpret the “factual” future tense was to reinterpret the semantics proper of operator F. I recounted the history of these attempts in section 5.3.4 of the previous chapter. Basically, the idea was to make semantics of F history independent by binding it with a single, distinguished history passing through a moment of evaluation. The routine technical tool was to incorporate a function trl f cn which assigns a history to each moment on a tree. Thanks to this function, we could get rid of the history parameter on the level of semantics proper: f-trl 0 0 f-trl m|= Fφ iff ∃m0>mm ∈ trl f cn(m)& m |= φ. In this semantics, there is no history parameter, so there is no need to initialize it. Therefore, we can accept a simplified postsemantics: f-trl f-trl c||− φ iff mc|= φ This procedure seems attractive on the first sight, but we have witnessed the problems that it generates. Therefore, I consider it to be a virtue of postsemantic futurism that it does not modify the semantics proper, but addresses the initialization problem on the postsemantic level. As I said, it is central to futurism that the context initializes both a possible moment and a possible course of events. This formal account is hardly an extravagant one. In fact, this kind of postsemantics for languages with modal and temporal operators has been recommended (and practically codified) by David Kaplan(1989, p. 547). Nevertheless, the standard treatment has been almost universally rejected in branching setting (from Thomason(1970) to MacFarlane(2014)). My purpose is to incorporate the orthodoxy into the branching setting. Thus, the lengthy metaphysical discussion was required to justify the most simple-minded solution to the initialization problem. Importantly, Belnap et al.(2001, p. 232) admits that “[i]t would clearly make technical sense to provide a context parameter ranging over histories.” However, they insist that “[i]ndetereminism, however, compels a view absolutely contrary to this” (p. 232). I have already mentioned their reason to believe this. They insist that “[a] single, well- identified context of use is typically part of a large variety of possible future courses of history” (p. 232). It is certainly so in the Genuine Realist setting. In actualism, however, the context is a part of a single world that linearly develops in time. Consequently, we can initialize the possible scenario as the history that is actualized by the world in which the utterance takes place (a possible moment of the context is a part of many possible histories, but the possible moment of the context is not the context). It is useful to contrast futurism with alternative postsemantic theories: 1. Contrary to modalism: “Will” is not semantically identified with “Inevitably will,” “Possibly will,” “Probably will,” or any other modal expression.

178 CHAPTER 6. BRANCHING ACTUALISM

2. Contrary to three-valued postsemantics: There are only two truth values for sentences in contexts. 3. Contrary to supervaluationism: For any sentence, and any context, either the sentence or its negation is true in the context. Classical connectives are truth- functional in context.

4. Contrary to extremism: Not all sentences about the contingent future are false in context. 5. Contrary to relativism: The meaning of the sentence and the features of the context are suﬃcient to establish the truth status of a sentence. No modal relativization is required (to a history, a context of assessment, a continuation, a set of transitions, etc.). 6. Contrary to STRL, it does not postulate non-actual contexts.

7. Contrary to trl f cn postsemantics: the past contexts initialize the same course of events as the present context.

A combination of these features is certainly acceptable (at the very least, it is consistent). In fact, it might be argued that futurism improves on the alternative accounts in the areas in which each of them was most problematic. I can use the words of Nicholas Rescher to summarize and advertise my views:

This approach preserves intact the standard group of logical and semantical concepts that cluster about the notion of truth and falsity. At the same time, it averts consequences of a necessitarian and fatalistic kind. The truth-status of a future contingent proposition is made to hinge upon what happens at that future time: there is no suggestion that its having a truth value, and an (ultimately) knowable one, in any way ﬁxes beforehand or predetermines what that truth value is to be. (Rescher, 1968, p. 215)

To further understand the speciﬁc character of postsemantic futurism and appreciate some of its virtues, it will be useful to confront it with the objections that Nuel Belnap and his collaborators (1994; 2001) raised against the TRL-theory. After all, futurism, just as the TRL-theory does distinguish one of the possibilities as the “red” possibility. However, we shall see that the actualist background of futurism helps it to get around Belnap et al.’s (2001) worries. Sections 6.3.1– 6.3.4 rely on the content discussed in (Wawer, 2014).

6.3 Response to objections 6.3.1 Metaphysics What in the structure of our world could determine a single possibility from among all the others to be “actual”? (Belnap et al., 2001, p. 162).

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The question has at least two natural readings. If it is to be read as “What in the structure of our world makes it necessary that h@ rather than some other possible history is actualized?” then the answer is: “Nothing!” The world is indeterministic and it can develop along any of the possible ways. It simply develops along one of them which we refer to as h@. However, if the question is to be read as, “What in the structure of the world makes it necessary that only one of the histories is actualized?” then the answer is, “The structure of the world itself.” According to actualism, the concrete universe simply does not branch in time. The histories in the branching tree represent all the ways the world might develop, but the world develops in one way only. Therefore, one (and only one) of the histories must represent the world as it actually develops. There is nothing deterministic about this result. The physical world “determines” the h@ in the very same way in which the “complex physical entity (‘the dice,’ thought of a a single object) (. . . ) and its actual position determines the actual state of the (two) dice” (Kripke, 1980, p. 17). To see that it is no mystery, compare Belnap et al.’s (2001) metaphysical objection with another puzzle. Let there be a fair lottery in which exactly one of the tickets is drawn and the drawn ticket wins, then we can paraphrase Belnap et al.’s (2001) question:

What in the structure of the lottery could determine a single ticket from among all the others to be “the winning one.”

If it means to ask, “What makes it necessary that ticket a rather than some other ticket wins?” the answer is, “Nothing!”; after all, the lottery is fair. If, on the other hand, the question is, “What makes it necessary that only one of the tickets wins?” the answer is, “The structure of the lottery itself”—we draw the tickets just once so one (and only one) of the tickets must win. This fact does not make the lottery deterministic.

6.3.2 Epistemology

[H]ow we could know whether we are on TRLabs. How could we ﬁnd out? (Belnap et al., 2001, p. 163)

First of all, let me notice that Belnap et al.(2001) use TRLabs to refer to one of the possible histories. Therefore, the objection presupposes that we can intelligibly ask if we, the concrete human beings, are parts of a possible scenario. This presupposition is questionable for at least some forms of actualism. Nonetheless it is, in another sense, a perfectly intelligible and interesting question: “How can we know that the world (with us in it) actualizes h@ rather than some other history?” It is evidently true that it does, but how do we know it? It is not quite by definition. We do not want to define h@ as “the actualized history” since then, the identity (h@=actualized-history) would be an analytic necessary truth. We would get the result that it is necessary that h@ (one specific history in the model) is the actualized history which contradicts the assumption that it is contingent which history is actualized.

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How do we know then that h@ is actualized? I propose to think about this issue in Kripkean (1980) terms. The description “the actualized history” should be thought of as a description which fixes a reference of the term h@ rather than its meaning (Kripke introduced the distinction at p. 55). When we look at the issue in this perspective, then the semantic value of the sentence, “The world actualizes h@,” is contingent. After all the world is indeterministic and it might actualize a different possibility than the one it in fact does. However, the sentence, “The world actualizes h@,” is true whenever used. Moreover, its truth relies on the linguistic conventions only. Therefore, we do not need to investigate to find out whether any use of the sentence “The world actualizes h@” is true. We know it, in a sense, a priori. Thus, the sentence, “The world actualizes h@,” is like the sentence “I exist,” or “I am here,” their semantic values are contingent but they can be known a priori (cf. Kripke, 1980, pp. 54ff.). That’s the answer to Belnap et al.’s (2001) puzzle, “[H]ow we could know whether the world actualizes h@?” Well, we know it a priori.

6.3.3 Actuality The TRL theory also has troubles with actuality. (. . . ) [T]his world’s being the actual world does not favor it over any others, but is just a reﬂection of the fact that this is the world at which we are conversing. To suppose that there is one from among the histories in Our World that is the absolutely actual history is rather like purporting to stand outside Lewis’ realm of concrete possibilia and pointing to the one that is actual. But this is wrong in both cases. (Belnap et al., 2001, p. 163).

There are two worries that might be extracted from this fragment. The first one resembles the metaphysical objection discussed above. This fragment just further reinforces my diagnosis that this objection is supported by the Genuine Realistic account of modality common to Lewis and Belnap et al.(2001). Another worry lurking in the quoted fragment originates in the observation that the actual world is the world at which we are conversing. The authors allude to the indexical theory of actuality. Adams(1974, p. 214) traces the origins of this idea back to Leibniz. In the 20th century it was discussed, and rejected, by Arthur Prior (see Lewis, 1970a, p. 185, n. 6), then articulated and ably defended by Lewis(1970a), formally developed by Kaplan(1989), and applied to Ockhamism by Belnap et al.(2001). The core of the idea is that words like “actually” or “actual” are structurally similar to indexical expressions like “now,” “here,” “I,” etc. The distinctive feature of these words is that their reference is sensitive to the context of use. Just as “here” refers to different places on different occasions of use, “actual” refers to different circumstances on different occasions of use. This linguistic idea is quite naturally combined with the philosophical picture presented by Genuine Branching Realism (and Lewisian Genuine Realism) since according to this position, all the possible circumstances have the same metaphysical status as the circumstance we are currently in. The only way to distinguish the latter is by using the phrase “The circumstances I am actually in.” A use of the word “actually” indicates the exact position on the tree of possibilities (or in the space of possible worlds).

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This account of actuality might seem to be at odds with the actualist ideology since the latter suggests that the actual world is absolutely distinct from the possible histories (including h@). I am going to show, however, that actualist futurism is compatible with the indexical notion of actuality. It shows that the indexical nature of the word “actually” does not imply modal neutrality. I am happy to share this conviction with Robert Stalnaker:

My point is that the semantical thesis that the indexical analysis of “actual” is correct can be separated from the metaphysical thesis that the actuality of the actual world is nothing more than a relation between it and things existing in it. (. . . ) [O]ne can accept the indexical analysis of actuality while excluding from one’s ontology any universes that are the way things might have been. (Stalnaker, 1976, p. 69). Proceeding formally, I adopt Kaplan’s (1989) treatment of indexicals. The lesson from Kaplan is that we need to somehow “store” the information at which context sentence φ is used, so we can utilize this information evaluating indexical expressions occurring in φ. So far, the context (through appropriate postsemantics) dictates where to start the process of semantic evaluation of the sentence. However, the value of the context-initialized index is not stored for further purposes. Therefore, if modal or tense operators shift the index to another position, we have no mechanism to “get back” to the initial, context-initialized position. To remedy this diﬃculty, we need to extended the semantic index. The exact shape of the extended index depends on which indexicals are present in the language (e.g., if we included the indexical like “here” we would need to add “the place of the context”). It also depends on which parameters are initialized by the context. In futurism, both possible moment and possible history are initialized, so the extended index has the form hmc/hc, m/hi. We can now phrase the natural semantic deﬁnition of the connective “it is actually the case that” (@).

Definition 6.2 (Futurist definition of actually:). mc/hc, m/h |= @φ iff mc/hc, mc/hc |= φ.

Thus, even if the index has been shifted by modal and temporal operators from the original position mc/hc to some other position m/h, then the operator @ brings them back to their initial value. Notice that both parameters are restored, so the operator “actually” is both modal and temporal operator.8 The purely temporal indexical “now” keep the history of evaluation untouched and shifts the moment of evaluation to the moment that is “co-present” with the moment of the context (the definition requires that the model is sliced into instants). | | Definition 6.3 (Now). mc/hc, m/h = @φ iff mc/hc, m(imc ,h)/h = φ.

8It is clear that “actually” has this double role if we consider a sentence (inexpressible in our limited language) like, “If I did not marry Ann, then, even in ten years I wouldn’t have as much money as I actually do.”

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The Branching Realists have always run into trouble defining the actuality operator. Thomason suggested that they should resort to the solution accepted by other brand of Genuine Realists: “See Lewis(1970a) and substitute ‘the actual future’ for ‘the actual world’ in what he says” (Thomason, 1984, p. 215, n. 14). But for Lewis, the actual world is the world at which we are conversing. So, if we wanted to use Thomason’s procedure, we should say that the actual history is the history at which we are conversing. But there is no such thing for the Branching Realist. In their opinion, we are conversing in many distinct histories. Therefore, there is a widespread belief in the branching literature that we cannot talk about “the history of the context.” Therefore, to understand the operator @, they had to deal with an impoverished index: hmc, m/hi. Only the assessment relativist can afford a slightly richer index and include both a possible moment of use (mu) and a possible moment of assessment (ma). Let me present some definitions available in the literature

Definition 6.4 (Relativist definition of Actually1:). 0 0 0 mc, m/h |= @1φ iff mc, mc/h |= φ for every h such that mc ∈ h (Belnap et al., 2001, p. 246).

Definition 6.5 (Relativist definition of Actually2:). 0 mc, m/h |= @2φ iff either (a) mc ∈ h and mc, mc/h |= φ or (b) mc < h and M, mc, mc/h |= 0 0 φ for every h such that mc ∈ h Belnap et al.(2001, p. 246). Definition 6.6 (Supervaluationist definition of Actually:). 0 0 0 mc, m/h |= @φ iff mc, mc/h |= φ for every h such that mc ∈ h (MacFarlane, 2008, p. 99).

Definition 6.7 (Assessment relativist definition of Actually:). 0 0 mu, ma, m/h |= @φ iff mu, ma, mu/h |= φ, for every h ∈ Hmu|ma (MacFarlane, 2008, p. 99, for definition of Hm1|m2 , see p. 84). The litmus paper that I am going to use to test the definitions is the initial-redundancy requirement for the actuality operator proposed by MacFarlane(2008). It is a postsemantic requirement which says that any sentence φ should be true at exactly the same context as sentence @φ: c φ iff c @φ The relativists believe that context of use is not sufficient to judge the truth value of a sentence in context. They demand that a modal parameter is specified (a history or a context of assessment). Hence, for relativists, the initial redundancy test is slightly different:

c/h||−hφ iﬀ c/h||−h@φ R R cu, ca||−φ iﬀ cu, ca||−@φ

The initial-redundancy seems like a reasonable demand. If it is true to say φ, it should be equally true to say @φ. For example, if I truly say “It rains” I could just as well (less economically, but more emphatically) say “Actually, it rains.” The same goes for sentences in future tense: it is true to say “It will rain” iﬀ it is true to say “Actually, it

183 CHAPTER 6. BRANCHING ACTUALISM will rain.” That is, the addition or removal of operator @ simply makes no difference for truth of sentences in contexts. This requirement might seem innocent but it takes a toll among the definitions of @. The first victim of the test is operator @1. Let us consider a sentence F p contingent at mc. Let h1 be such that mc, mc/h1 |= F p, then by history-relativist postsemantics h (def. 7.12, p. 232) mc/h1||−F p. However, since φ is contingent at mc, there is h2 such that mc, mc/h2 |= ¬F p which means that, by definition of @1, mc, mc/h1 |= ¬@1F p which implies in turn, by history-relativist postsemantics, that mc/h1 ¬@F p. Con- sequently, there is a sentence φ, a context mc, and history h, such that mc/h φ and mc/h ¬@1φ. So, @1 fails the initial redundancy test. With respect to some histories, we can truly say, “There will be a sea battle tomorrow and actually, there will be none.” This observation shows that Belnap et al.(2001) are not quite correct when they write “As always, Now: and the actuality connectives do very little work at the head of sentences considered as stand-alone, (. . . ) so that the shifting called for by Now: or an actuality connective is vacuous” (Belnap et al., 2001, p. 247). Belnap et al.’s (2001) alternative proposal (@2) does pass initial redundancy test but is not acceptable for independent reasons:

• If sentence φ is evaluated at a point m/h such that mc ∈ h, then @2 loses a part of its indexical nature. MacFarlane (2008, p. 99) even claims that Actually2 is simply redundant at such points. This is not quite accurate since Actuality2 functions as temporal indexical Now at these points. However, MacFarlane is partially right: at such points Actually2 loses its indexical nature as a modal operator. It is particularly visible at context-initialized indexes at which the sentence “Necessarily, there will be a sea battle if and only if there actually will be a sea battle” ((F p ↔ @2F p)) is always true.

• There is a sentence φ and a context-initialized index hmc, mc/hi at which P^F(φ∧ ¬@2φ) and @2φ are both true. So we can truly say “It might have been the case that there would be a sea battle and actually there wouldn’t be any. But actually, there will be one.” Such oddities follow because even if we use @2 twice in one sentence, it might behave as a modal indexical at ﬁrst occurrence and not as a modal indexical on the second occurrence.

I find these reasons sufficient to disqualify @2 as a candidate for a proper analysis of the word “actually.” Thus, we have eliminated two definitions of the operator @. However, the remaining three proposals still stand. All of them satisfy redundancy requirement. To differentiate between them, we need to devise a stronger test. One reasonable strengthening is a demand that not only uses of φ and @φ should be co-true at every context (as MacFarlane, 2008 insists), but they should also be co-false. Formally, the stronger requirement is that

(c ||− φ iﬀ M, c ||− @φ) and (M, c ||− ¬φ iﬀ M, c ||− ¬@φ)

I ﬁnd it quite reasonable. After all, “Actually” is a modal indexical, so it should be not only initially redundant but redundant also in scope of extensional connectives such as

184 CHAPTER 6. BRANCHING ACTUALISM negation. In particular, the sentence “It’s not the case that it will rain” should be true in the very contexts in which the sentence “It’s not the case that actually it will rain.” However, both supervaluationism and assessment relativism fail this requirement. Supervaluationism: For every contingent sentence φ used at c: c||−S ¬@φ and c||−/S ¬φ;

Relativism: For every sentence φ used at cu and still contingent while assessed at ca: R R cu, ca||−¬@φ and c, ca||−/ ¬φ. The futurist definition of “actually” I proposed satisfies the stronger test in full generality. In fact, operators ¬ and @ are mutually “transparent,” i.e., the equivalence ¬@φ ↔ @¬φ is a tautology under the futurist definition of @. We can propose an even stronger, and yet still quite natural version of the initial- redundancy requirement and demand that for every context c and every sentence φ, c φ ↔ @φ. It simply means that at any context, one is guaranteed to say the truth, claiming that there will be a sea battle if and only if actually there will be a sea battle. MacFarlane(2008) explicitly rejects this strengthening but he agrees that we need to “get over our qualms” to do so (p. 99 , n. 22). Both supervaluationism and assessment relativism falsify this equivalence whenever φ is a contingent sentence. At the same time, the equivalence φ ↔ @φ is true at every context in futurism. It is easy to understand why futurism constitutes such a friendly environment for the operator “actually,” while relativism and supervaluationism are hostile to it. In futurism, there is exactly one index initialized by any use of a sentence. Importantly, this index contains a specific history (hc) as its element. This index represents what has, is, and will in fact be the case, so it is quite evident that it should be utilized for interpretation of the operator @. Futurism is based on actualism which sharply distinguishes the actual from the possible, so the interpretation of @ is quite straightforward. The Genuine Branching Realists, on the other hand, reject any “actual” history. However, to retain the indexical meaning of @, they need to tie it to some feature of the context. The only available item seems to be “a unique causal past, and a unique future of possibilities, the whole of which is summed up by the moment of use” (Belnap et al., 2001, p. 226). As a result, Branching Realists tend to identify actuality with necessity. Belnap et al.(2001) even propose an intended reading of @ 1A to be: “It is settled true at this actual moment that A”(2001, p. 153). Consequently, Genuine Realists usually takes the sentence φ ↔ @φ to be true whenever used. So, for them it is always true to say “If it actually will rain tomorrow, it is settled that it will” (@F p → F p). Such observations reinforces the realists’ conviction that actuality is only a camouflaged form of necessity and whoever talks about the actual future is a determinist in disguise. Futurism disentangles the notions of actuality and necessity. In particular, it is not difficult to find a sentence φ and a context at which an use of @φ ↔ φ is false.9 A Branching Realists could escape all the problems, if only they incorporated the history of the context into the semantics index. This solution was abhorred, however, by most of the theorist writing on the subject, beginning with Thomason, through Bel- nap and his collaborators, ending with MacFarlane. They believed that incorporation

9A similar implication, @φ → @φ, is true whenever used, but it does not doom us to determinism (just as the truth of, “If it is raining now, then it will always be the case that it was raining now,” does not doom us to a ﬂood). It just witnesses to the indexical nature of the operator @.

185 CHAPTER 6. BRANCHING ACTUALISM of the actual history is entirely contrary to the spirit of indeterminism and branching possibilities. Against their warnings Roberto Loss(2012), has explored this idea and came up with what I think is the best treatment of actuality available in the realist setting. On the semantic level, he just accepted the futurist deﬁnition of @ (def. 6.2, p. 182). The ingenuity of his idea comes on the postsemantic level. To neutralize the imminent threat that the Branching Realists recognize in the history of the context, he modiﬁed the supervaluational postsemantics, so the history of the context10 becomes as “provisional” as the history of evaluation:

S + c||− φ iﬀ ∀h(h ∈ mc ⇒ M, mc/h, mc/h φ)

Since Loss’ semantics proper of @ does not differ from mine, his treatment respect all the intuitive truths which speak in favor of the futurist notion of @. I was not aware of the work of Loss when I was writing (Wawer, 2014) and I unjustifiably boasted that my definition offers the best semantic treatment of “actually” operator available for branching. Loss’ treatment is semantically indistinguishable from mine and it gives justice to initial redundancy of both affirmative and negated sentences (Loss calls this requirement initial-equivalence, p. 23). I also find Loss’ treatment compatible with the ideology of perspectival Genuine Branching Realism. If we agree to “provisionally” posit a history operator for the semantic analysis of F, we should not be much more worried with a “provisional” actual history. We should just insist, that from a perspective of a particular point at the tree, there always seems as if there were to be just one “actual” future, but it is a perspective-dependent, rather than an absolute fact. If anything should worry us about Loss’ idea, it is that he endorses supervaluational postsemantics and we have seen that this choice is not free of theoretical costs.

6.3.4 Semantics In section 5.3.4 of the previous chapter, I extensively analyzed a number of problems generated by various versions of the TRL semantics. We begun with the so-called “absolute TRL” (a-trl, see pp. 139, ﬀ.). This theory assumes that there is a single actualized possible history (TRLh) in a branching model and recommends to bind the interpretation of the future operator with this very history.

a-trl 0 0 0 0 a-trl M, m|= Fφ iﬀ ∃m (m < m ∧ m ∈ TRLh ∧ M, m |= φ)

This simple-minded theory is a very close predecessor of the equally simple-minded postsemantic futurism I presented. It is worthwhile then to investigate the a-trl got in trouble and how futurism avoids it. Let me recall the worry was raised by Belnap and Green:

Branching+TRL has the defect that it gives no account of the future tense relative to moments that do not lie on TRL.(Belnap and Green, 1994, p. 379).

10The author himself would not call this extra parameter the history of the context, however, it does the same semantic job—i.e., interprets modal indexical—thus, I think that the diﬀerence is mostly terminological.

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In futurism, we also presuppose that only one of the possible histories gets actualized, but the problem mentioned above does not arise. Futurism is a postsemantic, rather than a semantic theory, so, the semantics of F is not irrevocably tied to the actualized history. The job of the actual future is merely to initialize the semantic index to begin the process of evaluation. From then on, the semantic interpretation goes along the “kosher” Ockhamist rules. To test this procedure on a more complex case, let me recall the example of Belnap and Green

The coin will come up heads. It is possible, though, that it will come up tails, and then later (∗) it will come up tails again (though at that moment it could come up heads), and then, inevitably, still later it will come up tails yet again. (Belnap and Green, 1994, p. 379)

In the formalism of tempo-modal logic, we can encode the sentence as:

F p ∧ ^F(q ∧ ^F p ∧ F(q ∧ Fq)), where p stands for “The coin lands heads” and q for “The coin lands tails.” Belnap and Green notice that the a-trl renders the verdict that the embedded sentence F(q ∧ Fq) is false. So, according to a-trl, it is not possible for the coin to land tails twice in a row which is a strange verdict. To see that postsemantic futurism does not generate this problem, let us inspect a few steps of the computation of truth conditions:

• c||−f F p ∧ ^(F(q ∧ ^F p ∧ F(q ∧ Fq))) iﬀ

• mc/hc |= F p ∧ ^(F(q ∧ ^F p ∧ F(q ∧ Fq))) iﬀ

• mc/hc |= F p and mc/hc |= ^(F(q ∧ ^F p ∧ F(q ∧ Fq))) iﬀ • ...

The sentence is rather complex and so are its truth conditions. I have underlined the parts which are relevant for interpretation of the “problematic” sentence (*): “It will come up tails again.” Futurism dictates that this sentence is true at a possible moment m2 following mc in a possible history h1, if the coin lands tails at m2 and at a some moment m3, following m2 in h1, the coin lands tails once again. Thus, the truth conditions are as they should be. The semantic of Ockhamism oﬀers a handy procedure that dictates how to evaluate the the actually used sentences which describe what is possible and settled. Because futurism uses Ockhamism on the level of semantics proper, none of the semantic objections raised by Nuel Belnap and his collaborators (see section 5.3.4) is going to apply to postsemantic futurism.

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It is crucial that in the procedure above we do not need to decide which of the possible histories available at a possible moment m2 is the “then-actual” history. The interpretation of F at m2 just requires that we follow the possible history h1 that has been evoked by the ﬁrst possibility operator occurring in the sentence. This operator has “redirected” the evaluation process from the default, actualized history to one of the possible histories. However, once redirected, the process proceeds with no ob- structions. Therefore, we do not need to postulate the trl f cn to interpret F-sentences at possible moments. For the entire (post)semantic procedure to work properly, we just need to assume that exactly one of the possible scenarios is actualized by the world’s temporal development.

6.3.5 Postsemantics [Absolute Thin Red Line theory] gives no account whatsoever of predictive speech acts occurring at moments of use that lie oﬀ the TRLabs and is by so much useless. (Belnap et al., 2001, p. 162)

I have already explained in the previous chapter that this objection is signiﬁcantly distinct from the semantic objection discussed above. The problem raised here does not concern semantic interpretation of the future tense operator at moments outside of h@. It concerns postsemantic interpretation of sentences used at non-actual context. Above, I presented how futurism answers the properly semantic question. And what about the postsemantic question? Futurism, as it stands, does not explains which possible history to initialize, when a sentence is used by a non-actual Jack in a non-actual history. I explained, in the last chapter, why an actualist should not be much concerned with this diﬃculty. After all, according to the actualism, there are no people who live outside of the actual world. Therefore, there are no speech acts occurring at non-actual moments. Thus, an actualist can simply refuse to analyze sentences used at non-actual contexts, because there are no non-actual contexts. We can phrase the actualist attitude in the form of a simple requirement

∀mc mc ∈ h@. In fact, the whole idea of postsemantic futurism originated when I studied Belnap et al.’s (2001) semantic objections to different versions of the TRL theory. I realized that if only we assume that every context lies on the TRL, then the problems disappear. I initially suspected that it might be just a formal trick, but when I reflected more closely on the requirement I noticed that it can be naturally justified by modal actualism. Within this metaphysics, it is most natural to expect that the moment of the context should be a part of the actualized course of events. It is just an emanation of the actualist assumption that every speech act happens in the actual world. There are two ways of looking at the thesis that all predictions happen in the actual world. One can be described as “elitism,” and the other as “egalitarianism.” In elitism, we take assume the perspective of the actual history h@ and stubbornly insist that no moment outside of h@ can be reasonably conceived as the moment of the context. I have taken this approach in (Wawer, 2014). My reason was essentially metaphysical: there is only one world, which develops in one way only, and all the speech acts (and

188 CHAPTER 6. BRANCHING ACTUALISM their contexts) are parts of this unique world, or, as I have put it, “speech acts are concrete events and they happen in our world only and our world is not a branching structure” (Wawer, 2014, p. 387). There are no other worlds, “parallel” or “branched,” in which people make (true or false) assertions, so the postsemantic problem does not arise. The resulting formal model might be seen as a revival of the idea which accompanied Saul Kripke (1959; 1963), when he ﬁrst introduced the semantics for modal logic. In the models initially considered by Kripke, in contrast to what is usually called Kripke-models today, each model was equipped with a distinguished world, w∗. This world was meant to represent the actual world (it might be seen as an analog of my actualized “red” possibility TRLh). The distinguished world had a special role to play in semantics. It was a bridge between the theoretically useful notion of truth-at- a-world and the more fundamental notion of truth. In the models of Kripke(1963), a sentence is said to be true (simply true) if and only if it is true-in-the-actual-world. The futurist postsemantics quite faithfully embodies the Kripkean insight: a used sentence is true, if it is true-at-the-present-state-of-the world-in-the-actualized-course-of-events.

The other way of looking at the assumption ∀mc mc ∈ h@ is egalitarian. We accept that an arbitrary moment on the tree can be treated as a moment of the context. An egalitarian is ready to treat any moment as if it was an actual moment (i.e, the moment of the context). However, even within this liberal perspective, the actualist presupposition sneaks in. Yes, you can consider any possible moment as the moment of the context, nonetheless, whenever you do so, you need to assume that the possible moment is actualized. And if it is actualized, then it is a part of the actualized history. Therefore, one of the possible histories passing through this possible moment needs to be the actualized history h@. It is unintelligible to assume that the actualized history will be “missing” at a possible moment of the context, since it would be to assume that an utterance takes place “outside” of the world. Kaplan have argued that both approaches are equally legitimate. We study the same phenomenon from two distinct perspectives. He observed that in a typical model of modal logic (including the models he himself introduced), there are many possible worlds, seemingly on a par, and the truth value of a sentence varies from one world to another. However, he claims that we cannot conclude that truth is a relative notion that varies from world to world, since “truth, absolute truth in a model, is assessed at the ‘designated’ world” (p. 595). For Kaplan, the designated world is simply the actual world, which we can study “either in its guise as ‘world of the context of use’ or in its guise as ‘designated world.’ ” (p. 595). Thus, according to Kaplan, a sentence is (absolutely) true iﬀ it is true at the actual world, i.e., the designated world or the world of context of use. No matter which perspective you have—elitist or egalitarian—it is guaranteed that every prediction occurs in the actual world, i.e., mc ∈ hc. Hence Belnap et al.’s (2001) problem of predictions made at non-actual contexts does not arise. Importantly, one should never combine both these perspectives into a single theory. That is, one should never at the same time assume that one particular history represents the actual course of events (elitism), and then consider a possible moment outside of this history as moments of contexts (egalitarianism). Then, one either assumes a contradiction: that some moment is both actual and non-actual, or one ends up with a distorted version of the Thin Red Line, so suggestively criticized by Belnap et al.(2001).

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A postsemantic trl f cn theory presented by MacFarlane(2014) might be seen as an attempt to combine both these perspectives into a single picture. Thus, it should not be surprising that it fails. I have described MacFarlane’s problem in section 5.3.5.1, so let me just restate the main point. We are asked to consider a person, at m2, who is not in the actual future of her past. Then,

This fragment clearly indicates what goes wrong with the blend of “elitist” and the “egalitarian” perspectives. We are ﬁrst asked to assume that there is a distinguished history h1 in a model (elitist perspective). In the next step, we are asked to imagine someone, say Jill, at a possible moment m2, who makes an assessment of an earlier prediction (egalitarian perspective). It is crucial for the objection, however, that while we consider Jill who makes the assessment, we do not assume that her act takes place in the actual world. Then, we arrive at a peculiar situation, where an imaginary Jill, who experiences an imaginary rain is asked to assess a non-imaginary utterance that happened on the previous day. Moreover, when the imaginary Jill wonders whether the previous act was accurate, she needs to consult the real world, rather than the imaginary world she lives in. Well, it is no surprise that such a thought experiment produces rather controversial consequences. Observe that the diﬃculty would never arise if we stuck to one of the actualist perspectives. When we have the elitist attitude and we are asked to consider Jill who experiences rain, we need to object: there is no such Jill. The only Jill there is, is the Jill who lives in the actual world and experiences the sunny day. When we take the egalitarian stance and we are asked to take Jill who experiences rain, we need to forget that it is actually sunny and provisionally assume that the actual world is the rainy world. Then, Jill’s assessments depends on what is posited as actual, not on what is really actual. In particular, if we assume that the “rainy” possibility got actualized, then the prediction made on the previous day by Jack is inaccurate. Only if we mix the two perspectives, we run into trouble. I do not want to conclude, though, that trl f cn-postsemantics is fundamentally mis- conceived. I think it can be put to a fruitful philosophical use, but it has to be somewhat reconsidered. Observe that failure of trl f cn-postsemantics results from the fact that some contexts are not in the future of their past. This situation is as peculiar as the situation in which a context of utterance is not a part of the actual world. An actualist has every right to exclude both these options as conceptually unsound. Thus, just as a

“plain” futurist has the right to assume that ∀mc mc ∈ hc, so the “extended” futurist has a right to assume that

∀mc ∀m

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This requirement enforces that what happens now is in the actual future of the actual past. Already Thomason and Gupta(1980), who used trl f cn in their semantics (they call them future choice functions and symbolize by Fs), realized that such restriction is required. [W]e restrict ourselves to choice functions meeting a certain requirement: F must be normal at i, in the following sense. (3.2) F is normal at i iﬀ for all j < i, F j = Fi (. . . ) If F is normal at i, F treats i as “actual” from the point of view of moments in the past of i.(Thomason and Gupta, 1980, p. 79) Thanks to normalization, we can avoid MacFarlane’s postsemantic problem. When- ever we consider the context in which Jill assesses Jack’s previous utterance, we are forced to assume that this context is in the actual future of its past. When we restrict ourselves to trl f cn normal at mc, we can postulate the following postsemantics:

f + Definition 6.8 (Extended futurism). c||− φ iff mc/trl f cn(mc) |= φ. This postsemantic theory resembles the plain futurism I advocate, but is less parsimonious. Not only is the context rich enough to initialize the unique history which gets actualized, but it also initializes the unique possibility that would have been actualized at any possible moment. A parallel extension of futurism was considered in Christian theology. While William of Ockham argued that God knows the free choices that will be made, Luis Molina went further and postulated that God also knows which free choices would be made in counterfactual circumstances (for more details see Øhrstrøm and Hasle, 2011). As far as semantics of Ockhamism is concerned, the extra information is completely redundant. None of the operators resorts to trl f cn. However, it is easy to see that this information could come in handy for semantics of counterfactuals. After all, trl f cn dictates what would have been the case, had something else been the case (in fact, Thomason and Gupta used Fs for this exact purpose). I want to stress, however, that plain futurism is not by itself committed to the claim that counterfactual scenarios are also initialized. One can consistently believe that the world in which the sentence is used decides which of the possibilities is actualized, but it does not decide which of the possibilities would be actualized. In semantic vocabulary, it means that one may believe that “what will happen” can ground truth values of sentences about the future, but “what would happen” cannot ground truth values of sentences about counterfactual futures. It also means that a biconditional like “There will be a sea battle” is true iff there will be a sea battle, is legitimate while the biconditional “Had there been a sea battle, the Greeks would have won” is true iff had there been a sea battle, the Greeks would have won, is illegitimate. In (Malpass and Wawer, 2012), we argued for distinct semantic treatment of these two cases. I am not going to argue for this point here, I just want to stress that “plain” futurism that allows for truth of future contingents is not required to accept

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“plain” truth of contingent counterfactuals (in: Wawer and Wronski´ , 2015, we propose a theory in which contingent counterfactuals are neither true nor false at a context; this theory can be easily combined with futurism).

6.3.6 Possible predictions We have no way of getting a grip on “Had things gone otherwise, Jack would have asserted the following: ‘It will (eventually) rain.’ ” Given the context of Jack’s assertion, the TRL is no longer able to guide us in understanding his reference to his future. (Belnap et al., 2001, p. 162)

The objection might be understood as a postsemantic request to analyze sentences used at non-actual context. If that is the authors’ demand, it is unreasonable. A faithful actualist believes that there are no contexts outside of the actual world (hence every moment of the context is a part of the history of the context, i.e., of the actualized history h@). However, one can understand this objection in more moderate terms. The weaker challenge is to answer if the sentence that Jack would have said, would be true. This task can be divided into two separate sub-tasks. Firstly, it requires some analysis of counterfactuals, and secondly, an explanation of truth ascription to reported utterances embedded in scope of modal and temporal connectives. Belnap et al.(2001) do not oﬀer any semantics of counterfactuals, so it would be unfair of them to expect one from their opponents. Therefore, I will focus on the latter demand, and provide an actualist friendly explanation of truth ascription to reported predictions. It means that I will provide a formal analysis for sentences like:

1. Had there been a sea battle, Themistocles would have truly said, “The Greek ﬂeet will win.”

2. Themistocles could have truly said, “The Greek ﬂeet will win.”

The task is potentially problematic for an actualist. To give an appropriate semantic treatment of such examples of direct speech, Belnap et al.(2001, p. 174) and Belnap (2002b, p. 44) devised the operators which shift the moment of the context.11 However, actualists believes that the moment of the context needs to be a part of the actualized history, so they might ﬁnd Belnap’s procedure controversial. I will oﬀer two strategies to remedy this concern. Generally speaking, I think that truth ascription to a reported utterance embedded in a modal context is a much easier task than it might appear. It almost entirely parasites on the semantics of modal connectives that precede the report. For example, let us consider the sentence:

(A) Themistocles could have truly uttered, “The Greek ﬂeet will win.”

It is true in exactly the same circumstances as the sentence:

11The operators they propose are “α asserts ‘A’” and “Truly utters(α, β, ‘A’),” respectively.

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(A’) It could have been that (Themistocles utters “The Greek fleet will win” and it is true that the Greek fleet will win). Observe that the later sentence is by no means problematic for a futurist who believes in actual contexts only. (A’) is true in an actual context iff there was a really possible scenario in which Themistocles’ utterance was followed by a Greek victory. The analysis merely presupposes that a futurist is able to analyze the embedded modal and temporal connectives, and we have seen in section 6.3.4 that it can be easily accomplished. In general, the truth ascription to speech reports is very simple:

Definition 6.9. mc/hc, m/h |= α-truly-utters-‘φ’ iff mc/hc, m/h |= α-utters-‘φ’ and mc/hc, m/h |= Trφ. To simplify the matter, I assume that α’s language is a part of the metalanguage and that it does not contain the vocabulary which could generate semantic paradoxes. Let me restate the definition of the truth operator

mc/hc, m/h |= Trφ iﬀ mc/hc, m/h |= φ. We can apply this simple procedure to Belnap et al.’s (2001) example and identify the two: (B) Had things gone otherwise, Jack would have truly uttered “It will rain.” (B’) Had things gone otherwise, it would have been that (Jack utters “It will rain” and it is true that it will rain). The analysis of (B’) poses no problem for an actualist (who have decided for some treatment of counterfactuals available in the literature). She just needs to check if it is actually true that if things had gone otherwise, Jack’s act of assertion would have been followed by rain. She does not need to shift the context into non-actual circumstances for that purpose. So, against Belnap et al.’s (2001) worry, we have a perfect grip on Jack’s possible assertion, even if “TRL is no longer able to guide us in understanding his reference to his future.” The crucial point is that we do not need the actual history to interpret Jack’s utterance, because the relevant history has already been provided by an appropriate modal operator. There is a big diﬀerence between sentences that are used and sentences that could have been used. With respect to the actually used sentences, we need to initialize the history parameter to begin the process of semantic evaluation. In contrast, if we consider possible utterances, they are typically introduced by some sort of modal operator, such as “imagine that” or “let us assume that” or “it could have been that.”12 Then, the actualized history is no longer required, since the modal operator provides the possible scenario which is relevant for semantic evaluation and we do not need to decide which of the possible histories is the history that would have been actualized. To see how it works in practice, let us formally analyze the sentence

12Sometimes, the modal operator is not explicit, but then the context clearly indicates that we are asked to interpret the sentence in a posited, non-actual scenario. For example, if we play a game of Warhammer and the wizard says “You will die during the quest,” it is clear that his utterance is to be interpreted with respect to the imaginary world of the game.

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(A) Themistocles could have truly uttered: “The Greek ﬂeet will win.”

Let us encode it as P^F(Themistocles-truly-utters-‘F p’), where p stands for “The Greek ﬂeet wins.” The computation of (A) goes as follows:

1. c||−f P^F(Themistocles-truly-utters-‘F p’) iﬀ

2. mc/hc, mc/hc |= P^F(Themistocles-truly-utters-‘F p’) iﬀ

3. mc/hc, mc/hc |= P^F(Themistocles-utters-‘F p’ ∧ TrF p) iﬀ

4. ∃m

Thus the sentence, “Themistocles could have truly uttered ‘The Greek fleet will win,’ ” is true in an actual context if and only if there is a possible history which branched from the actualized history in the past, such that in that history Themistocles utters “The Greek fleet will win” and the Greeks later win. These are the desirable truth conditions of the sentence. They imply that if the outcome of the sea battle was contingent, then it is true that Themistocles could have truly uttered “The Greek fleet will win” (it is also true that Themistocles could have truly uttered “The Persian fleet will win”). As I explained, the relevant non-actual history h is provided by the modal operator ^, so we do not need to decide which of the possible scenarios following Themistocles’ possible utterance would be actualized. Thus, we can decide whether a sentence which would have been used could have been true without shifting the context of utterance. Thus, Belnap et al.’s (2001) challenge can be addressed—there is no conflict between the idea of a single actualized scenario and the claim that people could have uttered true or false sentences in possible circumstances. Actually, there is a complication which needs to be tackled. In the process described above, I simply checked if the sentence used by Themistocles is true in the possible circumstances. The procedure might fail, as our language contains indexical operators: Now and @. To see what I mean, imagine that during the battle the Per- sians tried to enclose the Greek fleet, Themistocles noticed it and said “The Persians are trying to set a trap now” On the following day, Aristides recounts the event

(C) Themistocles truly said “The Persians are trying to set a trap now.” My procedure dictates that (C) is true iﬀ it was the case that (Themistocles says “The Persians are trying to set a trap now” and it is true that the Persians are trying to set a trap now). But this is clearly wrong. The Persians are not trying to set a trap now (the

194 CHAPTER 6. BRANCHING ACTUALISM battle is over) and it is irrelevant for the truth of (C). What is relevant is whether the Persians tried to set a trap then, i.e., at the moment when the utterance was made. A similar problem can be raised for the other indexical. Consider a sentence: (D) If the Athenians did not send their ﬂeet, then Eurybiades would truly utter “The Persians will actually win.” What is relevant for the truth of the mentioned sentence is whether the Persians would then actually win, not whether they will actually win. An easy way to formally incorporate this observation is to follow the suggestion of Belnap(2002b) and redeﬁne the truth operator:

mc/hc, m/h |= Trφ iﬀ m/h, m/h |= φ

Then, everything works as it should. The moment and the history of the context have been replaced with the moment and the history at which the truth of the sentence is evaluated, so any indexical expressions occurring at φ will be interpreted with respect to the “new” context, rather than the actual context. However, this procedure might be questionable for puritan actualists. After all, it requires that we consider a possible moment m and a possible history h to be the moment and the history of the context, which suggests that there are non-actual contexts—an idea which actualists are reluctant to accept. There are at least two responses to the problem. Firstly, one can take an “egalitarian” stance and accept Belnap’s deﬁnition, but deny that involves any metaphysical commitment. One can insist that the shift of the context-parameters induced by the truth operator is harmless. In introduces a provisional, “as if” context, technically useful for interpretation of speech reports. An actualist can underscore that all contexts initialize one and the same history—the unique history which is actualized. Another history can become a history of the context only if it is explicitly introduced by a modal operator and a truth operator. So, non-actual histories and moments of the context should not suggest that there are non-actual contexts. Alternatively, one can take an “elitist” strategy and insist on keeping the original context parameters intact. Then, one can explicitly incorporate additional “quasi- context” parameters into the semantic index and use them for interpreting of the reported indexical expressions.13 I ﬁnd the solution quite convincing psychologically. When we hear a speech report we typically keep, in the back of our heads, the circumstances in which the sentence was used (the speaker, the time, the place, the possibility, etc.) and employ the information to interpret the indexicals used in the quoted material. The process partially surfaces in the natural language, when we translate direct to indirect speech. Observe that we can say either: John said yesterday, “I am hungry now.” or John said yesterday that he was hungry then.

13My strategy is inspired by the semantics proposed by Max Cresswell(1990). He devised his apparatus to analyze anaphoric usage of temporal and modal indexicals.

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The indexical expressions are syntactically transformed and semantically “redirected.” Their semantic value no longer depends on the actual context of the sentence, but on the “deferred” context, i.e., on the circumstances in which the reported utterance take place. Therefore, the procedure I propose might be seen as an improvement on Belnap et al.’s (2001) account. They remark that:

We might be thought to have made a “monster” (§6B.2) out of assertion, because the moment-of-use context parameter is no longer immobile. (. . . ) The monsterhood arises out of any treatment of direct discourse, and would disappear if we kept to indirect discourse, but we would not advise that course. As long as we are being explicit about direct discourse rather than “sneaking in a quotation device” (Kaplan, 1989, p. 511), all seems well; and in any case, better a monster than a fog. (Belnap et al., 2001, p. 174, n. 20)

The alternative treatment I propose does not require the shift of the context. My attempt might be seen as a partial analysis of reported speech. To convey my idea, I extend the index to mc/hc, ms/hs, m/h, where ms/hs represents the additional “quasi- context.” Then, I introduce the new pair of “quasi-indexicals,” Then and then-@, to our language: Deﬁnition 6.10 (Anaphoric indexicals).

| | mc/hc, ms/hs, m/h = Thenφ iﬀ mc/hc, ms/hs, mims /h = φ;

mc/hc, ms/hs, m/h |= then-@φ iff mc/hc, ms/hs, ms/hs |= φ. Thus, the definition of Then and then-@ are strictly analogous to the definitions of Now and @. The only difference is that the relevant moment and history of the “context” is ms and hs, rather than mc and hc. Thanks to this maneuver, we can interpret the occurrences of Now and @ in the uttered sentence with respect to what would then be present and actual. To accomplish this technically, we need to first replace every occurrence of Now and @ in the quoted sentence with Then and then-@, respectively. Let φs refer to the result of such substitution within the sentence φ. Finally, we require that the speech report “stores” the possible circumstances in which the speech act takes place as the quasi-context.14

Definition 6.11. mc/hc, ms/hs, m/h |= α-truly-utters-‘φ’ iff mc/hc, ms/hs, m/h |= α-utters-‘φ’ and mc/hc, m/h, m/h |= Trφs. Two things happen at the same time. Firstly, the semantic parameters m/h is stored in place of ms/hs and, secondly, φ is translated to φs, which means that all indexicals in the quoted sentence are appropriately modified to consult the stored quasi-context. Since the semantics is slightly changed, we need to appropriately modify the postsemantic convention:

f c||−φ iﬀ mc/hc, mc/hc, mc/hc |= φ.

14I suppose that relative pronouns like “that” or “which” naturally store the quasi-context.

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At the beginning of the process of evaluation, we identify the quasi-context parameter with the actual context parameter. It can be shifted only by some sort of “storing” operator. So equipped, we are ready to understand the truth ascription to possible predictions in full generality. To exercise the mechanism of the new connectives, let me present the last few lines of the computation of semantic value:

Themistocles could have truly uttered, “Now, the Persians hold sway, but actually the Greeks will win,” (P^F(Themistocles-truly-utters‘Nowp ∧ @Fq’)

1. c P^F(Themistocles-truly-utters-‘Nowp ∧ @Fq’) iﬀ

2. mc/hc, mc/hc, mc/hc |= P^F(Themistocles-truly-utters-‘Nowp ∧ @Fq’) 3.i ﬀ . . . iﬀ

0 0 4. ∃m∃h∃m0 (m < mc & h ∈ Hm & m > m & m ∈ h & 0 mc/hc, mc/hc, m /h |= Themistocles-truly-utters-‘Nowp ∧ @Fq’) iﬀ

0 0 5. ∃m∃h∃m0 (m < mc & h ∈ Hm & m > m & m ∈ h & 0 mc/hc, mc/hc, m /h |= Themistocles-utters-‘Nowp ∧ @Fq’ & 0 0 mc/hc, m /h, m /h |= Tr(Thenp ∧ then-@Fq))

To sum up, Themistocles could have truly uttered “Now, the Persians hold sway, but actually the Greeks will win” iﬀ

• There is a possible history h and a moment m0 in that history such that at m0 Themistocles utters: “Now, the Persians hold sway, but actually the Greeks will win”;

• The Persians hold sway at m0 in h;

• There is a moment m00, later than m0, in that history (i.e., in the history h introduced by the possibility operator) such that the Greeks win at m00.

The sentence is complex and so are its truth conditions. However, they turn out to be as expected. Additionally, the introduction of the new operators might be independently motivated. For example, they can be used for Cresswell-style analysis of anaphoric indexicals of the type:

Three years ago, I was about to become very happy, but I was miserable then.(P3Re f F(happy ∧ Then(miserable))

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Where Re f is an operator which stores a semantic index as a quasi-context. More interestingly for our purposes, the operators can be used to analyze a sentence like:

Had I tossed that coin, it could actually land heads and it could actually land tails. (p > (^Re f Fthen-@q ∧ ^Re f Fthen-@r))

Thus, we can represent the idea that every possibility could be actual. I also believe that we can use these operators to replace double-time reference in Belnap-style analysis of speech acts (see Belnap, 2002b) or to analyze historical counterfactuals in terms of purely temporal operators and standard historical modalities. I will not develop these ideas here, but they help to motivate quasi-context and demonstrate that it is not and ad hoc addition, introduced merely to solve the problem of possible predictions. The crucial point here is that we are able to ascribe truth values to possible utterances without shifting the context parameters. My procedure generates some theoretical costs; in particular, we need to incorporate two more parameters into the semantic index. But the mechanism is sound and precise. We can understand speech reports without context-shifting devices. Thus, we do not need a monster to dispel the fog.

6.4 Branching possibilities 6.4.1 Are actualist possibilities suﬃciently real? Futurism presupposes the actualist notion that the world develops linearly. There is a continuous line of distrust, however, towards the idea among the branching theorists. It has often been objected that it does not square easily with branching representation of indeterminism. There are many expressions of the distrust. Let me invoke several representative quotes:

If the determinist sees Time as a line, the indeterminist sees it as a system of forking paths. (Burgess, 1978, p. 159)

In this fragment (repeated in other works of the author), it is simply presupposed that the assumption of linear succession of events is tantamount to determinism. The claim is extended by Thomason:

The linear conception of time can countenance “alternative futures” as epistemic possibilities. (. . . ) For if a time α can have only one “real” future, times located in other alternative futures cannot really bear any temporal relation to α. They can bear an epistemic relation, being futures for a situation which for all we know is the actual one α.(Thomason, 1970, pp. 265, 270–271)

Thomason adds a characteristic objection that under the assumption that the before- after relation is linear, the possible courses of events are no longer metaphysically, but only epistemically possible; possible relative to our limited knowledge. A similar worry is raised by McArthur:

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[I]n our view, allowing any state to already be marked as (. . . ) that state which is (atemporally) actual, reintroduces the linear conception, because it denies that the other states are real alternatives. That is to say, under such a theory the additional alternatives become mere logical possibilities with no ontological claims whatever. (McArthur, 1974, p. 284–285).

The same intuition is revoked by Belnap and Green(1994). They define “open future” as a notion which excludes the idea of a single actual course of events. We shall call the view that in spite of indeterminism, one neither needs nor can use a Thin Red Line, the doctrine of the open future.(Belnap and Green, 1994, pp. 367) The crusade against the concept of a single actualized possibility, which began in (Belnap and Green, 1994), was continued in (Belnap et al., 2001), where the assort- ments of countering arguments was considerably extended. Partly due Belnap’s efforts, the current orthodoxy has it that “open future” is not compatible with the idea that only one of possible histories gets actualized.15 An influential author who revives the distrust towards the single, actualized history is John MacFarlane: Like Belnap and Green, I hold that positing a thin red line amounts to giving up objective indeterminism. The non-red branches in the tree are supposed to represent objectively possible futures, but their non-redness indicates precisely that they will not be the continuations of the history that include the utterance in question. (. . . ) In what sense, then, are the others really “possibilities”? They are possible in an epistemic sense: the utterer does not know which history is marked out with the thin red line. But objectively speaking they are not genuine possibilities at all. (MacFarlane, 2003, p. 325) There is a relatively easy way to deflect this kind of worries. The way is comparable to the Branching Realists’ solution to the problem of “multiple” futures. Remember that Lewis objected against Genuine Branching Realism, arguing that if worlds/histories branch, then “it will be both ways.” To answer this objection, many Branching Realist turn to logic. They propose a semantics (usually, the Ockhamist semantics) which renders the sentence F1 p ∧ F1¬p a counter-tautology. Since the sentence F1 p ∧ F1¬p is never true, they are no longer concerned about Lewis’ objection (see section 2.1, pp.9 ff. for an extended discussion). An actualist can use the very same strategy to show that linear time does not imply determinism. It is sufficient to propose a semantics which presupposes the idea that only one of the possible futures will be actualized, but one allowing for a contingent

15Todd(2015a) is a good example of an author who rea ﬃrms Belnap and Green’s view: The guiding thought behind the open future view . . . is this: there is, given indeterminism, no further way to narrow down the set of causally possible futures to a unique actual future; there is, then, on indeterminism, no unique actual future of the sort just speciﬁed. (Todd, 2015a, p. 12)

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future. In other words, we need a semantics which makes either F1φ, or F1¬φ true, but which falsifies F1 p → F1 p. We do not need to look very far for this kind of theory. Observe that in futurism, we assume that only one of the histories is actualized (the history of the context), but the sentence F p → F p is not valid. It might well be that F p is true, when ^¬F p is also true. Thus, if the realists were happy with a semantic response to the problem of many futures, they should be equally satisfied with a semantic response to the problem of necessary future. I am doubtful, however, whether they would by satisfied with the semantic line of response, just as I was unhappy with the semantic response of the Branching Realists (see sec. 2.1). Let us then look to the heart of their worry. Observe first that MacFarlane makes a surprising inference in this passage. He assumes that non-red histories “will not be the continuations” of the utterance and concludes, based on this assumption, that “they are not genuine possibilities.” Thus, he accepts the following implication: if a branch will not be actualized, it is not genuinely possible. Consider, however, a contraposition of his implication: if a branch is genuinely possible, it will be actualized. This principle is clearly wrong. Even if it is genuinely possible that the coin will land heads and that it will land tails, we should not conclude that it will actually land heads and it will actually land tails. MacFarlane’s line of reasoning, therefore, is not entirely convincing. Nonetheless, we have also seen Thomason infer that if only one of the possible futures will really happen, then other possibilities are not genuine. Therefore, I believe that there is a hidden premise which justifies the inference. I think that the inference gets much more intelligible if we assume Branching Realism. In this setting, a possibility is a genuine continuation of an event e iff there is a course of events that contains event e as its part. Then, when an actualists claims that such an event is a part of just one course of events, the realist naturally reads as the claim that there is only one real possibility. And if there is only one real possibility, then the world is ontologically deterministic. So, when actualists use branching structures, Genuine Realists surmise that they apply them to represent something like epistemic, doxastic, or logical possibilities. Actualists evade the difficulty by rejecting the assumption that events take place in many possible worlds (histories). An event takes place in no possible world. It takes place in a unique, actual world which is distinct from any of the possibilities and which actualizes one of them. So, when an actualist says that an event takes place in just one world, she does not mean to suggest that it takes place in just one possible world. The fact that an event is a part of just one world says nothing about the modal status of the world. The actualist can agree with Thomason that none of the possible results of a coin toss is temporally related to the event of the coin toss. However, she does not derive from this argument the conclusion none of these results is really possible. There is a clear and uncontroversial sense, indeed, in which the linear ordering does imply determinism. After all, the linear structure is an extreme case of a branching structure. Remember that according to actualism, the branching structure represents all the possible ways in which the world may develop. If all the possible ways in which the world may develop were accurately depicted by a linear structure, the world would be deterministic.

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However, such a reading of the actualist’s thesis is extremely uncharitable. Remem- ber that in actualism, the linear ordering16 describes how the world in fact develops. Reading the actualist thesis as a claim about the structure of possibilities is just as uncharitable as a depiction of a branching structure as the actually branching spacetime (i.e., as Naïve Branching Realism). Just as the latter makes “branching time look silly in a way that it surely isn’t silly,” (Belnap et al., 2001, p. 206), the former makes linear time look silly in a way that it surely is not silly. We need to remember that according to the actualists, the world is not identiﬁed with its possibilities. Thus, even when they claim that the events have a linear temporal structure, they can grant at once that the possible ways in which the world can develop do branch. A similar point has once been made by Bertrand Russell. He considered a deﬁnition of determinism according to which

A system is said to be “deterministic” when, given data, e1, e2,..., en at times t1, t2,..., tn respectively, concerning this system, if Et is the state of the system at any time t, there is a functional relation of the form

17 Et = f (e1, t1, e2, t2,..., en, tn, t). (Russell, 1953, p. 398)

At first, it might seem as a reasonable definition, especially if we think about f as a law of nature and about e1,..., en as the initial conditions which are supposed to jointly “fix” the later development. Russell proceeds to demonstrate, however, that the definition is in fact inadequate. In his demonstration, he implicitly assumes that: (A) For every time t, the material universe is in exactly one state at t. This seemingly innocent assumption, nevertheless, implies that “theoretically, the whole state of the material universe at time t must be capable of being exhibited as a function of t”(Russell, 1953, p. 401). After all, the function requirement is satisfied: one state is assigned to every moment in time. Let g be a function assigning to each time t, the state of the universe at t. We can easily transform g to a three-arguments function Et

Et(e1, t1, t) = g(t), where e1 is any piece of data concerning the system at t1. But then, function Et satisfies Russell’s definition, which implies that “our universe will be deterministic in the sense defined above” (p. 401). Observe, however, that assumption (A) is just a different ren- dering of the actualist assumption. Hence, the apparently innocent actualist assumption (A) leads to deterministic conclusions in just a few quick steps. Nonetheless, Russell did not abandon actualism. He concedes that he presents argumentum ad absurdum, but identifying as absurdity not the existence of the actualist function g, but the definition of determinism he initially proposed. He realized that if the mere existence of the appropriate function implies that the world is deterministic, then “no information is conveyed about the universe in stating that it is deterministic”

16Or a suitable modification of the thesis incorporating the relativity theory. 17Let me mention, as a piece of trivia, that when John Earman(1986, p. 10) quotes the definition, he skips the last argument t of the function f . It seems a trifle omission, but almost every time I discuss Earman’s text with the students, one of them accuses Russell of formal sloppiness or even a straightforward mistake.

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(p. 401). So, according to Russell, the fact that the temporal evolution of the world is linear, should not suggest the necessity of everything that happens. I agree with Russell’s diagnosis. I also believe that the notion of the universe being in one state at each time has little to do with the universe being deterministic. Observe that among the abstract objects, there are the functions which depict all the really possible ways in which the world can develop in time (you may think that these functions form a tree of physical possibilities). However, given the assumption (A), necessarily only one of these functions “gets things right.” We can think about the unique function as a depiction of the Thin Red Line, the unique actualized possibility. When we present the issue in this light, it is also visible, contrary to what Branch- ing Realists may suggest, that the actualized possibility is not metaphysically distinguished. It is metaphysically on a par with all the other possibilities. The single possibility which accurately captures the world is “made of the same stuff” as the possibilities that misrepresent it. In this sense, no possibility is “privileged,” “distinguished,” or “more real” in comparison to the other possibilities. The concrete world, on the other hand, is metaphysically distinct from all the possibilities. It is the object which actualizes one of the possibilities. The realists react to this line of reasoning with a claim that if alternative possibilities are not the same kind of things as the events that surround us, they are not “real enough.” Such possibilities might model logical, epistemic, doxastic, or other possibilities, but they are not “stuffy” enough to give the metaphysical indeterminism its proper due. This intuition is not specific to Branching Realism, it is inherent to Gen- uine Modal Realism, more generally. It was shared by its unquestionable champion, David Lewis, who, reflecting on the actualist accounts of possible worlds, concluded that: The actualized ersatz world is special, since it alone represents the one concrete world. And it is special not just from its own standpoint, but from the standpoint of any world. So it is noncontingently special, since contingency is variation from world to world. But it is part of the theory that the actualized ersatz world is the special one. So it seems to turn out to be a noncontingent matter which of the ersatz worlds is actualized. That is wrong, and needs explaining away. (Lewis, 1979, p. 533) Lewis thus shared the view that if we assume that only one of the possibilities is actualized, then other possibilities not only are not actualized but also, in an important sense, cannot be actualized.18 He later argues that only Genuine Realism is The Realism about possibility. Genuine Branching Realists seem to share his adamant conviction—that only their notion of possibility is good enough to capture the Real Indeterminism. They presuppose that any actualist account of possibility would be unsuccessful as an account of real possibility. However, as far as I can see, little positive argument has been given to support this conviction. Interestingly, David Lewis partially withdrew his accusation towards actualism.

18I think that Lewis’ objection rests on a confusion of “egalitarian” and “elitist” reading of actualism that I discussed in section 6.3.5, pp. 188 ﬀ. Observe that Lewis asks us to consider a standpoint of the world w1 and to simultaneously assume that another world, w2, is the actual world. No actualist should accept the kind of thought-experiment.

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In “Attitudes De Dicto and De Se” I objected that the ersatzer seemed to make it a non-contingent matter which ersatz world is actualized. I had not fully appreciated the diﬀerence between this theoretical setting and my own. I thank John G. Bennett for setting me right. (Lewis, 1986, p. 139, n. 2)

I share Lewis’ belief that when we fully appreciate the diﬀerence between Branch- ing Actualism and Branching Realism, it will also become clear that it is an entirely contingent matter which possibility is actualized.

6.4.2 Are genuine possibilities suﬃciently real? Undoubtedly, Branching Realism has some advantages. For example, we can appeal to the concrete events constituting Our World to ground modal truths. This feature made David Lewis think of himself as the real modal realist, in contradistinction to ersatz modal realists who substituted diﬀerent entities for the concrete worlds of Lewis. According to Lewis, actualists want “paradise on the cheap” (Lewis, 1986, p. 136). Interestingly enough, this very feature of Genuine Realism made some philosophers skeptical about Lewis’ project. They thought that the entities that Lewis proposes are, so to say, “too concrete” to play the role of modality. This point was particularly strongly stressed by Alvin Plantinga, who called David Lewis, slightly perversely, a paradigm example of modal anti-realist and reductionist:

I shall argue that Lewis is a modal realist and/or a realist about possible worlds in approximately the sense in which William of Ockham is a realist about universals: namely, not at all. (Plantinga, 1987, p. 189)

Plantinga is alarmed by the general idea of identiﬁcation of possibility with a collection of concrete objects:

First, Lewis is a modal reductionist: He offers reductive analyses of the phenomena of modality: he reduces possible worlds to maximal objects. (Plantinga, 1987, p. 213) and elsewhere, he comments on Lewis: On his theory, as I see it, there are no propositions, states of affairs, possible worlds, essences or objects with essential and accidental properties; what there are instead are concrete objects and set theoretical constructions on them. (Plantinga, 1987, p. 213) If the feature of Lewis’ theory is desirable or not calls for discussion. Lewis himself would be happy to admit that his theory is the only one which offers an entirely non- modal analysis of modal concepts. He accounts for modality in terms of entities which are traditionally more acceptable: concrete objects and sets. As a faithful student of Quine, he offers a “desert landscape” in place of a metaphysical jungle. The desert which Lewis presents is radically more extensive than the dessert of the traditional empiricists (e.g., there are infinitely many donkeys and flying pigs on Lewis’ desert).

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This feature of Lewis theory did raise many eyebrows, but it is fair to admit that he does not need to postulate propositions, properties, essences, or causes as the primitive elements of his ontology, which is a spectacular achievements. Regardless, Plantinga’s problem with Lewis’ analysis is not only that it is a reductive account of modality, but also that his analysis is simply wrong! He gives at least two reasons for his critical assessment. Firstly, he observes that a “possible world/history” is a technical term which can be explicated more naturally as a “way things could have been.” Then, he argues that these are akin to states of aﬀairs, properties, or propositions, or a set of propositions, but not to (maximal) concrete object. For example, the world we live in is not a-way-things-are. The latter is not the world, but a representation of the things in the world and relations among them. Hence, the crucial feature of a possible world, according to Plantinga, is that it needs to have the capacity to “describe.” It means that a possible world should posses certain representational capacities, so the things can be such-and-such according to a possible world.

[A] possible world represents things as being a certain way. But no concrete object or set theoretic construction does a thing like that. (Plantinga, 1987, p. 212)19

Secondly, Plantinga believes that existence or non-existence of the concrete maximal objects that Lewis postulates is entirely independent and irrelevant to the truth of modal claims:

There are objects that have properties contingently and propositions that are contingent; and that is true no matter how many maximal objects there are. I have the property of wearing shoes accidentally; the proposition Paul is over six feet tall is contingent; and this is so even if, as most of us believe, there is only one maximal object. So possible worlds can’t be maximal objects. Lewis’ theory, then, is not a realism with respect to possible worlds. (Plantinga, 1987, p. 212)

This particular belief was shared by another actualist, Saul Kripke: But when we talk in school of thirty-six possibilities (i.e., possible ways a pair of dice may land), in no way do we need to posit that there are some thirty-five other entities, existent in some never-never land, corresponding to the physical object before me. (Kripke, 1980, p. 17) I recapitulated Plantinga’s criticism of Lewis to shed a different light on the discussion of Branching Realism and draw attention to an interesting analogy. Branching realists join David Lewis in their firm conviction that the only kind of thing that is real enough to give account of real possibility and ontic indeterminism is a collection of concrete courses of events. Any account of possibility which postulates just one concrete course of events is dubbed “epistemic,” “logical,” or “linguistic.” Therefore, Branching Realists attack actualism in a fashion analogous to Lewis’ attacks on modal actualism in general.

19One could argue that thanks to their representational nature, possible worlds can serve as objects of propositional attitudes, as is sometimes assumed.

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This is a game that two can play, however. Instead of parrying the realists’ attack, Branching Actualists can counterattack with arguments borrowed from modal actualists. Let us then look at Branching Realism in the “Plantingian” manner. From this perspective, capturing possibilities in terms of the existence of a plethora of overlapping concrete courses of events, as the realists do, is an entirely reductionist approach to the real possibility. No concrete entity can play the role of possibility. Therefore, the realists get rid of possibilities and replace them with concrete entities connected by causal relations. We can follow Plantinga’s criticism further and assert that not only is the realist project reductionist, but it is also an inaccurate account of historical possibility. First, we can repeat his first argument and claim that no concrete event or series of events can stand for a-way-things-could-have-been, since “ways” have representational capacities which concrete objects lack. The other sort of criticism, Kripkean in nature, might be illustrated by the example: the mere fact that Donald Trump (as the realists would have us to believe) is an inhabitant of two overlapping spacetimes and runs for the President of the U.S. in one of them and does not run in the other, seems to be completely irrelevant to the fact that he could have chosen not to engage in politics. Furthermore, it seems that to understand the decision that Donald Trump faced, in no way do we need to posit that there are other actions of Trump, existing in other parts of Belnap’s Our World, analogous to the actions he has in fact taken. Thus, the numerous courses of events seem not to be directly connected to the modalities. Therefore, it seems that the Genuine Branching Realist approach to branching possibilities is fundamentally misconstrued and only the Branching Actualist can give a full credit to modality. Instead of adjudicating the quarrel, let me offer a bird’s eye view on the whole issue. The debate is fueled by two competing incentives. On the one hand, we can strive to limit the inventory of objects and events occupying the world to only the actual objects and events. The cost of such a limitation is that we either need to get rid of modal properties altogether (as David Hume would recommend), or to introduce an entirely new category of entities—modalities. We can think about them as a primitive category, we can introduce them in guise of dispositional/essential properties, or we can introduce them in the form of some sort of abstract entities like propositions, properties, or states of affairs. In any case, our ontological realm is going to swell. On the other hand, the realists can limit themselves to a one-dimensional picture. They accept a more excessive inventory of objects and events (i.e., they accept that all events on the tree exists), but in return, they can use these objects to get rid of the other ontological categories. We can explain modality in terms of existence of appropriately structured concrete events. We can also use these events to construct propositions, properties, states of affairs, etc. Ideally, we can also define dispositional/essential properties in terms of these events and constructions out of these events. Hopefully, at the end of the day we can get by without all these suspicious kinds of entities. It is an important advantage of the realist metaphysics. Thus, we face a difficult trilemma: either to dispose of modal notions altogether, or to introduce modalities as a new metaphysical category, or to accept a vast number of things and events into our ontology. As usual in philosophy, we need to choose the position which, all things considered, seems the least costly solution. I do not intend to marginalize the costs of the actualism which incorporates a primitive modality into metaphysics. Nonetheless, I claim that it is a

205 CHAPTER 6. BRANCHING ACTUALISM viable option in the branching setting. Most importantly, it is not as outrageous as is sometimes presented by its opponents. In any case, the situation is not as one-sided as the branching realists would want us to believe. Doubtlessly, the actualists face a challenge regarding the explication of the notion of historical possibility. They also need to justify why we should accept modalities in addition to ordinary events and objects. Nonetheless, none of these projects is doomed to failure at the outset. Thus, I want to undermine the prevalent conviction that giving up the commitment to plurality of concrete branching events amounts to giving up the real possibilities. Since it is not conclusively demonstrated in the ﬁrst place, that the realist can capture possibility and, secondly, that it is the only conception that can, we should not discredit actualism with one swift stroke of a hand.

6.4.3 The nature of branching possibilities According to Branching Realism, there is no absolute difference between what is actual and what is possible. All the events that we call possibilities are just as real and concrete (from their viewpoints) as are the events we participate in (from our viewpoint). Moreover, they are connected, via causal relations, to the events which took place in our past. The difference between possibility and actuality is a relative matter—different events are actual/possible from the perspective of different points in our world and no point of view is special. This vision offers a thoroughly ontic account of indeterminism. Our world is indeterministic, because it has concrete, mutually incompatible parts, which directly attest to the non-deterministic nature of reality. As we have seen, many of the realists insist that it is the only way to give proper due to ontic indeterminism. A characteristic line of thought is briefly summarized by Nuel Belnap:

If a certain possibility is real, then if it has any relevance at all for us, it must be part and parcel of Our World.(Belnap, 2006, p. 2, n. 2)

Hence, by contraposition, if a possibility is not a part of our world, then either it has no relevance, to us or it is not real. The actualists persistently insist, however, that possible events are not parts of our world. They constantly stress the only events that are part of our world are the events that actually take place. Thus, if they want to appeal to the notion of real possibilities, they should propose an alternative conception of ontic possibility. A notion which would not be committed to the concrete existence of alternative scenarios, but which would somehow “attach” possibilities to our world. I will not try to develop a comprehensive theory of so construed possibility and limit myself to a humbler project. I will brieﬂy point to three ways that a Branching Actualist can follow to incorporate the notion of real possibility into their worldview. First of all, an actualist can refuse to provide an analysis of real possibility. They can dig in their hills and insist that the distinction between possibility and actuality is conceptually primitive and not subject to further analysis. A trace of such attitude can be found as early as in Aristotle. When he introduces the notions of potentiality and actuality to his metaphysics, he makes a telling comment:

Our meaning [of potentiality and actuality] can be seen in the particular

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cases by induction, and we must not seek a deﬁnition of everything but be content to grasp the analogy. (Metaphysics, 1048a25–1048b9)

The mere fact of the actualists’ refusal to analyze the notions of possibility/actuality does not mean that they refuse to take them seriously (as is clear in the case of Aris- totle). The philosophers who embrace modal notions and take them as primitive are sometimes called modalists in the contemporary literature. They are especially reluctant to analyze modal claims in terms of an existence of some sort of objects (worlds, moments, histories, or possibilia). Their views is concisely summarized by Joseph Melia:

The modalist thinks that there is more to the world than is given by a description of what things there are, what categorical properties these things instantiate and what categorical relations these things bear to each other. But the modalist is sceptical about possible worlds: he does not accept worlds other than the actual one. The modalist accepts the objectivity of modal truth, but rejects the existence of possible objects.(Melia, 2003, p. 81)

For example, a modalist insists that it is an objective (and basic) truth about reality that it was really possible that there would be a third world war in the 20th century. Contrary to realists, however, they refuse to accept that this possibility is the same kind of event as WWII was. In other words, they refuse to accept the transition from “it is really possible that φ” to “there is a real possibility at which φ.” Since modalists are reluctant to understand modal claims in terms of existential claims, they are often skeptical about the analysis of these notions in terms of restricted quantifiers, as is common in the relational semantics for modal logics. The case of Arthur Prior is instructive, and very interesting. He was a pioneer in the field of temporal logic. In fact, he was one of the first to extensively use the techniques of the first order logic to semantically analyze various temporal and modal systems and he is often credited as one of the inventors of relational semantics for modal logic (see Goldblatt, 2006). However, Prior had always been skeptical about the philosophical significance of the techniques he used. He writes early on:

The interpretation of the PF-calculus within the 1-calculus is clearly a device of considerable metalogical utility. (. . . ) There are strong reasons, however, for refusing to attach this metaphysical signiﬁcance to the interpretability of the PF-calculus in the 1-calculus. (Prior, 1958, pp. 115–116)

PF-calculus is a system designed for temporal operators, while 1-calculus is its ﬁrst-order representation. Prior abides by the view until the end of his philosophical career. In the posthumously published book, he embraces modalism even more openly:

So, the original, normal or standard interpretation of the calculus sketched in 1.1 [a version of the modal system S5], i.e., the interpretation of it as a logic of necessity and possibility, can be presented as just a special case of

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the interpretation of it as a mildly odd formulation of the uniform monadic lower predicate calculus.20 It can be so presented. But do we illuminate the subject of modal logic by so presenting it? To this I want to say, No; or at all events, Not much. It is, if you like, formally, but not materially illuminating to present modal logic thus. (. . . ) [P]ossible worlds, in the sense of possible states of affairs, are not really individuals (just as numbers are not really individuals). To say that a state of affairs obtains is just to say that something is the case; to say that something is a possible state of affairs is just to say that something could be the case. (Prior and Fine, 1977, pp. 53–54)21

The modalist perspective on the notion of possibility can be readily applied to the historical notion of possibility. A historical modalist insists that among the truths which characterize reality are the truths about what might have been the case in the past and what might be the case in the future. This claim does not imply per se that we need to abandon the branching model of temporal possibility. It only means that we should not take it too seriously. According to a modalist, the tree is just a useful set theoretical (and diagrammatic) representation of the primitive modal facts about what might have been the case. Drawing trees is a convenient way of representing temporal possibilities on a piece of paper, but we should not conclude that the world is like a tree which has branches as its parts. After all, the temporal possibilities could also be represented (less informatively, perhaps) in a form of overlapping circles. Representation of possibilities in terms of treelike partial orders just turns out to be, to use Prior’s words, “a device of considerable metalogical utility.” An excellent example of how this device has been put to use is John P. Burgess’s (1980) proof of decidability of Prior’s Peirceanism. More generally, Mark Reynolds(2003) reports that decidability of various temporal logics follow from decidability of the second order monadic logic of trees. Therefore, the treelike representation is of great theoretical utility indeed, but it does not imply that reality literally forms a treelike structure. Modalism can readily be combined with actualism. We only need to add that the “indicative,” mooded facts have diﬀerent metaphysical status than “hypothetical,” mooded facts. Most simply, we can insist, as did Prior, that only the indicative facts obtain. It is a fact that you are reading this words and it is a fact that you could have been watching a movie right now. However, reading and watching are not instantiated by the concrete reality. Moreover, this is not a derivative of your particular perspective on reality. The fact that you are reading these words is absolute, since your modal perspective is the only modal perspective. Fine summarizes Prior’s views with a conjunction of two claims:

The ordinary modal idioms (necessarily, possibly) are primitive; Only actual objects exists. (Fine, 1977, p. 116)

20This “mildly odd formulation” is a result of the standard translation of the language of modal logic into the language of ﬁrst order logic. For an elegant exposition and discussion of the standard translation, see (Blackburn et al., 2001, pp. 83–85). 21A similar view has been (sometimes) suggested by Saul Kripke who writes in Naming and Necessity that “It is better still, to avoid confusion, not to say, ‘In some possible world, Humphrey would have won’ but rather, simply, ‘Humphrey might have won’ ” (Kripke, 1980, p. 48, n. 15).

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This attitude stands in contrast with the view of a Genuine Modal Realist, as is clear in case of Nuel Belnap who, writes that

If Our World contains some possibilities, then being a part of Our World cannot be a suﬃcient mark of actuality. (Belnap, 2006, p. 2, n. 2)

Modalism is the first of the ways to ground real possibilities within the actualist picture. It is to insist that only the actual events/facts/objects exist, but they there are primitive, and objective truths about what might have really existed. These truths about what might have been the case can be represented, in a theoretically useful way, in the form of branching structure. There is a closely related attitude towards real possibility available to actualism. It can be described as “dispositional” actualism.22 The dispositionalist actualist agrees with the modalist that there are objective truths about what is possible, but they venture to analyze them away. Nonetheless, they do not want to understand them in terms of existence of possible entities, but in terms of modal properties of actually existing objects. These properties come under various names, they are called “dispositions,” “powers,” “potentials,” or “capacities.” To begin with a very simple example, according to a dispositionalist actualist, it is true that I could have been asleep right now, because I was disposed to go to sleep an hour ago (or I had a causal power, or a potential, or a capacity to go to sleep). My disposition was not deterministic and it did not get realized, but it could really have. Some dispositions are temporally dependent on others. For example, my potential climbing of the Rysy mountain depends on my potential earlier excursion to the Tatra mountains, which in turn depends on a potential earlier decision to take time off work. The structure of temporal dependencies of potentials might be represented by a tree. Under this reading, the whole branching structure represents all the immensely intricate relations between the modal properties of actual objects. The exact form of the tree of possibilities is determined by the modal aspects of our concrete world and objects existing in it. It is more difficult to ground more complex modal truths, like that there could have been WWIII in the last century. This truth requires a whole lot more objects and their dispositions. For example, the appropriate leaders need to be disposed to give appropriate orders, the soldiers need to be disposed to follow these orders, the nuclear weapons need to be disposed to fire, etc. Thus, it is immensely difficult to actually carry out such a reduction of modal notions in complex cases.23 Things complicate further, when we realize that some objects (e.g., Nikita Sergeyevich Khrushchev), which might be required to ground modal truths no longer exist. In case of many future possibilities, they do not yet exist (for example, it is possible that the 76th President of the United States will be a Native American, but none of the Native Americans alive is disposed to be the 76th President of the United States). Thus, the holistic project to ground modal truths in modal properties of actually existing particulars is extremely complex. It is

22This view was recently dubbed “New Actualism” by Barbara Vetter(2011), but the name “New Actual- ism” was by Menzel(2015) to describe another theory, so I decided to use another term. 23In fact, it is diﬃcult to do the reduction even on an entirely abstract and formal level. The ideas presented in (Vetter, 2015) probably constitute the most mature and comprehensive attempt at such analysis to date.

209 CHAPTER 6. BRANCHING ACTUALISM diﬃcult on the formal level, but it also requires serious philosophical and conceptual eﬀort. I do not intend to address any of these complexities here. I only want to stress that dispositionalism is a particularly actualist-friendly account of modality. After all, the whole project aims to ground modal truths in actually existing objects and their properties. The actualist attitude is prevalent in the literate on the subject, to mention just a few examples:

Hardcore actualists think that what makes modal propositions true are ir- reducibly modal features of the actual world (such as laws of nature, dispositions, essences). (Contessa, 2010, pp. 341–2)

[W]hat is metaphysically possible is determined by dispositions found in the actual world. (Borghini and Williams, 2008, p. 21) p. 21)

[T]heir shared aim is to identify, within the actual world, the grounds, source or truthmaker of modal truths. (Vetter, 2011, p. 742)

This is an Actualist view of metaphysical possibility, since it proposes that all possible states of aﬀairs are grounded in the properties of actual objects. (Vance, 2014, p. 1112)

Dispositionalists are naturally disposed to “privilege” one of the possibilities— the actualized possibility. They are prone to distinguish it, since the actual objects and their properties are postulated as the “basis” of all the possibilities. Interestingly, dispositional actualism oﬀers a particularly branching-friendly account of modality. Chad Vance even contends that

[O]n New Actualist Dispositionalism, metaphysical possibility takes a branching structure (. . . ) Each decision that I make is a causal “node” of sorts, and alternatives are available to me at each of these nodes— alternatives which I am causally capable of actualizing. The result is that, on this account, metaphysical possibility takes a branching structure. (Vance, 2014, 1115–6)

As far as I know, no dispositionalist actualist gave a detailed description of how to “build” the branching structure of possibilities out of dispositions of individual objects.24 It is likely that a more detailed construction would need to overcome numerous obstacles (e.g., how to model mutually incompatible dispositions of distinct objects; how to account for object that come in and out of existence and dispositions thereof; or if we can guarantee that the resulting structure is going to formally resemble the traditional branching model). The most detailed and formally advanced study of these issues which I am aware is being currently conducted by Antje Rumberg as a part of her doctoral dissertation. I will not attempt to oﬀer a construction of my own. I only

24It is an active ﬁeld of research, however, so we might expect signiﬁcant progress soon. There are at least a few programs currently focused on the subject throughout Europe, e.g., in Berlin, Helsinki, Konstanz, Kraków, and Oxford.

210 CHAPTER 6. BRANCHING ACTUALISM want to stress that dispositionalism offers another actualism-friendly account of real possibilities. The view clearly endorses ontic possibilities, but it resists the idea that the possible branches are the same kind of entities as the reality surrounding us. Finally, a person who wishes to support the distinction between the actually existing reality and the realm of branching possibilities may resort to a more traditional genre of actualism and draw on the results of of five decades of philosophical reflection on the nature of possible worlds. The history offers many specimens of possible worlds theories, which are likely to encompass the realist intuition about possibilities, while rejecting their concrete existence. Among the most renowned and influential are those which construe possible worlds as: maximal possible states of affairs (e.g., Plantinga, 1970, 1974, 1987), ways things might have been (understood as properties or states in Stalnaker, 1976), or maximal consistent sets of language independent propositions (Adams, 1974). All of these offer a non-epistemic and language-independent notion of possibility. However, none is committed to a concrete existence of possible scenarios. Let us take the theory of Adams(1974) as our case study. In his view, the possible worlds are maximal consistent sets of propositions. Not all such worlds represents really possible scenarios in our sense (according to some of these worlds pigs fly faster than the speed of light, while it is not really possible). Therefore, we need to somehow distill the really possible scenarios. Preferably, we would like to somehow “attach” those possibilities to the concrete world and also to reconstruct their branching structure. A very natural construction recommends itself. To begin with, let us take an arbitrary time t (for convenience, consider your present time). Then, let us consider a complete history of the world up to time t. It is a large collection of propositions, let us call it Ht, which describes what happened in the world up to time t. Crucially, the description has to be carried out in what Rescher and Urquhart(1971) call “chronologically pure” terms. That is, the description of what happens at a time should not imply anything about what happens at any earlier or later time. To use Rescher and Urquhart’s example, “the collision of two automobiles occurs at t” is a chronologically pure description of an event while “the fatal collision of the first two automobiles produced in 1965 occurs at t” is not. Let us now augment Ht with another set of propositions R which encode the rules limiting how the world can really develop (for simplicity, I assume that R does not change in time). These rules limit the admissible transitions from one state to another. R can be thought as a collection of laws of nature or as a specification of essential properties. Then, to arrive at the collection of scenarios really possible at t (Pt), we 25 consider all the maximal extensions of the set Ht consistent with R. If David Hume was right and there are no necessary connections in nature, then set R is empty, and Pt contains all logically consistent scenarios. So, within the limits of logic, anything can really happen after t—the world is ultimately indeterministic at t. On the other extreme, if R is so rich that it allows for only one consistent extension of Ht, then the set Pt is a singleton and the world is deterministic at t.26 There is a huge range of in-between cases, where R excludes some logically consistent scenarios, but it does not limit their

25Let me note that the Genuine Branching Realists are traditionally very hostile to speciﬁcation of real possibility in terms of consistency with laws. Especially, if the laws are construed as nothing more than regularities or frequencies. 26I set aside intricacies related to gauge freedom.

211 CHAPTER 6. BRANCHING ACTUALISM number to only one. In such cases, we are dealing with limited (in)determinism. The process above generates all the possible scenarios available at t. To erect the 0 entire tree of possible scenarios P, we need to take a sum of all Pt0 , for any t < t, that S is P B t0

27A hint in the right direction is the “physically motivated models” of branching, recently introduced by Placek and Belnap(2012). They deﬁne a possible moment as a quadruple of real numbers paired with a collection of properties. A history is an appropriate maximal collection of such moments. So construed possibilities—complete distributions of properties over a spacetime—are good candidates for abstract possibilities of an actualist. We can say that the physical universe actualizes a possible history h iﬀ the concrete events which take place in the spacetime instantiate the properties that are attributed to the spacetime lo- cations in h. Placek informed me, however, that this interpretation is far from the authors’ intention. They thought that properties in each history are instantiated (in that history). They even refer to the properties attributed to the spacetime point x in a history γ as “the set of properties instantiated at spacetime point x in scenario γ.” (Placek and Belnap, 2012, p. 450).

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6.4.4 Divergence or branching? Let me comment on one more subject closely related to the notion of branching possibilities. The subject was famously introduced by David Lewis and often reoccurs in discussions on branching ever since. It can be most concisely described with the slogan: “divergence or branching?” Lewis argued that we should conceive of the treelike structures as a collection of diverging worlds. Two worlds w1 and w2 diverge at time t iff w1 and w2 match at t and always before, but never after. The notion of “matching” can be explicated in many alternative ways, e.g., in terms of identity, indistinguishability, or sharing the same primitive physical properties. The diverging structure resembles a rope more than it does a tree; a rope consisting of huge number of separate strands, brained together as long as they “match,” but at the moment when the matching stops, a cluster of strands is disentangled from the core of the rope and forms a thinner thread. In contrast, two world w1 and w2 branch at t iff w1 and w2 are identical at t and always before, but never after. Both notions are pre-relativistic and assume that the notion of the state of the world at t is well-defined, independent of the frame of reference. Since the notion of identity is stronger than the notion of matching, the thesis of divergence is weaker than the thesis of partial overlap (branching implies divergence, but not conversely). In particular, the worlds which diverge at t need not share any common part. The framework of divergent worlds is a hospitable environment for futurism. In this setting, it is clear why a context initializes a possible world (an utterance is a part of a possible world and just one possible world). Therefore, if one is inclined to futurism, but upholds modal realism, then divergence is a preferable set-up. Belnap et al.(2001) are realists who prefer branching to divergence. Their foremost concern regards the choice of primitive concepts of both theories. Belnap et al.(2001) begin with the “local” momentary possibilities and only later build larger constructions out of moments, using the causal relation ≤. In particular, the notion of a history, considered as a maximal collection of compatible possibilities, is a derivative, higher- order concept. When we “build” histories out of moments, and some moments have many alternative continuations, it is most natural to conclude that histories overlap. On the other hand, the conception of divergence requires quite different conceptual prerequisites. In particular, (i) the concept of possible world construed as a maximal object of some sort, (ii) the concept of temporal location which can be shared by events in distinct possible worlds, and (iii) the concept of two worlds matching at a given time. Belnap et al.(2001, pp. 197 ff.) have specific worries about each of these concepts, but more importantly, they share a deeply rooted conviction that representation of reality in terms of these concepts is fundamentally misconstrued, which is best summarized by the following words:

In particular, there seems to us no objective truth to all of those disjoint “worlds” in Figure 7.3 except to the extent that they jointly represent our one objectively real world with its concrete events in their indeterministic causal order giving rise to a system of overlapping, branching histories. We recommend not trusting diagrams like Figure 7.3 when they are not rooted in objective features of our only world. To put the matter another

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way, if what binds all the points on a certain level into the representation of a single concrete event is not sheer identity, then there is nothing else objective for it to be. (Belnap et al., 2001, p. 203) Another source of doubt regarding the divergent representation of possibility stems from the observation that if two possible worlds do not share a common part, there are no events that happen in both these worlds. Then, it is no longer clear if we can capture the intuition that a single concrete event has a number of possible continuations. To answer this worry, we can appeal to a Lewisian analysis in terms of counterparts, and say that the sentence, “The coin, which have landed heads, could have landed tails,” is true iﬀ in another world, there is is another coin, in an indistinguishable set-up, which is tossed and lands tails. Many philosophers share the intuition that this analysis distorts our modal intuitions. John Burgess writes, for example: “Tense logic insists, pace Lewis, that I am the very same person who could have gone to the shore; it’s not just someone like me who could have gone” (Burgess, 1978, p. 173). A very similar point is made by Saul Kripke: Thus if we say “Humphrey might have won the election (if only he had done such-and-such), we are not talking about something that might have happened to Humphrey but to someone else, a ‘counterpart.’ ” Probably, however, Humphrey could not care less whether someone else, no matter how much resembling him, would have been victorious in another possible world. (Kripke, 1980, p. 45)

Their arguments are not entirely convincing, since Lewis would happily admit that I am the very same person who could have gone to the shore and that we are talking about Humphrey, not someone else, when we say that he might have won the election. Lewis only believes that what might happen to Humphrey depends on what does happen to Humphrey’s counterpart, in another world. I think that this is the idea which in fact arouses Burgess’s and Kripke’s objection. They insist, in the Plantingian spirit, that whether there are other worlds and what happens in those worlds is irrelevant to what might happen to Humphrey. If this arguments holds good, then, by analogy, what will happen to another coin, in another world, is irrelevant to what might happen to the coin in our world and, thus, it cannot attest to this coin toss being indeterministic. This might be part of the reason why Belnap et al.’s (2001) Our World contains all the possible scenarios. Then, what might happen to a coin is a wholly “inner-wordily” issue. Observe that the whole issue of branching vs divergence is much less overwhelming (and much less controversial), if we think of possibilities in terms of abstract entities. For example, if we think of possibilities as sets of propositions, it is the most mundane to assume that two histories share a common part. As mundane as to assume that two sets have a non-empty intersection. The claim is entirely unquestionable and, more importantly, it does not in the slightest degree suggest that the physical world actually splits like an amoeba. When possibilities are conceived as abstract entities, the opposition of branching and divergence is no longer so militant. The physical reality does not branch, so the branching construction is nothing bat a pictorial representation of certain interrelations

214 CHAPTER 6. BRANCHING ACTUALISM between temporal possibilities. Whether we decide to represent them in terms of overlapping lines or in terms of diverging parallel lines is largely an aesthetic issue—it is a matter of drawing technique. Largely, but not entirely. After all, the fact that not all possibilities are physically realized does not imply that anything goes. Some representations grasp the structure of interrelations of temporal possibilities better than others. Interestingly, there is an important formal difference between divergence and branching. The difference significantly affects the behavior of modal notions interpreted in these two kind of structures. The difference was first noticed by Hirokazu Nishimura(1979). He gave an example of a formula in a tempo-modal language which is true in every branching model, but is falsified by some diverging models. At first, it might come as a surprise, since every branching model can be easily represented as a model of divergence. We only need to “detach” the overlapping histories (understood as maximal linearly ordered subsets of the branching structure), draw them as parallel possible worlds, and postulate that two possible worlds diverge at t if and only if the two histories which gave rise to the worlds branch at t. However, the reverse procedure is not always possible. It might happen that when we “glue” together diverging worlds, using the equivalence relation of “matching at t,” the resulting structure has some “extra” histories. Histories which have no representatives in the originating structure of diverging worlds. A picture might help. Let us take an infinite set of diverging worlds:

w1 w2 w3 w4 w5 w6 w7 ¬p ¬p ¬p p ¬p ¬p p ¬p ¬p p ... p

The blue dots represent times at which sentence p is true. The horizontal dotted lines represents the “matching” relation between regions of worlds (matching is an equivalence relation). Initially, all the worlds match, but, as time goes by, they gradually diverge . World w1 diverges ﬁrst, then w2, etc. As soon as a world diverges from the remaining bundle, p never reoccurs in that world. Thus, in each world, there is the last moment at which p is true. Let us now transform the collection of diverging worlds into a branching tree of overlapping histories, identifying elements of various worlds that match at a given

215 CHAPTER 6. BRANCHING ACTUALISM time. We end up with the following structure:

h1 h2 h3 h4 h5 h6 h7 hω ¬p ¬p ¬ p p ¬p ¬ p p ¬p ¬ p p

If we now consider histories to be maximal linear subsets of the branching structure, then Kuratowki-Zorn Lemma guarantees that the rightmost diagonal, “dotted” line is a maximal linearly ordered subset of the structure, hence, a history. The history has no counterpart in the original structure of diverging possible worlds. In this history, p keeps reoccurring. At ﬁrst sight, it might seem a technical nuance, but the nuance has important semantic consequences. To portray them, let us imagine that the structures above represent the following (idealized) game of coin tossing: if we toss a coin and it shows heads, we toss again; if it shows tails, the game stops and the coin is never tossed again. Let p stand for “The coin lands heads” and the blue dots represent the moments at which the coin lands heads. Consider the following piece of reasoning.28 Assuming that: (A) It is settled that when the coin lands heads, it might land heads again. (G(p → ^F p)) It follows that:

(B) It is possible that there will be no last time, when the coin lands heads. (^¬F(p ∧ G¬p))

In my opinion, the line of reasoning is valid (i.e. the implication G(p → ^F p) → ^¬F(p∧G¬p) is true). My opinion is shared by Thomason(1984, pp. 222) and Belnap et al.(2001, p. 201), but not with Øhrstrøm and Hasle(1995, p. 268). Given my understanding the the interaction of time and possibility, if it is always possible to prolong a series of coin tosses, it is possible to prolong it indeﬁnitely. Or, to put it diﬀerently,

28There are numerous similar examples discussed in the literature. This one was mentioned in (Reynolds, 2003, p. 361).

216 CHAPTER 6. BRANCHING ACTUALISM any person who believes (a) that any series of heads can be always prolonged and (b) that each series of heads will, sooner or later, come to an end, believes a contradiction. So, in my opinion, a faithful representation of the notion of historical possibility needs to guarantee the validity of the discussed reasoning. However, if you look carefully at the two kinds of structures depicted above, it soon turns out that it is settled, in the collection of diverging worlds, that that any series of heads can be always prolonged, but it is also settled that every series of heads will, sooner or later, come to an end. It is because the first structure represents all the finite sequences of the heads, but lacks the representation of the infinite sequence. Then, premise (A) is true, while conclusion (B) is false. I conclude that the structure of diverging worlds, interesting as it formally is, is not a representation of historical possibility. On the other hand, when we switch to the branching representation, the infinite heads-sequence naturally “emerges” out of the structure of possible coin tosses and this very emerging history guarantees that if the premise of the reasoning above is true, so is the conclusion. In every branching structure the reasoning from (A) to (B) is valid, which I consider a strong argument in favor of branching representation of historical possibility. You might, of course, disagree with my assessment of the argument though. It might not strike you as a valid principle as strongly as it strikes me. I will not try to convince you at all costs. Thankfully, nothing substantial hangs on it. As I have already mentioned, if you are an actualist, the issue of branching and divergence is not essential. Both views can be accommodated within this setting and you are not “forced” to accept the infinite-heads, ω-possibility as a real possibility. It stands in astonishing contrast with the realist account of modality. If you are a Branching Realist, your metaphysics almost forces you to endorse the inference.29 On this slightly more technical note, I conclude the discussion of different approaches to branching possibilities. In the next section, I want to address one more controversy concerning futurism and its link to actualist metaphysics.

6.5 Localism and trans-localism

I have argued earlier in this chapter that futurism squares well with modal actualism. If we metaphysically distinguish the actual world from the realm of possibilities, then one can quite naturally suppose that only one of the possible histories will be actualized. Owing to this supposition, we can ascribe truth values to sentences about contingent future. The diagnosis has not been shared, however, by the entire community of branching theorists. Most notably, Arthur Prior, who was a declared actualist, resisted the idea that future contingents can ever be true. [N]othing can be said to be truly “going-to-happen” (futurum) until it is so “present in its causes” as to be beyond stopping; until that happens, neither “It will be the case that p” nor “It will be the case that not p” is strictly speaking true. (Prior, 1968, p. 38)

29I say “almost,” since it is technically possible to redeﬁne the notion of history, so not all linearly ordered subsets of a tree are histories (i.e., to replace trees with so-called “bundled” trees, see e.g., Zanardo, 1996). Nonetheless, this move seems to be completely arbitrary within the metaphysics of Branching Realism.

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He developed Peirceanism to incorporate the insight. The name is rather fortunate, since Charles Sanders Peirce did reject futurism:

[N]either the being about to happen nor the being about not to happen has any reality at present; and the most that we can say is that the disjunction is true, but neither of the alternatives. (Peirce, 1958, 6.368)

Another philosopher of branching who combined (a version of) actualism with dedicated anti-futurism is Storrs McCall(1976). He conceives of a world as a constantly shrinking tree. In transition from one instant to another, the world “picks” one of the immediate possibilities, while the remaining possibilities “drop off.” The complete state-description of the universe, i.e., the universe-tree, is different at different times. The difference consists in this. If t2 is later than t1, the universe-tree of t2 is a proper subtree of the universe-tree at tl. (McCall, 1976, p. 343) I call McCall actualist, because he denies modal neutrality. He does not consider all possible histories to be on a par. It is an absolute fact that at a given time t, some of the possibilities have already uncoupled from the world and as the world proceeds in time, it converges to a single history (even if it never actually shrinks down to just one history). Nonetheless, he refuses to evoke the state of the universe at a later time to ascribe truth values to sentences used at an earlier time. He writes:

S is true [at time t, in a coordinate frame f ] if and only if there exists at t some condition (set of events) suﬃcient to make S true at the later time t0. (McCall, 1976, p. 355)

More recently, actualism was combined with anti-futurism by Oliver Pooley(2013). His branching world resembles this of McCall:30

The view has two essential elements. First, there is the idea that the future is genuinely open: at any instant, there are several possible ways that the world might develop. Second, there is the idea that only one of these possibilities in fact happens: as time passes, exactly one of the many possibilities becomes actuality, and the rest become mere might-have-beens. (Pooley, 2013, p. 337)

One could expect that the single possibility which becomes actuality could be used for semantic purposes. In particular, the truth values of statements about the future could be correlated with what will happen in the actualized possibility. But Pooley chooses another route:

[S]ince it is supposed to be genuinely unsettled, as of now, whether there will be a sea battle, neither of the following claims should count as true:

30They even choose a very similar formal representation of the dynamic world—a sequence of ever smaller trees. On the level of interpretation of the model, however, Pooley strongly distances himself from his predecessor.

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• There will be a sea battle tomorrow. • There will not be a sea battle tomorrow. (Pooley, 2013, p. 340)

Given that so many actualists openly opposed futurism, it would be certainly misleading to automatically associate these two views. I initially believed that the divergence among actualists results from distinct views on the metaphysics of time. After all, Prior was not only actualist, but also a presentist, i.e., he accepted that whatever exists, exists presently. It is a quite common inclination among the supporters of metaphysical presentism (or, the growing block) to conclude that if we can talk about the future at all, it is just the fragment of the future that already exists “in its origins” or “in its causes” and, as such, is already settled. Since the contingent future events do not yet exist in their origins, the statements about them cannot be true. The metaphysical motivation for modal realism and temporal presentism might be different, but their implications for semantics are similar: the context does not designate the single future continuation and, therefore, future contingents cannot be true. The affinity has been observed by David Lewis. In his attempt to understand the philosophers who claim that the future is unreal, he comments: Perhaps their meaning is clearer when they turn linguistic, and say that there is no determinate truth about the future. A modal realist who believed in genuine branching, in which worlds overlaps with others by having initial segments in common, could agree with that. To have determinate truth about the future, it helps to have a future; but also, it helps to have only one future. (. . . ) Against the common sense idea that we have one single future, advocates of many may join forces with advocates of none. (Lewis, 1986, p. 207) Thus, an actualist who rejects the notion of the history of the context, can help himself with the realist semantic strategies that I described in chapter4. Let me briefly investigate, by a way of an example, how the actualist can use supervaluations to account for future contingents. Let us recall the supervaluational postsemantics:

S c||−φ iﬀ ∀h(mc ∈ h ⇒ mc/h |= φ).

In actualism, mc is construed as the possible moment which is actualized by the world at the time of the context c. In ﬁgure 6.3, the linear series of actual events which happen in the successive instants of time is depicted on the right-hand side. The tree on the left-hand side depicts all the possible ways in which the world can develop. Let us assume that Jack utters the sentence, “It will be sunny,” in context c0, at time t0. At that time, the world actualizes the possible moment m0. At context c0, it is still open whether the world will be sunny or rainy on the following day. It is indicated in the picture by the fact that possible moment m0 is both a part of the “sunny” history h1 and the “rainy” history h2. Since the sentence, “It will be sunny,” is true at some histories passing through m0 and it is false at others, the sentence is neither true nor false at context c0.

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h1 h2 S c1||−It is sunny. m1: m2:! c1:

S c0||−/ It will be sunny. c0:# m0:#

Figure 6.3: Actualism + supervaluationism.

It turns out, on the following day, that it is actually sunny. At this later context c1, at time t1, the sentence, “It is sunny,” is true. Moreover, at this later context, the sentence “Yesterday, it was going to be sunny today” (P1F1 p) is true, because it is true at all histories passing through m1, ∀h(m1 ∈ h ⇒ m1/h |= P1F1 p). Also, if we study the S S sentence, “At instant t1, it is sunny,” Att1 (p), we see that c0||−/ Att1 (p), while c1||−Att1 (p). So, we can get a grip on the idea that sentences (or propositions) which are initially truth-valueless become true or false with the procession of events in time. Such a view often accompanies those who support presentism or the growing block in the metaphysics of time. I therefore initially considered it diﬃcult to square futurism with these positions. I even claimed in (Wawer, 2014) that futurism is best understood as underpinned by eternalist metaphysics. A longer reﬂection, however, shook my conviction. It may well be correct that there is nothing at present which could ground the truth of a future contingent like, “There will be a sea battle.” The present state of the world is compatible with two alternative scenarios. The second thought, however, is that the sentence, “There will be a sea battle,” is not about the present (the future tense is not negligible). It is about the future, so the present state of the world seems not to be relevant for the truth value of the sentence. What matters is whether the sea battle will happen at some future moment and even a presentist can agree that there will be future moments. If any of these future moments will hold a sea battle, then the sentence “There will be a sea battle” is true; otherwise, it is false. Therefore, it seems that we do not need to resort to eternalism to justify futurism. The link (if there is any) between metaphysics of time and semantics is more intricate. A similar realization came to Storrs McCall. He changed his view on the future truths without changing his view on metaphysics of time. In 1984, he writes:

When I wrote my 1976 paper I was dead set against the idea that one of the future branches could be singled out in any way, even by the notion of truth, and went to considerable lengths to construct a ’temporal’ theory of truth which allowed many sentences about the future to be neither true nor false. Now I believe I could have saved myself the trouble. (. . . ) The notion of truth, so to speak, bakes no bread, it simply ﬂoats on top of

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whatever events occur or will occur and in no way constrains or aﬀects the possibility of any of them occurring. This ‘supervenient’ character of truth is no threat to the branched universe, and in particular does not succeed in reducing the number of branches to one (. . . ) What is true depends upon what will exist and what will occur, but what will exist and will occur does not depend upon what is true. (McCall, 1984, p. 176)31

Other authors also insisted that the link between metaphysics of time and semantics is not that close. For example, Alex Baia(2012) notices 32 that presentism seems at odds with truths about past and future, given that we accept the following grounding principle:

(G1) Every true proposition depends for its truth on how the world is.

He then observes that in our world the proposition that there were dinosaurs is true, while in the skeptical, Russellian world, which looks exactly like ours, but popped into existence only five minutes ago, the same proposition is false. Therefore, the truth of the proposition that there were dinosaurs cannot be explained in terms of what presently exists. Consequently, it seems that we either need to sacrifice presentism, or past (and future) truths. Baia goes on to argue that the conflict is apparent. He points out that the conflict is generated by (G1) and we can avoid it if we replace (G1) with its tensed variant:

(G2) Every true proposition depends for its truth on how the world was, is, or will be. (Baia, 2012, p. 345)

If we accept (G2), the presentist can explain why it is true in our world that dinosaurs existed, while it is false in the the Russellian world. So, presentism per se does not exclude true sentences about the future. Some additional semantical principles, like (G1), are required to prove the conﬂict. It is also arguable whether non-presentism can automatically grant truth values to sentences about the future (or past). Oliver Pooley, for example, rejects presentism, associates with those, who “deny that any one time is absolutely privileged” (Pooley, 2013, p. 334, see also pp. 347–348). Nonetheless, he denies any true future contingents, since he thinks that: [O]ne does not look to other elements of the model in order to deduce the tensed facts that hold relative to a given element. (Pooley, 2013, p. 348, see also p. 342)

31He develops this view and applies it to the theological issue of Divine foreknowledge in McCall(2011). There, he repeats that what is true depends in turn on what obtains, including what capacities obtain, in the domain of events. To suppose the opposite, and maintain that the way the world is depends on what is true, or on what God knows, violates supervenience by reversing the dependence relation. To repeat, truth depends on events, and events do not depend on truth. (McCall, 2011, p. 504)

32I should note that I grossly oversimplify Baia’s argument. An idea similar to Baia’s is defended by Sven Rosenkranz(2012).

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actualizes The coin will The coin will land heads at t land heads at t1 1

Possible courses of events

actualizes The coin will land heads at t1

Possible courses of events

Figure 6.4: At t0, the utterance takes place. At t1, the coin lands heads.

Hence, it is also arguable whether a “democratic” treatment of all temporal instants guarantees that future-tense statements have a truth value. An extra semantic principle is necessary to ensure this feature. The semantic issue seems orthogonal (or at least indirectly connected) to the position in the metaphysics of time. The reflection on the history led me to thinking that, within the actualist setting, the debate on the semantics of future contingents is less a debate about the nature of time and more about the nature of truth. Both parties agree that the truth value of the sentence should depend on the world, but they disagree about the details of this dependence. Let me illustrate the difference using figure 6.4. Let us say that at t0, Jack utters a sentence (S), “The coin will land heads at t1,” and that it is undetermined at t0 if it will happen. At time t1, the coin does in fact land heads. Can we say that the sentence uttered by Jack is true at t0? Opinions differ. One brand of actualism, which I call localism, proclaims that the truth value of a sentence (S) at t0 should depend entirely on what is the case at t0. As the upper frame of the picture indicates, there is no trace, at t0, of how the coin will land at the later moment. There is only Jack, holding a fair coin and intending to toss it. The physical properties of the coin (and Jack’s hand) at t0 allow both for a heads- and a tails-possibility. Based on what the world is like at t0, we can figure out what the possible options are (we can reconstruct the tree of possibilities), but it is impossible to figure out how the coin will in fact land. The localists conclude that the sentence “The coin will land heads at t1” is

222 CHAPTER 6. BRANCHING ACTUALISM

33 therefore not true at t0. There is a contrasting trend among actualists, which I call trans-localism. In this view, the truth value of the sentence, “The coin will land heads at t1,” uttered at t0, does not depend on what is the case at t0, but on what will be the case at t1. Pictorially, in trans-localism, the truth value of a sentence uttered in the first frame of the figure 6.4 depends on what happens in the second frame of figure 6.4. The conflict between the two competing approaches is nicely exemplified by the early twentieth century dialogue of two Polish logicians and philosophers—Tadeusz Kotarbinski´ and Stanisław Lesniewski´ . A few years before Jan Łukasiewicz proposed his three-valued logic, his student, Tadeusz Kotarbinski´ (1913), wrote a very elegant essay, “The problem of the existence of the future.” In the text, he argues that the future, contrary to the past and the present, does not exist. More precisely, he believes that the future is “gappy.” The part of the future predetermined by the past and the present does exist, while the fraction of the future that is within the human power to influence, does not (Kotarbinski´ , 1913, pp. 78–79). He offers two arguments to support his claim. Both of them take for granted that people are free and creative beings, a conviction which Kotarbinski´ shared with Łukasiewicz. The first of the arguments relies on the notion of creation. Kotarbinski´ observes that to create means to bring into existence. Kotarbinski´ agrees that some portion of the future, like sea tides or motions of celestial bodies, already exist and, therefore, cannot be created. If the whole of future already exists, however, nothing can be brought into existence. Thus, no creation is possible, which contradicts Kotarbinski’s´ assumption that we, humans, are capable of creation. Therefore, Kotarbinski´ rejects the assumption that the whole future exists. He believes that his thesis requires revision of classical logic, because he intimately links truth with existence:

An object exists, when the proposition asserting the object is true and conversely: the proposition asserting the object is true, when the object exists. (Kotarbinski´ , 1913, p. 75, all translations mine)34

Kotarbinski´ concludes that, since some bits of the future do not exist, some propositions about those bits of future are not true (“We truly create only when we create the

33 Actually, most actualists take a slightly different stance and argue that what is true at t0 does not depend on what is the case at t0, but on what is settled at t0. Assuming that the past of t0 is settled at t0, the sentences about the past are either true or false, even if there is nothing, at t0, to ground the truth value of these sentences. A notable exception is Jan Łukasiewicz, who was faithful to a thoroughly local notion of truth and found the truths about the past as suspicious as truths about the future. We should not treat the past differently from the future. If the only part of the future that is now real is that which is causally determined by the present instant, (. . . ) then only those parts of the past are at present real which still continue to act by their affects today. Facts whose effects have disappeared altogether, and which even an omniscient mind could not infer from those now occurring, belong to the realm of possibility. (Łukasiewicz, 1970b, pp. 127–8)

34Kotarbinski’s´ article has been translated to English in (Kotarbinski´ , 1968). However, I feel that some intricacies of Kotarbinski’s´ argument are lost in translation, so I stress the logic of the argument in my translation.

223 CHAPTER 6. BRANCHING ACTUALISM truth” Kotarbinski´ , 1913, p. 80). In fact, he argues that they are not false, either. There- fore, he concludes that free creation contradicts classical logic. In particular, he thinks that bivalence does not apply to propositions regarding contingent future (p. 85). He also admits that if a proposition is neither true nor false, then its object neither exists nor does not exist (p. 87). Another argument questioning the existence of the future relies on the intricate interrelations of truth, existence, necessity, and action. Kotarbinski´ ﬁrst assumes that to be possible is to be consistent with the set of all truths. He concludes that Every truth is necessary, every falsity—impossible. (Kotarbinski´ , 1913, p. 88)

Then, Kotarbinski´ assumes a principle resembling Frankfurt’s (1969) principle of alternative possibilities: The only subject of action is what might be and might not be, what is neither necessary, nor impossible, what lies in the sphere of double-sided possibility. (Kotarbinski´ , 1913, p. 89)

The two principles are jointly sufficient to conclude that Action ends where the truth begins, the truth ends where action begins. (Kotarbinski´ , 1913, p. 89)35 Given that the truth is intrinsically connected to existence, we can safely conclude that existence also ends where action begins. Therefore, the notion of action also excludes future existence. Kotarbinski´ ’s essay is much richer than my brief sketch, but the few remarks I have offered need to suffice. Kotarbinski´ ’s essay provokes a passionate reaction of Stanisław Lesniewski´ (1913). The primary purpose of the essay is to defend the view that every proposition is either true or false, including propositions about the future. Nonetheless, Lesniewski´ does not conclude, as Kotarbinski´ thinks he should, that the future exists. In fact, he defends an even more parsimonious metaphysics. No future object exists at the present, for it is not present similarly no past object exists exists now since it is not present either. (Lesniewski´ , 1913, p. 108)

To endorse unrestricted bivalence, Lesniewski´ then rejects the simple truth-existence principle. He writes, contra Kotarbinski´ :

35He later adds: “An omniscient being could create nothing, an omnipotent being—could know nothing.” It is not an equivalent formulation, since knowledge implies truth, but not conversely. Therefore, it is possible that sentences about the future are true at t, even though no one can knows, at t, that they are true. In any case, it is interesting to observe that Kotarbinski´ ’s views on mutual limitation of knowledge and action accord with the thesis later expressed by Arthur Prior: It may be said that contemplation without action is impotent and action without contemplation is blind; but such impotence and blindness are inescapable if contemplation is to be contemplation and action action. (Prior, 1968, p. 49).

224 CHAPTER 6. BRANCHING ACTUALISM

A judgment asserting an object can be true not only at the time when this object exists. If this is so, then it does not follow from the fact that a judgment asserting a future object is already now true, that this future object exists at present. (Lesniewski´ , 1913, p. 97)

If truth-existence principle is rejected, bivalence does not stand in conﬂict with creation. We can create, i.e., bring into existence, objects, even if it is already true that the object which we will create will exist.36 I do not want to suggest that Lesniewski´ advocates some non-classical notion of truth that detaches truth from existence. Far from it; he clearly indicates that truth relies on being. He writes for example:

Since I was indeed sad on Feb. 26, each of the judgments; “I was sad on Feb. 26” that I have uttered is (at the very moment when it is present) true. (Lesniewski´ , 1913, p. 95) 37

Thus, according to Lesniewski´ , the truth of a sentence uttered at one time evidently depends on what exists, it just depends on what exists at another time. In particular, if a sentence is about what happens on February 26, 1913, its truth value depends on what happens on February 26, 1913. In this light, it is clear why the truth value of a proposition P about what happens at t does not change in time. The truth value of P at any time t0 parasites on its truth value at t which, in turn, parasites on what happens at t.38 It applies to truths about the past just as much as to truths about the future:

Let us further assume that, being able to do either “by a free act,” I will “by a free act” conclude this paper. The judgment asserting the possession by me the property of having concluded this paper will then [be]39 true. However, in view of the fact that all truth is true without a beginning, this judgment is already true now. (Lesniewski´ , 1913, pp. 107–8)

A very similar was been advocated, over half a century later by Nicholas Rescher (1968) (he uses symbol Tt(p) to indicate that proposition p is true at time t):

For what we are proposing to do is to construe the present assertion of a future-oriented proposition, for example,

Ttoday(p-tomorrow)

36I will not recount the details of Lesniewski´ ’s response to Kotarbinski´ ’s second arguments, although it does deserve a thorough exposition, as it is truly unique. At the crucial point of his counter-argument, Lesniewski´ points out that necessary truth of A does not imply that A is necessary. 37By “judgment” Lesniewski´ means a speciﬁc, spatiotemporally limited (interpreted) linguistic item. For a detailed exposition of Lesniewski’s´ semantic principles see (Betti, 2006, sec. 4). 38Given the assumption it is easy to understand why Lesniewski´ claims that Thus, if the judgment “A is B” is true at the present time, we must conclude that the contradictory judgment “A is not B” is always (. . . ) false. (Lesniewski´ , 1913, p. 97) He appeals to the thesis a number of times to refute Kotarbinski´ ’s ideas. 39The translator chose the term “become,” which is contrary to the Polish original and to the spirit of the text.

225 CHAPTER 6. BRANCHING ACTUALISM

as amounting to a future assertion of a present-oriented proposition, viz.

Ttomorrow(p-today) Our semantical perspective is to let the issue of the truth or falsity of a chronological proposition hinge entirely upon how matters turn out at the time at issue, so that the allocation of a truth-status to future-contingents is perfectly innocuous, because it prejudges nothing. (Rescher, 1968, pp. 214–215)

Thus, trans-localism disagrees with localism with regards to interactions between time and truth. Localists rejects Rescher’s identiﬁcation. In their opinion, truth and temporal operators do not commute. They think that Ttoday(p-tomorrow) is a stronger claim than Ttomorrow(p-today). As a consequence, they are willing to accept

Ttomorrow(p-today) ∨ Ttomorrow(¬p-today), while they reject

Ttoday(p-tomorrow) ∨ Ttoday(¬p-tomorrow). I have already explained their reasons. They think that truth is temporally local: what is true at a time depends on the state of the world at the time. The local notion of truth indeed has deterministic connotations. Intuitively, the present and the past state of the world cannot be altered. Only the future can be affected; what happens right now (or has happened in the past) is settled. Then, if the truth of a sentence can be grounded in what presently exists or existed in the past, then whatever is true, is indeed necessary. If we agree with Prior that “nothing can be said to be truly ‘going-to-happen’ until it is so ‘present in its causes,’ ” then we also need to agree with him that [I]n an important sense of “truths” there are no contingent truths; once a thing reaches the status of “truth” there can be no going back on it. (Prior, 1968, p. 42) In contrast, the trans-localist insists that in an important sense of “truths,” there are contingent truths and some things can be said to be truly “going-to-happen” even if they are not “present in their causes.” If we assume that the only truths are settled truths, surely no future contingent is true, but this does not need to be assumed. Localism is not the obligatory semantic option and it requires independent justification. One way to justify localism is to say that the truth has a kind of “binding force.” The binding force of truth cannot mean, however, that the truth of the sentence, “There will be a sea battle tomorrow,” somehow forces the seamen to fight. To think this way is to treat the truth of a sentence as a physical fact which has a causal influence on events in the world. But it is surely a wrong way of thinking about the notion of truth. Truth of a sentence, if it can be called a fact at all, is a “soft” fact (Øhrstrøm and Hasle, 2011, discuss the distinction between “soft” and “hard” facts). It is a kind of fact that does not affect “ordinary” facts like sea battles and human actions. The point was eloquently stated by Tomasz Placek:

226 CHAPTER 6. BRANCHING ACTUALISM

[A] proponent of truth-makers may mistakenly see in the innocuous DS 40 thesis a claim that for any true sentence pT , any corresponding sentence of the form “it is true at t that pT ” has a truth-maker that obtains at the time t. Firstly, this reading of the DS thesis results from mistaking truthmakers for causes. Secondly, the truth-maker of the sentence “It is true at t that pT ” is a fact obtaining at T rather than at t. The proponent of truthmakers assumes that between sentences and their truth-makers a relation hods. This, however, is a relation and not an interaction, though the name suggests that truth-makers make something or act on something. (Placek, 2006, p. 180) Some philosophers tried to derive the binding force of truth from the law of non- contradiction. For example both Kotarbinski´ (1913) and Le sniewski´ (1913) accepted the following inference:

φ and ¬φ cannot be both true (¬^(Trφ ∧ Tr¬φ)) φ is true (Trφ) ¬φ cannot be true (¬^Tr¬φ). But the argument is invalid in any typical modal logic. We could derive the conclusion, if we read the second premise as Trφ. That is, if we assumed that truth is equivalent to necessary truth. However, it is exactly this point that the argument was meant to establish, so it would be evidently circular. Alternatively, the inference would be valid if we encoded the first premise as (Trφ → ¬^¬Trφ) ∧ (Tr¬φ → ¬^Trφ), but it is an assumption much stronger than the law of non-contradiction. If we so rendered the first premise, we would in fact assume that truth implies necessary truth and our argument would also be circular. Actually, Aristotle devoted a large portion of the famous fragment of De Inter- pretatione to argue that the reasoning presented above is not valid. A very elegant reconstruction of Aristotle’s point is presented by Rescher(1968). Briefly, Aristotle recognizes that the argument is based on an unjustified distribution of necessity over disjunction, suggested by syntactic ambiguity. “Necessarily A or ¬A” might be read as (A ∨ ¬A) or as A ∨ ¬A. The latter does not follow from the former.41 One more argument in favor of localism relies on a “logical” definition of possibility. We have already witnessed this definition, as it was accepted by Kotarbinski´ . According to this definition, to be possible, a scenario has to be consistent with all the truths. If the set of all truths contains the truths about what will happen, then there is only one possible scenario. Any alternative would need to contradict the “true” scenario at one point or another. The definition certainly limits the set of possibilities to only one. Kotarbinski´ thought that it is induced by the law of non-contradiction, but we have just seen that it is not. In my view, the definition violates the spirit of historical possibility. To be historically possible at time t is not to be foreclosed by the state of the world at t. No reasonable description of the state of the world at t should include the truths about what happens at times later than t.

40 DS: if pT , then it is true at any time t prior to T, that pT . The sentence pT states what happens at T. 41A similar ambiguity is detectable in alternative formulations. For example, “If A, then necessarily not non-A” has two readings: (A → ¬¬A) and A → ¬¬A.

227 CHAPTER 6. BRANCHING ACTUALISM

A more fortunate phrasing of the logical definition of possibility would say that a possible scenario is consistent with all the truths about the past and present, where truths about the past do not contain truths like “It was the case yesterday that there would be a sea battle two days later.” If we phrase the logical definition of historical possibility this way, the truths regarding the future by no means limit the number of available possibilities. Therefore, neither physics, nor logic dictates the local notion of truth. It has to be independently motivated. As far as I can see, the local notion of truth is mostly suggested by the linguistic practice of assertion. Normally, we are not allowed to assert that a sentence is true unless our assertion is appropriately supported. A reasonable way to support assertions about the future is to point to the presently existing “origins” or “causes” of the future which guarantee the predicted outcome. Therefore, it seems that assertions regarding the future cannot be correct unless the appropriate ground already exists. I discuss this issue in section 4.3 and 4.6, so let me just note that the link between an assertion being correct in a context and a sentence being true in a context is delicate. We need to be very careful not to mistake pragmatic considerations specific to the practice of asserting for semantic considerations regarding truth. We also need to remember that the truth often does not behave as settled truth, when embedded in complex constructions like “It might be true,” “It is either true of false,” “What you said was true,” “If it will happen, then what you say is true,” etc. Also, speech acts other than assertion (bet, guess, order, promise etc.) seem not to rely on the local notion of truth. I have mentioned in the introduction that the tension between the two notions of truth has been recognized already by Aristotle. He contrast two competing intuitions:

• For if every aﬃrmation or negation is true or false, it is necessary for everything either to be the case or not to be the case. (. . . ) It follows that nothing either is or is happening or will be or will not be, by chance or as chance has it, but everything of necessity. (Metaphysics, 18a34–18b9) • Nor, however, can we say that neither is true. (. . . ) [I]f it neither will be nor will not be the case tomorrow, then there is no “as chance has it.” Take a sea-battle: it would have neither to happen nor not to happen. (Metaphysics, 18b17–18b25)

In the end, he admits that truth can be identiﬁed with necessity in case of sentences about the past and the present. With regard to sentences about the future, he oﬀers a somewhat ambiguous resolution:

[I]t is necessary (. . . ) for one [of the contradictories] to be true rather than the other, yet not already true or false. (Metaphysics, 19a23–19a39)

I read this statement as a distinction of two notions of truth: “plain” truth (P-truth) and the “already” truth (A-truth).42 P-truth has no necessitarian connotation, while A- truth does imply necessity. Aristotle previously admits that the past and the present are necessary (18a28–18a29), so it is clear to the reader why “already” has a necessitarian

42Such distinction and its inﬂuence on the semantics of future contingents is discussed in an excellent essay of Georg von Wright(1984).

228 CHAPTER 6. BRANCHING ACTUALISM overtone. The distinction of two notions of truth goes hand in hand with Aristotle’s earlier metaphysical claim that “not everything that is, necessarily is; and not everything that is not, necessarily is not” (19a23–19a39). Then, we can say that the things which are necessarily are the things that are already; assertions about these are already true. In contrast, the things that are, but are not necessarily, are the things that are not already, i.e., the things that will be in the future; assertions about these are true, but they are not already true. Importantly, A-truth should not identified with with present P-truth or with P-truth- now. If “the chance has it” that there will be a sea battle tomorrow, then the sentence, “There will be a sea battle tomorrow,” is P-true-now, but it is not A-true. The second notion has a stronger meaning: a sentence is A-true iff it is “made true by the things that already exist(ed).” The notion of A-truth is local, while P-truth is trans-local. This implies that an instance of the T-schema: The sentence, “There will be a sea battle,” is true iff there will be a sea battle, is correct in case of P-truth, but is is questionable in case of A-truth. Also, statements about the contingent future are universally either P-true, or P-false, but they are neither A-true, nor A-false. Aristotle probably took P-truth to be conceptually primitive and A-truth derivative. It is supported by Sorabji’s (1980, pp. 94–95) observation that Aris- totle endorses bivalence at many places in his writing, but never makes the proviso for sentences about the future. Futurism is based on the idea that the truth of a sentence in a context should be identified with “plain” truth, rather than “already” truth. The futurist admits that possibilities are dynamic (the same event may be contingent at one time and necessary at another). Nonetheless, in futurism, truth does not inherit the dynamic nature of possibility: if an event happens, then it was always plainly true that it would happen and it will always be plainly true that it had happened. A futurist can admit that people sometimes use the term “true” in the stronger sense of A-truth, but he takes this notion to be derivative. It can by analyzed in terms of necessary-P-truth. Observe that necessary-P-truth has the dynamic nature of A-truth. It is necessary-P-true that you are reading this words, but in 2010 it was not necessary-P-true that you would. Thus, the futurist can model the modally loaded meaning of true, while also explain why in many context the truth has a simple, factual meaning. The distinction between the two notions of truth can help to explain why actualists disagree about the truth status of future contingents. They disagree whether what will happen in the future is relevant for the ascription of truth at present. The localists think that it is not. They insist that whether a sentence is (already) true at a given moment should depend on what is necessary, given what the world is like at this moment. The trans-localists disagree and think that what will actually happen is highly relevant for the truth value of sentences about the future. They deprive the notion of truth of the necessitarian overtones. Futurism appeals to the second of these notions, which allows for truth without necessity. The futurist postsemantics I propose develops the Lesniewski´ –Rescher path of thinking about truth and time. A path previously walked by many modern modern, medieval, and ancient philosophers, possibly beginning with Aristotle.

229 Summary

This work weaves together two threads present in philosophy of branching—semantics and metaphysics. I relate the semantic theories introduced in the branching setting to their underlying, metaphysical considerations. In my view, they key metaphysical assumption inﬂuencing these semantic considerations is modal neutrality. According to modal neutrality, all elements of the structure are equally real. In particular, no part of the structure can be absolutely distinguished as the actual. Actuality is a property pertaining to every possibility relative to itself. I call the brand of theories that accept modal neutrality Genuine Branching Realism. Genuine Branching Realists sometimes describe the elements of the tree as concrete events linked by causal relations. This description might suggest a distorted reading of the branching structure as the actual (space)time splitting like an amoeba. I termed the distorted view Naïve Branching Realism. To dismiss the oversimpliﬁed theory, Genuine Branching Realists typically point to a modal character of the branching tree. It is a mistake on behalf of Genuine Branching Realists, however, to suggest that the elements of their branching structure are possible events (or so I claim). I argued, following Aristotle, that the notion of possibility is comprehensible only if complemented with the notion of actuality. Therefore, as soon as we call the elements of the structure possible events, we need to be ready to contrast them with actual events, a step that the Genuine Branching Realists were particularly reluctant to take. The elements of the Genuinely Realistic structure can be called proto-possible/proto-actual. Every one of them has been possible relative to every other and actual relative to itself. I have argued that the Genuine Branching Realist set-up is particularly hostile to futurism (where futurism is a thesis that some future contingents are true in some contexts). In such metaphysics, the act of prediction of a contingent future event is a part of more then one distinct scenario. Therefore, we cannot distinguish one of the scenarios, as the scenario in which the utterance takes place. Also, we cannot distinguish one of the scenarios as the actual scenario, since in Genuine Branching Realism, every scenario is actual relative to itself and none of them is absolutely actual. Therefore, to call a sentence that predicts a contingent event true, rather than false, is completely arbitrary. Genuine Branching Realists devised various semantic techniques to analyze sentences about the future, given that for them, there is, objectively speaking, no such thing as the future. The techniques range from extremism, and modalism, through many-valued semantics and supervaluationism, ending with various brands of relativism. The Thin Red Line theory, as I understand it, took upon itself the unpromising

230 CHAPTER 6. BRANCHING ACTUALISM task of introducing futurism into the Genuinely Realistic worldview. According to the theory, we can distinguish the scenario relevant for semantic evaluation as the scenario that will actually take place. Thus, the Thin Red Line theory presupposes that we can divide the events on the tree into the actual events and the possible events. Nonetheless, TRL relies on the Genuine Realist account which has it that all the events on the tree are equally real (sometimes expressed in form of the idea that the world is branching). So, the Thin Red Line generates a tension. On the one hand, events are differently real (some are actual and some are possible), but on the other hand, all are equally real (all of them are actual-to-themselves). I discussed a number of futuristic TRL-theories and explained how the tension inherent in the TRL view can be exploited to attack this theory on metaphysical and semantic grounds. I conclude that the only way to defend futurism in the context of branching is by rejecting modal neutrality. To uphold that some future contingents are true, we need to accept that there is an absolute difference between what is possible and what is actual. I call this view Branching Actualism. In Branching Actualism, contrary to Branching Realism, we interpret the tree as a structure of possible events and contrast them with the actual events. In the procession of time, the actual events actualize the possible events. Then, we can conditionalize the truth value of sentences about the future on what will actually happen in the future. If we accept Branching Actualism, we can subsume to futurism without bothering about the objections which threatened the TRL theory. I originally expected to cover a more extensive philosophical territory in this work. I wanted to study in more detail the classical arguments supporting the so-called logical determinism. The last section might be seen as a prelude to this kind of detailed investigations. I planed to devote one more chapter to pragmatics of assertion. I was particularly interested in instances, when people make claims about the future, while being aware that what they say is not really definitely settled (like “Our train will arrive in 2 hours”). I wanted to study how this kind of behavior can be explained by different semantic theories of future contingents. I was also interested in a formal analysis of speech acts other than assertion in the context of indeterminism. I also intended to supplement the analysis of future contingents with an analysis of what might be called counterfactual future contingents (like “Had I tossed a coin, it would have landed heads”). I wanted to substantiate the opinion I expressed in section 6.3.5 that although the theories of future contingents and of counterfactual future contingents are often conflated in the literature, they are formally and philosophically independent. I also contemplated generalizations of the branching structure allowing for temporal loops or backward branching and studied their influence on the Ockhamist semantics. Great part of these research projects are in quite advanced phase, so I hope that I will have an opportunity to accomplish some of them in the actual future. Nonetheless, due to limitation of time and space, and to preserve the integrity of this work, I decided to confine my discussion to the core metaphysical and semantic issues concerning future contingents. Within the limited scope, I tried to shed new light on some controversies and misconceptions which accompany the notion of branching possibilities. I also intended to restore the plausibility of the actualist insight. I have little faith that my work will convince all philosophers to consider futurism as a viable semantic position, but I do hope that it will at least help us to agree on what we disagree.

231 APPENDIX

7.6 History relativism as extreme assessment relativism

This section elaborates on the idea that history relativism of Belnap et al.(2001) might be seen as an extreme version of MacFarlane’s assessment relativism. Otherwise, Mac- Farlane’s relativism is a generalization of Belnap et al.’s (2001) relativism. Observe that relativizing the truth value of a sentence to a history resembles assessing the sentence, as if from the perspective of the end of time in this history. Such an observation was made already by Prior:

[T]he Ockhamist seems to treat what is still future in a way in which it would only be proper to treat what has been future—he views it as it would be proper to view it from the end of time. (Prior, 1967, pp. 130–131)

This line of thinking about Ockhamism seem to have been discarded in the later development of the theory. It might be used to establish a new understanding of history relativism, however. I intend to develop the idea that when history relativists relativize the truth value of a sentence at context to a history, they metaphorically situate themselves at the transcendent end of the history. Then, they indeed view a course of events “as it would be proper to view it from the end of time.” In the spirit of Prior’s philosophy, I think that one of the best ways to understand a thesis in philosophy of time is to give it a formal account. To grasp Prior’s insight I intend to apply relativism of John MacFarlane and demonstrate that the notion of truth at context relative to a history, coincides with truth at a pair of contexts. We shall see that, in accordance with Prior’s insight, the context of assessment needs to be situated at the end of time. Let me first encode the postsemantics of history relativism in form of the definition: Definition 7.12 (History relativism postsemantics). m/h||−hφ iff m/h |= φ. It conveys the idea that the sentence can be called true at context only if a history is specified. When it is already specified, it is sufficient to use ordinary Ockhamist semantics to establish the truth value of the sentence. Let me begin the investigations with a simple example. Let us take a branching model M in which there is the maximal element in every history—“the end of time” in this history (there is at most one such element, given that histories are linearly ordered). Let us call mh the maximal moment in history h. Let us first observe that

232 APPENDIX

Lemma 7.1. Let M be a model in which there is a maximal element mh in every history h, then ∀h(Hmh = {h})

Proof. Take an arbitrary history h ∈ M, mh ∈ h, so h ∈ Hmh , therefore, {h} ⊆ Hmh . To 0 0 0 0 prove that Hmh ⊆ {h}, assume, for reduction, that ∃h h , h & mh ∈ h . Since h , h , 0 0 0 0 0 0 ∃m0∈h0 m ∈ h & m < h. Since mh, m ∈ h and h is linearly ordered, there are two options: 0 0 1. m < mh. By no-backward-branching and maximality of h, m ∈ h, which contradicts our assumption.

0 2. mh ≤ m . Since mh is the maximal element of h and h is a maximal, linearly 0 0 0 ordered subset of M, mh ≮ m . Thus, m = mh, but then m ∈ h, which contradict our assumption. A simple proof is sufficient to establish that for any m in a limited model M described above, a sentence is true relative to a history h iff it is assessed as true from perspective of moment mh, i.e., from the end of time in the history: h R Fact 7.1. m/h||−φ iff m, mh||−φ

Proof. Since m, mh ∈ h and mh is the maximal element of h, then m ≤ mh. From this we can conclude that Hm|mh = (Hm ∩ Hmh ) = Hmh . Hence, Hm|mh = Hmh . By lemma 7.1,

Hmh = {h}. Therefore, Hm|mh = {h}. R 1. m, mh||−φ iﬀ

2. m/h |= φ, for every h ∈ Hm|mh iﬀ 3. m/h |= φ iﬀ (by def. 7.12) 4. m/h||−hφ. Thus, in the “upper-bounded” model, to substantiate Prior’s claim is easy. Appli- cation of Ockhamism is not limited to such models, howeverl. It might well be that some (or even all) of the histories in a model never end. In such cases, what would it mean for an Ockhamist to view future “as it would be proper to view it from the end of time”? I propose to read it along the following lines: an Ockhamist views future as it would be proper to view from a hypothetical end of time. To give formal meaning to the maxim, I construct what I call a doomsday extension of a branching model. Let M B hM,

(i) ∀h∃!mh∈MD ∀m∈hm < mh

(ii) ∀mh∈MD ∃!h∀m∈h(m < mh ⇔ m ∈ h) It means that we attach a single extra moment on top of every history in the original D model M. I will call such an extended structure M , and mh the moment which is attached on top of history h. Let me pause to show that model MD is still a model of branching. Its ordering relation is evidently a partial order, so let me just check if it is a connected order without backward branching.

233 APPENDIX

Fact 7.2 (MD is a BT model).

Connectedness ∀m,n∈M∃o∈mo ≤ m & o ≤ n

D Proof. The only interesting case is when we pick mh1 , mh2 ∈ M , but even then D we just need to choose any moment m ∈ h1 and n ∈ h2. By deﬁnition of M ,

m < mh1 and n < mh2 , and since m and n are connected and ≤ is transitive, mh1

and mh2 are also connected.

No-Backward-Branching ∀m1,m2,m3(m1 ≤ m3 & m2 ≤ m3) ⇒ (m1 ≤ m2 ∨ m2 ≤ m1)

Proof. We just need to check if it is satisﬁed for every m3 = mh ∈ MD. Take an arbitrary mh ∈ MD, then, by condition (ii), we have that all the moments below mh are in a single history. And since every history is linearly ordered, there is no danger of backward branching.

Before I proceed, let me observe that the construction of the MD guarantees that there is the maximal element in every history. Therefore, lemma 7.1 applies, and we D have that in M , ∀hHmh = {h}. Let us investigate the relations between history relativism and assessment relativism in the doomsday model. Observe ﬁrst that there is M and φ such that:

h D R M, m/h||−φ & M , m, mh||−/ φ

There are two possible kinds of reasons for the failure.

“Material” There may be φ which is false everywhere in h, but true at mh. (“Four horsemen of the Apocalypse are riding their horses” is a good candidate for φ). Then G¬p is true at any moment in h in the base model, but false in the doomsday extension of the model. “Structural” Addition of the doomsday signiﬁcantly modiﬁes the structure of the histories. Most evidently, seriality no longer holds and thus Gφ → Fφ is not valid in the extended model.

It is then a valid question whether we can give a formal credit to Prior’s insight in the general case. I propose a relatively easy solution: to limit the range of the future operator such that it does not reach all the way to the doomsday. The new deﬁnition of F in doomsday model should be modiﬁed as follows:

D 0 0 0 D 0 Deﬁnition 7.13. M , m/h |= Fφ iﬀ ∃m0 (m ∈ h & m > m & m , mh & M , m /h |= φ). By duality of F and G, we obtain that

D 0 0 0 D 0 M , m/h |= Gφ iﬀ ∀m0 (m ∈ h & m > m & m , mh) ⇒ M , m /h |= φ

234 APPENDIX

Therefore, we end up with a model which has an extra element on top of every history, but the element is not attainable by the connective “it will be the case that.” So, the doomsday is in one sense at the end of time, but at another, it is outside of time. I am not sufficiently versed in theology to give a convincing account of this idea, but I am confident that it was entertained at some point in the history of human thought. Importantly for us, this modification allows to prove an analogue of fact 7.1 in full generality: Fact 7.3. Let M be an arbitrary branching model and MD its doomsday extension, and let m ∈ M, then:

h D R M, m/h||−φ iﬀ M , m, mh||−φ Proof. By induction on complexity of φ, in particular: 1. M, m/h||−hFφ if (by def. 7.12) 2. M, m/h |= Fφ iﬀ (by def. of F)

0 3. ∃m0|m0∈h & m0>mM, m /h |= φ iﬀ (by def. 7.12)

0 h 4. ∃m0|m0∈h & m0>mM, m /h||−φ iﬀ (by inductive assumption)

D 0 R 5. ∃m0|m0∈h & m0>mM , m , mh||−φ iﬀ (by def. 4.7)

0 0 0 D 0 9. ∃m0 (m ∈ h & m > m & m , mh & M , m /h |= φ) iﬀ (by def. 7.13) 10. MD, m/h |= Fφ iﬀ (by Lemma 7.1)

D 11. ∀ ∈ M , m/h |= Fφ iﬀ (since m < m , by def. 4.6, p. 84) h Hmh h D 12. ∀ ∈ M , m/h |= Fφ iﬀ (be def. 4.7) h Hm|mh

D R 13. M , m, mh||−Fφ Thanks to modiﬁcation of the truth clause of F in the doomsday model, we can give full credit to Prior’s insight. The Ockhamist does look at the future as if it has been future, that is, from the perspective of the end of time. A necessary addition to vindicate this insight is that in the models in which time has no end, the end of time is “beyond time.”

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7.7 Counterfactual accuracy ascription

Let me present one more argument against the postsemantics of assessment relativism. I will argue that if we agree that supervaluationism violates our intuitions concerning retrospective accuracy assessment, then we need to admit that relativism violates our intuitions concerning counterfactual and possible retrospective accuracy assessment. It is particularly interesting, given that such a failure was used by MacFarlane, pp. 209–211 to criticize another postsemantic theory—the Thin Red Line. Let me return to our simple meteorological model. This time, Jake enjoys a delight- ful sunbath at m1, content with the accuracy of the assertion he made the day before. To kill some time, he is reﬂecting on his good luck. Well, it is sunny and he was accurate to say yesterday that it would be. Nonetheless, he realizes that it could have been rainy, and then it would have been accurate to say the opposite. If it was rainy, then yesterday it would have been accurate to say that it would be rainy.

If it was rainy today, then yesterday it would have been accurate to say that it

would be rainy today. h2

m1: h1 m2:!

Given relativist semantics and postsemantics, we cannot convey this idea. It turns out that, for a relativist, it would not have been accurate to assert that it would be rainy, even if it was rainy. In addition, although it might have been rainy, it might not have been accurate to assert that it would be. It means that it might have been “not sunny,” but my assertion that it would be sunny must have been accurate. These are the bizarre consequences which we need to accept if we follow assessment relativism. To prove it formally, let us focus on the sentence: (CA) If it was rainy today, then yesterday, it would have been accurate to say that it would be rainy today. Intuitively, this sentence should be true; in fact, it sounds as an analytic truth. I intend to demonstrate that it can be false in relativism. To derive this consequence, we need some semantics to express counterfactuals in the branching setting. There is a disagreement in the field as to which exact definition is best to express time-sensitive counterfactuals (cf. Thomason and Gupta, 1980; Placek and Müller, 2007; Wawer and Wronski´ , 2015). However, in our simple model, all definitions give the same results, so we do not need to worry about the details.

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Let us then assume the simplest definition in the spirit of Stalnaker(1968): Definition 7.14. M, m/h |= φ > ψ iff M, s(φ, m/h) |= ψ. Where s(φ, m/h) is, intuitively speaking, the “closest” moment/history pair at which φ is true. All that matters is that, in our tiny model, s(rainy, m1/h1) = m2/h2. Now, Consider the sentence, “If it was rainy today, then yesterday, it would have been accurate to say that it would be rainy,” uttered at m1. I shall formally represent it as rainy > P1AccRF1rainy. To simplify the exposition, I will use the supervaluationism- friendly account of relativism that I argued for in section 4.6. Nothing substantial depends on my choice, as the whole argument can be rephrased in terms of double- relativized notion of truth at a context. Having said so much, let me calculate the relativist truth value of this sentence, asserted at m1.

S 1. m1||−rainy > P1AccRF1(rainy) iﬀ (by def. 4.3)

2. ∀h(m1 ∈ h → m1, m1/h |= rainy > P1AccRF1(rainy) iﬀ (since Hm1 = {h1})

3. m1, m1/h1 |= rainy > P1AccRF1(rainy) iﬀ (by def. 7.14 of >)

4. m1, m2/h2 |= P1AccRF1(rainy) iﬀ (by def. of P1)

5. m1, m0/h2 |= AccRF1(rainy), then (be deﬁnition 4.10)

6. ∀h(h ∈ Hm0|m1 ⇒ m1, m0/h |= F1(rainy)) iﬀ (since Hm0|m1 = {h1})

7. m1, m0/h1 |= F1(rainy) iﬀ

8. m1, m1/h1 |= rainy.

As the last sentence is false, so is the ﬁrst. It means that, assessed from the perspective of the sunny day, it is not true that if it was rainy today, then in would have been accurate to say yesterday that it would be rainy today. In fact, from the perspective of m1, under relativism, you can truly say that even if it was rainy today, it would sill have been accurate to say yesterday that it would be sunny today (i.e., S m1||−rainy > P1AccRF1(sunny)). The crucial step in this reasoning is obviously the step from 5 to 6 in which I assume that “it is accurate to say that” is context sensitive. The relativist needs to assume such context-sensitivity to explain retrospective accuracy claims. In case of counterfactual accuracy, however, context-sensitivity strikes back. Indeed, the context of the whole counterfactual should not be relevant for the counterfactual accuracy assessment. The context relevant for assessment should “follow” the counterfactual and shift to moment m2. But a relativist cannot allow for that. If he allowed the context of assessment to follow the point of evaluation, then in case of “simple” retrospective assessment (P1AccRF1(sunny)), the context of assessment would need to be shifted to yesterday. But in the context yesterday, it is not accurate to say that it will be sunny on the next day. It seems that the very maneuver which relativism applied to better supervaluationism leads relativism to trouble with counterfactual assessment claims. Let me discuss, and block, several possible ways to save relativism.

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No-Go ways out

Modal operators shift the context of assessment The most immediate response that comes to mind is that counterfactuals are a special kind of connectives which shift the context of assessment. So, when we judge the accuracy of a counterfactual, we should shift the context of assessment to the circumstances in which we evaluate the consequent of the counterfactual. Some remarks of MacFarlane(2014) suggest that he might approve of this general idea

[T]he relativist can say not only that yesterday’s prediction of sunshine was accurate, but also that it wouldn’t have been accurate if it had rained today. (That is, it isn’t accurate as assessed from a context on the other branch.) (MacFarlane, 2014, p. 228)

Well, in case of our example, the solution would indeed wor, but it has a number of drawbacks. First of all, it seems entirely ad hoc. When relativists works with temporal connectives, they need to explicitly prohibit the context of assessment to follow the point of evaluation to the past. To obtain the desired results, relativists evaluate the sentence used in the past, but they keep the context of assessment ﬁxed in the present. Now, when the counterfactuals complicate the picture, the context of assessment is allowed to be moved to an alternative history. To see how arbitrary this is, let us look at the above-mentioned quote. It is completely unclear why the proviso in parentheses is introduced in the case of a counterfactual, but it is not introduced in the case of the past tense; i.e., it is unclear to me why MacFarlane recommends that the sentence

The prediction wouldn’t have been accurate if it had rained today.

should be understood as

The prediction is not accurate as assessed from a context on the other branch.

While at the same time, the sentence

Prediction of sunshine was accurate.

should not be understood as

The prediction is accurate as assessed from the context of the past moment.

Besides being arbitrary, the solution has other drawbacks. For example, MacFar- lane(2008) independently argues that an interpretation of the modal indexical “actually” requires counterfactuals to not shift the context of assessment. For example, if I say, “If it was rainy, the weather would be worse than how it actually is,” then the proper interpretation of the sentence requires “actually” to refer to the sunny context of assessment, rather than to the rainy context of assessment. So “accurate” requires the counterfactual to shifts the context of assessment while “actually” requires it not to shift it. Therefore, to get “AccR” right, we would need to sacriﬁce “actually.”

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The ﬁnal problem of this solution is that it works only in the case of counterfactuals. It cures this particular symptom, but it does not aﬀect the core of the disease. We can construct similar counterexamples that do not use counterfactual constructions. For example, let us assume that at m0 I say, “It will be rainy.” At moment m1, I assess that my assertion was inaccurate. However, given the proof of Mike, the Director of the Bureau of Quantum Weather, I know that it could have been rainy. So I want to say, at m1, “My assertion might have been accurate” (or, more pedantically, “It might have been the case now that one day ago it was accurate to assert that it would rain on the next day”).43

(MA) P^NowP1AccRF1(rainy) S It turns out that under relativism, it cannot be said (m1||−/ P^NowP1AccRF1rainy,I leave to the reader the details of the computation). The rain today was a possibility yesterday, but it was not possible to be accurate predicting the rain today. In fact, if only it is sunny, the assertion that it would be sunny was necessarily accurate (pedantically, “It must have been the case now that one day ago, it was accurate to say that it would S be sunny on the next day,” is true at m1, m1||−HNowP1AccRF1(sunny)). It means that if only I was right yesterday about the weather today, then I must have been right. These are the curious results closely related to the problem of counterfactual accuracy. However, in these cases, the answer is not as simple. We cannot just say that the modal operator of possibility shifts the context of assessment, because to obtain the correct prediction, the context of assessment would need to trace the point of evaluation from m1/h1 to e.g., m0/h1 (following operator H), then trace it to m0/h2 (following ^), then trace it to m2/h2 (following Now), and then suddenly stop and do not trace it back to m0/h2 (following P1), where AccR operator is evaluated. Not is the behavior of the context of assessment extremely strange, but also it forces us to break the principle that the context of assessment is not changed by tense connectives.

Hidden connectives that shift the context of assessment Let us explore another possible solution to the problem. On the formal level, it is possible to introduce an operator which generates the desirable results. We could introduce a semantic device shifting the context, similar to a connective Ref, discussed by Max Cresswell(1990). In our case, the appropriate semantics for a relativist Ref should be:

Deﬁnition 7.15 (Ref). mc, m/h |= Re f φ iﬀ m, m/h |= φ. Now, we can reformulate the logical form of (CA) as follows:

(CA’) rainy > RefP1AccR(F1rainy)

And (MA) turns out to be:

(MA’) P^NowRefP1AccR(F1rainy) I leave it to the reader to calculate that such sentences have the truth conditions which coincide with the intuitive judgment.

43I understand “might-have-been” as P^Now. I borrow the deﬁnition from Belnap et al.(2001, p. 245).

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The solution is formally available to a relativist, but it comes with a price. First, we introduce a connective which Kaplan(1989) would call a “monster,” i.e., a connective which shifts the context. Second, there is no independent justiﬁcation to place the operator Ref where I placed it—I did it only to achieve the desired result.44 Third, the solution generates a problem of proper interpretation of the operator Actually, in case of which, as I have already explained, we should not shift the context of assessment. Ultimately, there is no trace of the operator Ref on the surface of (CA) or (MA). Therefore, we can conclude again that we just gerrymander the logical form of these statements and add a mysterious operator to rescue relativism and obtain the results we desired.

Overt connective shifting the context of assessment In fact, it is possible, in fact, to reformulate (CA) and (MA) so that the introduction of operator Ref is well motivated. Namely, we can replace (CA) and (MA) with: (CA’) If it rained, it would be assessed that yesterday it was accurate to say that it would rain today.

(MA’) It might have been assessed that yesterday it was accurate to say that it would rain today. In these cases, the operator “it is assessed that” takes the place occupied by Ref in the formal versions of CA’ and MA’ presented above. Thanks to the maneuver, we can justify the modiﬁcation of the logical form along the lines suggested in the previous section. However, it is clear that (CA’) and (MA’) express diﬀerent thoughts than (CA) and (MA), and we do not need relativism to explain the appeal of (CA’) and (MA’). If relativism is to be an attractive theory, it should be able to express (CA) straightaway, rather than just to replace it with (CA’).45 I conclude that the problem of counterfactual accuracy assessment is not easy to solve (which is ironic, since MacFarlane(2014, p. 210) uses a similar problem—of imaginary accuracy ascription—as a counterargument to another postsemantics, see sec. 5.3.5.1).

7.8 Modal would and modal will

In this section, I recount the two preliminary theories which preceded the Supervalua- tional Thin Red Line.

Modal would

Initially, we accepted the TRL1 reading of F, i.e. we kept the future focused solely on the TRL:

44 After all, why would we encode the counterfactual accuracy ascription as rainy > RefP1AccRF1rainy and we would not encode retrospective accuracy ascription as P1Re f AccRF1rainy? 45Also, in personal conversation, John MacFarlane expressed his dissatisfaction with the translations above as a potential way to save relativism.

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Definition 7.16 (“it will be the case that φ” is true at moment m). a-trl 0 0 0 a-trl M, m|= Fφ iff ∃m0 (m > m & m ∈ TRL & M, m |= φ. To deal with predictions made off the TRL we created a new, and somewhat artificial, tense operator “would.” So, if England score before half time, they will go on to win the match. In contrast: had England scored before half time, they would have gone on to win. We wanted a definition of “would” which does not require moving the TRL. There are two ways to express modal strength of “would” (W and WF ), which correspond to whether or not to include the TRL (via the F operator) in the definition: Definition 7.17 (“it would be the case that φ” is true at moment m).

• M, m|=a-trlWφ iff ∀h(m ∈ h ⇒ ∃m0(m0 ∈ h & m < m0 & M, m0|=a-trlφ)); • M, m|=a-trlWF φ iff (M, m|=a-trlWφ or M, m|=a-trlFφ). Both versions behave in the same way on merely possible moments; they inherit the meaning of the future oriented necessity operator— F . However, they differ when F evaluated on the thin red line, W still takes the meaning of F while W follows the actual future. This distinction is subtle, but important. Let us consider the following example to illustrate the difference. John and Anna played a game of chess (on the TRL) and Anna won. However, these two are more or less equally skillful chess players, both could have won. Michel, unaware of the fact that the game was played, formulates the following judgment, which he considers to be counterfactual: “Had John and Anna played chess, Anna would have won.” Is he right? Let us consider the moment m when John and Anna are beginning the game. At this moment, it is true they both might win, and it is true that Anna will actually win. Then, WF (Anna is winning) is true, but W(Anna is winning) is not. The intuition behind WF is that if an antecedent of the counterfactual is actually true, then “would” should behave as “will.” On the other hand, W has a constant, modally strong meaning, even for the actual moments. Since the intuitions about the truth of the sentences as Michel’s above are shaky, we decided to include some investigation of both options.

Problems with modal “Would(s)” When one carefully investigates the consequences of the semantics above, one ﬁnds that it fails rather spectacularly. I include below a list of a few of its shortcomings, it is by no means exhaustive. They point to counter-intuitive interaction between the connectives of our language under the proposed semantics. The problematic examples be naturally divided into the groups depending on some properties of an evaluation point:

Oﬀ the TRL, for every m, for every STRL-model M:

1. M, m6|=a-trlFφ, even if φ is ψ ∨ ¬ψ, i.e., M, m6|=a-trlF(ψ ∨ ¬ψ);

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2. M, m|=a-trlGφ, for arbitrary φ (where G : = ¬F¬); 3. M, m|=a-trlFφ ↔ F¬φ;

a-trl 4. M, m|= Fφ → F ¬φ.

Oﬀ the TRL, for some m, for some STRL-model M:

5. M, m6|=a-trlWF (φ ∨ ψ) → (WF φ ∨ WF ψ); 6. M, m|=a-trlφ ∧ ¬HWF φ;

7. M, m6|=a-trlWF φ∨WF ¬φ (even though WF φ∨¬WF φ is valid. Clearly then, WF ¬φ and ¬WF φ are not equivalent);

a-trl 8. M, m6|= F φ → Fφ (however, at any m ∈ TRL we have that a-trl M, m|= F φ → Fφ → F φ).

On the TRL, for some m for some STRL-model M:

9. M, m6|=a-trlFφ → Wφ

a-trl 10. M, m|= Fφ ∧ F φ ∧ ¬ F Fφ

For some m (on or of the TRL), for some STRL-model M:

11. M, m6|=a-trlφ → HWφ. But the weaker version φ → H¬W¬φ still holds.

a-trl 12. M, m6|= F φ → φ

a-trl 13. M, m|= F ¬φ ∧ F φ

Additionally, F and F are not duals. The dual of F is a “weak” G which says that “it is possible that it always is going to be the case that” and the dual of F is a “strong” G saying that “it is necessarily always going to be the case that.”

Modal “Will” One way around some of the troubles pointed above is to modify the meaning of the future connective F so that it gets the meaning which was previously reserved for WF . We call this the “modal-will.” This move allows to retire the “would” operator completely. The “modal-will” semantics for F is as follows: Deﬁnition 7.18 (“It will be the case that φ” is true at moment m).

a-trl 0 0 0 0 a-trl a-trl M, m|= Fφ iﬀ ((∃m m > m & m ∈TRL & M, m |= φ) or M, m|= F φ).

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The idea here is that “it will be that φ” is true at m iff either m is in the TRL, and φ is true later in the TRL, or m is not in the TRL and φ is inevitable at m. The gain of this move is quite significant. First of all, we do not need to introduce an independent “would” operator which would take care of predictions made at non- actual moments. The semantics of “will” is sufficiently rich to cover such cases. As a result, we can consent to the thesis that “would” is simply a superficial, grammatical modification of “will” and not an independent operator, which seems intuitively right. Secondly, acceptance of this new definition of F rescues us from many of the intuitive difficulties which affected the semantics enriched with “would” which resulted from assenting to the strict, puritan reading of future operator. Not all the logical problems disappear though. When WF is replaced with F, we are left with:5,6,7, 12, 13(10 is true in some discrete models). Also, G is not the dual for F and also the quasi-deterministic Fφ → F φ is always true off the TRL. This list of semantic problems is not the only thing that discouraged us from endorsing the idea of modal-will. A more repellent factor is conceptual in nature. Given the semantics of modal-will, the meaning of “It will be the case that” changes depending on the circumstances. Therefore, it is a purely contingent factor that “will” means plain, non-modal “will” rather than “necessarily-will.” It just so happens, in the actual course of events that it does not have this necessitated meaning. This idea does not match well with our intuitions. What makes the difference between actual and non- actual predictions is not that they mean something else, but that only in the former case the world ultimately settles whether they are true or not.

7.9 Ockhamist validity and STRL validity

This section contains the results of research on the notion of STRL validity. Definition 7.19 (STRL validity). Formula φ is STRL-valid, ||−strlφ, iff for every STRL- model M: = hM, <, TRL, Vi and every moment m ∈ M, m||−strlφ. Definition 7.20 (Ockhamist truth). Formula φ is Ockhamist true, |= φ, iff for every branching model M: = hM, <, Vi and every m ∈ M and h ∈ Hist, M, m/h |= φ. Theorem 3. For any sentence φ,

||−strlφ iﬀ |= φ.

Proof. Given that every TRL-structure is based on an ordinary branching structure, this fact is a straightforward consequence of fact 7.6 below. Thus, on the most general level, the truths of Supervaluational Thin Red Line coincide with the logic of Ockhamism which is a good thing since I have already argued a number of times that Ockhamism is a safe position as far as the logic of combination of time with possibility is concerned. However, the general theorem above conceals some intricacies of the relation between STRL logic and Ockhamism. Surprisingly, when we focus on a particular structure, it might happen that a sentence which is STRL-valid in this structure, is not Ockhamist true in the underlying structure.

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Deﬁnition 7.21 (STRL validity in a structure T). Formula φ is valid in a TRL-structure T = hM, <, TRLi, T||−strlφ, iﬀ for every model M = hM, <, TRL, Vi based on T and every moment m ∈ M, m||−strlφ.

Deﬁnition 7.22 (Ockhamist validity in a structure F). Formula φ is Ockhamist valid in a structure F = hM,

F |= φ ⇒ T||−strlφ

Proof. The proof is a straightforward consequence of the notion of Ockhamist truth (definition 3.2, p. 33), supervaluational TRL-truth (definition 5.14) and validity (definitions 7.21 and 7.22). Assume that φ is Ockhamist valid in a structure F. This means that for any moment/history pair m/h and any model M based on F, M, m/h |= φ. Now, it is sufficient to analyze the notion of truth that Malpass and I adopted (def. 5.14, p. 155) to notice that for every TRL-model N based on T and every moment m, N, m||−strlφ. Hence, φ is TRL-valid. The converse does not hold in general. Fact 7.5. There is a branching structure F = hM,

T||−strlφ and F 6|= φ

The counterexample is not that easy to find though. The definitions guarantee that for any TRL-model N = hM, <, TRL, Vi and any m < TRL, if N, m||−strlφ, then in BT- model M = hM, <, Vi “underlying” N, for any h such that m ∈ h, M, m/h |= φ . Additionally, if m ∈ TRL, then N, m||−strlφ implies that M, m/TRL |= φ. Therefore, a counter-example can be found only at m/h pairs such that m ∈ TRL and h , TRL. We found such counterexample. Let us consider a rather unusual TRL-structure T such that the ordering < is dense on the TRL and it is also dense “outside the TRL” (i.e. ∀m1, m2((m1 < TRL & m2 < TRL & m1 < m2) ⇒ ∃m3 m1 < m3 < m2). Now, assume that there is at least one discrete “jump” from the TRL; i.e. that the following condition holds ∃m1, m2(m1 ∈ TRL & m2 < TRL & m1 < m2 & ¬∃m3 m1 < m3 < m2). In such a peculiar structure it would be TRL-valid that T||−strlFφ → FFφ even though in the underlying BT-structure F, F 6|= Fφ → FFφ (to see that, we just need to pick a moment m on the TRL and a moment n ∈ h immediately after m outside the TRL and assume that M, n/h |= φ and n is the only such moment in h then M, m/h 6|= Fφ → FFφ). This TRL-structure generates additional problems: even though in our structure T||−strlFφ → FFφ, T||−/strlH(Fφ → FFφ), so the latter is not an TRL consequence of the former (see definition 7.25, p. 246). Interestingly, if Pφ → PPφ is valid in TRL-structure T = hM, <, TRLi, then relation < is dense. To see this, assume otherwise, i.e. (a) T||−strlPφ → PPφ and (b) < is

244 APPENDIX

not dense, that is, ∃m1∃m2(m1 < m2 & ¬∃m3 m1 < m3 < m2). Now, consider a model N = hM, <, TRL, Vi such that V(q) = {m1} for some q ∈ Atom. Finally, examine two possible cases:

strl strl • m2 ∈ TRL. Since T||− Pφ → PPφ, we have that N, m2||− Pq → PPq. By deﬁnition 5.14, this implies that N, m2/TRL |= Pq → PPq. Since, V(q) = {m1} and m1 < m2, we have that N, m2/TRL |= Pq which implies (by deﬁnition 3.2, p. 0 0 0 33) that N, m2/TRL |= PPq. It follows that ∃m m < m2 and N, m /TRL |= Pq. 0 Since V(q) = {m1}, we have that m1 < m < m2 which contradicts assumption (b).

strl • m2 < TRL. Again, we have that N, m2||− Pq → PPq. Consequently, N, m2/h |= Pq → PPq, for arbitrary h such that m ∈ h. Then, we pick any such h and reason just as in the previous case to derive contradiction with (b). So, there is an asymmetry between P and F in STRL; in particular Fφ → FFφ is an STRL consequence of Pφ → PPφ (consult deﬁnition 7.25), but the converse does not hold. However, it is enough to consider all STRL models based on F for the two notions of validity to coincide. Fact 7.6. Let F = hM,

7.10 Postsemantic consequence in STRL

This section contains the results concerning preservation of STRL-truth-at-context which we have established in (Malpass and Wawer, 2012). Firstly, if one accepts a properly semantic notion of consequence (i.e. preservation of truth at an index), then STRL consequence is just Ockhamist consequence. After all, STRL postsemantics uses Ockhamist semantics as its base. Deﬁnition 7.23 (Ockhamist semantic consequence). Let Γ be a set of sentence. We say that sentence φ is a semantic consequence of Γ, Γ |= φ, iﬀ for every model M:

∀m∀h (∀ψ ∈ Γ M, m/h |= ψ ⇒ M, m/h |= φ).

245 APPENDIX

The first advantage of this approach is that it preserves all the classical rules of inference. Moreover, the notions of consequence and implication are very closely connected: φ |= ψ iff |= φ → ψ. Another benefit of this definition, sometimes neglected in the discussion of supervaluationism, is that it preserves a natural analogy between the notions of truth and consequence. If “super-truth” is defined in terms of truth at every precisification, then “super-consequence” should be derived from consequence at every precisification. We should notice though, that such definition of consequence is not always accepted in the context of supervaluationism. The particularly common alternative is the preservation of postsemantic truth at context: Definition 7.24 (Supervaluational consequence). Let Γ be a set of sentences. We say that a sentence φ is a supervaluational consequence of Γ, Γ||−S φ, iff for branching model M: ∀m(∀ψ ∈ Γ M, m||−S ψ ⇒ M, m||−S φ). In the context of branching, such definition was endorsed by Thomason(1970). Someone who believes that this is an accurate supervaluational definition of consequence might urge that the TRL version should be analogous: Definition 7.25 (STRL consequence). Let Γ be a set of sentences. We say that the sentence φ is an S TRL-consequence of Γ, Γ||−strlφ, iff for every S TRL model M:

∀m(∀ψ ∈ Γ M, m||−strlψ ⇒ M, m||−strlφ).

An interesting fact about STRL-consequence is that it behaves considerably better than its supervaluational cousin. To see it, let us remember that Timothy Williamson (1994) showed that the supervaluational notion of consequence leads to unnatural conclusions, even to, “in a sense a violation of classical propositional logic” (1994, 151). He shows that the inference rules of contraposition, conditional proof, argument by cases, and reductio ad absurdum are not valid given the supervaluational notion of consequence. Pleasingly, our TRL-supervaluationism does not suffer from these problems. The TRL actually comes to the rescue, as we shall go on to demonstrate. Let me remind one of the arguments against supervaluationism to give a sense of the problem, and to see how we escape from it. Contraposition says that if ψ is a consequence of φ, then ¬φ is a consequence of ¬ψ. The supervaluational version of this argument does not always work though. For instance, if we substitute F p for φ and F p for ψ, then F p||−S F p holds while ¬F p||−S ¬F p does not hold. The problem does not arise in STRL. Unlike Thomason’s supervaluationism, we allow that ∃M∃m (M, m||−strlφ and M, m||−/strlφ). This is a consequence of the second disjunct in the clause defining STRL-truth; the sea battle might only happen on the TRL and no other branch. In this situation, the prediction is history-independently true, but the sea battle is not inevitable. It means that in our account Williamson cannot make the first step of his argument (φ||−strlφ does not hold in general). The contingent sea battle is a counter-example to this schema. The way of avoiding each of Williamson’s three other attacks is the same; each time it is the difference that the TRL makes which helps out (see section 4.5 79 of (Williamson, 1994), especially pages 151–152 for descriptions of the other arguments).

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Another problem for the supervaluationist notion of global consequence was raised by Tweedale(2004) who noticed that φ, ^ψ||−S ^(φ ∧ ψ) is a valid inference rule in standard supervaluationism. It is a bad result since if we substitute F p for φ and ¬F p for ψ we get that F p, ^¬F p||−S ^(F p ∧ ¬F p). So, we can infer a logical impossibility from logically possible set of assumptions. Again, STRL avoids this problem, φ, ^ψ||−strl^(φ ∧ ψ) is false. To see that, just imagine that φ is a future contingent true at m ∈ TRL, then we have that m||−strlφ, m||−strl^¬φ but m6|=TRL^(φ ∧ ¬φ). The fact that our STRL defends itself against these arguments seems to be a benefit of our theory. However, despite STRL consequence gets around the problems that the ordinary supervaluationist consequence suffers, I would not recommend this notion of consequence. My reason stems from the observation that even though the arguments, as stated in the literature, do not harm our proposal, they can be reformulated in an arguably less persuasive, but still quite severe form. Our postsemantics avoids the standard problems of supervaluationism because it works at some moments in non- supervaluational manner. Nonetheless, we need to remember that at all moments outside the TRL the semantics is entirely supervaluational. One might try to exploit its partially supervaluational character and reconstruct the arguments in a moment-oriented fashion: Definition 7.26 (STRL consequence at moment m). Let T = hM, <, TRLi be a TRL- structure and let m ∈ M. We say that φ is an S TRL consequence of a set of sentences strl Γ in structure T at moment m, Γ||−m,Tφ, iff for every TRL-model M = hM, <, TRL, Vi based on T we have that

∀ψ ∈ Γ M, m||−strlψ ⇒ M, m||−strlφ.

It is not the notion often met in the literature, but one can give it some intuitive reading. It encodes a postsemantic consequence at a given point of a structure, independently of how the valuation function works. Since our language is tensed and the structure might differ from point to point, this notion might be helpful. For example, at any maximal moment m of a structure T, we have that for any φ and ψ, strl strl F(φ ∨ ¬φ)||−m,Tψ, F(φ ∨ ¬φ)||−/ ψ in general. Importantly, this notion of semantic- consequence-at-a-moment enables us to exploit the supervaluational characteristics of non-actual moments and establish moment-dependent arguments à la Williamson or strl Tweedale. For example, for some m outside TRL of a structure T: φ||−m,Tφ, but strl strl ¬φ||−m/,T¬φ and F p, ^¬F p||−m,T^(F p ∧ ¬F p). These arguments do not seem to be as strong as their more general versions, but I still find them worrisome. To sum up, the notion of Ockhamist consequence recommends itself since it is natural and free of logical charges. However, even the postsemantic STRL consequence is untouched by arguments against supervaluational consequence which is a fair result. It is important to stress that it is the existence of TRL that takes the wind out of the critics’ sails. Only a rather unusual moment-dependent modification of the notion of postsemantic consequence raises some worries.

247 APPENDIX

7.11 Truth operator(s)

There is always considerable amount of worry while introducing “it is true that” and “it is false that” operators. In case of our theory, besides of a danger of semantic paradoxes, we need to deal with uncertainty of the intended interpretation of these operators. Since there is a semantic and a postsemantic level of analysis, there are two alternatives notions of truth at play. Let me first discuss the option that attempts to mimic the postsemantic notion of truth at the context on the properly semantic level: Definition 7.27 (“It is true that φ” is true at m/h). Let φ be a sentence and M a TRL- model, then: M, m/h |= Trφ iff M, m/TRL |= φ, if m ∈ TRL, or ∀h0(m ∈ h0 ⇒ M, m/h0 |= φ), if m < TRL. We define “it is false that” (Fl) as Flφ := Tr¬φ; and “it is undetermined that” (Und) as Undφ := ¬Trφ ∧ ¬Flφ. There are some seemingly pleasing consequences of such defined notions. For example we have that for every m ∈ TRL and every sentence φ, m||−strlTrφ ∨ Flφ, while for every future contingent evaluated at m < TRL, we have that m||−strlUndφ. However, the controversial outcomes far outgrow the merits. The first set of problems has to do with the consequence relation. I have just recounted that STRL, contrary to classical supervaluationism, does not generate some troublesome results. Only a moment-dependent notion of consequence has proven to be problematic. However, if we endow our basic-level, Ockhamist semantics with the above-defined truth operator, we import the problems of the supervaluational consequence to the STRL consequence. For example φ||−strlTrφ always holds while ¬Trφ||−strl ¬φ does not; similarly, φ, Und¬φ||−strlUnd(φ ∧ ¬φ) is true. Another set of problems is generated by the fact that Tr behaves differently at actual and non-actual moments. The result is quite bizarre. For example, for some contexts out of the TRL, the following sentences are not true:

• Tr(φ ∨ ψ) → (Trφ ∨ Trψ); • φ ↔ Trφ; • Trφ ∨ Tr¬φ;

• FTrφ → TrFφ; • Trφ ∧ HTrFφ. All of them behave well at any m ∈ TRL, however even at some m ∈ TRL we have that m||−strl^(φ ∧ ¬Trφ). All these oddities ensue because the meaning of truth operator is different at actual and non-actual moments. To avoid this phenomenon, we decided to define the truth operator in a unified manner throughout the whole domain:

248 APPENDIX

Definition 7.28 (“It is true that φ” is true at m/h). Let φ be a sentence and M a TRL- model, then: M, m/h |= Trφ iff M, m/h |= φ. Operator “It is false that φ” is defined as Flφ := ¬Trφ. It is a definition proposed already by Thomason(1970) and it is hard to imagine a more straightforward and intuitive one. It simply says that the truth conditions of Trφ are exactly those of φ. By the same token, “it is false that” behaves just as negation (in contrast to definition 7.27 where falsity of a sentence is identified with the truth of its negation, rather than negation of its truth). Under this semantics of Tr, the virtues of STRL are restored. As far as the STRL- consequence is concerned, the incorporation of the truth operator is no longer harmful. In particular, none of Williamson’s or Tweedale’s counterarguments apply to STRL consequence, so it regains its advantage over traditional supervaluationism. By the same token, the intuitive validities are restored. All of the sentences above, Tr(φ∨ψ) → (Trφ ∨ Trψ), φ ↔ Trφ, Trφ ∨ Tr¬φ, FTrφ → TrFφ, and Trφ ∧ HTrFφ are always true.

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