From Language to Thought: on the Logical Foundations of Semantic Theory

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From Language to Thought: On the Logical Foundations of Semantic Theory

Dissertation

Presented in Partial Fulllment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University

Giorgio Sbardolini, MA

Graduate Program in Philosophy

The Ohio State University

2019

Dissertation Committee:

Stewart Shapiro, Advisor Øystein Linnebo William Taschek Neil Tennant © Copyright by

Giorgio Sbardolini

2019 Abstract

Sentences have meanings: the things we say, and the things we believe. Semantics is the theory of meaning, and thoughts, i.e. the meanings of sentences, are among the objects of semantic theory. But what are meanings? What is the place of meaning in the natural world? In the discussion below, I shall motivate formal constraints on the logical and metaphysical foundations of semantic theory.

Some philosophers have suggested that semantics is a piece of modal metaphysics.

The modal approach to meaning covers a lot of empirical and conceptual ground, but imposes a sharp separation between epistemology and metaphysics. The modal approach ultimately fails, since the mechanism invoked to recombine the metaphysical with the epistemic leads to inconsistency. The lesson is that semantics is not modal metaphysics, and that sameness of meaning is a hyperintensional notion.

Other intensional paradoxes follow more generally from assuming that thoughts can be individuated to a more or less precise degree, i.e. as the only thoughts having a certain property. These assumptions are often very plausible. Some contemporary accounts of the intensional paradoxes save consistency at the cost of rejecting these plausible assumptions. This puzzling situation leads naturally to wonder about the conditions for referring to thoughts: how do we individuate them?

Reference to abstract objects may be achieved by singular terms whose semantic properties are established by abstraction. On this proposal, reference is explained by a criterion

ii of identity for the referents, which is in turn established by an abstraction principle on which sameness of meaning is equated with hyperintensional equivalence. Such notion cannot be as ne-grained as contemporary accounts of structured propositions take it to be, on pain of ruling out a compelling pragmatic account of redundancy in the use of language: speakers naturally take certain pairs of sentences to have the same meaning, so that syntactic complexity does not inevitably amount to semantic dierence.

Any plausible hyperintensional notion of equivalence faces, in higher-order logic, the

Russell-Myhill paradox. However, consistency can be restored by a dynamic understanding of abstraction, on which the truth conditions of identity statements are dened in- crementally. From this perspective, there are dierences between what is absolutely true, i.e. for any renement of the identity relation, and what is potentially true, i.e. for some accessible renement. This distinction tames Russell-Myhill while still leaving room for hyperintensionality, and accommodates plausible assumptions about the individuation of thoughts.

On the resulting picture, thoughts are “shadows of sentences”, to use an image of W.

V. O. Quine, and quantication over thoughts is understood predicatively. This is the logic and metaphysics for the foundations of semantics.

iii Acknowledgments

I am grateful to many people who made my time as a graduate student an excellent experience, both intellectually and personally. Special thanks to my adviser, Stewart Shapiro, who has always been insightful and supportive. I have never left Stewart’s oce after a philosophical discussion feeling that no progress had been made, and I have always had the encouraging awareness that the development of my research was my responsibility, whether the mistakes and blunders, or the achievements. It is a privilege to be Stewart’s student.

Most of the research I have done has been provoked by reection on topics to which members of my dissertation committee have spent many years contributing. Although it is sometimes said that there is no progress in Philosophy, there is no denying that the present work builds on top of recent scholarly conversations in which Øystein Linnebo,

William Taschek, and Neil Tennant have been primary participants. I strive to further develop this conversation—hopefully in directions that they are sympathetic to. I want to express my gratitude for your guidance and mentoring, and for the time and energies you have dedicated in helping me.

I have beneted from conversation with many others, in many dierent ways: some with challenging questions, some with advice while I was writing, some for help on earlier drafts, all from whom I have learned something in the course of a few intense and excit- ing years. I would like to thank Francesco Berto, David Braun, Ben Caplan, Ben Lennertz,

iv Tristram McPherson, Chris Pincock, Carl Pollard, Graham Priest, Craige Roberts, Richard

Samuels, Kevin Scharp, Gabriel Uzquiano, Tim Williamson, the faculty members and the graduate students of the Ohio State Philosophy department, the faculty members and the graduate students of the Ohio State Linguistics department, and audiences at the univer- sities of Milan, Virginia Tech, Connecticut, Stockholm, Munich, Padua, Oslo, Amsterdam,

St. Andrews, and Bualo, where parts of this work have been presented, invariably to receive terric feedback. I also would like to thank anonymous reviewers who have given helpful comments on parts of this dissertation.

I gratefully acknowledge the generous support of the Ohio State Graduate School for a Presidential Fellowship for the year 2019, and the combined support of the Ohio State

Graduate School for a Matching Tuition Award and of the ConceptLab at the University of Oslo for a Collaborative Fellowship for the year 2017.

Finally and most importantly, heartfelt thanks to my friends, in America and in Eu- rope, and to my family, for help in life beyond Philosophy.

v Vita

April 21, 1988 ...... Born - Monza, Italy

2010 ...... B.A. Philosophy, Università degli Studi di Milano 2012 ...... M.A. Philosophy, Università degli Studi di Milano 2013-present ...... Graduate Teaching Associate, Ohio State University

Publications

Research Publications

Two-dimensional Paradox. Australasian Journal of Philosophy, 2018. with S. Negri: Proof Analysis for Lewis Counterfactuals. The Review of Symbolic Logic, 9, 1: 44–75, 2016.

Fields of Study

Major Field: Philosophy

vi Table of Contents

Page

Abstract ...... ii

Acknowledgments ...... iv

Vita...... vi

1. Introduction ...... 1

1.1 Propositional variables ...... 5 1.2 Propositional attitudes ...... 15

2. Two-dimensional paradox ...... 20

2.1 Kaplan’s principle ...... 21 2.2 Two-dimensionalism ...... 22 2.3 Diagonals and Antidiagonals ...... 27 2.4 Beyond intensionality ...... 37

3. Prior’s gambit ...... 43

3.1 Prior’s paradox ...... 43 3.2 Prior’s theorem ...... 48 3.3 A most paradoxical argument ...... 54 3.4 Coherentism about thoughts? ...... 59

4. Reference to Thoughts ...... 63

4.1 Puzzles about quantication ...... 63 4.1.1 Background ...... 66 4.2 The dilemma about mathematical truth ...... 71 4.3 A Metasemantic Benacerraf ...... 73

vii 4.4 Descriptive Singular Terms ...... 77 4.4.1 A Fregean argument ...... 81 4.5 “Standard names” ...... 83 4.6 Begrisschrift §3...... 86

5. Wittgenstein’s fundamental thought ...... 95

5.1 Structured propositions ...... 95 5.2 On redundancy ...... 100 5.3 Principles of conversation ...... 102 5.4 A look ahead ...... 110

6. Metasemantic Predicativism ...... 112

6.1 Aboutness paradox ...... 113 6.1.1 Neither truth nor sets ...... 114 6.1.2 The structure of reality ...... 118 6.2 Predicative logic ...... 124 6.2.1 Dynamic abstraction ...... 128 6.2.2 Revised Prior’s theorem ...... 135 6.3 Addendum ...... 139

7. Conclusions and further work ...... 152

Bibliography ...... 158

viii Chapter 1: Introduction

An utterance of the English sentence ‘Snow is white’, relative to a context, expresses the thought that snow is white. That thought is what I come to believe, if you tell me that snow is white. We can describe our agreement that snow is white, by saying that our mental states have that thought in common. Thoughts are universals of semantics, and they can be employed to account for the contents of sentences, various aspects of our mental lives, and epistemic notions like agreement and disagreement.

For these purposes, thoughts are assumed to be true or false with respect to how things are in the world: there is thus a direct connection between the world on one side, and meaning and belief on the other. But if thoughts are true or false, it follows that at least some logical relations hold among them: material equivalence (being both true or both false), truth preservation (if these thoughts are true, so is this), and satisability

(these thoughts can be true together). What else can be said about the logic of thoughts?

And what are they, that their nature might support such logical structure?

These two questions, about the logic and the nature of thoughts, are the topic of the present work. I shall address them from two sides. First, the problem of paradox. In- tensional paradoxes arise fairly quickly when we have the linguistic resources to discuss formal theories of thoughts, threatening to reduce them to triviality. The logic of thoughts

1 should at least avoid triviality, and yet, the situation here being no dierent than other domains of inquiry troubled by the paradoxes, it is controversial what measures to adopt in response. Secondly, the problem of propositional identity. Material equivalence does not suce for the identity of thoughts: the thought that snow is white is true if and only if the thought that the sky is blue is true. Indeed, they are both true. Certainly however, those are not the same thought. Moreover, the thought that snow is either white or not, and the thought that the sky is either blue or not, are necessarily such that the former is true if and only if the latter is. Indeed, they are both necessarily true. However, they are not the same thought. So neither does necessary equivalence suce, as a condition of identity for thoughts. The quest is for the right notion of equivalence, in order to dene propositional identity. A discussion of the problems of paradox and of propositional identity leads to rich and complex philosophical debates on synonymy, the nature of intensionality and hyperintensionality, and the logic of propositional attitudes.

In my view, thoughts are part of the natural world, and the project of this dissertation falls within the broader project of explaining how thoughts come to occupy a certain place in the world. I now want to give a sketch of the philosophical conception of thoughts that serves as background to my discussion, although I shall not defend it here.

All begins with human agents, communicating with each other by the use of signals in a game of senders and receivers—roughly along the lines of David Lewis (1969). Their actions may at rst be quite chaotic, and the success of communication may result from chance just as often as from understanding. Soon enough though, certain regularities will begin to crystallize, driven by the agents’ ability to learn from experience, and by a widespread interest for a successful outcome to their linguistic interaction. Seman- tic regularities, such as that signal α carries message p, are emergent properties of a

2 self-organizing system of communicative strategies dened on a population of linguistic

agents. Signals are utterances, namely the minimal syntactic units that carry informa-

tion. Thoughts are the messages carried by the signals. As the association of a signal to

its message stabilizes in a robust convention, it becomes possible to dene thoughts by

abstracting on equivalent sentences: intuitively, sentences that in the same situation may

be used to say the same thing.

The abstraction of thoughts is the basis of my response to the problems of proposi-

tional identity and of paradox. It is an application of the technology of abstraction prin-

ciples, developed by Crispin Wright (1983) and many others from ideas rst discussed by

Gottlob Frege in Die Grundlagen der Arithmetik (1884). Thus the ensuing debates on abstraction principles have rich and profound philosophical consequences, demonstrating the fruitfulness of these ideas. The resulting conception of thoughts merges two powerful views: that conventional meaning derives from regularities in the use of language, and that abstract objects can be dened from meaningful linguistic expressions. Here I shall focus on the second part, but in the background lies a generalization of Frege-inspired abstractionism (which tends to assume that linguistic expressions have conventional meanings). The broader philosophical conception of abstract objects is naturalistic: it is rooted

in the linguistic and cognitive practices of human agents, and on our understanding of

what makes communication possible.

The methodological and conceptual boundaries I have just described are not widely

shared nor rmly held among philosophers. Theories of thoughts appear to be motivated

by a rich variety of philosophical commitments, often depending on the philosopher. For

this reason, chapters 2, 3, 4, and 5 clear the ground from the most popular existing theories.

3 The positive proposal is put forward in chapters 4, 5 and 6. In more detail, the plan is as follows:

• Chapter 2 is dedicated to the intensional view, on which thoughts are sets of possi-

ble worlds. Its most sophisticated version, Stalnaker’s (2006) and Chalmers’s (2006)

Two-dimensionalism, fails by reason of paradox. I conclude that propositional iden-

tity cannot be as coarse-grained as necessary equivalence: the logic of thought

ought to be hyperintensional.

• Chapter 3 discusses intensional paradoxes more generally, and explores a view in-

spired by Williamson’s (2013) abductive methodology in Metaphysics. On this view,

consistency is restored despite Prior’s (1961) paradox, but at the cost of a surprising

result known as Prior’s theorem. I try to raise doubts about this view, which gives

answers to questions that logic ought to leave open.

• Chapter 4 presents a challenge, somewhat reminiscent of Benacerraf (1973), to ex-

plain how reference to thoughts is possible. I develop an account of reference to

thoughts on which sentences are “canonical names” of thoughts, based on an ab-

straction principle according to which two sentences ‘α’ and ‘β’ express the same

thought just in case they are hyperintensionally equivalent.

• Chapter 5 shows that theories of structured propositions, on which propositional

identity is characterized as strict equivalence (Church, 1974), are not empirically

adequate. Consequently, hyperintensional equivalence is neither as coarse-grained

as necessary equivalence, nor as ne-grained as strict equivalence.

4 • Finally, chapter 6 presents a predicative account of the Russell-Myhill paradox (Rus-

sell, 1903; Myhill, 1958), of which an original interpretation is defended. The solu-

tion is motivated by a dynamic understanding of abstraction (Linnebo, 2018). Its

value over alternative views is that it allows the theory of thoughts to be both con-

sistent and hyperintensional. Further benets are that Prior’s paradox is no longer a

threat, and that Prior’s theorem is reduced to triviality. An Addendum summarizes

the logic of thought, including a semantics and (sketches of) proofs of soundness

and Henkin completeness.

So the solution I present to the problem of paradox is predicativity, which is motivated by abstractionism and supported by many ensuing benets. While I will not give a - nal answer to the problem of propositional identity, I shall reduce it to a “Goldilocks” problem: propositional identity, characterized as hyperintensional equivalence, is intermediate between necessary and strict equivalence. The search space for a nal answer is thus properly delimited, and some insight into how further progress could be made is given. Conclusions and further work are discussed in chapter 7.

In the rest of this Introduction, I shall set up various denitions and stipulations about the formalism that will hold throughout, and introduce the problems of paradox and of propositional identity in more detail.

1.1 Propositional variables

Thoughts are semantic objects, often also called ‘propositions’, or ‘sentential contents’ in the literature, as I shall call them myself from time to time for consistency. The word

‘thought’ has a mentalistic avor, for although thoughts are abstract objects, they are obviously related to the mind. In this work, I will not discuss details of this relation,

5 nor be concerned with the stylistic dierences between the terms ‘thought’, ‘proposition’,

‘sentential content’, and so on.

I will also not pay attention to context-dependence or to non-declarative sentences, and so the relative qualications will simply be omitted everywhere. I shall make the idealizing assumption that the discussion takes place in the sterilized setting of higher- order logic. After all, context-dependence and illocutionary force are matters of language use, and the paradoxes I shall discuss arise regardless of anybody actually saying anything.

I assume throughout that thoughts are in some sense objective, and in some sense part of reality. According to Prior (1971, 6), this perspective may be traced back to Bernard

Bolzano. In particular, my work is very inuenced by the early Russell of Principles of

Mathematics (1903), one of the main protagonists in the intellectual tradition that began with Bolzano. This is in part because Russell’s work took place long before many choices had been made, in particular about higher-order logic and the logic of attitudes, that have allowed us to isolate what are now well-understood extensional logic principles, but have also hidden from view an intensional perspective on logic. I shall follow Russell, Prior, and others, in assuming that thoughts are objective and real at least insofar as they may be quantied over by expressions of the appropriate type.1

The possibility of quantifying over thoughts immediately allows for the formulation of famous paradoxes like the Liar:

When a man says “I am lying,” we may interpret his statement as: “There is a proposition which I am arming and which is false.” ... in other words, “It is not true for all propositions p that if I arm p, p is true.” The paradox

1Later in his career, Russell became suspicious of abstract entities, and nally came to reject thoughts in Russell (1918). It is also worth mentioning that Frege (1918) too regarded Gedanken (the forerunners of present-day thoughts) as in some sense objective and real. However, the further commitment that we can quantify over them doesn’t square with some of Frege’s other pronouncements (e.g. about the notorious “concept horse puzzle”), and is clearly not an option in his thoroughly extensional logical language. Some of the subtle questions involved here are taken up below, mostly in chapter 4.

6 results from regarding this statement as arming a proposition, which must therefore come within the scope of the statement. (Russell, 1908, 224)

Let us call this man Epimenides, and consider the circumstances in which he is arming only that not for all p, if Epimenides is arming p, then p is true. Is Epimenides arming something true on this occasion? If he does, then something he is arming is not true.

But since we are assuming that he is arming nothing else, that which he is arming must not be true. Hence he is not arming something true. So everything he is arming is true, i.e. that which he is arming. So Epimenides is arming something true if and only if he is arming something not true. There are many variations on this theme.

Interestingly, Russell mentions no names of sentences, as has become common practice in presenting the Liar paradox since Tarski (1936). Instead, he uses a variable bound by a quantier, intended to range over thoughts, or propositions. The dierence is not insignicant: inferences that lead to contradiction appear to be not those licensed by the truth predicate, but by the laws of quantication. Moreover, while the ability to refer to sentences does not seem to raise any special philosophical challenge, reference to thoughts may look suspicious to nominalistically inclined philosophers, of which Quine

(1970) is perhaps the main example. I shall discuss some of these worries below. Paradoxes engendered by quantication over thoughts are known as the paradoxes of intensionality

(Tucker and Thomason, 2011; Uzquiano, 2015b), and how to solve them is one of the main questions around which this dissertation is organized.

There are various options to formalize Russell’s informal reasoning. Following Prior

(1961) and Kaplan (1995), we may consider a propositional variable ‘p’ and a predicate

‘Q’ that takes a sentential complement. The interpretation of ‘Q’ may be, intuitively,

7 ‘Epimenides is arming that’—but any predicate of the right type would suce.2 Then the formula ‘ p Qp p ’ expresses the thought that for any thought, Epimenides is arming it only∀ ( if it→ is¬ not) the case. Contradiction follows from assuming that such thought is the case, and the only one armed by Epimenides. This is the simplest version of the paradox, often called Prior’s paradox or the Intensional Liar paradox.

Propositional variables, such as ‘p’, look unusual. Apparently, they may both be bound

by a quantier ( p...p...) and combine with a sentential operator ( p). Usually, it is indi-

vidual variables∀x, y, ..., that may be bound by the rst-order quantiers,¬ while formulas

combine with the sentential operators (e.g. the connectives). It might seem just a confu-

sion to mix things up.

However, there is no confusion. Propositional variables are variables occurring in sen-

tence position, while the individual variables of rst-order logic are nominal variables, or

variables occurring in name position. The syntactic dierence between the two is a matter

of which positions they are allowed to occupy in a sentence, aecting their combinatorial

properties. Individual variables combine with predicates to form sentences, and sentences

combine with sentential operators to form more complex sentences. In contrast, variables

in sentence position can combine with sentential operators. There are some options to

make sense of them, but I intend to set aside doubts about propositional variables in this

Introduction.

A rst reaction to accommodate worries about propositional variables is to insist that

they are just a special sort of rst-order variables. One then needs at least a truth pred-

icate ‘is true’ to combine with propositional variables, so that the sentential compound

2Strictly speaking, ‘Q’ is a compound expression that includes at least temporal and agential information. But since the subject and the time do not matter at all in the discussion below, I shall treat it for simplicity as a monadic predicate of expressions of sentential type, and call its denotation an attitude operator.

8 ‘p is true’ may sensibly occur in the scope of the connectives. Intuitively, this strategy amounts to analyzing ‘Grass is green’ as ‘The thought that grass is green is true’, where

‘The thought that grass is green’ is a noun phrase denoting a thought. The analysis is not very illuminating and in some sense gets things backwards, but need not be inco- herent. However, as Asher (1990) points out, an intriguing feature of the paradoxes of intensionality is that they appear to depend neither on truth-theoretic nor on disquota- tional principles. Bringing back all that machinery, part of the special interest of these paradoxes would be lost.3

I shall avoid the truth predicate and take the more plausible view that propositional variables are not rst-order. The language I shall use is the language of Russellian Inten- sional Logic (Church, 1973b). The only basic type is e, for individuals. The only formation rule for higher types is that, for 0 i j, νi, ..., νj are types only if νi, ..., νj is a type.

The latter is the type of functions≤ from≤ νi, ..., νj to thoughts, entities( that Russell) called propositional functions. The type of thoughts is , i.e. setting i j 0. The language of

RIL contains countably many individual constants() (a, b, ...), and=n-ary= individual predicates (F, G, ...), to which countably many propositional variables (p, q, ...), and countably many (monadic) attitude predicates (P,Q, ...) and attitude predicate variables (X, Y, ...) are added. Metavariables for closed formulas (sentences) are Greek letters from the be- ginning of the alphabet, and metavariables for expressions of functional type are Greek letters from the middle of the alphabet. The sentential operators are negation , implication , and metaphysical possibility , the rest being dened as usual. More¬ will be added in→ the course of the discussion. n

3It is important to notice that the correspondence between truth-theoretic and intensional paradoxes is tight: it is straightforward to dene a truth predicate given a relation E between a formula and the thought it expresses, and quantiers over thoughts: just set ‘T ϕ’ as ‘∃p(ϕEp∧p)’. Unfortunately, I have not worked out the details of the truth predicate implicitly dened by the account of the quantiers I present below.

9 Elements of the various types are assigned semantic values from three mutually dis-

joint domains: an arbitrary non-empty set E of individuals, and sets T and F of truths

and falsehoods respectively. A domain of worlds W is added for a Kripkean interpretation

of the modal quantier (Kripke, 1963), with an ordinary S5 semantics (or weaker systems,

should one so prefer). I assume that thoughts exist necessarily, and that each world has

the same domain (the world-relativity of the interpretation function aects only which

properties things contingently have). Expressions are assigned a semantic value relative

to (a world w and) a variable assignment σ: for each world, expressions of type e are

interpreted in E, expressions of type are interpreted in Dp T F , and expressions

Da of functional type a are interpreted() in D(a) Dp f D= a ⋃ Dp , where a may be any type. Interpretation( ) is then extended consistently= = { to∶ all formulas:↦ } for example, requiring that if the interpretation of α is in T , the interpretation of α is in F , and so on. Notice that the connectives are therefore relations on Dp. By doing¬ things this way,

the logic could be described as a calculus of relations, somewhat along the lines of the

Principles of Mathematics, and not so much as a truth-functional calculus—indeed, there

are no truth-functions in RIL. (For complete details of the system sketched here, see the

Addendum.)

Importantly, types in RIL are in general not extensional. In other words, Extensional-

ity is not an axiom, and indeed holds restrictedly to propositional functions of individuals,

thoughts, and propositional functions (here I follow Church’s trick to use boldface expres-

sions to stand in for expressions of any syntactically appropriate type):

(ext) xi, ..., xj ϕ xi, ..., xj ψ xi, ..., xj ϕ ψ

∀ ( ( ) ↔ ( )) → = with i, j 1. ext states that materially equivalent propositional functions of the same arguments≥ are identical. It is surely a simplication to hold that propositional functions

10 of thoughts are extensional, but importantly, ext fails for the case in which i j 0, since materially equivalent thoughts are need not be identical. Failures of ext are= the= hallmark of intensional logics. Indeed, Russell and Whitehead (1927, §C*20) dene ‘intensional’ as not extensional.

There are other ways to study intensional logic. In Montague’s Intensional Logic

(Montague, 1974; Gallin, 1975), thoughts are intensions, i.e. functions from possible worlds to truth values—or, equivalently, their characteristic sets. Quantication over thoughts may be simulated by quantication over sets of worlds. In this framework ext holds for all types, since intensions are set-theoretic entities: thus the type theory can’t distinguish thoughts that are true at all and only the same worlds. As Muskens (2007, 631) points out, to call this logic ‘intensional’ is a bit of a misnomer, for the resulting understanding of intensionality is not quite as inclusive as what is given by the Principia Mathematica denition. Still, some of the paradoxes of intensionality may be formulated in MIL: for instance, Kaplan’s (1995) paradox is what Prior’s paradox looks like in MIL, as we shall see in the next chapter. Others, however, lose bite: as we shall see in chapter 6, MIL comes with an account of propositional identity that undermines the intuitive grip of one of the premises of the Russell-Myhill paradox (Russell, 1903; Myhill, 1958). It is then easy to reject it, but this is only because the logic makes fewer distinctions than we would like to make.

A more general approach is Church’s Intensional Logic (Church, 1951, 1993), also called the Logic of Sense and Denotation, which is what results from a Fregean extensional logic, like the logic of Grundgesetze, expanded in order to allow for terms referring

11 to “senses” and for quantication over them.4 The intensional paradoxes can be straightforwardly formulated in LSD. Compared to the perhaps more orthodox (certainly more

Fregean) LSD, the system RIL I adopt here can be seen as a simplication, in which a distinction between sense and denotation is abandoned. This allows to considerably simplify the syntax, avoiding Church’s “delta” operator ∆ that links senses to their denotations, and the notion of predication, avoiding the bizarre Fregean doctrine that expressions in the scope of attitude operators denote their senses and not their ordinary denotations. All expressions of RIL denote their “senses”, and there is only one kind of predication.5

Having presented the formal system and some alternatives, let us return to initial doubts about propositional variables. Even though the appearance of intensional paradoxes in LSD shows that they cannot quite be blamed entirely on the availability of devices for variable binding in sentence position (for there are none in LSD), one could still hold that quantication over thoughts is somehow misguided. For example, English and other natural languages have nothing like the propositional quantiers of Prior and Russell.

Such radical departure from home may look suspicious to some (Hodes, 2015, 384).

The easiest way to make sense of variables in sentence position is to think of them as 0-place predicate variables. The quantier in ‘ p Qp p ’ is thus a second-order

quantier, and ‘Q’ a third-order predicate. In an intensional∀ ( → ¬ second-order) language we

have variables in predicate position, varying on relations with any number of argument

4For a general presentation of Church’s LSD, see Klement (2002) and Anderson (1998). 5A qualication is needed, for many have found Frege’s disjunctive doctrine of predication not at all bizarre, contrary to what I said, but rather compelling. Part of the issue depends on the signicance of so-called “slingshot” arguments. An example is a collapsing argument intended to show that the semantic values of sentences are truth values, if anything at all. Church (1956) claims that such an argument can be extracted from Frege (1892). A similar argument is also sketched by Gödel (1944). If the slingshot works, Frege’s doctrine commands acceptance. However, the signicance of slingshot arguments is a matter of considerable controversy, for they appear to rest either on a rather loose notion of synonymy, or on an analysis of descriptions as singular term (Stoutland, 2003; Neale, 1995). I do not make either assumption.

12 places. We may then think of sentences as 0-place predicates, over which we quantify. A net advantage of the second-order interpretation of propositional variables is that Prior’s paradox is shown to arise directly in second-order logic with a single third-order predicate

(and Russell-Myhill arises directly in third-order logic with a single fourth-order predicate). So these paradoxes are of great interest and generality. Furthermore, the second- order interpretation shows that anybody prepared to theorize in a higher-order language should nd no reason to complain about the language of RIL. Doubts about the intelligibility of propositional variables become doubts about the intelligibility of higher-order variables, but higher-order logic has been powerfully defended by Shapiro (1991), Boo- los (1999), and others. As to the absence of propositional variables in English, and the alleged unintelligibility that would result from this, Williamson (2003, 459) remarks that, although second-order formulas are often only misleadingly glossed by English sentences

(e.g. ‘ F.F Socrates ’ as ‘There is something Socrates is/does’), there is still no reason why one∃ couldn’t( learn) a second-order language by immersion, just as every other language can be learned. If a language can be learned, it is intelligible.

The second-order reading of RIL is the easiest way to make sense of propositional variables, but it is not the only way. Another option is to interpret the quantiers in sentence position substitutionally, rather than objectually. This move would indeed appease the nominalist. The distinction in this case concerns two dierent kinds of truth conditions for sentences containing the quantiers:

OQ: If quantiers are interpreted objectually,

‘ xφx’ is true i some element of the domain D belongs to the extension of φ, which

is∃ a set of elements of D.

13 SQ: If quantiers are interpreted substitutionally,

‘ xφx’ is true i some expression t of the substitution class C of x is such that the

result∃ of replacing t for x in ‘ xφx’, namely ‘φt’, is a true sentence.

∃ OQ denes truth relative to a domain D of objects, while SQ denes truth relative to a class

C of linguistic expressions and to the truth of some other sentence. Use of ‘ ’ according to

OQ commits one to the existence of objects, whereas use of ‘ ’ according∃ to SQ commits

one at most to a set of expressions (Quine, 1961). In eect, on∃ the substitutional reading

the formula ‘ p Ap p ’ does not literally say that nothing Epimenides says is true, but

rather that there∀ ( is no→ expression¬ ) in the substitution class of p such that a true sentence is

formed by replacing it for p in ‘ p Ap p ’. A classic discussion, and defense against

criticism of the substitutional reading∀ ( is→ Kripke¬ ) (1976).

Notice that if D and C do not match in a certain way, substitutional and objectual

quantication may come apart. This may happen if there are empty terms in C, like sin-

gular terms with an empty extension (e.g. ‘Sherlock Holmes’), or if there are unnamed

objects in D (objects that are not the extension of any singular term). In the rst case

‘ xφx’ may be true on the substitutional reading, even though there is no object that satises∃ φ. In the second case ‘ xφx’ may be false on the substitutional reading, even though there is some object that satises∃ φ. Perhaps the lesson is that quantication is, on the substitutional reading, in some sense language-relative. Perhaps truth is, consequently, language-relative as well.

Should one be willing to face these consequences, it seems that there are ways for propositional variables to make sense even to the satisfaction of nominalist qualms. This should be enough to oset initial doubts about the proper use of the language of RIL.

The paradoxes, of course, do not disappear just by choosing to interpret propositional

14 variables substitutionally. Having agged the substitutional reading as an option, the rest of this work is best understood along the lines of the more standard objectual reading of the quantiers.

1.2 Propositional attitudes

The question how to interpret variables in sentence position is not independent of the question what propositional attitudes are. These are supposed to be relations between agents and their thoughts: the contents of their beliefs, hopes, and claims. Propositional attitude reports are sentences like ‘Epimenides asserts that the sky is blue’, where an agent (Epimenides) is reported as standing in a certain relation (assertion) to a thought

(the thought that the sky is blue).

There are two main kinds of syntactic analyses of attitude reports, at least in Logic and Philosophy—that is, setting aside the syntax of natural languages.6 According to the predicate analysis of propositional attitude reports, a report such as (1a) is analyzed as

(1b):

(1) a. Plato believes that Socrates is honest

b. [S [NP Plato] believes [NP that [S Socrates is honest]]] with a relation, ‘x believes y’, anked by a name on the left, ‘Plato’, and a singular term on the right, ‘that Socrates is honest’ (so the syntactic type of ‘x believes y’ is S/NP NP).

The compound ‘Plato believes y’ is a predicate, hence the name of the analysis. On the predicate analysis the complementizer ‘that’ works in eect as an “abstraction” operator

6For completeness, I should mention that there is at least a third historically important analysis of attitude reports, rst presented in Russell (1906): the multiple relations theory of judgement. This theory is widely regarded as unsuccessful, and is the source of many interpretive issues in Russell’s philosophy.

15 that takes a sentence and yields a singular term denoting the thought expressed by the

sentence.

The predicate analysis is very common among philosophers. King et al. (2014), for

example, frequently adopt it on account of the ease with which it handles inferences such

as the following:

Plato believes that Socrates is honest. Aristotle believes that Socrates is honest. Therefore, there is something that Plato and Aristotle both believe.

On the predicate analysis this is just an instance of -introduction. Variable-binding on

the predicate analysis may be understood as never being∃ on anything but the position of

nominals, whether ‘Plato’ or ‘that Socrates is good’: these are just NPs. So propositional

variables aren’t genuinely occupying sentence position on this analysis. This consider-

ation seems to support a rst-order reading of propositional variables, which, for the

reasons given above, is not desirable.

Another option, rst advocated by Prior (1971, 18), is the prenective analysis of attitude

reports—see also Quine (1960, 216), Tennant (1977, 421-2), and more recently Trueman

(2017). Accordingly, (1a) is analyzed as (2):

(2) [S [NP Plato] believes that [S Socrates is honest]] with a relation ‘x believes that s’ (of type S/NP S) between a name, ‘Plato’, and a sentence,

‘Socrates is honest’. This relation (as well as the syntactic analysis based on it) is called prenective, because it’s both a predicate with respect to its left argument, and a connective with respect to its right argument. As far as the formalization of basic quanticational inferences go, the prenective analysis scores just as well as the predicate analysis, provid- ing we have a means of binding variables that have the logical type of sentences—which,

16 as I said above, we have. My preferred package of views for the interpretation of RIL consists in an objectual reading of the quantiers and a prenective analysis of the syntax of attitude reports.

Propositional attitudes are thus relations between an individual and the semantic contribution of a sentence. If sentences denote truth values, as in ordinary rst-order logic, variables in sentence position range over a domain of two objects, namely the truth values (unless we are prepared to countenance more truth values than those). In a way, this is like assuming that there are only two thoughts to distinguish: a true one and a false one. That is, of course, implausible. We can get more interesting models if we let there be domains of thoughts, and the type of sentences be complicated accordingly. This can be done smoothly in an intensional setting. But the problem of dening domains of thoughts requires an account of propositional identity: to dene the right conditions for sameness of thoughts expressed. This is the old question of dening synonymy for arbitrary sentences.

The reason why distinctions among thoughts should be more ne-grained than the distinctions denable only on the basis of truth and falsity is familiar, and was briey mentioned above. Some sentential operators, like the connectives, are extensional. Ma- terially equivalent sentences can be replaced salva veritate in the scope of dominant occurrences of the extensional operators. Other sentential operators are modal, like , and materially equivalent sentences cannot be replaced salva veritate in the scope of n. So is not an extensional operator, i.e. it cannot be analyzed as a function of the truthn valuen of its sentential argument. Moreover, if sentences in a modal language express thoughts, which of course they do, sameness of truth value does not suce for sameness of thought expressed.

17 However, necessarily equivalent sentences can be replaced salva veritate in the scope of modal operators. Relying on the familiar analysis of modality in terms of possible worlds, one could thus imagine that thoughts are—not truth values, but—distributions of truth values across a space of possible worlds: intensions. However, the reasoning may be repeated. Sentences formed by applying an attitude operator to necessarily equivalent sentences may dier in truth value. For example, ‘Everything is self-identical’ and

‘Arithmetic is incomplete’ are necessarily equivalent, and yet (3a) and (3b) dier in truth value:

(3) a. Nobody ever doubted that everything is self-identical.

b. Nobody ever doubted that arithmetic is incomplete.

The former is probably true, and the latter is certainly false. So necessarily equivalent sentences cannot be replaced salva veritate in the scope of attitude operators—the argument in this form is due to Benson Mates (1950). By parity of reasoning, we should conclude that attitude operators are not modal, i.e. that they cannot be analyzed as functions of the intensions of their sentential arguments, and that necessary equivalence does not suf-

ce for sameness of thought expressed. However, this conclusion is controversial, and is resisted by those who wish to hold on to the identication of thoughts with intensions

(Stalnaker, 1984; Lewis, 1986). The next chapter is dedicated to this view.

Finally, two possible developments should be mentioned. In some respects, the apparent distinctness of the thoughts expressed by necessarily equivalent sentences seems to be one of several logical omniscience puzzles, for it generalizes to the problem of explaining the apparent failure of closure of attitude operators under logical consequence (Parikh,

1987; Tennant, 1977). On the other hand, logical omniscience also appears to introduce new diculties, related to the vague notion of an “easy” consequence (Jago, 2014).

18 Furthermore, the problem of propositional identity might be related to Frege’s puzzles,

about the substitution of co-referential terms in attitude reports (Tichý, 1988; Salmon,

1986). A delightful introduction to Frege’s puzzles is Russell’s:

If a is identical with b, whatever is true of the one is true of the other ... Now George IV wished to know whether Scott was the author of Waverley; and in fact Scott was the author of Waverley. Hence we may substitute Scott for the author of ‘Waverley’, and thereby prove that George IV wished to know whether Scott was Scott. Yet an interest in the law of identity can hardly be attributed to the rst gentleman of Europe. (Russell, 1905, 47)

Again, the question seems to be about apparent failures of truth-preserving substitution in the context of a propositional attitude, in this case for coreferential singular terms.

On the other hand, maybe Frege’s puzzles are a dierent matter, depending on whether thoughts are partly individuated by referential relations (Soames, 2015).

Benson Mates’s argument, the puzzles of logical omniscience, and Frege’s puzzles, have all been considered evidence that the logic of thoughts is hyperintensional, i.e. that

the conditions of identity of thoughts are more ne-grained than necessary equivalence.

However, it is controversial whether and how they might be related, and so I will set

aside logical omniscience and Frege’s puzzles. As I mentioned, some authors do not even

regard Beson Mates’s argument as conclusive evidence for a hyperintensional theory of

thoughts. In the next chapter I shall consider the most sophisticated attempts to reconcile

the intensional view with Benson Mates’s observation. These attempts fail by reason of

paradox.

19 Chapter 2: Two-dimensional paradox

One of the most successful theories of thoughts is the intensional theory, according to which thoughts are sets of possible worlds or, equivalently, intensions, i.e. functions from worlds to truth values (Lewis, 1986; Stalnaker, 1984). Despite its conceptual clarity and far-reaching empirical applications, the possible worlds theory comes with a less than satisfactory resolution of the problem of propositional identity. It is held that any necessarily equivalent sentences express the same thought, but this entails that there is only one necessary truth, and only one necessary falsehood—a predicament that derives from the extensionality of intensions (intensions are identical just in case they have the same members).

Two-dimensional analyses of speech and thought attempt to respond to the diculties within the limits of the possible-worlds framework. However, given some natural assumptions to which two-dimensionalists are committed, the two-dimensional approach is provably inconsistent. It should therefore be jettisoned. The lesson is that “[s]emantics, the study of linguistic meaning, is not a branch of modal metaphysics” (Almog, 2009, 6).

An intensional logic for the foundations of semantics ought to be genuinely intensional, i.e. not extensional—in conformity with the original denition of ‘intensionality’ in Prin- cipia Mathematica (Whitehead and Russell, 1927, §C∗20).

20 2.1 Kaplan’s principle

Kaplan (1995) considers an intuitively plausible bridge principle between modality, thoughts,

and attitudes:

(K=) p q Qq p q

∀ n ∀ ( ↔ = ) For any p, it is possible that p, and only p, is Q-ed. Here ‘Q’ is schematic for a sentential

operator like ‘Epimenides believes that’. In some cases, whether K= has true instances

will be controversial. For instance, some thoughts might be impossible to know, because

they are necessarily false. Set these cases aside. For attitudes that, intuitively, do not

track the truth of their argument, e.g. ‘believe’, ‘assert’, or ‘imagine’, K= appears to be less

problematic. I shall simply take belief as the preferred interpretation for Q.

Kaplan insists that whether or not K= is true for some Q should depend on our meta-

physics, and that K= should not be deemed inconsistent by the logic of attitudes. However, contradiction follows from K= and the widely held assumption that a thought determines a set of possible worlds. For K= entails that there are at least as many worlds as thoughts,

hence at least as many worlds as sets of possible worlds. However, there are more sets of

possible worlds than worlds by Cantor’s theorem. This is Kaplan’s (1995) paradox.

Principles like K=, and even weaker than K=, generate a number of puzzles, all in the

family of the paradoxes of intensionality (Tucker and Thomason, 2011). These paradoxes

are very interesting, because they appear not to originate from a naïve conception of truth

(Asher, 1990), and because their study might lead to a better understanding of the logic

of attitudes (Thomason, 1986).7

7Other paradoxes in this family include Prior’s (1961) paradox, the Russell-Myhill paradox (Russell, 1903), the Knower’s paradox (Kaplan and Montague, 1960), and perhaps Kripke’s paradox about time and thought (Kripke, 2011). The rst item on this list is the topic of chapter 3, and the second of chapter 6.

21 Many have taken the lesson of Kaplan’s paradox to be that there are cardinality problems with the domain of possible worlds (Davies, 1981). Unsurprisingly, the paradoxes of intensionality crop up in two-dimensional semantics as well (Whittle, 2009), but I shall present an inconsistency derived from K= and assumptions which two-dimensionalists are independently committed to. This conclusion does not turn on cardinality considerations.

So two-dimensional analyses of speech and thought face a particularly strong challenge, and they don’t seem to oer any insight into a possible solution.

2.2 Two-dimensionalism

Two-dimensional semantics developed from Kaplan’s Logic of Demonstratives (Kaplan,

1989), through the work of Stalnaker (1978), Evans (1979), Davies and Humberstone (1980),

Chalmers (1996), Jackson (2000), and others. One of the most famous applications of the two-dimensional framework is the account of Kripke’s (1980) distinction of epistemic notions like a priority and a posteriority from metaphysical notions like necessity and contingency. At least Stalnaker (1984) extended the account to cover puzzles of logical omniscience. In spite of many dierences, both Stalnaker and Chalmers subscribe to this account—they are the main target in what follows, because their interpretation and application of the two-dimensional framework clearly brings out the diculty I raise below.

Two-dimensional accounts of Kripke’s examples, and more ambitiously of all cases of apparently distinct but necessarily equivalent thoughts, consist in analyzing them as utterances, or sentence-tokens, whose interpretation depends on two thoughts: one that bears modal properties, like necessity, and one that has a certain epistemic prole, like aposteriority. The epistemic achievement reported by an attitude report is thus an appreciation that a thought with certain modal properties is expressed by a particular sentence.

22 A posteriori knowledge of metaphysically necessary truths (such as that Water is H2O) is thus a sort of semantic knowledge. This strategy allows Stalnaker and Chalmers to square

Kripke’s separation of necessity from apriority with the assumption that if a world is not a genuine metaphysical possibility, then it’s not a way the actual world might turn out to be a priori. Chalmers (2011) calls this assumption Metaphysical Plenitude.

According to two-dimensionalists, the interpretation of an utterance depends, rst, on a thought assigned to it by the conventional semantics. This is known as ‘C-intension’,

‘secondary intension’, or, as I will call it, ‘horizontal proposition’. Secondly, interpretation depends on a thought that accounts for the utterance’s cognitive value, and is known as

‘A-intension’, ‘primary intension’, ‘epistemic intension’, or, following Stalnaker, ‘diagonal proposition’. Dierences in terminology indicate subtle dierences in how the various authors understand the underlying notions, but for the moment we can safely neglect these dierences.

An assumption that Stalnaker and Chalmers share is that horizontal and diagonal propositions are intensions. Thus the distinction between two propositions, a horizontal and a diagonal, yields a distinction between two kinds of possible worlds. Worlds are “considered as actual” or “considered as counterfactual”, depending on whether they determine the content of an utterance, when considered as actual, or on whether they determine the truth of a content, when considered as counterfactual. Thus each utterance is associated with a ‘propositional concept’ (Stalnaker’s phrase), also called ‘two- dimensional intension’ (by Chalmers). This is a function similar to a Kaplanian character that for an utterance of ‘Hesperus is Phosphorus’ may be represented by the following table:

(H) Hesperus is Phosphorus.

23 H w1 w2 w1 true true w2 false false

The space of possibilities consists of just two worlds w1 and w2, for simplicity. On the

top row, the possible worlds represent variation among the facts that determine truth:

these are the worlds considered as counterfactual, i.e. the metaphysical possibilities at

which the truth of a content is evaluated. On the leftmost column, the worlds represent

variation among the facts that determine which content a sentence has: these are the

worlds considered as actual.

Assume for illustration that w1 is the actual world. At w1 both ‘Hesperus’ and ‘Phos-

phorus’ refer to Venus, perhaps because the descriptive properties associated with the

names, presumably responsible to x their reference, happen to both hold uniquely of

Venus in our world, or perhaps because the causal histories of uses of the names go back

to Venus. Whatever the theory of reference, if w1 is considered as actual, H expresses the

horizontal proposition that Venus is identical to Venus. Since at no world Venus fails to

be self-identical, the horizontal proposition expressed at w1 has true on all squares. So an utterance of H at our world expresses a necessarily true proposition. Let’s imagine furthermore that at w2 ‘Hesperus’ refers to Venus and ‘Phosphorus’ to Mars, perhaps because the descriptive properties associated with ‘Phosphorus’, i.e. something like being the brightest heavenly body visible in the morning sky, hold uniquely of Mars at w2, or perhaps because the name ‘Phosphorus’ in w2 has a dierent causal history. If w2 is considered as actual, H expresses the horizontal proposition that Venus is identical to Mars.

Since no metaphysically possible world is one in which distinct objects are identical, the horizontal proposition expressed at w2 has false on all squares.

24 According to proponents of two-dimensional semantics, the contingency of the diag-

onal proposition of H captures the non-trivial “epistemological signicance” of an utter-

ance of H, i.e. that H is not a priori knowable. The diagonal is the function taking the rst

element of the sequence of pairs w1, w1 , w2, w2 to truth, and the second to falsity.

Equivalently, it’s an intension mapping⟨⟨ a world⟩ ⟨ x to⟩⟩ the truth value obtained by evaluat-

ing at x the horizontal proposition expressed by the utterance at x considered as actual.

In our example, this is a contingent proposition. So if speakers come to believe that the

diagonal is true then they come to believe that the actual world is w1, and not w2.

Stalnaker and Chalmers disagree on the interpretation of the two-dimensional frame-

work. On Stalnaker’s metasemantic interpretation, the semantic content of an utterance

is given by the horizontal proposition, and the diagonal proposition is related to the hor-

izontal via Gricean conversational principles. Given any utterance, for every world con-

sidered as actual (these are worlds that might be the actual world, for all the interlocutors

know at this stage in the conversation), there is always at least an intension, characterized

by diagonalization, that is pragmatically relevant for the purposes of interpretation. The

diagonal provides a description of the content that the speaker might have intended to

communicate by using the sentence she used, given the common ground.8

Chalmers prefers to characterize diagonals through heuristics given by the intuitive

assessment of a priori entailments (Chalmers, 2002). According to Chalmers, each world

wi considered as actual can be completely characterized by a description Di that a pri-

ori entails the truth of H’s primary intension at wi considered as counterfactual. To

8Stalnaker’s interpretation of the two-dimensional framework has changed over the years. Importantly, he has become skeptical that a priori contingent statements can be accounted for by the diagonalization strategy. This change of mind reects a dierent understanding of the framework, but these are issues that we may set aside. See Stalnaker (2006) for discussion.

25 avoid regress, the description Di is supposedly formulated in a neutral (or “non-twin- earthable”) language that includes at least logical and mathematical terminology, the vocabulary of fundamental physics, and phenomenological vocabulary, for which, according to Chalmers, there are no Kripkean phenomena of the sort two-dimensionalists are trying to explain.

In any case, Chalmers claims that the intension obtained via Stalnaker’s diagonalization method is “straightforwardly equivalent” to his epistemic intension (Chalmers, 2006,

104). On Chalmers’s rationalist interpretation of two-dimensional semantics, the seman-

tic content of an utterance of H encompasses the information represented by the whole

two-dimensional intension. As a result, Chalmers’s two-dimensional intensions are not

context-sensitive in the way Stalnaker’s propositional concepts are. This result is cru-

cial for Chalmers’s claim that the diagonal proposition is necessary if, and only if, it is

a priori—a biconditional that he labels “Core Thesis”, for the importance it plays in his

conceivability argument against materialism (Chalmers, 1996, 2002, 2005a, 2006).9

Thus the ability to individuate propositions by diagonalization plays a crucial role

in both Stalnaker and Chalmers’ philosophical projects. The many dierences between

them are noteworthy, but they don’t matter for present purposes. Paradox is generated by

assumptions that both Stalnaker and Chalmers agree upon: (a) that the diagonalization

strategy characterizes a proposition, (b) that the proposition thus characterized can be

believed (or asserted, imagined, and so on), and (c) by Metaphysical Plenitude:

(MP) The set A of worlds considered as actual is a subset of the set C of worlds considered as counterfactuals.

9There are other distinctions between Stalnaker and Chalmers on two-dimensionalism, including on the semantics of proper names. For further discussion, see Chalmers and Jackson (2001), Chalmers (2005b), and Stalnaker (1999, 2003, 2006). For an in-depth criticism of the two-dimensional approach to Kripke’s puzzles, see Soames (2005).

26 By MP, there are no worlds in A that are not in C: no worlds that might be the actual world are metaphysically impossible. As Soames (2005) argues in detail, two-dimensionalists like

Chalmers and Stalnaker are committed to MP. Indeed, at least Chalmers (2002) explicitly endorses the stronger claim that he calls Modal Rationalism:

(MR) The set A of worlds considered as actual is the set C of worlds considered as counterfactual.

MP entails that if a world w is not a genuine metaphysical possibility, then it is not a way the actual world might turn out to be. This leads to so-called illusions of possibility

(Yablo, 2006). For instance, there is no metaphysically possible world at which Hesperus isn’t Phosphorus. Therefore when we seem to be conceiving a scenario in which Hespe- rus isn’t Phosphorus, we are in fact misdescribing an apparent epistemic possibility as a genuine metaphysical possibility. This conclusion is perhaps the most central result in two-dimensionalist modal epistemology.

2.3 Diagonals and Antidiagonals

In the example above, the two-dimensional intension given by the diagonal of H could be seen as associating each world considered as actual to a proposition that has the property, roughly, being expressed by an utterance of H at that world. In other words, in the two- dimensional intension of H the semantic contribution of the predicate ‘An utterance of H expresses that’ takes each horizontal proposition as argument at each world considered as actual.

Using the formal apparatus developed by two-dimensionalists, we may generalize the construction of a diagonal for any predicate ‘Q’ of the right type:

(D) There is a proposition that, for any world x, is true at x i the Q-ed proposition at x is true at x.

27 Thus taking ‘Q’ to be being expressed by the (relevant) utterance, Stalnaker denes a diag-

onal as “the proposition that is true at x for any x if and only if what is expressed in the

utterance at x is true at x” (Stalnaker, 1978, 81). Notice that the diagonal is well-dened so long as something like MP is true, so that the world determining what’s Q-ed may also

be a world at which a proposition is evaluated.

However it seems that, under the same assumptions, we can also dene an antidiago-

nal proposition: (AD) There is a proposition that, for any world x, is true at x i the Q-ed proposition at x is not true at x.

We obtain an antidiagonal by construction, simply taking the diagonal proposition char-

acterized by some Q, and then switching its truth values at all worlds. So long as D denes

a proposition, its two-dimensional negation is a proposition, and is dened by AD.

Antidiagonals lead to inconsistency, however. In particular, AD is incompatible with

Kaplan’s principle K=: that for any proposition p, it is possible that p, and only p, is Q-ed.

To see this, let ‘Q’ be being expressed by an utterance of H and take the antidiagonal for

that attitude. Call it r: by AD, for any world x, r is true at x i the proposition expressed by an utterance of H at x is not true at x. Assume the instance of K= for that same attitude.

So for every proposition p, it is possible that p, and only p, is expressed by an utterance of H. So it is possible that r, and only r, is expressed by an utterance of H. Consider now the possible world, call it w, at which r, and only r, is expressed by an utterance of H. It follows from AD that r is true at w i the proposition expressed by an utterance of H at w is not true at w. Since r is the only proposition expressed by an utterance of H at w, r

is true at w i r is not true at w. This is the two-dimensional paradox.

As I mentioned, any attitude operator except those that track the truth of their com-

plement is a possible value of ‘Q’ in K=. Likewise, diagonals are dened by an attitude

28 operator, following Stalnaker’s recipe D. For concreteness, I chose ‘An utterance of H ex-

presses that’, which is modeled after Stalnaker’s own example. But the derivation does

not depend on this particular choice of ‘Q’. For an unproblematic case, consider Plato’s

beliefs. Instantiating K=, we derive that for any proposition p there is a possible world

at which p is the only proposition believed by Plato. The antidiagonal dened by ‘Plato believes that’ is the proposition that is true at a world i the proposition believed by Plato at that world is not true there. Such an antidiagonal is an inconsistent instance of K= on

Plato’s belief.

One initial worry could be that if, for whatever reason, the horizontal proposition

corresponding to a world considered as actual is undened at that world, then the an-

tidiagonal is also undened. This worry is easily disposed of. For we could stipulate that

the negation in AD is external (also called ‘exclusion’ or ‘presupposition-cancelling’ nega-

tion): it maps falsity to truth, truth to falsity, and undenedness to falsity. Alternatively,

following a strategy that is familiar from discussions of revenge problems for semantic

paradoxes, we can reformulate the antidiagonal so that it is true just in case the Q-ed proposition is either false or undened.

Another worry might concern the possibility operator in K=. What kind of possibility is it? I am assuming that K= reads: for any p, there is a world such that p is identical to whatever is Q-ed at that world. This seems a reasonable way of understanding Kaplan’s principle in a two-dimensional setting. Of course K= was not originally formulated in two-dimensional semantics. On the current understanding, the possibility operator in K= ranges over the set A of worlds considered as actual. So it binds a variable for a world considered as actual that determines the content of ‘Q’. Technically, this doesn’t make

29 much of a dierence if we assume MR, as Chalmers does: in that case, we work with a

single batch of possible worlds.

A way to look at this contradiction is the following: suppose we can list all propo-

sitions p1, p2, .... Each corresponds to a possible world w1, w2, ... such that, if wi turns

out to be actual, pi is Q-ed. This list is a two-dimensional intension for Q. By AD, we

can construct an antidiagonal proposition r, and we can show that it’s not on the list. It

is dened as follows: r is true at wi i pi is not true at wi. Then r is dierent from all propositions in the list: it is dierent from p1 because they have dierent truth values in

= w1, from p2 because they have dierent truth values in w2, and so on. From K it follows that for any proposition q, there is a wi at which the corresponding pi is identical to q.

Contradiction immediately follows.

This Cantorian reformulation of the two-dimensional paradox suggests that we are

dealing with a genuine paradox. We can also see that the paradox ts Priest’s Inclosure

Schema (Priest, 1994b,a). According to Priest, the Schema represents the structure of the

paradoxes of “the limits of thought”. The Schema is set up by means of a domain Ω, two properties ψ and φ, and a diagonal function δ, satisfying the following conditions:

1. Ω x φ x exists, and ψ Ω .

= { S ( )} ( ) 2. If x Ω and ψ x then

⊆ ( ) (a) Transcendence: δx x

(b) Closure: δx Ω. ∉

∈ Clearly, a contradiction follows from Transcendence and Closure as soon as one substitutes

Ω for x in (2).

30 The two-dimensional paradox is formulated in the set-theoretic expansion of a set W

of possible worlds. As domain Ω, consider the powerset PW , and for properties ψ and

φ, simply take the same property I of being an intension. Clearly, both PW and all its

subsets are intensions. Thus, the conditions are:

1. PW exists.

2. If Π PW then

⊆ (a) Transcendence: δπ Π

(b) Closure: δΠ PW∉.

∈ One might deny (1), but hardly anyone would, prior to encountering the paradoxes. So

the null hypothesis is to take condition 1 to be satised. Let Π be the set of all propositions

such that there is a world at which they are uniquely Q-ed. To simplify notation, I will

write ‘wQ!p’ to mean ‘p is uniquely Q-ed at w’. So let Π p PW w W wQ!p .

About the logic of ‘Q!’, I shall assume numerical uniqueness,= { in∈ keeping∶ ∃ with∈ the( assump-)}

tions above: (UNI) w p, q wQ!p wQ!q p q

If p and q are∀ uniquely∀ ( Q at∧w, then) →p is=q. As we shall see below, this is a simplication: the argument works even if we simply assume material equivalence, and thus replace the identity sign by a biconditional in UNI. In any case, since every member of q is a member of PW , q PW .

It remains⊆ to be shown that Transcendence and Closure hold. Dene δ as the diagonal function over Π, namely the function returning for Π the set of worlds such that there is a proposition in Π that is uniquely Q-ed at them but to which they don’t belong:

δΠ w W p Π wQ!p w p

= { ∈ ∶ ∃ ∈31 ( ∧ ¬ ∈ )} This is precisely the set-theoretic construction that corresponds to the diagonal proposi-

tion determined by ‘Q’.

It is easy to show that δΠ Π. Suppose the opposite for reductio. Then w wQ!δΠ

by denition of Π, and let¬w be∈ such world, so that wQ!δΠ. By denition of∃δ, w( δΠ if,)

and only if, p Π wQ!p w p . Suppose rst that w δΠ. Then p Π w∈ Q!p

w p and∃ thus∈ (p Π w∧Q ¬!p ∈ w) p . Hence wQ!δΠ¬ ∈w δΠ and¬∃ so contradiction∈ ( ∧ results¬ ∈ ) by modus ponens.∀ ∈ ( Therefore,→ w∈ )δΠ. Then p Π→wQ∈!p w p) by denition of δ. Take p as a witness, and so wQ!p∈ w p.∃ By∈ UNI,( since∧w ¬Q!∈p and wQ!δΠ, it follows that p δΠ. So w δΠ, and this∧ is ¬ a contradiction.∈ So δΠ Π. This establishes

Transcendence.= ¬ ∈ ¬ ∈

On the other hand, K= gives us a direct argument for Closure. For K= entails that δΠ is such that w wQ!δΠ . Hence δΠ Π, hence δΠ PW .

Priest claims∃ ( that the) inclosure schema∈ “cuts at∈ the joints” in the nature of paradoxicality. Indeed, of all paradoxes that philosophers are interested in, all but Curry-style contradictions t Priest’s schema. There is considerable debate about the centrality of the schema in our thinking about the paradoxes,10 but the fact that the two-dimensional paradox satises the schema is at least some evidence that the two-dimensional paradox is a genuine paradox.

It is important to appreciate that nothing as strong as identity in Kaplan’s principle is required. Recall that K= is formalized as follows:

(K=) p q Qq p q

10As I remarked,∀ n ∀ the( Schema↔ seems= ) to involve some unnecessary structure if it is to t the two- dimensional paradox. A more compelling criticism of the Schema is that it completely fails to capture the paradoxicality of Curry-style paradoxes. Tennant (1995) advances the non-existence of a reduction pro- cedure in an argument to absurdity as an alternative criterion of paradoxicality, but I have not tested it on the paradoxes discussed here.

32 Stalnaker and Chalmers, and many others, think that identity between propositions is

simply necessary equivalence, because intensions are extensional entities. Thus K= really is K◻: (K◻) p q Qq p q

∀ n ∀ ( ↔ ◻( ↔ )) but an even weaker principle suces for contradiction: (K) p q Qq p q

∀ n ∀ ( ↔ ( ↔ )) K is asymmetrically entailed by K= or by K◻. K is the claim that for every p, it is possible that any Q-ed thought is materially equivalent to p. The analogous move to weaker principles has been made by Bacon, Hawthorne, and Uzquiano (2016) regarding Prior’s paradox, and it is important for two reasons: rst, it shows in both cases that the paradoxes derivable from K do not depend on a conception of propositions as rst-order objects, for which the identity relation may be regarded as undened; secondly, it shows in both cases that these are not cardinality paradoxes, because no assumptions about numerical uniqueness need to be made.

The latter point is established about the two-dimensional paradox by noting that there is no need to assume numerical uniqueness in AD either: existence is enough. Recall AD, modeled after Stalnaker’s denition of a diagonal proposition: (AD) There is a proposition that, for any world x, is true at x i the Q-ed proposition at x is not true at x.

Since the right-hand side of the biconditional in AD contains a denite description, it is easy to see that AD entails AD :

(AD ) There is a proposition∃ that, for any world x, is true at x i there is a Q-ed proposition at x that is not true at x. ∃ The two-dimensional paradox, in the derivation I presented above, had AD and K= as premises. The paradox still follows from the weaker AD and K, dropping numerical

33 ∃ uniqueness from both premises. Assume AD and K. Let r be the witness for the truth

of AD , so for any world x, r is true at x if, and∃ only if, there is a Q-ed proposition at x that is∃ not true at x. K entails that q Qq r q . Let w be the world at which

q Qq r q . Then in particular,n∀ since( r↔is( materially↔ )) equivalent to itself, it follows that∀ ( r is↔Q(-ed↔ at ))w. Furthermore, suppose r is true at w. Then by AD there is a Q-ed proposition at w that is not true at w. Let m be a witness for that claim:∃ m is Q-ed at w and it is not true at w. But by K, m is Q-ed at w if, and only if, r is materially equivalent to m. So r is materially equivalent to m, r is true at w and m is not true at w. Contradiction.

Therefore, r is not true at w. Then r is Q-ed at w and not true at w. So there is a Q-ed propositions at w that is not true at w. So r is true at w. Contradiction again.

Since numerical uniqueness is not required anywhere in the derivation, there are no cardinality constraints imposed by AD and K on W . Unlike Kaplan’s paradox, the two- dimensional paradox does not depend on∃ a cardinality problem.

In discussion of Kaplan’s paradox, Chalmers (2011) suggests that the space of possible worlds may be too large for there to be a set of all worlds. In particular, he argues that

Kaplan’s paradox is not a threat because Cantor’s theorem fails for W . But this diagnosis does not get at the heart of the matter. For instance, suppose that there are only 3 worlds: w1, w2 and w3. For simplicity, for a sentence S, let there be a Q-ed proposition at w1 that is necessarily true, and a Q-ed proposition at w2 that is necessarily false. (So far, this is analogous to the propositional concept we had above for ‘Hesperus is Phosphorus’.) If at w3 the antidiagonal of S is Q-ed, then it will be false at w1, true at w2, and at w3 it will be true just in case it is false.

34 S w1 w2 w3 w1 true true true w2 false false false w3 false true !?!?

Nothing interesting would happen in this case if we expand the table so that there are too

many worlds to form a set. So Chalmers’s suggestions regarding Kaplan’s paradox doesn’t

help here. As I emphasized, the problematic assumptions are that diagonalizing charac-

terizes a proposition, that this proposition can be Q-ed, and Metaphysical Plenitude. So

the two-dimensional paradox is specic to two-dimensionalism, since it is possible that,

even if the set of worlds was constrained in such a way as to avoid Kaplan’s paradox,

contradiction would still follow by the two-dimensional paradox.

A slightly more formal version of the two-dimensional paradox can now be given.

I state the premises using quantiers over possible worlds, instead of modal operators.

Since any non-empty set W can play the role of the set of worlds, the proof below shows

that the two-dimensional paradox follow generally in higher-order logic (taking the inter-

pretation of propositional constants and variables to be dened over elements of PW ).

The assumptions which it depends on are perhaps not beyond revision, but they do seem

to be made by those who subscribe to the two-dimensional framework.

I shall represent the distinction between variables for worlds considered as actual and

variables for worlds considered as counterfactual in the syntax. In ‘wQp’, read ‘p is Q- ed at w’, w is related to a proposition by an attitude operator ‘Q’, which could receive

any appropriate interpretation. So world variables in ‘wQp’ are for worlds considered as

actual. World variables occurring on the left of the membership relation, as in ‘w p’,

are for worlds considered as counterfactual. Membership represents the truth predicate∈

35 in possible worlds semantics, since ‘w p’ models p’s truth at w. With these stipulations

in place, the initial assumptions are: ∈

(K=) p w q wQq p q

∀ ∃ ∀ ( ↔ = ) (AD) p w w p q wQq q′ wQq′ p q′ w q

∃ ∀ ( ∈ ↔ ∃ ( ∧ ∀ ( → = ) ∧ ¬ ∈ )) K= and AD unproblematically entail, respectively:

(K) p w q wQq v v p v q

∀ ∃ ∀ ( ↔ ∀ ( ∈ ↔ ∈ )) (AD ) p w w p q wQq w q

∃ ∃ ∀ ( ∈ ↔ ∃ ( ∧ ¬ ∈ )) For elimination, assume:

∃ (a) w w r q wQq w q

∀ ( ∈ ↔ ∃ ( ∧ ¬ ∈ )) Instatiate K on r, and take a witness u:

(b) q uQq v v r v q

∀ ( ↔ ∀ ( ∈ ↔ ∈ )) By Metaphysical Plenitude we may instantiate the universal in (a) on u:

∈ ↔ ∃ ( ∧ ¬ ∈ ) Assume u r. It follows from (c) that q uQq u q . Take a witness q∗ for that

existential,∈ which yields uQq∗ and u ∃q∗(. However∧ ¬ ∈ from) (b) it follows that uQq∗

v v r v q∗ , from which contradiction¬ ∈ easily follows. Hence u r. Since uQr↔

follows∀ ( ∈ obviously↔ ∈ from) (b), we have uQr u r, hence the right hand¬ side∈ of (c). Hence

u r. Contradiction again. Two-dimensional∧ ¬ semantics∈ is inconsistent.

∈

36 2.4 Beyond intensionality

There are a few options to deal with this result. The rst option is to embrace the contra-

diction. Priest (1991) presents a inconsistent account of the paradoxes of intensionality,

and one might think that antidiagonals are true contradictions. An argument in support of

this decision might begin with the Principle of Uniform Solution (Priest, 1994b): roughly,

like paradoxes should get like solutions. The two-dimensional paradox ts the inclosure

schema, and since Priest’s account applies to all inclosure paradoxes, it ought to apply to

the two dimensional paradox too. These considerations are preliminary: the Principle of

Uniform Solution has been called into question (Smith, 2000), and Priest’s acceptance of

true contradictions remains controversial. In addition, two-dimensionalists like Stalnaker

and Chalmers might prefer a consistent solution.11

The two-dimensional paradox could be blamed on some false premise, or on some invalid inference. As usual, we would like a logical trick that restores consistency, and is independently motivated. We would also like an explanation why the assumptions we made looked acceptable even though they were not.

The premises are K and AD . The latter is a consequence of AD. Rejecting AD doesn’t seem promising for two-dimensionalists.∃ AD denes a proposition so long as D does, assuming that the negation of a proposition is a proposition. Since two-dimensionalists are committed to the proposition dened by D to account for Kripke’s a posteriori necessities, they are committed to the one dened by AD.

11Priest’s (1991) account of paradoxes like Prior’s (1961) works only insofar as the paradox is not reformu- lated as a Curry-style contradiction, which is easy enough to do. This limit highlights the fact that Priest’s account depends on a certain perspective on the truth-conditional clauses for some connectives, negation in particular. Since the paradoxes of intensionality do not turn on naïve assumptions about truth, Priest’s account of the paradoxes of intensionality does not seem well motivated.

37 Another option is to reject K, which follows from K=. These Kaplanian principles might seem reasonable and intuitively motivated, but Chalmers and Stalnaker may not be committed to them. Besides, Kaplan’s paradox gives us more reason to think that these principles are not to be trusted. Kaplan does insist that K= shouldn’t be deemed logically

false, but perhaps the right course of action is the one advocated by Anderson (2009),

Bueno et al. (2013), and Bacon and Uzquiano (2018), in regards to Kaplan’s paradox, who

claim that paradoxes following from K= should be considered as a reductio of K=.

However, rejection of K is in tension with two-dimensionalism’s other commitments.

If K is false, then there is a proposition that is impossible to Q uniquely up to material

equivalence, and if K= is false then there is a proposition that is impossible to Q uniquely.

In particular, there is a proposition that can’t be believed (or armed, imagined, or ex-

pressed by an utterance) up to material equivalence. What is this ineable proposition?

A logic of thought on which it turns out that it is impossible for a thought to be the object

of arbitrary propositional attitudes looks unsatisfactory (Tucker and Thomason, 2011). In

the present setting, however, perhaps the most plausible candidate to point to as a coun-

terexample to K would be the antidiagonal itself. If anything, that could be a proposition

that is impossible to believe uniquely (up to material equivalence). Therefore,

(i) either antidiagonals are not believable at all,

(ii) or believing antidiagonals is hard, so that they are never uniquely believed.

Stalnaker and Chalmers may not be committed to K, but they are committed to the claim

that diagonals are knowable. This is because they think that some diagonals are know-

able a posteriori: according to two-dimensionalists, a posteriori necessities are sentences

38 associated with a necessary horizontal proposition, and a contingent diagonal. A posteriori knowledge of the diagonal explains the aposteriority of such sentences. If diagonals are knowable they are believable (and imaginable, thinkable, and so on). Plausibly, if a proposition is believable, so is its negation. Therefore, antidiagonals are believable too, contrary to (i). So Stalnaker and Chalmers may be left with (ii). As above: if antidiagonals can’t be uniquely believed, it would seem that diagonals can’t be uniquely believed either.

But what would it be to claim that we know a proposition that we can’t believe uniquely

(nor imagine uniquely, nor think uniquely, and so on)? And what are these rather mysterious propositions that get in the way of our rmly grasping diagonals? Do we know all the propositions that we grasp in the frustrated attempt to grasp a diagonal and nothing else? These seem to be awkward questions for a theory of thought, and indeed questions that one would be better o without. Maybe these diculties can be made to disappear, but at the moment the picture of modal epistemology coming out of two-dimensionalism seems rather obscure. For these reasons, perhaps Stalnaker and Chalmers might want to stick with K.

Commenting on Whittle’s (2009) version of Kaplan’s paradox, Chalmers maintains that, on certain ways of presenting Kaplan’s paradox, the puzzle comes down to a version of the Liar paradox. Kaplan himself points that out. Chalmers goes on to dismiss

Whittle’s puzzle since, he claims, the Liar is a problem for everyone. This reply is hardly satisfactory. An antidiagonal is indeed, in some sense, like a Liar. So a diagonal is in some sense like a Truth Teller. One should then say that diagonals and antidiagonals are both pathological propositions, though only antidiagonals are paradoxical. Chalmers even suggests, following a popular strategy to handle semantic paradoxes, that the Liar is truth-valueless; but then it appears that the same holds of diagonals and antidiagonals.

39 Presumably, diagonals will be truth-valueless in every possible world, because the contradiction doesn’t depend on any specic scenario. So diagonals can’t have the cognitive prole that two-dimensionalists attribute to them.

A family of options are available to reject the reasoning. Some of these options are articulated in the discussions of Prior’s paradox by Bacon et al. (2016), and by Tucker and

Thomason (2011). The options concern restrictions on the quanticational inferences that the two-dimensional paradox depends on. These involve both quantiers over thoughts and quantiers over worlds.

There may be more than one reason why Universal Instantiation fails, but for present purposes one could speculate that the modal quantiers range over the domain of all worlds. After all, in two-dimensional semantics we distinguish a set A of worlds considered as actual and a set C of worlds considered as counterfactual. So one might naturally think that we don’t have unrestricted quantiers over all worlds because there is no set that includes all worlds, both those considered as actual and those considered as counterfactual.

Still, on this picture the prospects for two-dimensionalism aren’t promising at all. For this strategy amounts to a denial of Metaphysical Plenitude: the claim that if a world is an a priori candidate for the actual world, then it’s metaphysically possible. For there would be worlds in A, witnessing for the truth of K=, that are not in C. Since Metaphysical Plen- itude follows from Chalmers’s “rationalist” commitments, this option is not available to him. Likewise, it is not clear what would be possible worlds that are not metaphysically possible, on Stalnaker’s version of two-dimensionalism. As Soames (2005) points out, one of the main motivations behind the approaches of Chalmers and Stalnaker to two- dimensionalism is the denial that there are genuine possibilities that are not metaphysical

40 possibilities. Intuitively, if there were genuine possibilities that aren’t metaphysical possibilities, such as perhaps the possibility that Hesperus isn’t Phosphorus, there wouldn’t be the need for a two-dimensional analysis of a posteriori necessities. Notice that if we were willing to give up Metaphysical Plenitude, then a way out of this puzzle would immediately be available: this shows that it is the two-dimensionalists’ own assumptions that are responsible for the appearance of a paradox in the two-dimensional framework.

There are more options, but I refrain from second-guessing what a contextualist, contingentist, free-logical, or ramied solution to the two-dimensional paradox might look like.12 These options require very dierent pictures of the metaphysics of worlds and propositions, and for each option there are very dierent ways to implement it. I am not sure that the eort to salvage two-dimensionalism, itself born to salvage the intensional view, is worth the trouble. Better to start over, rather than adding epicycles.

Two-dimensionalists like Stalnaker and Chalmers are forced either to reject diagonal propositions, thus losing one of their most important explanatory tools, or to revise the logic and the resulting metaphysics of propositions on rather unconvincing grounds. The paradox indicates that diagonals are pathological, raising serious doubts about the legitimacy of their employment in theories of speech and thought. Two-dimensionalism, as an attempt to accommodate hyperintensional phenomena in an intensional setting, fails. The lesson seems to be that semantics, as Joseph Almog remarks, is not modal metaphysics

(Almog, 2009, 6): its foundations require more than the resources of Montague’s intensional logic (Montague, 1974; Gallin, 1975), namely possible worlds and set theory. The

12Burge (1984), Glanzberg (2004), and Lindström (2009) oer contextualist solutions to the Liar. Details dier but all of them detect a context-shift in the derivation. Contingentism is briey considered by Tucker and Thomason (2011) in discussion of Prior’s paradox, whereas Bacon et al. (2016) discuss the prospects for free logic and ramication. The latter is tentatively put forward by Kaplan (1995), and mentioned elsewhere as a general solution to the intensional paradoxes (Church, 1976; Anderson, 1980)—including of course Russell and Whitehead (1927).

41 task is set for the next chapters: to develop a hyperintensional solution to the problem of propositional identity, and a consistent logic of thoughts.

42 Chapter 3: Prior’s gambit

The two-dimensional paradox is related to some other paradoxes of intensionality, Prior’s

paradox in particular, that depend on principles like K= or weaker than K=:

(K=) p q Qq p q

∀ n ∀ ( ↔ = ) For any thought p, it is possible that p, and only p, is Q-ed. In this chapter, I discuss

Prior’s gambit: the proposal, advanced rst by Arthur Prior (1971), to restore consistency

by refuting K=, passing its denial as a truth of logic. I shall generalize Prior’s paradox to the innitary case, to show that this kind of strategy populates metaphysics with an embarrassing abundance of thoughts. I conclude with some reections on ontology and metaphysical methodology.

3.1 Prior’s paradox

Suppose that Epimenides believes that nothing he believes is true, and believes nothing else. Is what he believes a truth? If it is, then it is not, and if it is not, then it is. This is an informal version of Prior’s (1961) paradox, sometimes presented as an intensional version of the Liar paradox. The reasoning can be formalized in an intensional language that allows for bound variables in sentence position. Expressions like ‘Nothing believed by Epimenides is true’ are taken at face value, as containing quantiers over thoughts.

43 The formalization of inferences involving the propositional attitudes is thus particularly

straightforward.

More formally, Prior’s paradox follows from asking about the truth of the thought

that p Qp p , i.e. that nothing believed by Epimenides is true. Let us abbreviate

‘ p Qp∀ ( p→’¬ as) ‘ϑ’. The simplest version of the paradox follows from the assumption that∀ ( Epimenides→ ¬ ) believes nothing else but ϑ, a premise that we may formalize as K=*:

(K=*) q Qq ϑ q

∀ ( ↔ = ) Notice that this formula, and all its cognates below, is schematic for ‘Q’. Prior’s paradox does not depend on any particular choice of predicate, which for illustration I read as

‘Epimenides believes that’. Predicates that take arguments of the type of sentences are called attitude predicates.

K=* is likely false, as a matter of fact, because Epimenides believes many things besides ϑ, if he even believes that at all. However, it seems that K=* could be true. Hence

q Qq ϑ q : it is possible that Epimenides believes nothing but ϑ. Plausibly, this is n∀not just( something↔ = ) special about ϑ, but it is generally so. This line of reasoning supports a principle rst isolated by Kaplan (1995):

(K=) p q Qq p q

∀ n ∀ ( ↔ = ) Kaplan shows that K= is inconsistent with Cantor’s theorem in Montague’s intensional logic (Gallin, 1975). The reasoning is straightforward: consider a domain of possible worlds W and suppose that thoughts are members of PW , i.e. sets of possible worlds.

Interpreting the language of K= in the standard possible-worlds semantics, K= asserts that for every set of worlds there is a world that satises a certain condition. Thus, K= projects

44 an injection from PW to W . By Cantor’s theorem, there is no such function. This is

Kaplan’s paradox.

As it happens, weaker principles than K= are just as problematic. In chapter 2, I have shown that K, the principle obtained by weakening ‘=’ in K= to a material biconditional, is inconsistent with the existence of diagonal propositions in two-dimensional semantics.

Similarly Anderson (2009), and Bacon et al. (2016), note that Prior’s paradox too follows in second-order logic without identity. In this case we assume that, for any thought p, it is possible that Epimenides believes that p and, if anything, only thoughts materially equivalent to ϑ:

(K*) q Qq ϑ q

n ∀ ( ↔ ( ↔ )) which can be readily seen as an instance of a more general principle:

(K) p q Qq p q

∀ n ∀ ( ↔ ( ↔ )) For any thought p, it is possible that if any q is Q-ed, q is materially equivalent to p.

K describes an apparently possible scenario in which any Q-ed thoughts have the same truth value. Recall that ‘ϑ’ is ‘ p Qp p ’. Prior’s original version of the paradox is derived using only Universal Instantiation∀ ( → ¬ and) classical logic (parametric occurrences are boldfaced):

1. q Qq ϑ q From K* 2. ϑ Reductio assumption 3. Qϑ∀ ( ↔ϑ ( ↔ )) 2, UI 4. Qϑ ϑ ϑ 1, UI 5. ϑ → ¬ 2, 3, 4 6. p Qp↔ ( p↔ ) 5 7.Q¬ p p elim 8.Q∃ p( ∧ϑ ) p 1, UI 9. ϑ ∧ ∃ 7, 8 ↔ ( ↔ ) 45 Contradiction at lines 5 and 9. This proof is strictly classical (for the step from 5 to 6).

There is also a minimal version, in the sense of Johansson’s (1936) minimal logic: intu-

itionistic logic without explosion. Let ‘η’ abbreviate ‘ p Qp p ’:13

1. q Qq η q ∃ ( ∧ ¬ ) From K 2. η Reductio assumption 3.Q∀ p( p↔ ( ↔ )) elim 4.Qp η p 1, UI 5. η ∧ ¬ ∃2, 3, 4 6. p Qp↔ ( ↔ p) 5 7. Qη¬ η 6, UI 8. Qη∀ ( →η ¬¬η ) 1, UI 9. η→ ¬¬ 7, 8 ↔ ( ↔ ) Contradiction at lines 5¬¬ and 9. Besides minimal logic, and K which I shall discuss later, Prior’s paradox requires only standard rules for the quantiers and a comprehension axiom schema. Taking propositional variables to be 0-place second-order variables, in the manner of Williamson (2003), these may be stated as versions of ordinary quanticational principles of higher-order logic:

(UI) pϕ ϕ q p

(IComp) ∀p p→ α[ ~ ]

∃ ( = ) where ‘q’ is a propositional variable free for p in ϕ, and ‘ϕ q p ’ is the result of replacing all occurrences of p in ϕ with q. In IComp, ‘α’ is a sentence,[ ~ ] hence it contains no free variables. IComp is impredicative, since there are no restriction on formulas eligible as instances in place of α. A recent defense of impredicative comprehension principles in metaphysics is Williamson (2013).

In IComp, ‘ ’ is Church’s (1984) connective of propositional identity, called strict equivalence. The job= of a comprehension principle is to license substitution of variables in the

13Bacon and Uzquiano’s (2018, fn. 5) claim that Prior’s paradox requires “classical propositional logic” is thus strictly speaking incorrect.

46 scope of predicates of the right type, in this case substitution of propositional variables in the scope of attitude predicates. Notice that the straightforward analog of standard second-order comprehension for 0-place variables, namely

p p α

∃ ( ↔ ) would not work, because material equivalence does not license substitution salva veritate for thoughts. Suce it to say that strict equivalence is provably a congruence on the domain of attitudes given Church’s axioms. Reasoning schematically for simplicity, we can conclude:

(Cong ) p, q p q ϕp ϕq

= ∀ ( = → ( → )) Hence strict equivalence in IComp does what it is supposed to do. The notion of propositional identity it captures will be the topic of chapter 5.

Let us now turn to K, a general claim about the relation between thoughts, modality, and propositional attitudes: for every thought p, it is possible that Epimenides believes nothing but p up to material equivalence. Faced with Prior’s paradox, and bound perhaps by a commitment to impredicative comprehension, some philosophers have felt compelled to reject K (Lewis, 1986; Slater, 1986; Anderson, 2009; Bueno et al. 2013; Bacon and Uzquiano, 2018). It was Prior himself who rst recommended this strategy, although quite grudgingly, as we shall see.

47 3.2 Prior’s theorem

The strategy is to interpret the argument above not so much as a paradox, but rather as

a reductio proof. I shall call this strategy veridicalism. Quine (1966) calls ‘veridical para-

doxes’ true conclusions established by purportedly valid proofs, which however sound

absurd. He gives only a couple of examples:

Frederic, protagonist of The Pirates of Penzance, has reached the age of 21 after passing only ve brithdays. Several circumstances conspire to make this possible. Age is reckoned in elapsed time, whereas a birthday has to match the date of birth; and February 29 comes less frequently than once a year. ... In a certain village there is a man, so the paradox runs, who is a barber; this barber shaves all and only those men in the village who do not shave themselves. Query: does the barber shave himself? (Quine, 1966, 1–2)

And of course the answer to that question is that the barber shaves himself if and only if he does not. So, Quine says, we conclude that there is no such barber, and no village.

Surprising as this might be, this conclusion must be true. So is the conclusion that Frederic can be 21 after only 5 birthdays. Prior’s strategy is to take Prior’s paradox to be veridical, establishing as their true conclusion that K is false.

It should be said that the category of veridical paradoxes is not perspicuous. Frederic’s story seems no more than a clever party joke, and as to the barber, Lycan comments:

the Barber is classied as veridical because its conclusion is the truth that there is no barber who shaves all and only those who do not shave themselves; but turn it on its head, so that its conclusion is rather the absurdity that there is a barber who both does and does not shave her-/himself. There is still no fallacy, but only the innocent-seeming premiss that there is a barber who shaves all and only those who do not shave themselves. (Lycan, 2010, 616)

The trade-o is between apparently good reasoning with a true but surprising conclusion, and apparently good reasoning with an absurd conclusion and innocent premises. Thus

48 veridicality of a paradox comes down to whether denial of one of the premises is more

or less plausible than logical revision. So long as we are talking about leap years, where

we have a clear sense of what’s plausible and what’s not, everything is ne. But should

controversy arise, the category of veridical paradoxes fails to be neutral.

Taking the veridicality line on Prior’s paradox, we conclude that there are thoughts

that necessarily can’t be believed without believing something else materially distinct.

Since thoughts that are materially distinct are also numerically distinct, it follows that

K= fails as well, and for some thoughts, necessarily if we believe them we also believe something else. Recall that the paradox does not depend on the interpretation of the attitude predicate, so there are also thoughts that can’t be asserted, or imagined, in isolation.

There are, as Gabriel Uzquiano puts it, “elusive” thoughts.14

A number of philosophers have followed Prior, but for a number of dierent reasons: there is no consensus on why K fails, and no accepted explanation why K appears to be true at least for some choice of the attitude operator. Intuitions that K is true can always be overridden, of course, but only by stronger countervailing considerations.

Recall that Kaplan’s paradox is the derivation of an inconsistency from K= and the thesis that each thought determines a set of worlds. David Lewis (1986) thinks that each thought is a set of worlds, so Kaplan’s paradox is a serious concern. According to him, there is a trade-o between taking thoughts to be sets of possible worlds, as it were by denition, and taking them to be eligible objects of attitudes. In the latter case, thoughts should fulll some functional role, but there is no reason to think that there are as many mental functions as sets of possible worlds, since according to Lewis mental functions are relatively natural properties, and natural properties are relatively sparse. On the other

14G. Uzquiano, Elusive Propositions. Talk at the University of St. Andrews, June 17, 2018.

49 hand, if we stipulate that thoughts are sets of worlds, we should not expect K= to hold.

Given our cognitive limitations, there is nothing surprising about thoughts that cannot be believed, or asserted, imagined. Some thoughts, for example, are too complex or ger- rymandered for our nite minds to grasp—like innite conjunctions, perhaps.

Lewis’s argument has not managed to convince many, as Tucker and Thomason (2011) point out. First of all, Epimenides’s thought ϑ is certainly not too complex or gerryman- dered to be believed, so Lewis’s considerations are not clearly relevant to Prior’s paradox.

Secondly, Lewis’s estimate of the alleged scarcity of propositional attitudes is based on his functionalism about the mind, and on his theory of natural properties. It is unlikely that an uncontroversial solution to the intensional paradoxes could depend on such controversial assumptions. Finally, it should be noted that Prior’s paradox does not depend on the claim that thoughts are or determine sets of worlds. So Lewis’s considerations simply miss the mark.

Prior himself was reluctant to embrace veridicalism, presumably because independent motivation is hard to come by. Indeed, he seems to be making dierent suggestions at dierent times. The rst is this:

The fact is that not only the truth or the justication of what we say, think, or fear may depend on facts outside our assertions, thoughts, and fears—on any realist view, this is only to be expected—but the very possibility of making certain assertions, and the very possibility of having certain thoughts and fears, may depend on certain external things being or not being the case. (Prior, 1971, 88)

Perhaps “making certain assertions” is impossible in unlucky circumstances. This apparently means that it is impossible for Epimenides to believe that nothing he believe is true— almost as if his brain would short circuit. Since nothing in particular has been assumed about belief, we might conclude as well that there are thoughts that cannot be asserted,

50 or imagined, by somebody in some circumstances. As Tucker and Thomason (2011) point

out, it is denitely unclear why this should be the case, given how little Prior says. At the

very least, the claim that a paradoxical thought like ϑ can’t be asserted or imagined seems

plainly incompatible with the evidence we have (Burge, 1984). For it looks as though we

just did assert, and imagine, what supposedly Epimenides could neither assert, nor imag-

ine! And it does not seem to be less of a cognitive burden for us to do so than it would be

for him. So the cost of following Prior’s rst suggestion is rather impressive.

Prior’s second suggestion is perhaps more promising. Since K leads to contradiction,

we conclude its negation by reductio, and then the following (‘Anti-K’) by classical steps:15

(AK) p Qp q Qq p q

∃ ◻ ( → ∃ ( ∧ ¬ ↔ )) i.e. there is a thought p such that necessarily it is believed only if some materially distinct

q is believed. While Prior’s rst suggestion was that some thought can’t be believed by

Epimenides, the second is that some thought is such that if Epimenides believes it, he

manages to believe something else as well.

AK is logically equivalent to a surprising result known as Prior’s Theorem:16

(PT) p Qp q Qq q q Qq q

15For the proof, notice rst∃ that◻ (∀p n→(Qp(∃∧( ∀q(Qq∧ →)(∧p ∃↔(q)) is∧ logically ¬ ))) equivalent to K. Reasoning through the paradox, we then conclude ¬∀p n (Qp ∧ ∀q(Qq → (p ↔ q)). Pushing the negation inward we get AK. 16Proof of PT↔AK. For the left-to-right direction, assume PT and assume K for reductio. Fix a witness p for PT. By K, n∀q(Qq ↔ (p ↔ q)). So at some world, ∀q(Qq → (p ↔ q)), and so Qp, and by PT, ∃q(Qq ∧ q) and ∃q(Qq ∧ ¬q). Fixing some more witnesses, Qs, s,Qq, ¬q. Moreover, Qs → (p ↔ s), and Qq → (p ↔ q). Hence p ↔ s and p ↔ q. Since s and ¬q, contradiction follows. Therefore, AK follows. For the right-to-left direction, assume AK. Taking a witness for the existential and eliminating the necessity, Qp → ∃q(Qq ∧ ¬p ↔ q). Assume Qp, and take a witness for the inner existential. So Qq ∧ ¬p ↔ q. Either p or ¬p. If the latter, then q, and then we have Qq ∧ q, and Qp ∧ ¬p. Hence both ∃q(Qq ∧ q) and ∃q(Qq ∧ ¬q). If the former, then ¬q, and so we have Qq ∧ ¬q and Qp ∧ p. Hence both ∃q(Qq ∧ q) and ∃q(Qq ∧ ¬q) either way. By conditional reasoning, and then closing modally and existentially, PT follows.

51 i.e. there is a thought that, necessarily, is believed only if a truth is believed, and a false-

hood is believed as well. So PT can be established via AK from the Prior’s paradox rea-

soning. Alternatively, it is instructive to follow Prior’s (1971, 87) original proof, and prove

PT directly in classical second-order logic with necessitation. Recall that ‘ϑ’ abbreviates

‘ p Qp p ’:

∀1.( → ¬ ) Qϑ Assumption 2. ϑ Reductio assumption 3. Qϑ ϑ 2, UI 4. ϑ 1, 3 5. p Qp→ ¬ p 1, 4, UI 6. ¬p Qp p 4 7. Qϑ∃ ( ∧p ¬ Qp) p p Qp p 1, 5, 6 ∃ ( ∧ ) A step of necessitation→ ( and∃ ( of existential∧ ¬ ) ∧ ∃ closure( ∧ yield)) PT.17 So there appears to be a direct

argument from logic alone to PT, and then from that to the negation of K. The package of

claims in support of Prior’s gambit looks solid.

However, it is hard to put one’s philosophical conscience to rest in this situation. Prior

himself writes:

At this point I must confess that all I can say ... is that so far as I have been able to nd out, my terms are the best at present oering. I have been driven to my conclusion very unwillingly, and have as it were wrested from Logic the very most that I can for myself and others who feel as I do. (Prior, 1961, 32)

What’s missing is an illuminating account of the truth of PT or, which is just the same, of the falsity of K. PT does follow from standard logical principles, but so does the paradox, give or take the truth of K. So, absent independent reason to regard PT as true, the alleged validity of PT is at best only a question-begging reason for the invalidity of K. The demand for a good philosophical account of PT is pressing.

17A version of PT can also be established in minimal logic, on which if ϑ is Q-ed, then something false is Q-ed but not everything Q-ed is false: Qϑ → (∃p(Qp ∧ ¬p) ∧ ¬∀p(Qp → ¬p)).

52 Prior’s proof of PT shows that ϑ is a thought such that necessarily its being believed entails that two thoughts are believed, a truth and a falsehood. Moreover, the assumption that ϑ is true reduces to absurdity under the assumption that it is believed, witness the reasoning in Prior’s paradox. So if ϑ is believed, it is not true. Hence there is an untrue thought believed by Epimenides. So far so good: but PT says that there are two thoughts believed by Epimenides. Which one is the other one?

Slater (1986) suggests that ‘ p Qp p ’ expresses simultaneously the truth and the mysterious falsehood. So ‘ p ∀Qp( →p ¬’ is) ambiguous. However, there is no evidence

(not even from natural language,∀ ( → notoriously¬ ) replete with ambiguities) that sentences

may have more than one semantic value. There is plenty of evidence, of course, that

more than one thought might be relevant for competent language use, but in standard

semantic theories, sentences have only one semantic value. Moreover, intensional ver-

sions of Nixon-Dean cases (Cohen, 1957; indeed, Kripke’s (1975) original case illustrates

the point, provided one takes surface grammar at face value) would force one to accept

a massive amount of ambiguity. In eect, the ambiguity view amounts to the claim that

interpretation is not a function, and this is a massive cost.

There is something uncomfortable about PT, and in particular about resting one’s case

for consistency on the existence of a true thought believed by Epimenides, about which we

say no more. Of course, PT is derived in classical logic, so it should be no surprise that we

can establish an existence claim without exhibiting a witness. I say, that isn’t surprising:

the question is not about the validity of non-constructive reasoning; it is about, however,

the explanatory adequacy of veridicalism as a way out of paradox.

Let’s put this point in perspective. The lesson of Russell’s paradox is, according to

many, a negated existence claim: there is no set of all non-self-membered sets, and to

53 suppose otherwise is absurd. Yet even to explain why there is no such thing as Russell’s

set requires some sophistication. Philosophical accounts of the cumulative hierarchy do

exactly this: they explain why there is no set of all non-self-membered sets (Boolos, 1971).

Here we have the opposite situation: there are more thoughts oating around than one

might have expected, and to suppose otherwise is absurd. But how could this be? Prior’s

strategy does restore consistency, but it should come with a proper explanation.

3.3 A most paradoxical argument

The gist of PT is that one can’t believe only falsehoods if the thought that all one’s beliefs

are false is one of them. Since dierence of truth value entails numerical dierence, it is

also possible to conclude that if one believes that all one’s beliefs are false, one can’t have

only one belief. In more detail: if ϑ is Q-ed, then a truth is Q-ed and a falsehood is Q-ed

(recall that ‘ϑ’ abbreviates ‘ p Qp p ’).

Qϑ∀ ( →p ¬Qp) p p Qp p

This has been proved by Prior.→ Identity(∃ ( implies∧ ¬ ) ∧ sameness ∃ ( ∧ of)) truth value, which we can

state as a principle:

(Equi) p, q p q p q

We may now reason as follows: ∀ ( = → ( ↔ )) 1. Qϑ Assumption 2. ϑ Reductio assumption 3. Qϑ ϑ 2, UI 4. ϑ 1, 3 5. p Qp→ ¬ p 4 6.Q¬ p p elim 7. ∃p(p ∧p ) p p Equi, UI 8. p ∧ϑ 4, 6,∃ 7, UI 9. ∀p(Qp= p→ (ϑ ↔ )) 5, 6, 8 10. Qϑ=~ p Qp p ϑ 1, 9 ∃ ( ∧ =~ ) → ∃ ( ∧ =~ 54) If ϑ is Q-ed, something else is Q-ed as well. Certain thoughts cannot be believed, or asserted, or imagined, in isolation, as it were. Bacon and Uzquiano (2018) generalize this result to any nite number of thoughts. It suces to assume that ϑ is materially equivalent to a disjunction of n thoughts p1, ..., pn, for any nite n. The proof proceeds much as above:

1. ϑ p1 ... pn Assumption 2. Qϑ Assumption 3. ϑ ↔ ( ∨ ∨ ) Reductio assumption 4. Qϑ ϑ 3, UI 5. ϑ 2, 4 6. p Qp→ ¬ p 5 7. ¬ p1 ... pn 1, 5 8. ∃p(1 ...∧ ) pn 7 9.Q¬(p ∨p ∨ ) elim 10. ¬p p∧ p∧ ¬ p p Equi, UI 11. p ∧ϑ p p1 ... p pn 8, 9,∃ 10, UI 12. ∀p(Qp= →p ( ϑ↔p ))p1 ... p pn 9, 11 13. Qϑ=~ ∧ p =~Qp ∧ p ∧ϑ =~ p p1 ... p pn 2, 12 ∃ ( ∧ ( =~ ∧ =~ ∧ ∧ =~ )) If ϑ is Q-ed, there→ ∃ is a( Q-ed∧ ( thought=~ ∧ dierent=~ ∧ from∧ =~ϑ, and)) dierent from any of p1, ..., pn, for any nite number n of thoughts. There are more Priorean phenomena of this kind, more and more bizarre.

Suppose it’s the end of a long a day, during which Eubulides had a certain number of thoughts. So far, just as many true thoughts as false ones. A moment before he falls asleep, Eubulides thinks one more thought: Most of my thoughts today are false. Is it true that most of Eubulides’s thoughts today are false, or is it false? Suppose it’s true. Then

Eubulides’s true thoughts are one more than the false ones, so it’s not the case that most of his thoughts are false. Contradiction. So his thought that most of his thoughts are false is false. So his false thoughts are one more than the true ones, hence most of his thoughts are false after all. Contradiction again.

55 This is a slightly dierent version of Prior’s paradox, of course, with ‘most’ instead of

‘every’. I will take some liberties in formalizing the reasoning, because ‘most’ cannot be

dened in predicate logic (Barwise and Cooper, 1981). Comprehension over propositional

functions lets us dene T and F :

p T p Qp p p F p Qp p

∀ ( ↔ ∧ ) ∀ ( ↔ ∧ ¬ ) T and F may be the sets of Eubulides’s true and false thoughts, respectively. (As above, choice of ‘Q’ is arbitrary.) It can easily be shown that p T p F p .

Assume standard truth conditions for ‘most’ (Peters∀ and( Westerståhl,↔ ¬ ) 2006), ‘Most H are G’ means that the number of H-and-Gs is greater than the number of H that aren’t

G. If the domain is nite, this comes down to saying that ‘Most H are G’ is true just in case the number of H-and-Gs is greater than half the number of H. More precisely:

Most H,G H G H G

( ) ↔ S S > S − S This cannot be dened in predicate logic. Instead, I will stipulate a convenient simplication to derive the contradiction, and then note that the proof generalizes to all nite numbers. The stipulation is to assume that Eubulides has just one true thought and just one false thought, except possibly his thought that most of his thoughts today are false.

Abbreviating ‘Most(Qp, p)’ by µ, I shall assume the following about µ:

¬ µ !pF p p q T p T q p q

→ ¬(∃ ∧ ∃ ∃ ( ∧ ∧ =~ )) !pT p p q F p F q p q µ where ‘ !pϕp’ is dened,( as∃ usual,∧ as ∃ ∃p (ϕp ∧ q ϕq∧ =~p ))q→ . So these two conditionals read: if∃ most of Eubulides’s thoughts are∃ ( false,∧∀ then( it→ is not= the)) case that Eubulides thinks exactly one falsehood, and (at least) two truths. Moreover, if Eubulides thinks exactly one

56 truth and (at least) two falsehoods, then most of Eubulides’s thoughts are true. These

conditionals are true, given the truth conditions of ‘most’ and the simplifying stipulation

I made. The latter can easily be removed later, in order to generalize the reasoning to all

nite numbers.

Finally, let us assume that, with the exception of µ, Eubulides thinks just a truth and

just a falsehood:

!p T p p µ !p F p p µ

Finally, assume Qµ, i.e. that∃ Eubulides( ∧ =~ thinks) ∧ ∃µ.( Suppose∧ =~µ.) Since µ and Qµ, T µ by the denition of T given above. Thus p q T p T q p q , since !p T p p µ . Since

p T p F p , it is easy to show∃ that∃ ( !pF∧ p. Contraposing∧ =~ ) on∃ the( rst∧ µ-conditional=~ )

∀above,( ↔!pF¬ p ) p q T p T q p q ∃ µ. Thus by modus ponens, µ. Since Qµ

and µ,(∃F µ by∧ the ∃ above∃ ( denition.∧ ∧ = So~ ))p →q ¬F p F q p q . Moreover it¬ can be shown

that¬ !pT p by the same reasoning as above.∃ ∃ ( Hence∧ by∧ modus=~ ) ponens on the second µ-

conditional,∃ µ follows. Contradiction.

At this point I simply remark that, helping ourselves with numerical quantiers de-

ned in the usual manner, the truth conditions of ‘most’ justify all instances of the fol-

lowing conditionals:

µ !npF p n+1pT p

→!n¬pT(∃ p n+1∧pF ∃ p µ)

and instead of assuming, as we just(∃ did, that∧ ∃ except for) →µ, Eubulides thinks exactly a truth

and exactly a falsehood, we may assume at the outset that:

!np T p p µ !np F p p µ

∃ ( ∧ =~ ) ∧ ∃ ( ∧ =~ )

57 that is: except for µ, Eubulides thinks just as many truths as falsehoods. The rest of the proof stays the same.

To restore consistency, one could reject the assumption that µ is Q-ed, but it is just as implausible as Prior’s rst option was. Another option is to reject the assumption that except for µ, Eubulides thinks just as many truths as falsehoods. So we can establish, for any n, that if Eubulides thinks µ then he thinks exactly n truths just in case he does not think exactly n falsehoods. This is consistent, of course:

Qµ !np Qp p !np Qp p

→ (∃ ( ∧ ) ↔ ¬∃ ( ∧ ¬ ))) Either µ or µ. In the former case, holding xed that Eubulides thinks n truths other than

µ, it must be¬ that there are at least m falsehoods he thinks, with m n. In the latter case, holding xed that Eubulides thinks n truths other than µ, it must be> that there are at least m truths he thinks, with m n.

It is very unclear why all> of this should be true. By necessitation and closing existentially, one could however conclude that there are thoughts such that, if they are Q-ed, then true and false Q-ed thoughts cannot be in one-to-one correspondence. There must be some “elulsive” Q-ed thought that breaks the correspondence. This is very implausible because one could set up a correspondence as follows: for any thought he thinks, Eubu- lides thinks its negation. Under classical assumptions, it follows that Eubulides thinks just as many truths as falsehoods.

Another option comes forward: in the innite case, everything is ne. If Eubulides thinks countably many truths, and nitely many falsehoods, then it is false that most of his thoughts are false. If he thinks countably many falsehoods, and nitely many truths, then it is true that most of his thoughts are false. Finally, if he thinks countably many truths and falsehoods in one-to-one correspondence, it is false that most of his thoughts

58 are false, but adding one more false thought to a set of innitely many falsehoods does not break down the correspondence. He still has just as many true thoughts as false ones.

This would be an innitary analog of Prior’s theorem. There, the claim was that in a scenario in which Epimenides only has false beliefs, believing that all his beliefs are false entails that there is also a true belief he has. Now, the claim is that in a scenario in which

Eubulides thinks just as many truths as falsehods, thinking that most of his thoughts are false entails that he thinks innitely many thoughts.

Prior advises that “there are surprises in logic” (Prior, 1971, 80).18 It is hard to believe, in Prior’s case, that believing that all my thoughts are false entails that I’m believing some extra thought I wasn’t even aware of, provided all my other beliefs, if any, are false. It is even harder to believe, in the present case, that believing that most of my thoughts are false entails that I have innitely many thoughts, provided all my other beliefs, if any, are just as many true as false. These consequences elicit incredulous stares.

3.4 Coherentism about thoughts?

I would like to conclude with some reections on the alleged logical status of the claims made so far. What kind of thinking about existence would support the veridicality of

Prior’s paradox and of the ‘most’ paradox?

Many philosophers don’t think that logic alone can prove the existence of much. There are exceptions though: God’s existence is allegedly proved by the ontological argument.

That, of course, is very controversial. More interesting is coherentism, a view in philosophy of mathematics championed among others by David Hilbert. In a famous letter to

Frege, he writes:

18This is a quip on Wittgenstein’s remark that “there can never be surprises in logic” in proposition 6.1251 of the Tractatus (Wittgenstein, 1921).

59 if the arbitrarily given axioms do not contradict each other with all their consequences, then they are true and the things dened by them exist. This is for me the criterion of truth and existence. (Frege, 1980, 39-40)

According to coherentism, the consistency of the axioms suces for existence of certain objects they characterize. Coherentism nds support among mathematicians and philosophers of mathematics. Contemporary versions of coherentism may be found in various places. An example is Stewart Shapiro’s (1997) structuralism, which decouples coherentism in a combination of views, while also being permissive about the notion of

‘coherence’ (Hilbert’s passage may suggest a syntactic notion of coherence):

sometimes we use the “is” of identity when referring to oces, or places in a structure. This is to treat the positions as objects, at least when it comes to surface grammar. ... Places in structures are bona de objects. (Shapiro, 1997, 83) The main principle behind structuralism is that any coherent theory characterizes a structure, or a class of structures. (Shapiro, 1997, 95)

So any coherent theory characterizes a structure, and its places are objects. Structuralism could oer a model of existence on which we might make sense of the abundance of thoughts that Prior and his veridicalist followers invite us to accept. After all, veridicalism is coherent as far as it goes.

In mathematics, it is tempting to think that existence is relatively easy: perhaps coherence is enough, as Hilbert and Shapiro suggest. Outside of mathematics matters are dierent. The clearest disanalogy is with concrete objects: the coherence of imagining a

$100 bill in my pocket sadly does not suce for the existence of that $100 bill, as Kant famously pointed out with regard to the ontological argument.

60 However, thoughts are abstract, and perhaps one could think of thoughts as positions in a structure.19 Thoughts stand in certain relations with one another: truth-conditional and modal relations at least, but more ne-grained relations as well, like aboutness (Yablo,

2014). Maybe the structure of thoughts can be characterized precisely enough for us to say that thoughts are positions in it.

The structure of natural numbers, Shapiro holds, is exemplied by any system of objects that satisfy the axioms of second-order Peano Arithmetic. These could be Zermelo sets, von Neumann sets, instants of time, and so on: there have to be enough of them and they have to form a sequence, which is to say that some relation among them must obtain to model the successor function. Perhaps the structure of thoughts could be exemplied by any system of objects that satisfy the axioms of the logic of thoughts. There have to be enough of these objects and they have to stand in a relation that models the structure of thoughts. What kind of relation could be a model of that?

A system of objects doesn’t have to be very complicated to exemplify the sequence of natural numbers. In fact, any system of objects for which we can identify something playing the role of 0 and on which something plays the role of the ‘is the successor of’ relation, is a system of objects that exemplies the natural number structure. There are many candidates. Indeed, in Shapiro’s work, structuralism is motivated in part as a response to Benacerraf’s (1965) famous puzzle about what numbers could (not) be. Benacerraf already provides multiple candidate systems of objects that could be the natural numbers, and the structuralist response is that any of them are candidate position-llers for the natural numbers.

19McDaniel (2015) defends the view that propositions are positions in a structure of true-in-virtue-of relations. I nd this view quite mysterious, not only for the hardly systematic use of ‘in virtue of’ in metaphysics, but also for the hardly rigorous characterization of the alleged structure.

61 It is hard to imagine a Benacerraf’s 1965-style puzzle about thoughts. Indeed, it is hard to imagine what relations there could be that are rich enough to model relations among thoughts. Importantly, these relations should be characterized independently of our understanding of the thought-candidates themselves, just as the sequence of natural numbers can be characterized independently of the natural number-candidates. I doubt that there is even an independent characterization of necessary truth preservation, i.e. independently of what thoughts are. What in the world could possibly stand in a relation of necessary truth preservation, other than thoughts themselves? Things get only more complicated if we move past necessary truth preservation.

If so, there are no systems of objects that could be candidates to exemplify the structure of thoughts. A Benacerraf’s 1965-style puzzle doesn’t get o the ground, and this cast doubts on the plausibility of a structuralist account of thoughts. The strategy would be to give priority to the structure in order to reduce questions of identity and existence to questions about the structure. However, this strategy saddles us with an extremely inationary ontology of ‘elusive’ thoughts, and a methodology of dubious explanatory value.

62 Chapter 4: Reference to Thoughts

In this chapter I shall address the explanatory problem of reference to abstract objects, thoughts in particular. This is where I think the value of Prior’s gambit gives out. To this end, I shall present some puzzles about quantication, including a metasemantic version of Benacerraf’s (1973) famous challenge about mathematical truth. The challenge is to explain what makes reference to thoughts possible. As a reply, I shall sketch a general account of singular thought about abstract objects. The account evades the challenge thanks to an explanatory reduction of questions of reference to a criterion of identity, introducing an abstraction principle for thoughts. Thus, we may recover from an externalist perspective the important Fregean insight that reference to abstract objects is easy.

4.1 Puzzles about quantication

Prior’s theorem states that if Epimenides believes that everything he believes is false, then he believes a truth as well as a falsehood (Prior, 1971, 87):

(PT) Q p Qp p q Qq q q Qq q

∀ ( → ¬ ) → (∃ ( ∧ ) ∧ ∃ ( ∧ ¬ )) This surprising result holds for any interpretation of ‘Q’, i.e. any property of the right type: to be believed by Epimenides, or asserted, imagined, and so on. This is counterin- tuitive: what’s so hard about believing or asserting just one thought (at a time)? Prior’s

63 gambit is to pass PT as a logical truth, despite its counterintuitiveness. Among Prior’s followers are Lewis (1986), Slater (1986), Anderson (2009), Bueno et al. (2013), and Bacon and

Uzquiano (2018), although there is little agreement on what the lesson of PT is supposed to be.

I think that the lesson of PT is best appreciated against the background of an account of reference to thoughts. As I will show in the next chapter, PT is not a logical truth, but a quirk that follows from not adopting the proper logic for thinking about thoughts.

The proper logic is a predicative logic, on which PT is made harmless, for reference to thoughts is established by an abstraction principle which, as many other principles of this kind, ought to be understood predicatively if the paradoxes are to be avoided. It will take this chapter and the next to fully spell out these considerations. I shall begin by saying what is problematic about reference to thoughts, in connection with PT.

The rst puzzle is about identity. Suppose that Aristotle dreams that nothing he dreams is true. It follows by PT that Aristotle dreams at least a truth and a falsehood: i.e. letting P be ‘Aristotle dreams that’, p P p p and p P p p . The false dream is that nothing Aristotle dreams is true,∃ for( if ∧p P) p ∃ p( and∧P ¬ )p P p p then by UI and MP, p P p p , hence p P p∀ ( p →by¬reductio) . There∀ ( remains→ ¬ ) some truth that is dreamt¬∀ ( by Aristotle.→ ¬ ) Notice¬∀ that( the falsehoods→ ¬ ) that Epimenides and Aristotle believe and dream, respectively, are dierent: for one is that p Qp p and the other is that p P p p . However, what are the mysterious truths∀ ( that→ PT¬ asserts) to exist?

Is the truth∀ ( dreamt→ ¬ by) Aristotle, if he dreams that nothing he dreams is true, identical to the truth believed by Epimenides, if he believes that nothing he believes is true? Or do they dream and believe dierent truths? Any answer to this seems arbitrary. If so, then

64 it is arbitrary to even say how many thoughts there are in given circumstances—this is certainly not encouraging.

There is another puzzle in the vicinity. Consider again the mysterious truth believed by Epimenides if he believes that p Qp p . We may ask: did Parmenides teach it or did he not? And what about∀ whether( → Zeno¬ ) understood it, or whether Pythagoras proved it? In other words, in a scenario in which Q p Qp p , is it the case that

R p Qp p , or S p Qp p ? I shall call this the∀ ( predication→ ¬ ) puzzle. Again, any answer∀ ( seems→ ¬ arbitrary.) ∀ ( The→ predication¬ ) puzzle hints at a violation of a constraint on reference articulated by Gareth Evans, which points us in the right direction:

if a subject can be credited with the thought that a is F, then he must have the conceptual resources for entertaining the thought that a is G for every property of being G of which he has a conception. This is the condition that I call ‘The Generality Constraint’. (Evans, 1982, 104)

Evan’s Generality Constraint is controversial, for it seems to entail that “category mistakes” such as ‘The number 2 is cold’ express entertainable thoughts (Camp, 2004). Re- gardless of the status of the Generality Constraint, it is important to highlight that the predication puzzle, like the identity puzzle on cardinality questions, brings out the tight connection between the possibility of singular thought and quanticational reasoning.

It may be not surprising that there are diculties in an account of reference to thoughts, but this does not depend on validating PT. After all, how is reference to thoughts supposed to work, in general? Even more dramatically: what makes reference to abstract objects possible in the rst place? These questions are Kantian in spirit: they are questions about the necessary conditions for the possibility of a certain cognitive act. Before we can conclude that Prior’s lead has taken us to a dead end, we should say what a proper account of reference to abstract objects might look like, and to thoughts in particular.

65 4.1.1 Background

Part of Quine’s rejection of intensional notions is based on diculties in the interpretation of variable binding in “indirect” or “opaque” contexts, i.e. in the scope of modal and propositional attitude operators. For Quine (1956), the primary contention is about terms that purport to refer to individuals in indirect contexts. It is clear however that terms that purport to refer to thoughts should have the same destiny. One way to present the diculty is to ask how (4) may be true, on its de re reading, even if Ralph would condently answer No to the question whether Ortcutt is a spy:

(4) Ralph believes that Ortcutt is a spy.

In the relevant scenario, Ralph does believe of him, namely Ortcutt, that he is a spy, though he would not assent to the thought that Ortcutt is a spy. Substitution puzzles of this sort led Quine to conclude that thoughts are “creatures of darkness” (Quine, 1956, 186), to be eliminated from a respectable ontology.

David Kaplan’s (1968) famous reply had been to show that a semantics for quantication in the scope of propositional attitudes is available to Quine, eectively by reducing problematic occurrences of referring expressions in the scope of propositional attitude operators to existential claims about “senses” that pick out the referents in the proper way (senses are considered unproblematic from the Fregean perspective Kaplan argues to be implicit in Quine’s discussion). So an occurrence of ‘Ortcutt’ in the de re reading of (4), may be analyzed roughly as follows:

(5) s B(s is a spy) ∆ s, Ortcutt

∃ ( ∧ ( ))

66 where s is a sense that selects Ortcutt in the appropriate way, and ‘B’ is the unproblematic de dicto relation of belief between Ralph and a Fregean thought, and Church’s (1951) ‘∆’ is the relation between a sense and its referent.

Kaplan’s account of “quantifying in” requires appropriate constraints on the selection of senses for the individuation of referents. On this point, Kaplan makes the intriguing suggestion of “standard names”. Intuitively, while ‘3 4’ and ‘12’ both refer to 12, only the latter is a standard name for 12. The purpose of× standard names is to open “a road back from reference to meaning” (Parsons, 2009, 56): Ralph’s relation to Ortcutt in a de re attitude report like (4), is simulated by a relation to a Fregean thought partly constituted by a standard way of picking out Ortcutt. If only in a Fregean setting, Kaplan has shown how to makes sense of singular thought on Quine’s watch.

Quine’s arguments are now generally regarded as overreaching: the lesson of substitution failure seems to be simply that the logic of propositional attitudes is not extensional, hence coreference is not sucient for substitution. However, questions about the connection between singular thought and quantication in a non-extensional setting still remain.

The Quine/Kaplan exchange rst put into focus the question of interpreting free variables in the scope of attitude operators: in particular, to explain the meaning of ‘Ralph believes that x is a spy’ under an assignment that maps x to Ortcutt. There is of course an analogous question about free variables ranging over thoughts. For example, we want to be able to explain what is the meaning of ‘it’, in an utterance of (6):20

20Not by chance but by design, (6) is a Donkey sentence: the pronoun occurs outside the syntactic scope of the operator that binds it. On my understanding of these topics, dynamic binding can be considered as (at least close to) the linguistic counterpart of singular thought. I shall not discuss dynamic theories of quantication in details for reasons of space. An account of dynamic binding that is particularly friendly to the assumptions I make below is Evans (1977).

67 (6) Kant asserted that arithmetic is known a posteriori. Frege denied it.

Intuitively, the second sentence in (6) means that Frege denied the thought asserted by

Kant, but (6) is true even if Frege had no notion of Kant nor of any of his assertions. In other words, the second sentence in (6) is true just in case Frege denied that arithmetic is known a posteriori. The question whether is a way of characterizing Frege’s relation to the thought that arithmetic is known a posteriori that bypasses descriptive means of reference to it is analogous to the problem of characterizing Ralph’s relation to Ortcutt in a way that avoids descriptive ways of picking him out.

The strategy I intend to follow is similar to Kaplan’s. Following an insight from Tyler

Burge (2009, 312), I shall take sentences to play the role of “standard names” for thoughts.

Ironically, we can thus recover another of Quine’s remarks, putting it to work against

Quine’s intentions: according to Quine (1970, 10), thoughts are “shadows of sentences”, and he didn’t think much of the shadows. My proposal will be to exploit the idea that thoughts are, in a sense, specied by the sentences that express them, like a shadow is

“specied” by the opaque body that projects it, to explain how it might be possible to refer to thoughts. Unlike Kaplan, I shall not make Fregean assumptions about reference, nor shall I assume that thoughts are Fregean structures composed of senses.

Motivation for the account sketched below derives from a challenge, that I call Metase- mantic Benacerraf for its similarity with Benacerraf’s challenge about mathematical truth

(Benacerraf, 1973). In rough outlines, the challenge is to provide a compelling account of the conditions under which reference to abstract objects might be determined, from an externalist perspective. I argue that these conditions are satised by specifying thoughts.

Specication involves: (1) reference to a thought by means of a sentence that express it.

68 So even though all of the following expressions (and possibly more) might pick out the

same abstract object, the third does not specify it in the relevant sense:

(7) a. That arithmetic is known a posteriori

b. The thought/proposition/state of aaits that arithmetic is known a posteriori

c. What Frege denied

Secondly, specication involves (2) a criterion of identity. For there is no guarantee of

reference (singular thought) until it is determinate whether, e.g.

(8) a. 2 2 4.

b. 4 + 2 = 2

are the same− thought.= Many authors accept that there may be multiple ways of referring

to thoughts, e.g. on the model of (7), including King (2002) and Schier (2003). The latter

in particular argues that reference to thoughts is involved in what he calls something-

from-nothing transformations: Rupert believes that the Earth is at. Therefore, there’s something Rupert believes.

These approaches agree with mine that the problem of reference to thoughts is solved

(or at least improved) by having reference determined by sentences which specify the thought. However, they all fall short of responding to the Metasemantic Benacerraf challenge, presented below, for they fail to indicate what are the identity conditions of the things thus specied. (An identity criterion is indeed oered by both King and Schier, on which the syntax determines the identity conditions of thoughts. That criterion will be shown to be inadequate in chapter 5.)

Given a sentence, one can form a singular term referring to the thought it expresses.

For this purpose, I shall exploit the machinery of Descriptive Singular Terms (like Evans’s

69 (1982) ‘Julius’), namely singular terms whose semantic properties are xed descriptively as a matter of metasemantic theory. This does not entail that reference presupposes quantication. It only follows that our denition of singular terms that refer to thoughts presupposes quantication over them, and there is no circularity involved here: we can certainly quantify over all real numbers prior to having dened names for all of them.21

The denition of this class of singular terms, as well as the specication of a thought by means of a sentence, opens up the space for a compelling Fregean argument that reference is explanatorily dependent on a criterion of identity. On this account, reference is easy: it suces that the conditions of identity of the referents be specied.

dsts explain the possibility of reference, but not reference itself: descriptions could fail to pick out anything. The task is, as it were, to “anchor” dsts to their referents more directly, and this is what is done by a sentential specication. To this end, I dene canonical dsts, which are something like “standard names” for thoughts formed by means of the sentences that express them. We can thus establish the identity conditions of thoughts by setting the truth conditions of identity statements in which ‘=’ is anked by canonical dsts. The proposal is thus to adopt an abstraction principle for thoughts, drawing from the rich literature on Neologicism (Wright, 1983; Hale and Wright, 2001). Abstraction allows one simultaneously to explain the coreferentiality of canonical dsts, and to dene the ontology of thoughts.

As we shall see, canonical dsts have many of the properties that most philosophers think proper names have. Rigidity derives from the metasemantic xing, essentially for the reasons Kripke gives in Naming and Necessity. Moreover, the account of reference

21As has been pointed out, this position certainly requires more argument. I am currently not in a position to provide more considerations in support of the view on this point, to avoid charges of explanatory circularity. I leave this discussion for further work.

70 is directly referential in the sense that thoughts expressed by means of dst are singular, or object-dependent. Some other alleged aspects of direct reference fail to be part of the account. To begin with, I draw no conclusions about acquaintance, i.e. a special epistemological relation by which a subject is put “en rapport” with an object, despite acquaintance being sometimes considered a necessary component of singular thought. My account is also silent on the metaphysical underpinnings of direct reference, which according to some must rely on structured propositions. These views, on acquaintance and on structured propositions, are inspired by Donnellan (1966), Kaplan (1978), and others. Push back to separate acquaintance and structured propositions from the topic of singular thought is more recent (Armstrong and Stanley, 2011; Stalnaker, 2012; Hawthorne and Manley,

2012).

4.2 The dilemma about mathematical truth

A proper account of reference to thoughts will require some choices in metasemantic theory: the theory or collection of theories that explain why certain expressions have the semantic value they have.22 These choices are motivated by a certain worry, closely related to a famous worry raised by Benacerraf (1973), that I will introduce below. Let me

rst briey review the original.

Benacerraf’s dilemma about mathematical truth is an epistemological worry for mathematical realism. The dilemma is fairly well understood: either we accept a standard semantics for mathematical language, and in particular a Tarskian treatment of quantication, or a causal theory of knowledge for mathematical belief. The former requires

22For an introduction to the topic, see Yalcin (2014). This section is inspired by Heck (2011), especially ch. 8. I also think that a great deal of what I have to say here is foreshadowed by Wright (1983) and several others in the literature on Neologicism.

71 acceptance of a domain of mathematical objects, platonistically conceived, and the latter requires that mathematical objects be causally related to our beliefs about them. For simplicity, we can say that a platonistic conception comprises at least the claim that mathematical objects are not part of the causal explanatory order: mathematical objects may

gure in explanations of physical or cognitive events, but not in causal explanations. (For example, the divisibility of 36 by 3 partly explains why Ada, Ben, and Chris could divide 36 candies evenly, even though that mathematical fact doesn’t causally explain the partition event.) So the truth conditions of mathematical sentences posit a realm of objects that are not part of the causal explanatory order, but a causal theory of knowledge requires them to be part of that order. This formulation of the dilemma is close to the original and, as it is well known, it can be both improved and generalized.

For the improvement, we reckon that nobody is really going to revise the standard semantics. Attempts have been made, of course. But these attempts have never managed to impress a large enough portion of the philosophical community—large enough to shift the standard. So one of the two horns of the dilemma simply drops, and the dilemma becomes a challenge: to explain how we can have a plausible epistemology of mathematical objects, platonistically conceived.

For the generalization, we reckon that the scope of Benacerraf’s challenge is broader than causal theories of knowledge. Field (1989, 25) presents it as “a challenge ... to explain the reliability” of our mathematical beliefs (Field’s emphasis):

We start out by assuming the existence of mathematical entities that obey the standard mathematical theories; we grant also that there may be positive reasons for believing in those entities. ... [Benacerraf’s challenge] is to provide an account of the mechanisms that explain how our beliefs about those entities can so well reect the facts about them. (Field, 1989, 25-26)

72 Field’s reconstruction has several merits, but in particular it claries that the challenge

targets any epistemology on which some of our beliefs about mathematical objects are

reliable—so, pretty much any plausible epistemology:

Given what science tells us about our own cognitive apparatus and our ways of nding out about the world around us, some of the rich ontological claims of ordinary discourse look distinctly problematic, leading us to ask, but how could beings like us come to have reliably true beliefs about objects like that? (Leng, 2016, 242)

There are dierent ways to respond to Benacerraf’s challenge. They range from the denial that there’s a genuine challenge (Clarke-Doane, 2016), to the denial that there are mathematical objects and consequent nominalistic reinterpretation of mathematical language (Field), to various attempts to develop an epistemology of abstract objects that is compatible with some form of platonism (Hale and Wright, 2002). Benacerraf’s epistemo-

logical challenge is only indirectly relevant to Prior’s work. However, I think that there

is a closely related metasemantic challenge that bears directly on the topics of reference, predication, and identity.

4.3 A Metasemantic Benacerraf

Benacerraf’s challenge derives from two ingredients: the abstractness of mathematical objects, i.e. their being removed from the causal order, and the reliability of our mathematical beliefs. In epistemology, there is nothing particularly puzzling about numbers or sets themselves.23

Versions of Benacerraf’s challenge apply to abstract objects outside of mathematics: any objects that are cognitively removed from us in a way that makes it at least initially

23In ontology, there is something puzzling about numbers and sets (Benacerraf, 1965). I discuss the analogous questions, regarding thoughts, in the nal section of chapter 3.

73 puzzling to explain how we might have reliable beliefs about them. There will be a dierence in force among dierent versions of Benacerraf’s challenge, due to the dierences in strength of our commitment to various theories of abstract objects: if there is no reason to think that theories of made-up metaphysical spooks are reliable at all, there is no relevant explanation to give.

Just like numbers and sets, thoughts are usually taken not to be part of the causal explanatory order. Our commitment to thoughts is to theoretical posits that play the role of contents of declarative sentences and objects of attitudes. Beliefs formed on the basis of semantic theories are often reliable, just like beliefs formed on the basis of (proofs from) mathematical axioms. This is true both of the rigorous theories of Linguistics, whose empirical predictions are routinely tested, and of ordinary speakers’ interpretations of attitude ascriptions, from which they get valuable information about their environment— about what their interlocutors believe, assert, or fear.

So it seems reasonable to say that a version of Benacerraf’s challenge applies, and with some force, to abstract objects quantied over in semantic theories, hence to thoughts in particular. The analogous question is how we come to form reliable beliefs about thoughts.

Presumably, there are possible replies to this version of the challenge parallel to replies available for the original case. As I said, in this chapter I shall not discuss the epistemological question of reliability, but a closely related question of reference.

There are at least two aspects that distinguish Benacerraf’s original challenge from the challenge I shall now articulate. The rst is that while the former is about mathematical objects, the latter is about thoughts. This, as I have just pointed out, is not a serious point of disanalogy: it’s the abstractness that matters. The second distinction is that while the former, as Field says, is “a challenge to explain the reliability of our beliefs”, the latter is

74 a challenge to explain our ability to refer to thoughts, i.e. to single them out whether by

linguistic or cognitive means. The Metasemantic Benacerraf challenge is the challenge to

provide an explanatory account of reference to thoughts, platonistically conceived.

To appreciate the force of the Metasemantic Benacerraf, let us take a step back, and ask,

more generally, what explains our ability to single out anything in speech and thought.

What, for instance, grounds the ability to refer to concrete individuals? On Kripke’s (1980)

picture, our ability to refer to Napoleon by the name ‘Napoleon’ is grounded in part in

a deferential intention to use the name as other members of one’s linguistic community

have done, in a history of uses of ‘Napoleon’ that goes back, ultimately, to a perceptual

link with Napoleon himself—this is when, according to philosophical folklore, we imagine

Napoleon’s parents having initiated a convention of use of ‘Napoleon’ by dubbing the

future emperor: “You shall be called Napoleon”, while pointing at him, or something like

that. As Tyler Burge (2009) explains:

Applying a name like “Aristotle” to the most famous Aristotle requires one’s usage to connect to a historical chain that must be characterized in terms of a set of very particular applications. ... ultimately going back to initial applications of the name (or a cognate) to a perceived individual. (Burge, 2009, fn. 10)

Note the mention of perception. This picture is not unproblematic, but many philosophers

nd it compelling.

It seems reasonable to suppose that the case of abstract objects is similar in certain respects. In particular, reference to abstract objects is neither self-explanatory nor primitive, but somehow grounded in facts about use. As in the case of individuals, reference to abstract objects plausibly depends, in part, on the deferential intentions of speakers and thinkers with respect to their linguistic community, so that reference is partly a matter

75 of membership in the right causal-historical chain. The role of intentions is also crucial, for the initial “baptism” ction, in coordinating on the same referent: Napoleon’s parents point to Napoleon and name him ‘Napoleon’, they don’t point to an undetached

Napoleon-part.

For these reasons, we can agree with Kripke that reference is transmitted by means of deferential intentions. But how is it determined? It is implausible to assume that reference to an abstract object is ultimately grounded in any kind of perceptual link with the referent—as Kripke’s ctional dubbing ceremony seems to suggest. Kripke’s story is apparently about the use of proper names to refer to concrete objects in our perceptual environment, though perhaps remote in space and time. Such objects are not altogether causally removed from us. The challenge is to explain how the Kripke/Burge picture applies outside the realm of causal ecacy. For it seems that an explanatory account of reference to abstract objects cannot be causal all the way down—that is, at least in its initial point, where the chain latches on to its origin, the ground for reference cannot be causal. But if not in perception, we may ask, how are abstract objects ultimately “given to us” (Frege, 1884, 73)?

The challenge is not limited to proper names, for we can certainly refer to abstract objects demonstratively (Kaplan, 1989), and it is not limited to the use of linguistic expressions, for we can certainly single things out in thought—indeed the cognitive aspect of reference seems, if anything, to have a certain priority over the linguistic aspect: thus according to John Perry (2009, 195), “[o]ur ability to think about particular things depends on our having some connection to them. Such connections are involved in various kinds of thinking and are exploited when we refer to the things”.

76 The plan is to see what must be modied about Kripke’s picture to get to a compelling

metasemantic account of reference to abstract objects. The argument that follows from

here, if correct, presents a response to the Metasemantic Benacerraf in several steps: rst, I

shall introduce Descriptive Singular Terms (dsts) as a concrete proposal for singular terms that purport to refer to abstract objects—terms like ‘Julius’, i.e. whoever invented the zip if anyone did (Evans, 1982). The possibility of reference by means of a dst depends on which domain the variables of their reference-xing description range over. An account of which domain that is, in turn, depends on which members it has, because domains are extensional. Therefore, the possibility of reference to abstract objects by means of dsts is

(partly) explained by the identity-conditions of such objects. Thus questions of reference are explanatorily reduced to a criterion of identity for the referents. This is a Fregean partial answer to the Fregean question I asked above (“how are abstract objects given to us?”), sketched against the background of a thoroughly un-Fregean, externalist, theory of reference. What will be left to do is to explain reference itself, not just its possibility. This,

I shall do by letting thoughts be specied by sentences.

4.4 Descriptive Singular Terms

It seems that we can clearly talk about ‘The truth asserted by Epimenides’, talk about it, and disagree about it. Such expressions are perfectly intelligible. We don’t have conventional proper names for thoughts as we do for individuals, but it seems plausible to suppose that descriptions are the default means of reference for our talk and thought about thoughts. Likewise, presumably, for other abstract objects, like the rst irrational number to be discovered, or my favorite plane gure: descriptions are a very general means of picking out objects, not necessarily concrete ones.

77 The instincts are right of those who think that the answer to the Metasemantic Benac-

erraf lies in the theory of descriptions. However, the evidence seems to be that descrip-

tions are not devices of reference (Neale, 1990, 2005): intuitively, ‘the F ’ is syntactically

and semantically like ‘every F ’ and ‘some F ’, and not like a proper name. This observation

lines up with Kripke’s (1980) famous arguments that descriptions do not have the seman-

tic and epistemological properties of proper names. Thus it seems best not to assume that

descriptions are singular terms.24

Descriptions have a dierent role to play in the theory of reference. Linguistic expres-

sions like ‘the F ’ don’t have to be singular terms to single things out. Kripke himself, of

course, allows for the reference of a singular term to be “xed” by a description. After all,

we do have a practice of stipulating arbitrary constants as names of objects that are sin-

gled out by descriptions: let ‘a’ refer to the F . Specically, a formula ‘The F is G’, if true,

has F and G satised by the element of a singleton set, whereas ‘Some F is G’, if true, has

F and G satised by the elements of a nonempty set. So the satisfaction conditions of a

descriptive formula do single something out. Moreover, since descriptions are not singu-

lar terms, descriptive xing of reference involves no regress, for quantication does not

presuppose reference (it is not the case that every object in a domain over which we can

quantify must have a name). The strategy is to dene singular terms under a descriptive

metasemantic rule that assigns them a unique referent. A quantier picks out something

that is stipulated to be the referent of a Descriptive Singular Term (dst).

24This perspective oers a good ground from which to reject “slingshot” arguments which, as I mentioned in the Introduction, have been wielded against theories of thoughts (Neale, 1995). Moreover, it is independently supported by linguistic evidence. In natural language, denite noun phrases, like ‘the F ’, dier from indenite noun phrases, like ‘an F ’, only in their presuppositional requirements—not in their syntax or semantics (Coppock and Beaver, 2012).

78 For concreteness, the interpretation function Iσ,c,w, relative to an assignment σ of elemnts of a domain D to free variables, a context c, and a world w, maps each dst t in DST to an object in the domain D w in a model M. (For reasons discussed in the

Introduction, I henceforth omit the relativization( ) to a context and a world.) The semantic properties of elements of DST are dened by the following rule:

(DST) For all t DST , Iσ t a D relative to a model M, if for some F , a is the only object in D that satises F in M. ∈ ( ) = ∈ The Existence and Uniqueness condition is a reference-xing descriptive clause. As for

Russell’s (1905) clever denition of English ‘the’, the proviso in eect says that the referent of a dst ‘t’ is exactly the a in the domain of the model that falls under F —only some values of ‘F ’ are appropriate, as I shall discuss later. Notice rst that the referent of t is given directly by the semantic rule. In other words, the semantic prole of dsts is completely exhausted by their role as referring expressions, as they contain no descriptive information about their referents: the referent is just a thing, though the metasemantic rule to pick it out it descriptive.25

If dsts are at least conceptually possible, they suce to show how dene linguistic expressions that refer to abstract objects. The conceptual possibility of dsts is enough for a rational reconstruction of an account of reference to abstract objects, to be assessed on the ground of plausibility, if not of empirical adequacy. Whether there are dsts in natural language is a question outside the scope of my discussion.

25Predelli (2016) denes “Russellian names” by means of a “context of association”. A semantic rule attached to a singular term maps a context of association to the expression’s Kaplanian character (Kaplan, 1989), which in turn is a function from the context of utterance to the expression’s intension. Thus, in a sense, the singular term can be thought of as abbreviating a description with which it is associated by the lights of the appropriate context—in this sense, these names are Russellian. Predelli’s account is similar to mine, but I refrain from shoving a metasemantic condition into a semantic rule by multiplying contexts, as Predelli does. Positing a “context of dubbing” for Kripkean singular terms, and a semantic rule that maps the context of dubbing to an expression’s character, would equally seem somewhat articial, and lead to complications about competence and cognition that are best set aside.

79 Examples of dsts may be familiar. ‘Julius’ refers to the entity, if it exists, that satises the open formula ‘ y F y x y ’ where F is ‘invented the zip’ (Evans, 1982, 31). That

is, we may gloss ‘Julius’∀ ( as↔ a dst= referring) to ‘whoever invented the zip, if anyone did’.

According to Kripke (1980), ‘Neptune’ was introduced by Leverrier as a singular term for

‘whatever is responsible for anomalies in the orbit of Uranus’, and ‘Vulcan’ for ‘whatever

is responsible for anomalies in the orbit of Mercury’. As has been argued at length by

Evans, Kripke, and others, ‘Julius’ and ‘Neptune’ fully qualify as rigidly designating and

directly referential singular terms. As for ‘Vulcan’, reference is in that case merely possible,

since there is no Vulcan: the existence condition fails.26

Therefore, a uniqueness condition associated with the referent is not enough to pick

out a referent. The existence condition in DST is required. A well known consequence of

potential failure of reference is that the inference from ‘F Vulcan ’ to ‘ xF x’ may not be

truth-preserving. If we adopt the model of dsts for reference( to abstract) ∃ objects, we ought

to work in something like a free logic—a logic in which quanticational inferences require

explicit existence assumptions (Evans, 1982). In practice, the ordinary rule of Universal

Instantiation is replaced by Free Universal Instantiation:

(UI) xF F t x

(FUI) ∀xF → [y~y] t F t x

∀ → (∃ ( = ) → [ ~ ]) where ‘F t x ’ is the result of replacing a singular term ‘t’ for ‘x’ in ‘P ’, and ‘ y y t ’ is an existence[ ~ ] claim for t. dsts provide a solution to the Metasemantic Benacerraf,∃ ( = but) consequently the logic of thought should be one in which the existential import of singular terms can be controlled.

26These popular philosophical tales may well be historically false: see fn. 2 of (Tennant, 2010). As far as ction goes, it illustrates the point.

80 For the reasons given so far, existence assumptions should be restricted in the logic

of thought. In particular, the theory of quantication over thoughts employs (a version

of) free principles like FUI. As pointed out by Bacon et al. (2016), in a logic of this kind

Prior’s theorem does not follow. Thus we get the added benet of avoiding the identity

and predication puzzles discussed above, and the obscurities of Prior’s theorem.

4.4.1 A Fregean argument

The use of dsts as a model for reference to abstract objects suggests an intriguing argument to reduce reference to identity. The reduction is explanatory (i.e., neither metaphysical, nor ontological, nor epistemological): besides the ordinary conditions on the appropriate use of a singular term, such as the speakers’ deferential intentions towards other members of their linguistic community, what remains to be explained about the possibility of reference to abstract objects is their identity conditions. Deferential intentions determine how reference is transmitted, and I shall now argue that reference by means of dsts is established by the identity conditions of the referents.

Consider whether it is possible to refer to some a by a dst. According to DST, it is well

within our competence to stipulate that a singular term ‘t’ refers to a provided that, for some F , a is the only element of the domain of the model that satises F . Such descriptive clauses are like a shot in the dark: if they hit a target, it depends on what’s in their range.

As shown by the contrast between ‘Neptune’ and ‘Vulcan’, reference depends on whether the quantiers in clauses of the form ‘whatever satises F ’ range over a domain that includes the relevant entity. So questions about whether reference by means of ‘t’ is possible reduce to questions about which domain the quantiers in the reference-xing descriptive clause range over. In other words, an explanation of how reference by means

81 of a dst works takes us to an account of the domain which the descriptive clause quanties over.

A domain is a set-like object. Questions about which domain is the domain of a description reduce to questions about which members it has: dierences between any two domains come down to there being members of one that are not members of the other.

So an explanation of which set-like object a certain domain is, depends, in turn, on an account of the identity of its members. The possibility of reference by means of dsts is

thereby explained by a criterion of identity for a domain of objects, possibly including

its referent. In other words, the possibility of reference by means of a dst reduces to an account of the identity-conditions of the objects in the domain of quantication of the reference-xing description—together with an account of speakers’ deferential intentions and referential intentions for the determinacy of reference. Thus, criteria of identity for abstract objects of the relevant type suce to explain the possibility of referring to them.27

Descriptive reference by means of dsts has in common with proper names on Kripke’s story that the practice of referring is ultimately explained by, among other things, facts about the referents themselves (not concepts, nor senses, nor modes of presentation).

According to Kripke’s story, a baptism is a way of attaching the practice of naming to that very individual which is named. According to the argument I have just given, the identity conditions of things in the range of quantiers in a description suce for the description to x the reference of a dst. Identity is matter determined by the things themselves, not

27Questions about what explains reference should be sharply distinguished from questions about the cognitive states of speakers as they perform acts of reference: metasemantics is not epistemology. Gareth Evans (1982) defends an epistemological condition on reference, in some sense parallel to the metasemantic condition I defend here, on which reference depends on the identication of the referent by the speaker: some kind of ‘knowing which’. On the externalist account presented here, the doxastic states of speakers play no role in explaining reference: the story about dsts is compatible with all sorts of errors and miscon- ceptions on the part of speakers, as it should be—and as is the lesson of classic arguments by Putnam (1975); Kripke (1980); Burge (1979).

82 by concepts, senses, or modes of presentation. Hence whether you refer to an object by a name or by a descriptive singular term, reference depends at least in part on the identity of the referent.

This conclusion establishes a tight connection between identity and the possibility of reference. This is a remarkably Fregean conclusion: a reduction of reference to criteria of identity is often advocated by defenders of Frege’s strategy for deriving an ontology by abstraction. Thus, in a sense, the argument I have presented here falls well within the scope of Frege’s strategy. It is however importantly dierent from existing Fregean accounts: (i) unlike the account in Hale and Wright (2001), I have avoided controversial appeals to Frege’s notion of ‘recarving’ (Frege, 1884, 74-75); (ii) unlike the account in Linnebo (2018), and the one by Hale and Wright, I have avoided Frege’s conception of reference as mediated by a concept, which is problematic given widely accepted externalist arguments. What then is reference mediated by, if anything? The answer I shall give in the next section is not general, but a proposal made about thoughts only: reference to thoughts is mediated by the sentences that express them, and by which thoughts are specied. (Plausibly, the proposal may apply to all “language-dependant” abstract objects, i.e. all the objects of semantics.)

4.5 “Standard names”

I have argued so far that questions about reference to abstract objects by means of dsts reduce to questions about their identity-conditions. The argument is fully general and applies to all sorts of abstract objects.

There are two things left to do to complete the account. First, we need to give a criterion of identity for the referents; second, we need to reconsider the question that I

83 set aside in the previous section, about what kind of predicate might occur in a reference-

xing clause. The obvious diculty is that some predicates specify more than one thing, and others nothing at all. So for example it would not be acceptable to dene a singular term ‘t’ that refers to an object a provided that a exists and uniquely satises x x, for this would single out every object. Likewise one cannot dene a singular term that= refers to the unique existing x such that x x. The two questions for this section, what is a criterion of identity for the referents¬ and= what properties can be used to single out the referents, are of course closely related. While the argument I gave so far is fully general,

I shall now focus on thoughts only.

Recall (6), repeated here as (9):

(9) Kant asserted that arithmetic is known a posteriori. Frege denied it.

Intuitively, we want to say that Frege denied the thought that arithmetic is known a posteriori. As noted above, Frege need not know anything about Kant’s assertions, in order for

(9) to be true. Indeed, Frege could be under all sorts of mistaken beliefs about Kant and his assertions, without this compromising his chances to bear an attitude directly about that thought. The structure of the case is parallel to Quine’s Ralph/Ortcutt example, though here the target is a thought and not an individual (or a sense).

The metasemantics of dsts, properly restricted, has a solution. Intuitively, sentences are something like the privileged vehicles for our cognitive access to thoughts: when we refer to the thought that S, or the proposition expressed by ‘S’, thoughts are specied by the sentences that express them. Quine was right, after all, when he said that thoughts are

“shadows of sentences” (Quine, 1970, 10). We can think of sentences as “canonical names” for thoughts, roughly in the sense of Kaplan (1968). This insight is foreshadowed by Tyler

Burge:

84 When I think I (or you) believe that not all people are great pianists, I must think the representational thought content that not all people are great pianists in the course of attributing it. I also canonically name or designate the representational thought content via a singular term, the analog in thought of a that-clause. [Footnote, TB: A special feature of these (e.g., that-clause) canonical content-names is that mastering them requires mastery of the named or referred-to contents themselves. So there is, in a certain way, an even more intimate relation between this sort of canonical name and its named contents than there is between a canonical number name like “2” and the number. Here one literally must understand the denotation (the customary content or sense) before grasping the content of the name or individual concept that canonically names it. ... ] My relation to the referent is not purely descriptive. It is true that the canonical specication is ability general and conceptual. But the specication is backed by comprehension of the referent. Comprehension is at least as direct and noninferential, psychologically and epistemically, as perceptual relations. Comprehending a representational content is exercising an ability that is constitutively associated with inference. But it is not itself inferential or descriptive. I think that comprehension is a direct intellectual capacity that when constitutively combined with reference can make de re reference possible, when reference is carried out in a canonical way. (Burge, 2009, 311-312)

Perception plays a role in establishing the reference of a name, in the ordinary Kripkean account. Burge suggests that a singular term, “the analog in thought of a that-clause”, might refer to a thought—and that the cognitive achievement that perception leads to in the ordinary case, is here obtained by “comprehension”. The question of what comprehension amounts to is a topic for elsewhere.

85 To streamline these remarks, let be a many-one from sentence to thoughts.28 Intu- J K itively, for a sentence α , the expression⋅ ‘ α ’ can be read ‘The thought that α’ or ‘The J K thought expressed by α#’. Canonical Descriptive Singular Terms are any dsts such that:

(CDST) Iσ t a #D, relative to a model M, provided that for some α , a is the only element of D specied by α . ( ) = ∈ # # ‘ ’ is not the interpretation function: it is a symbol of the language, that takes a sentence J K as⋅ argument and forms a singular term referring to the thought it expreses. In eect,

canonical dsts are dsts in which choice of a reference-xing property is restricted to

properties of the form being the thought that α, or being the thought expressed by ‘α’, for

some α.

4.6 Begrisschri §3

By means of canonical dsts we can state a criterion of identity for thoughts such that

at least one and at most one thought satises the reference-xing condition. Employing

resources that may be familiar from the rich literature on Neologicism, I shall take the

identity conditions of thoughts to be established by abstraction on the sentences that

express them. The benet of an abstractionist approach is that questions about what

thoughts are bottom out here (Rayo, 2013).29

28The argument of ‘ ⋅ ’ is a “sentence”, by which I mean a sentence-type, or Chomskyian Logical Form: an input to semantic interpretationJ K that yields thoughts as output. Needless to say, sentence-types are also abstract objects (though, if Chomsky is right, they are “generated” by a mental faculty. The proposal I sketch below is therefore to abstract thoughts from things that are themselves abstract objects. There is no regress here, because the relation of a sentence/Logical Form to its concrete instances (an utterance) is that of a type to its tokens. The metaphysics of type/token abstraction is perhaps not quite the same as in the abstraction of objects from equivalence classes. For an account of the former that could serve as background to my discussion, see Linnebo (2012b). 29For a start on Neologicism and Abstractionism more generally, see the Stanford Encyclopedia article by Tennant (2017), and the works of Wright (1983); Hale and Wright (2001); Burgess (2005); Heck (2011); Ebert and Rossberg (2016); Linnebo (2018). On identications, see Linnebo (2006) and Dorr (2016).

86 There are many examples of abstraction principles. In Grundlagen §64, Frege presents an abstraction principle for directions (Frege, 1884, 74). For any lines l1 and l2, the di-

rection of line l1 is identical to the direction of line l2 just in case l1 is parallel to l2. In formulas, read ‘Dir l1 ’ as ‘the direction of line l1’. Abstraction is given by a relation of parallelism ‘ ’: ( )

~~ l1, l2 Dir l1 Dir l2 l1 l2

In this abstraction, directions∀ are( specied( ) = by parallel( ) ↔ lines,~~ in) the sense that the obtain-

ing of a relation of parallelism between lines suces to explain reference to their shared

direction. The relation of parallelism between lines provides a criterion of identity for

directions.

Perhaps the most famous abstraction principle is Hume’s Principle, considered and

rejected by Frege (1884), and later revived by Neologicists (see fn. 29). Let ‘F ’ and ‘G’ be

concepts (i.e. functions from individuals to truth values), ‘#F ’ a singular term read ‘the

number of Fs’, and ‘ ’ the relation of equinumerosity between concepts:

∼ (HP) F,G #F #G F G

∀ ( = ↔ ∼ ) For any F and G, the number of F s is the number of Gs just in case F is equinumerous

with G. Just as the direction principle denes identity-conditions for directions, HP de-

nes identity-conditions for numbers. Whereas directions are specied by lines, numbers

are objects specied by concepts. The fascination with HP derives from Frege’s theorem:

a result establishing that the axioms of Peano Arithmetic can be derived from HP alone,

The proposal to resolve questions of identity and existence about thoughts by abstraction is, in my view, a theoretical choice motivated by considerations sketched at the end of chapter 3. We should not expect that all kinds abstract objects are delivered by an appropriate abstraction principle. As far as the present work is concerned, thoughts are.

87 together with some denitions, in second-order logic (Heck, 2011).30 If HP is a priori

knowable or conceptually true, as some neologicists have argued, Frege’s theorem shows

that the truths of arithmetic are a priori knowable or conceptual truths.

Abstraction principles are powerful metaphysical gears. An equivalence relation on

the elements of some domain (e.g. lines or concepts) suces to dene the identity condi-

tions of abstract objects that are specied relative to that domain. Directions are specied

by lines that are parallel to one another, and numbers are specied by equinumerous con-

cepts. Likewise, we might say that thoughts are specied by the sentences that express

them. Specication makes a thought cognitively available for reference and quantica-

tion, in the way discussed above. Specication is not a causal relation, like the percep-

tual relation between speakers and referents that ultimately grounds uses of the name

‘Napoleon’. It’s the metaphysical counterpart of Burge’s cognitive notion of comprehen-

sion. The hypothesis that thoughts are the abstracts of sentences is compatible, and indeed

supported, by an account of reference that satises the conditions for singular thought,

and overcomes Benacerraf’s metasemantic challenge.

The project of abstracting thoughts was inaugurated by Frege himself, in §3 of Begri-

sschrift: A distinction between subject and predicate does not occur in my way of rep- resenting a judgement. To justify this I remark that the contents of two judgments may dier in two ways: either the consequences derivable from the rst, when it is combined with certain other judgments, always follow also from the second, when it is combined with these same judgments, [and conversely,] or this is not the case. The two propositions ‘The Greeks defeated the Persians at Plataea’ and ‘The Persians were defeated by the Greeks at Plataea’ dier in the rst way. Even if one can detect a slight dierence in meaning, the agreement outweighs it. Now I call that part of the content that is the same in both the conceptual content. Since it alone is of signicance for

30The equinumerosity of concepts, F ∼ G, can be stated in second-order logic as follows (Boolos, 1987): ∃R(∀x(F x → ∃!y(Gy ∧ xRy)) ∧ ∀x(Gx → ∃!y(F y ∧ xRy))).

88 our ideography, we need not introduce any distinction between propositions having the same conceptual content. (Frege, 1879, 12)

According to Frege, the thoughts (‘conceptual contents’) of two sentences are identical just in case whatever follows from the rst, together with some assumptions, follows from the second, together with the same assumptions, and vice versa. Let α, β, γ, be metavariables for sentences, and U a (possibly empty) set of sentences. Let the sequent sign ‘ ’ formalize Frege’s ‘following from’. An equivalence relation roughly meaning

‘having⊢ the same consequences as’ can be dened from Frege’s remarks (Tennant, 2003):

(FP) α β U α, U γ β, U γ J K J K = ↔ ∀ ( ⊢ ↔ ⊢ ) Frege’s Principle asserts that the thought that α is identical to the thought that β just in case whatever follows from α together with some assumptions, follows from β together with the same assumptions. FP is a proposal for a conceptual identication: to equate sameness of thought expressed with having the same consequences.31 Frege certainly

31This was Frege’s view around 1879. Around 1906, Frege endorsed the Equipollence principle (Frege, 1979, 197), eectively a version of indiscernibility:

two sentences A and B can stand in such a relation that anyone who recognizes the content of A as true must thereby also recognize the content of B as true and, conversely, that anyone who accepts the content of B must straightaway accept that of B.(Equipollence). ... So one has to separate o from the content of a sentence the part that alone can be accepted as true or rejected as false. I call this part the thought expressed by the sentence. It is the same in equipollent sentences (Frege, 1979, 197-198) A tempting suggestion is that Frege abandoned the project of abstracting thoughts after the discovery of the distinction between sense and reference. However this might be too hasty: in the quoted passage Frege mobilizes abstraction to “justify” his rejection of the subject/predicate analysis of judgment, which certainly remained a cornerstone of his logic throughout. Since he did not change his views, there is reason to think he did not abandon the project of thought abstraction (though of course since, after the sense/reference distinction, his logic became thoroughly extensional, that project shifted into the project of abstracting truth-values, i.e. the only kind of contents for sentences allowed in Grundgesetze). It should be noted that abstraction is not incompatible with Equipollence: it depends on how ne-grained the equivalence relation on the right-hand side is. In a hyperintensional logic, the identity conditions of thoughts established by abstraction could satisfy Equipollence (in fact, they do in the theory I present here, see the Addendum).

89 took ‘ ’ to be reexive and transitive. If so, the right hand side of FP can be simplied:32

⊢ (BFP) α β α β β α J K J K = ↔ ( ⊢ ∧ ⊢ ) The thought that α is identical to the thought that β just in case α and β follow from one

another. In this case, the proposed identication is between the concepts of sameness, for

thoughts, and mutual implication. BFP is Burali-Forti’s principle, for Cesare Burali-Forti

who made this suggestion about 15 years after Frege (translation by Mancosu (2016, 94)):

If A and B are propositions, the relation A B.B A or, A is equivalent to B, is reexive, symmetric, and transitive ... We obtain then from each proposition A the abstract entity value of A or→ meaning→ of A: and we say that “The meaning of A is equal to the meaning of B, just in case A is equivalent to B”. (Burali-Forti, 1894, 147)

Thus Frege and Burali-Forti had in mind the same principle. In light of the discussion

above, I take thoughts to be dened by abstraction on equivalent sentences. Mine is not a

proposal that follows the details of Frege and Burali-Forti, whose interpretation is not my

aim. I do follow, however, their general project. In particular, I shall introduce a symbol

for hyperintensional equivalence, whose properties are dened and motivated in the next⇔

chapter, which holds among sentences just in case the identity relation holds between the

thoughts they express. The principle is given in schematic and axiomatic form, and some

remarks on the dierences between these two statements is given below:

(TA) α β def. α β J K J K

(TA) p, q= p ⇔q ( def.⇔ p) q J K J K 32BFP implies FP if ‘⊢’ is reexive∀ and( transitive.= Assume⇔ ( BFP⇔ and)) its left hand side. Then α ⊢ β and β ⊢ α. Then α, U ⊢ γ only if β, U ⊢ γ for β ⊢ α and ‘⊢’ is transitive. Likewise for the other direction. Assuming the right hand side of FP and BFP, taking U = ∅, the left hand side of BFP follows by instatiating γ as β by the reexivity of ‘⊢’, and vice versa. Thus FP follows. Moreover FP implies BFP by essentially the same reasoning.

90 In its schematic form, Thought Abstraction asserts that the thought that α is the thought

that β just in case ‘α’ and ‘β’ are hyperintensionally equivalent. In its axiomatic form,

Thought Abstraction asserts that for all thoughts are identical just in case they are hy-

perintensionally equivalent. I shall stipulate for the moment that ‘ ’ is an equivalence

relation, and regiment it properly in the next chapter. So hyperintensional⇔ equivalence

suces for the abstraction of thoughts. Hyperintensionally equivalent sentences express

the same thought. The qualier ‘hyperintensional’, as the discussion in the next chapter

will clarify, indicates that unlike material or necessary equivalence, ‘ ’ is the analog of

identity, and supports substitution of equivalent sentences in the scope⇔ of attitude oper-

ators. (Its logic will be, accordingly, quite weak—but not so weak as to yield structured

propositions, as we will see.)

Usually, among the two sides of an abstraction principle one nds a material bicon-

ditional, as in e.g. ‘ F,G #F #G F G ’. One would then expect to nd, e.g.

‘ α β α ∀ β ’.( But= this only↔ says∼ that) the left-hand side holds just in case J K J K the right-hand= ↔ side( ⇔ holds.) Presumably if the logic contains modal operators, Neologicists would be inclined to strengthen the connection between the two sides to a necessary equivalence: necessarily the left-hand side holds just in case the right-hand side holds.

Since however only ‘ ’ suces to state sameness of thought, it seems more approprate to formalize TA as I did.⇔ Sameness of thought is presumably an adequate condition for abstraction more generally, e.g. the thought that the number of F is the number of G is the same thought as the thought that F is equinumerous with G. In any case, since hyperintensional equivalence does not hold if material equivalence fails to hold, TA implies the weaker equivalence.33

33See the Addendum for details. Notice furthermore that since hyperintensional equivalence suces for substitution salva veritate in the scope of propositional attitude operators, as I shall argue in the next

91 There is a worry about explanatory circularity, concerning TA, which is especially salient in case one considers the axiomatic version. In a denition of a referential device for thoughts (such as the one dened on the left-hand side of TA), is it acceptable to assume that one can quantify over thoughts (on the right-hand side)? Doesn’t this mean that we are assuming what we are supposed to show? This problem is too hard to be solved in a brief comment, and deserves further investigation. For present purposes, suce it to notice that TA is very similar to what the situation is with HP: for on HP, one quanties on the right-hand side over all objects that can be counted. This includes numbers, i.e. the objects referred to on the left-hand side by a referential device dened by the abstraction itself. Of course, many other objects besides numbers can be counted. Imagine now that numbers were the only things in the range of the rst-order quantiers on the right-hand side of HP, and the situation is structurally parallel to TA: only thoughts are in the range of the quantiers on the right, even though reference to them is dened on the left. This is not circular, for quantication is prior to reference.

TA indirectly denes the ontology of thoughts and their identity conditions, by dening directly the truth conditions of identity statements in which thoughts are referred to by canonical descriptive singular terms, i.e. singular terms of the form α for some ‘α’. J K As Burgess (2005) shows, an abstraction principle like TA can be decoupled in a theory of abstraction, consisting of an existence claim and an identity claim (I present this for the chapter, one cannot believe that ‘α’ and ‘β’ are hyperintensionally equivalent if one fails to believe that they express the same thought. In my view, this is compatible with, but in no way entails, the claim that TA is a priori, or that it is conceptually true, for speakers hardly have any conception of hyperintensional equivalence, and of sameness of the thoughts expressed by sentences.

92 schematic version only, the axiomatic version being handled in analogous way):

(EX) p p α J K (ID) ∃p,( q =p α) q β p q α β J K J K ∀ ( = ∧ = → ( = ↔ ( ⇔ ))) The combination of EX and ID highlights the existential claims that can be derived by abstraction. Notice that EX is true just in case the existential condition in CDST, i.e. the denition of canonical dsts, is satised. Notice further that ID is true just in case the uniqueness condition in CDST is satised, because of the functionality of ‘ ’. In eect, J K since the position marked by ‘α’ in EX is implicitly bound by a universal quantier⋅ (and explicitly so in an axiomatic version of the principles above), the metasemantic underpinnings of TA implement the assumption that every sentence whatsoever expresses a thought and exactly one: the thought that, as I argued, it species. While this result completes the account of reference, it is the source of paradoxes, as we shall see in chapter

6.34

The ontology of abstraction is relatively well understood. Following Linnebo (2012a), abstractionist projects can be described as implementing a form of metaontological minimalism: little is required of the world for the existence of the abstracted material. On the other hand, precisely because the demands on reality are modest, abstractionist views tend to support forms of ontological maximalism (Eklund, 2006): there are a great many

34Some philosophers hold that there are inexpressible thoughts (Wrigley, 2006), perhaps on the basis of informal cardinality consideration. I don’t think that this, or other arguments I have seen for the existence of inexpressible thoughts are conclusive. von Fintel and Matthewson (2008) take the assumption that all meanings are expressible to be a natural hypothesis for the empirical study of natural language, so I take the position that are no inexpressible thoughts to be a comfortable default. Proper argument shall be given elsewhere.

93 objects of the kind so dened. The abstraction of thoughts falls under these general principles, and is compatible with dierent ways of developing the metaphysics of abstraction discussed in existing literature.35

35TA inherits some merits and some defects of other abstraction principles, but I leave a detailed discussion for another place. Besides the Caesar Objection and Bad Company (about which, see the works cited in footnote 29), there are also original diculties, as I mentioned in footnote 34.

94 Chapter 5: Wittgenstein’s fundamental thought

The “fundamental thought” of the Tractatus, according to its author, is that the logical constants do not represent. Failures to comply with Wittgenstein’s fundamental thought pay a high price: the standard Gricean explanation for some observable facts about language use is no longer available. Theories of structured propositions fail to comply. Therefore, thoughts cannot be as ne-grained as structured propositions. Thus, propositional identity is a “Goldilocks” problem: thoughts are more nely individuated than intensions, but less nely individuated than structured propositions.

5.1 Structured propositions

According to theories of structured propositions, the content of a (declarative) sentence

(relative to a context) is a structured object. Structured propositions have been proposed as metaphysical candidates for thoughts. Simplifying away from various distinctions made by dierent authors, which are irrelevant to the argument below, we may follow

Salmon (1986) and others, and represent the content of a subject/predicate sentence of the form ‘Fa’ as a structure o,being F consisting of the semantic value o of the singular term ‘a’ and a property being< F expressed> by the predicate ‘F’ (where o may be an individual, or an individual “under a mode of presentation”, depending on whether the theory under discussion makes Fregean or Millian assumptions about names).

95 Theories of structured propositions have been motivated by questions about reference and propositional attitude reports, and enjoy wide appreciation in part because of the straightforward metaphysical hypothesis on aboutness they articulate: that we may understand a sentence ‘Fa’ as being about the semantic value o of ‘a’ in terms of o’s being a constituent of the structured proposition expressed by ‘Fa’. Theories of this kind have been defended by Salmon (1986), King (2013), Soames (2015), Hanks (2015), among others.

The dierent authors disagree about several matters of detail, but the point I would like to make holds regardless.

A more technical motivation for structured propositions is that they seem to provide a simple metaphysical model of the “senses” of sentences, or thoughts, in Alternative (0)

(Church, 1973a, 1974). Church’s Logic of Sense and Denotation admits of various formula- tions, depending on dierent criteria of propositional identity, or synonymy. Alternative

(0) implements the strictest criterion (Anderson, 1980, 221): intuitively, no two syntactically distinct sentences express the same thought, except for (i) replacement of expressions that are stipulated to be synonymous, and (ii) renaming of bound variables.

Church (1984) introduces ‘strict equivalence’ as a two place connective, ‘ ’ that represents identity of thoughts (in the language of Principia Mathematica (Whitehead= and Rus- sell (1927)), identity is undened for complex formulas). Intuitively, ‘α β’ just in case the thought expressed by ‘α’ is the thought expressed by ‘β’. Axioms for= strict equivalence that correspond to Alternative (0) would include at least the following:

(10) a. p p p

b. ∀p,( q =p ) q p q

c. ∀p, q((p = q) = p= q)

d. ∀pψp(( =qψq) → = )

∀ = ∀ 96 The rst three axioms capture the idea that strict equivalence is a higher-order analog of

identity, and they entail symmetry and transitivity. The last line is an axiom schema for-

malizing the criterion of synonymy for Alternative (0) informally described above. Struc-

tured propositions are identical if the sentences that express them are strictly equivalent

in Church’s sense (for the “only if” direction of this claim, further assumptions are needed,

but to avoid complications let these suce for now).

It is helpful to organize the discussion of dierent criteria of propositional identity

around the abstraction principle whose adoption I motivated in chapter 4. Prima facie,

there is a range of options, depending on the choice of the equivalence relation among

sentences on the right hand side. At one extreme is the thesis that sentences express the

same thought just in case they are necessarily equivalent. Let ‘πI ’ map a sentence to its intension. We can state this principle schematically as follows, for sentences α, β:

πI α πI β α β

= ↔ ◻( ↔ ) The intension of α is the intension of β just in case α is necessarily equivalent to β.

This is true, and it is straightforward to see that a model for this equivalence is given by the powerset PW of the set of all possible worlds W . However, taken as a proposal for propositional identity, this equivalence fails. Notoriously, replacement of necessarily equivalent sentences within the scope of propositional attitude operators is not truth- preserving.

Church’s Alternative (0) is the other extreme. Let ‘πS’ map a sentence to the structured proposition it expresses:

α β πSα πSβ

( = ) → =

97 The converse implication, as I said, requires assumptions that are no longer shared among dierent theories of structured propositions. However, on this proposal, we avoid failure of substitutivity for obvious reasons: for a propositional attitude operator ‘Q’, there are no counterexamples to the entailment from Qα and α β to Qβ, precisely because strict equivalence is so strict. Clearly, the right explication= of ‘ ’ falls somewhere between necessary equivalence and strict equivalence. I shall argue⇔ that it falls properly between: that (i) it is more ne-grained than necessary equivalence but also that (ii) it less ne- grained than strict equivalence.

Evidence in favor of (i) is given by the failure of Two-dimensionalism to accommodate hyperintensionality—as I argued in chapter 2. The argument for (ii) relies on what

Wittgenstein calls “his fundamental thought” in the Tractatus Logico-Philosophicus, Propo- sition 4.0312:

My fundamental thought is that the ‘logical constants’ do not represent. That the logic of the facts cannot be represented. (Wittgenstein, 1921)

Compare Wittgenstein’s “fundamental thought” with Nathan Salmon’s Thesis IV and The- sis VI, of his theory of structured propositions (at least in Salmon’s Frege’s Puzzle, these theses are not defended by an argument):

Thesis IV. Any expression may be thought of as referring, with respect to a given context, time, and possible world, to its information value with respect to that context. (Salmon, 1986, 187) Thesis VI. The information value, with respect to a given context, of an n-adic sentential connective, is an attribute, ordinarily of the sort of things that serve as referents for the operand sentences. (Salmon, 1986, 187)

Together, IV and VI entail that the sentential connectives (Wittgenstein’s logical constants) refer to “attributes” (i.e. properties) of structured propositions. Properties are representational if anything is—there are many ways to understand this claim, but it is the

98 nature of properties to represent something as being a certain way: being wise is a way

Socrates is. Furthermore, the sharing of properties is taken to explain similarity: Socrates

and Aristotle are similar in at least one respect, for they share the property being wise.

There are many metaphysical models of properties: sets of possibilia, concepts, univer-

sals, tropes, and so on, on which the notion of “sharing” and the representationality of

properties are cashed out dierently. However we think of properties, though, it appears

that according to Salmon the logical constants do represent, contra Wittgenstein. Other

theories of structured propositions agree with Salmon on this (see e.g. Soames (2015, 30)

and Hanks (2015, 98-108), who also do not argue for this assumption, as far as I know).

The logical constants represent, according to these theories, for they contribute ma-

terial to the proposition expressed by a sentence in which they occur. In particular, con-

junctions and disjunctions work out as follows:

πS α β conj, πSα, πSβ

( ∧ ) =< > πS α β disj, πSα, πSβ

( ∨ ) =< > where conj and disj are the “attributes” of propositions contributed, to the structured propositions expressed by complex sentences, by conjunction and disjunction respectively. Roughly speaking, these attributes can be thought of as the properties being conjoined and being disjoined. What exactly these properties are is a matter that depends on the dierent authors, but all defenders of structured propositions agree that conjoined and disjoined sentences express structures that have a certain complexity, reecting the syntactic complexity of the sentences. (Something similar goes for negation and other connectives, but I will focus on disjunction and conjunction for now.) Thus, theories of structured propositions egregiously violate Wittgenstein’s fundamental thought.

99 5.2 On redundancy

The diculty for theories of structured propositions is to capture a signicant notion of redundancy. We want to be able to say that (11a) is redundant with respect to (11b):

(11) a. Linda is dancing.

b. ??? Linda is dancing and Linda is dancing.

(11b) is infelicitous, as indicated by the question marks: ordinary speakers don’t go around asserting things like that. (If you don’t believe me, add more self-conjunctions, as many as you need to feel uncomfortable.) Of course, (11b) is syntactically well-formed and has a perfectly intelligible meaning. So the infelicity of (11b) is not syntactical, nor grammatical more generally. Moreover, the unacceptability of (11b) is not due to its length or complexity: two or three self-conjunctions are enough to be unacceptable, and yet conjunctions of two or three conjuncts are perfectly ordinary.

The case of disjunctions is perfectly analogous, and again the infelicity must have a pragmatic origin (as above, feel free to add as many self-disjunctions as you need to get the point):

(12) a. Laura is swimming.

b. ??? Laura is swimming or Laura is swimming.

These contrasts are nothing extraordinary. Indeed, there is a very straightforward Gricean story to explain the markedness of (11b) and (12b), and there is more than one way to tell it. In general, the story is about redundancy (Ciardelli and Roelofsen, 2017).

The pairs (11a)/(11b) and (12a)/(12b) are pairs of alternatives. In each pair, the ‘b’ sentence is redundant with respect to the ‘a’ sentence. Intuitively, speakers are never in a position to assert the longer and more complex alternative, since doing so accomplishes

100 nothing more than what is accomplished by an utterance of the simpler alternative, and

the simpler alternative is always available. Principles of cognitive or conversational econ-

omy rule out the redundant alternative. In terms of the quasi-formal notion of ‘what is

said’ (Grice, 1989), the explanation can be put as follows: what is said by the redundant ‘b’

forms is no more and no less than what is said by the ‘a’ forms, and speakers who follow

the maxims get to say what they need to say by the simpler form. There is no point in

uttering the redundant alternative, which is thus ruled out on pragmatic grounds.

Consider the structured propositions expressed by a sentence and its self-conjunctions

or self-disjunctions. Let a sentence ‘α’ express a structured proposition πSα, so that sentence ‘α α’ expresses conj, πSα, πSα . These two objects, πSα and conj, πSα, πSα ,

are distinct:∧ if nothing< else, because the> former is a proper part of the< latter. Likewise,>

πSα and disj, πSα, πSα are distinct, also by reason of mereology. Hence the structured

propositions< of a sentence,> of its self-conjunction, and of its self-disjunction, are all dis-

tinct objects. They are distinct just in the same sense in which the structured propositions

of (13a), (13b), and (13c) are distinct:

(13) a. The sky is blue.

b. Susan thinks that the sky is blue.

c. The sky is blue and snow is white.

The structured propositions of (13a), (13b), and (13c), are three dierent objects, the rst

of which is a proper part of each of the other two.

If ‘α α’ and ‘α’ don’t express the same thought, it seems no longer possible to claim

that what∧ is said by ‘α α’ is no more and no less than what is said by ‘α’. These sentences

just say dierent things,∧ like (13a), (13b), and (13c). Metaphysically, it is the same type of

dierence, so if a structured proposition is what a sentence says, there is no denitional

101 acrobatics that will allow us to say that sentences that express dierent structured propositions may nonetheless share their ‘what is said’, on pain of drawing the same conclusion for (13a), (13b), and (13c). So self-conjunctions are no longer redundant with respect to their conjuncts. Likewise for self-disjunctions.

Theories of structured propositions make the logical constants representational, against

Wittgenstein’s fundamental thought, and the cost is that an intuitive account of redundancy appears to be no longer available. So there is independent reason to follow Wittgen- stein’s dictum. The more general consequence is that egregious violations of Wittgen- stein’s fundamental thought, such as those incurred by theories of structured propositions, make these theories incompatible with independently motivated parts of Linguis- tics (in this case, Gricean pragmatics). The next section brings out this more general consequence.

5.3 Principles of conversation

We can certainly come up with contexts in which self-conjunctions and self-disjunctions are assertable. One could imagine, for example, a lousy dance party in which most guests are outside smoking, and someone starts to wonder: if so many people are out smoking, who’s inside dancing? Ben peeks inside very quickly and says:

(14) Linda is dancing... [looks to the left, looks to the right] ...and Linda is dancing.

That’s it! No one else is dancing. Or consider a frustrated teacher telling her student:

(15) Look. Either you turn in the nal paper, or you turn in the nal paper!

There’s no way around turning in the paper! In the right contexts, and with the right prosody and intonation, self-conjunctions and self-disjunctions are assertable after all. So

102 it’s not as though there is never any use for such sentences. But in (14) and (15) a lot of

assumptions have been built in, about emphasis, stress, prosody and mood, which cannot

typically be relied on for general semantic purposes. These assumptions are needed to

restore the possibility of using these constructions, pragmatically defective by default. In

ordinary contexts, if (11b) and (12b) are uttered with no particular emphasis, they are just

unassertable.

The informal explanation I gave earlier for the markedness of the ‘b’ sentences was

based on Grice’s notion of ‘what is said’. In more detail, an utterance of (11b) results in

a violation of the Maxim of Manner. (All of the following holds for (12b) too, mutatis

mutandis.) Manner is the requirement to “Be perspicuous” (Grice, 1989, 27), and Grice

famously illustrates it by the following contrast:

(16) a. Miss X sang “Home Sweet Home.”

b. Miss X produced a series of sounds that corresponded closely with the score

of “Home Sweet Home.”

The speaker who utters (16b), rather than the “nearly synonymous” (16a) (Grice’s phrase),

does so presumably “to indicate some striking dierence between Miss X’s performance

and those to which the word singing is usually applied. The most obvious supposition

is that Miss X’s performance suered from some hideous defect” (Grice, 1989, 37). An utterance of (16b) therefore implicates this obvious supposition.

An utterance of (16b), despite its “near synonymy” with (16a), does not result in pragmatic infelicity because outing Manner results in the derivation of an obvious supposition, namely that what Miss X did with the song was not quite what we would ordinarily

call singing it. The derivation of implicatures is the standard Gricean route to reconcil-

ing the (apparent or genuine) violation of a maxim with the assumption of cooperativity

103 among interlocutors. In the (11a)/(11b) contrast, violation of Manner remains unresolved,

because (in typical contexts) there are no communicative eects the interlocutors might

exploit. For example there is no sense in which what Linda did, according to (11b), is not

quite what we would ordinarily call dancing.

The relevant sense in which (11b) is redundant with respect to (11a) is what could be

called informational redundancy: the informational value of the longer sentence is just

the same as that of the shorter alternative. If the sentences were nearly synonymous, as

Grice says, but not informationally redundant, as (16a) and (16b) are, outing Manner

would lead the listener to derive some relevant implicature. The implicature in this case

conveys the gap in informational value between singing “Home Sweet Home” and doing

something that is not quite good enough to count as singing “Home Sweet Home”. But

there is no such gap between (11a) and (11b).

The existence of redundancy constraints about information value in natural language

follows from general principles of communication. According to Horn’s (1984, 13) Neo-

Gricean reformulation of the maxims, ecient communication optimizes two principles:

(Q) Say as much as you can (given R)

(R) Say no more than you must (given Q)

Cooperative interlocutors are required by Q to maximize informativity, and by R to min- imize cognitive eort. Q works for the benet of the listener, and R for the benet of the speaker. The Q and R principles can obviously pull in dierent directions (the listener would like to receive more information, and the speaker would like to undertake little cognitive work). Standard Gricean reasoning can be recovered by appeal to Q and R.

104 Horn’s account therefore subsumes Gricean pragmatics under principles of communicative economy of great generality.36

We can start from the observed infelicity of (11b) and work towards the notion of informational redundancy in Horn’s framework. We would like to explain why (11b) is unassertable (in ordinary contexts, and without special repair mechanisms such emphasis, special prosody, and information about the context of utterance). The reason (11b) is unassertable, intuitively, is that its informational value is just the same as that of (11a).

So speakers can say as much as they can, satisfying Q, by uttering either of (11a) or (11b).

However, (11b) says more than they must. An utterance of (11b) asks the speaker for more cognitive eort adding no informational value, thus violating R. Self-conjunctions like

(11b) impose production costs that are not compensated for by additional value, for exactly the same informational value is conveyed with less eort by an utterance of (11a). Since this relation between (11a) and (11b) is systematic, and does not depend on accidental features of the utterance or of the context, utterances of (11b) are systematically marked.

This is a powerful and well-motivated explanation, which follows from general principles of cognitive economy in the linguistic domain.

As I pointed out above, (11a) and (11b) express dierent structured propositions. That is, (11a) and (11b) represent dierently, because according to theories of structured propositions, the logical constants are representational. Sentences that represent dierently may do so in two ways: either their informational value is dierent or they dier in some other way, i.e. broadly speaking pragmatically. However, merely pragmatic dierences,

36Horn’s theory based on Q and R has been related to Zipf’s (1949) principles of linguistic economy, and to Optimality Theory (Prince and Smolensky, 1993), in the context of a general theory of complex cognitive systems, such as the human mind.

105 cannot plausibly be considered a model of the dierence between (11a) and (11b). For example, (11b) is not a way of asserting the semantic (informational, at-issue) content that

Linda is dancing, plus the unasserted, not-at-issue content that the property being conjoined is predicated of the thought that Linda is dancing taken twice. Such a view would be disastrous as an account of the most boring conjunctions, like (13c). So, since (11a) and

(11b) represent dierently, but they do not represent dierently in a pragmatic sense, it must be that their information value is dierent (lest we posit another, as yet nowhere detected, dimension of meaning).

Therefore, on theories of structured propositions, it is no longer the case that an utterance of (11b) has more production cost for the speaker, compared to an utterance of

(11a), without extra informational value for the listener. According to these theories, there is extra informational value because there is an additional representational component (a predication of the property being conjoined) that is part of the asserted (at-issue, semantic) content. So the explanation of the markedness of (11b) based on Horn’s principles breaks down: a speaker can reduce eort by uttering the shorter sentence, but at the price of not expressing the same structured proposition. Theories of structured propositions violate

Wittgenstein’s fundamental thought about the logical constants, and so fail to deliver the appropriate notion of redundancy. It would seem that they cannot be integrated with very general and independently motivated principles of communication.

Notice that the account of redundancy based on Horn’s principles works well, even for theories of structured propositions, for other cases of redundancy, such as the discourse in (17):

(17) ??? Linda is dancing. Linda is dancing. Linda is dancing.

106 The discourse in (17) is defective. Speakers don’t utter the same thing over and over. The reason is that, roughly for the reasons given above, their communicative goals are already accomplished by an utterance of the rst sentence in that list: to say ‘Linda is dancing’ is enough to say what must be said, satisfying Q. To add more of the same would gratuitously violate R, with nothing gained. Theories of structured propositions can deliver this result, because each sentence in (17) expresses the same structured proposition. This is precisely what goes wrong with self-conjunctions and self-disjunctions: according to theories of structured propositions, self-conjunctions and self-disjunctions are not synonymous with their conjuncts and disjuncts.

The phenomenon of redundancy is consonant with the remark, made elsewhere in the literature on propositions, that there exists such a thing as an upper bound to how many distinctions a theory of thoughts should make. For example, from a Millian perspective on names, it is straightforward that some sentences express contents that should not be distinguished (for example, ‘Hesperus is bright’ and ‘Phosphorus is bright’). The point

I made in this chapter is broader: certain sentences have contents that should not be distinguished, (not just from the perspective of additional theoretical commitments, but) if we are to explain how language is used. Absent an account of redundancy phenomena, theories of structured propositions cannot be integrated with independently motivated parts of Linguistics to deliver general explanations of observable data about language use. This is a striking failure.

The challenge for theories of structured propositions is to nd a notion of redundancy that applies to (11a)/(11b) and does not require identity of proposition expressed. No doubt, further stipulations can be superimposed on the structured propositions framework to avoid the problem. Of course these stipulations would have to be well motivated:

107 it would be preposterous to just add to the theory that ‘α α’ is redundant with respect

to ‘α’. The question is what the source of these redundancy∧ facts might be, given that it is not the nature of structured propositions. Furthermore, it should be pointed out that whatever stipulation we wish to add, it is a further stipulation: additional commitments that should be taken on, because the theory is not explanatorily sucient.37

Perhaps redundancy can be characterized without appeal to ‘what-is-said’ or seman-

tic content. After all, theories of structured propositions can appeal to some notion of

logical equivalence: (11a) and (11b) are, quite obviously, logically equivalent. Those sen-

tences are easy model-theoretic and proof-theoretic consequences of each other. Speakers

somehow might exploit some fact of this kind in processing the utterances. Accordingly,

the required notion of informational redundancy is logical redundancy—a notion that can

be understood in dierent ways, but that can be independently characterized with no

diculty. The notion of redundancy that I called ‘informational’, and that explains the

markedness of sentences like (11b), therefore, is determined by logic.

This suggestion could be elaborated in more detail, but it is not promising enough to be

worth the eort. If we take on board the commitment to lter out of the realm of asserta-

bility any syntactically complex constructions that are logically equivalent to a simpler

alternatives, it is easy to derive disastrously wrong empirical predictions. The reason for

this is that standard predicate logic is notoriously blind to aboutness constraints—which,

as you recall, are an important motivating force for theories of structured propositions.

37Compare structured propositions with Jago’s (2014) hyperintensions, Yablo’s (2014) partial contents, Leitgeb’s (2018) HYPE, or Ciardelli et al.’s (2017) inquisitive logic: all of these theories provide a model of thoughts in which propositional identity is hyperintensional, but on which conjunction and disjunction are idempotent. This is not particularly surprising: idempotency follows immediately if the metaphysical model for coordination is not mereological constituency but set-theoretic membership.

108 Consider what we do when we explain paradoxes like Russell’s. We assume that there

is a set of all non-self-membered sets, call it the Russell set, and we conclude a contradic-

tion. The contradiction is usually simply indicated as ‘ ’ in logic texts, but we might as well state it more vividly as, for example, the claim that⊥ something is not self-identical, or any other absurdity for that matter. Thus a clever philosopher, having explained Russell’s paradox, concludes by saying:

(18) There is no Russell set, or else something is not self-identical.

However, on any plausible formalization, (18) is logically equivalent to (19) because the second disjunct is logically false:

(19) There is no Russell set.

Any orthodox notion of model-theoretic or proof-theoretic consequence yields this result.

So if redundancy were a logical notion, as the suggestion I’m considering takes it to be,

(18) should be odd or unassertable. (18) is logically redundant compared to (19), so it is expected to be ltered out and marked. It is, of course, nothing of the sort. Instead, it is a perfectly ordinary way to conclude a reductio proof. If redundancy were a matter of logic, we should expect the more syntactically complex alternative (18) to be systematically marked, as the simpler but logically equivalent (19) is available. This is not the case.

There is no easy x for structured theories of content. The required account of redundancy should come from the theory of thoughts: sentences may represent no more and no less than other sentences with a dierent syntactic structure. Failure to satisfy

Wittgenstein’s fundamental thought is a mistake, and so thoughts are not as ne grained as the sentences that express them.

109 5.4 A look ahead

The equivalence relation between sentences on the right hand side of an abstraction prin-

ciple for thoughts is the analog of the identity relation. This equivalence establishes the

identity conditions of the thoughts expressed by the sentences it relates. Formally, equiva-

lence licenses inferences by substitution. Leibniz’s law states that identity is a congruence:

(Cong=) p, q p q X Xp Xq

∀ ( = → ∀ ( → )) which is a statement of the indiscernibility of identicals in the domain of thoughts and

propositional attitudes. The problem of propositional identity that I laid out in the Intro-

duction is to dene a notion of congruence on the domain of thoughts. It is well known

that necessary equivalence fails, for the following formula is false:

p, q p q X Xp Xq

∀ (◻( ↔ ) → ∀ ( → )) It suces to consider two necessarily equivalent thoughts, like the thought that Socrates is

self-identical and the thought that Arithmetic is incomplete: while ‘Nobody ever doubted

that Socrates is self-identical’ is presumably true, ‘Nobody ever doubted that arithmetic

is incomplete’ is false (Mates, 1950).

Church’s proposal in Alternative (0) leads to a correct statement of congruence:

(Cong ) p, q p q X Xp Xq

= ∀ ( = → ∀ ( → )) However, the argument in this chapter shows that the model of thoughts determined by taking ‘ ’ as the analog of identity is too strict. Progress is made if we dene an equivalence relation= on thoughts, ‘ ’, that satises all the axioms for Church’s Alternative (0)

(Church, 1973a, 1974; Anderson,⇔ 1980), plus at least idempotency axioms for conjunction

110 and disjunction:

p p p p p p p p

⊢ ∀ ( ∧ ⇔ ) ⊢ ∀ ( ∨ ⇔ ) It is still true that ‘ ’ is adequate, in that it licenses quanticational inferences by substitution (see the Addendum):⇔

(Cong) p, q p q X Xp Xq

∀ ( ⇔ → ∀ ( → )) I have shown already how to abstract thoughts, by means of TA, given .

Therefore, the problem of propositional identity has been signicantly⇔ constrained, and the search space for the exact relation among thoughts that explains their identity conditions considerably reduced. This is progress, even if it’s not the last word. Im- portantly, I have also illustrated how inquiry could proceed from here: the idempotency axioms are motivated empirically, by looking at how logical parts of language like the connectives are used in speech. As we rene our understanding of ‘ ’, the underlying abstractionist account need not be revised.38 ⇔

38For limits of space, I mention briey without discussion a generalization of the argument presented above. There is linguistic evidence that a conjunction is equivalent to its weaker conjunct, and that a disjunction is equivalent to its stronger disjunct. Intuitively, an utterance of ‘John is 30 years old and more than 25 years of age’ is marked, because the left conjunct asymmetrically entails the right conjunct, and so the utterance is redundant compared to ‘John is 30 years old’. This has rst been noticed by Hurford (1974), and further discussed and generalized by Katzir and Singh (2014) and Ciardelli and Roelofsen (2017). Hence a hyperintensional logic should validate absorption laws: if α ⊢ β then both ⊢ α ∧ β ⇔ α and ⊢ α ∨ β ⇔ β. It is worth emphasizing that some hyperintensional logics, like Kit Fine’s (2017) “exact” truthmaker logic and Steve Yablo’s (2014) logic of aboutness, do not validate the absorption laws. Others do, such as inquisitive logic (Ciardelli et al., 2017) and HYPE (Leitgeb, 2018), but also Fine’s “inexact” theory.

111 Chapter 6: Metasemantic Predicativism

On a plausible reconstruction, the Russell-Myhill paradox is a paradox of aboutness, and a challenge for any hyperintensional theory of thoughts. This reconstruction shows that the Russell-Myhill is a counterexample to Ramsey’s long-standing classication of the paradoxes in two categories: the truth-theoretic and the set-theoretic ones. Moreover, it contradicts a recent interpretive line, which takes the Russell-Myhill to be a reductio of theories of structured propositions. Instead, Russell’s own conclusion, namely that the logic of thought should be predicative, is more plausible. (1) Predicativism allows the theory of thoughts to be both consistent and hyperintensional: no other accounts of Russell-Myhill can do that. (2) Predicativism is well motivated, since the principles leading to contradiction follow from an hitherto unrecognized abstractionist approach to thoughts, and predicativism oers a principled approach to abstraction, understood as a dynamic process. (3) Moreover, Prior’s paradox does not follow in a predicative logic. (4)

The nal advantage is that Prior’s theorem is reduced to a harmless triviality. Predica- tivism thus oers a unique, comprehensive, and systematic perspective on the realm of thoughts. In an appendix to this chapter, I shall summarize the framework of predicative Russellian intensional logic, and sketch proofs of soundness and completeness on a sequence of models.

112 6.1 Aboutness paradox

Everything there is, it seems, may be the object of a thought. For example, for every x, there is the thought that x is self-identical. More generally, for every x there is a thought about x. This thought need not be an identity, and x itself need not “a constituent” of it.

Furthermore, it is plausible to hold that thoughts about distinct things are distinct.

Since 2 is not 3, the thought that 2 is self-identical is distinct from the thought that 3 is self-identical. Thoughts discriminate nely among parts of reality. Contraposing, if a thought solely about x is identical to a thought solely about y, then x y.

Consider now classes of thoughts: the previous remarks would seem= to hold in the higher-order domain just as well. For every class of thoughts there is a thought about that class, and for distinct classes there are distinct thoughts about them. These two premises appear plausible regardless of whether ‘classes’ are understood to be sets or, as I shall assume below, pluralities.

The puzzle I want to discuss is that these claims are mutually inconsistent, at least when ‘M’ and ‘N’ are higher-order variables ranging over classes of thoughts:

1. For every M there is a thought about M;

2. For every M and N, a thought about M is a thought about N only if M N.

= Premises 1 and 2 establish a one-to-one correspondence between thoughts and classes of thoughts, but that can’t be, for familiar Cantorian reasons. A diagonal argument brings up the inconsistency. Notice, rst, that some thoughts about classes belong to the classes they are about, others do not. For example, a thought about abstract objects belongs to the class it is about (if thoughts are abstract objects). Consider now the class W of thoughts that don’t belong to the classes they are about, and ask whether a thought p about W

113 belongs to W . If it does then p is one of the thoughts that do not belong to the class

they are about, hence it doesn’t belong to W . If p does not belong to W , then p is one of

the thoughts that do not belong to the class they are about, hence it belongs to W . The

contradiction appears unavoidable.

This result is not new: it is a version of the Russell-Myhill paradox—also called Ap-

pendix B paradox or Russell’s paradox of propositions (Russell, 1903, Appendix B). Indeed,

I would claim, at its core, Russell-Myhill is a paradox of aboutness. This may not be clear

from Russell’s original discussion (with ‘proposition’ for ‘thought’):

If m be a class of propositions, the proposition “every m is true” may or may not be itself an m. But there is a one—one relation of this proposition to m: if n be dierent from m, “every n is true” is not the same proposition as “every m is true”. Consider now the whole class of propositions of the form “every m is true”, and having the property of not being members of their respective m’s. Let this class be w, and let p be the proposition “every w is true”. If p is a w, it must possess the dening property of w; but this property demands that p should not be a w. On the other hand, if p be not a w, then p does possess the dening property of w, and therefore is a w. Thus the contradiction appears unavoidable. (Russell, 1903, 527)

The Russell-Myhill paradox can be derived adding a single predicate about to (impredicative) plural logic, subject only to the principles Russell assumes, that is 1 and 2 above. Fol- lowing a suggestion by Uzquiano (2015a), plural logic interprets Russell’s class-theoretic language while discharging Russell’s unnecessary commitment to an ontology of classes

(or sets) of thoughts. (See Linnebo (2017) for an introduction to plural quantication.)

6.1.1 Neither truth nor sets

Let’s pretend for a moment that thoughts are objects. This assumption is not necessary for the paradox, as shown by Goodman (2017), and I shall drop it later; however the pretense highlights the point that the Russell-Myhill paradox is not about classes (or sets). Indeed,

114 as Russell clearly saw, the simple type-theoretic restrictions of the set-theoretic universe

do nothing to prevent this contradiction. (More precisely, what Russell clearly saw is that

the simple type theory of the (broadly speaking) Fregean logic in which his Principles of

Mathematics were written did nothing to prevent the contradiction from arising.)

The Russell-Myhill paradox can be formalized in plural logic, with variables p, q, ...

ranging over thoughts and plural variables xx, yy, ... ranging plurally over any things

including thoughts. We need a single predicate to express the aboutness relation between thoughts and pluralities: ‘about p, Ψ, xx ’ is to be read ‘p is the thought about xx that

they are Ψ’. Here choice of Ψ is inessential:( ) it could express being self-identical as in the

example above, or being all true as in Russell’s example. It only matters that Ψ is held constant throughout. So we could bind the argument position of Ψ in an about formula

by a λ operator, or more simply x a Ψ, and dene a two-place predicate ‘about p, xx ’

to be read ‘p is about xx’, leaving choice of Ψ presupposed. I shall assume that such( choice)

is held xed throughout (or, equivalently, that the λ-expression is never converted).

Premises 1 and 2, previously stated informally, are an existence claim and a claim

about the identity of thoughts. Russell implicitly assumes the rst premise in the opening

line of the passage above, as revealed by his use of the denite article (“if m be a class of

propositions, the proposition...”), and explicitly assumes the contrapositive of the second:

“if n be dierent from m...” With the above stipulations in place, and replacing plural

variables xx and yy for Russell’s class-terms m and n, the premises are:

(EX-Plur) xx p.about p, xx

(ID-Plur*) ∀xx,∃ yy p, q about( ) p, xx about q, yy p q xx yy

EX-Plur looks∀ unproblematic:∀ ( for( any) things,∧ there( is) a→ thought( = → about= them.)) As for ID-

Plur*, notice that it can be simplied since pluralities are extensional. The one of relation

115 ‘ ’ holds between a thing and the things which it is one of. Hence:

(ID-Plur)≺ xx, yy p, q about p, xx about q, yy p q r r xx r yy

∀ ∀ ( ( ) ∧ ( ) → ( = → ∀ ( ≺ ↔ ≺ ))) ID-Plur asserts that any thoughts p and q about any things xx and yy (respectively) are

the same only if xx are yy. (Recall that such thoughts about things are all of the same form: that the things are Ψ for some Ψ.) Finally, consider the “Russell plurality” rr of all

and only the thoughts p such that, for some things zz, p is about zz but it is not one of

them. The existence of the Russell plurality follows from an impredicative instance of the

plural comprehension scheme:

(Comp-Plur) xx p p xx ϕp

∃ ∀ ( ≺ ↔ ) Each instance of Comp-Plur asserts that there are some things consisting of all and only those thoughts satisfying a formula that goes in place of ‘ϕ’. This principle is intuitively very plausible.39 An instance of Comp-Plur denes the Russell plurality rr:

(R) p p rr zz about p, zz p zz

∀ ( ≺ ↔ ∃ ( ( ) ∧ ≺~ )) We may now reason as follows (parametric occurrences of terms are boldfaced):

1. about p, rr EX-Plur 2. p rr Assumption 3. zz about( )p, zz p zz 2, R 4. about≺ p, rr about p, zz 1, 3 5. ∃r r( rr ( r zz) ∧ ≺~ ) 4, ID-Plur 6. p rr ( p )zz∧ ( ) 5 7. p∀ (rr≺ ↔ ≺ ) 3, 6 8. zz≺ about↔ ≺p, zz p zz 1, 7 9. p ≺~ rr 8, R ∃ ( ( ) ∧ ≺~ ) 39Comp-Plur corresponds to≺ impredicative class comprehension in a formulation of Russell-Myhill in a theory of classes, which would be closer to Russell’s text. Deutsch (2014) presents such derivation in Morse-Kelley class theory (Kelley, 1955).

116 Contradiction at lines 7 and 9. In presenting a plural version of the paradox, I am following a remark made by Uzquiano (2015a, 329). Given Russell’s text, the suggestion could reasonably be made that absurdity follows from the illegitimate assumption that there is a set (or a class) of all thoughts (see e.g. Menzel (2012)). But Russell’s class-theoretic terminology is inessential—as Russell himself came to realize (Klement, 2002, 178). Pluralities are not ontologically committing in the way sets are, but the resources deployed in the reasoning suce to derive a contradiction.

Moreover, the plural interpretation helps oset Deutsch’s (2014) response to Russell-

Myhill, according to whom the paradox shows that some thoughts are not members of classes. As Uzquiano remarks, “to claim that some propositions are never members of classes is to claim that some propositions never stand in the one of relation to some propositions”, but this move seems (to him, and I agree) “very costly” (Uzquiano, 2015a, 333).

This situation helps us derive a rst conclusion on the broader topic of the nature of paradox. Going back to Frank P. Ramsey (1926), there is a tradition for thinking that all paradoxes are either set-theoretic like Burali-Forti, or semantic like the Liar. Ram- sey’s distinction is usually understood as tracing the contradictions to two sources: naïve set theory, or semantic ascent. Paradoxes of the former kind are handled by placing appropriate restrictions on set-theoretic existence assumptions. But the ontology of set (or class) theory is not essential to the Russell-Myhill paradox, and neither is Russell’s class theoretic language. There is thus evidence that the Russell-Myhill is not a set-theoretic paradox.

Paradoxes due to semantic ascent are handled by a sophisticated theory of truth, or else by modifying the classical truth-conditional clauses for the connectives. The derivation above does not employ truth-theoretic terminology, and in particular it does not turn on

117 any instance of Tarski’s T-schema (Tarski, 1936). So there is reason to think that Russell-

Myhill is not a truth-theoretic paradox either.

Thoughts, of course, have truth-conditions, but they also have aboutness-conditions.

However, aboutness is not a truth-theoretic matter (Ryle, 1933). Some thoughts are true at all and only the same possible worlds, hence they are truth-conditionally equivalent, and yet they are about dierent things. Such is the case with the thoughts that 2 is self- identical and that 3 is self-identical. The notion of aboutness has been recently discussed in the inuential work of Yablo (2014) and Fine (2016)—see also Hawke (2018) for an overview.

So the Russell-Myhill is a counterexample to Ramsey’s distinction. If quantication over or reference to thoughts suces for qualifying as a paradox of intensionality, the

Russell-Myhill is a good case in favor of Tucker and Thomason’s (2011) argument that the paradoxes of intensionality falsify Ramsey’s classication of the nature of paradox. (For further criticism of Ramsey’s classication, see Tucker (2010).)

6.1.2 The structure of reality

The Russell-Myhill paradox by no means depends on assuming that thoughts are objects.

As we just saw, it is certainly important to appreciate that, even if they were, the paradox is not in violation of set-theoretic constraints. However, if thoughts are not objects, they don’t form pluralities, and yet the paradox can still be derived in a higher-order setting.

Let us take propositional variables to be 0-place second-order variables, following Prior

(1971) and Williamson (2003), and as I have been doing throughout this dissertation. If thoughts are not objects, some proposal should be made to interpret occurrences of the identity sign among propositional variables, but on all of the most plausible proposals, the

118 paradox is still very compelling. A rst option is to follow Goodman (2017), and assume

that identity among thoughts is third-order indiscernibility: ‘p q’ in ID-Plur is then short for ‘ Z Zp Zq ’. Since the only assumption needed in the= derivation about identity or its∀ analog( ↔ is reexivity) (for the step from line 4 to line 5 above), Goodman’s suggestion makes clear that the Russell-Myhill is derivable in higher-order logic without any particular assumptions about the nature of thoughts. However the suggestion does require that we quantify over enough of the third-order domain to distinguish thoughts that are distinct, and this is potentially controversial. Alternatively, we may adopt Church’s (1984) strict equivalence ‘ ’ axiomatized as in Alternative (0), or even better my hyperintensional equivalence ‘= ’, which as I argued in chapter 5 is overall a better characterization of propositional identity⇔ than ‘ ’. Indeed, is a congruence on the domain of thoughts

(for details, see the Addendum):= ⇔

(Cong) p, q p q Z Zp Zq

∀ ( ⇔ → ∀ ( → )) and so a formulation in terms of hyperintensional equivalence, is at least as good as Good-

man’s formulation. Neither option relies on a controversial rst-order interpretation of

propositional variables.

The use of ‘about’ (as well as, optionally, the three-place ‘about’) allows us to artic-

ulate the aboutness constraints that motivate EX-Plur and ID-Plur, without hiding them

inside syntactic structure. This move clearly goes beyond Russell’s original formulation

and motivation, but has two important benets: (i) it makes the role of aboutness explicit

in the derivation, illustrating the signicance of Russell-Myhill in our understanding of

the nature of paradox—a point I have made at the end of the previous section; (ii) it shows

that there is no need for propositional variables occupying sentence position.

119 Aboutness is certainly, in part, a matter of syntactic structure. Sentential subjects are

typically subject matters—at least in atomic sentences: the subject matter, or topic, of an

utterance of ‘F a’ is typically a. However, aboutness can be understood independently of syntactic structure, as it has been in recent work (Yablo, 2014; Fine, 2016). (Among contemporary accounts of aboutness, Perry (1989) is perhaps the most directly inspired by the intuitive idea that sentential subjects are subject matters.)

There are two adjustments to make if we wish to get closer to Russell’s original: (i) assume that propositional variables can occupy sentence position; and (ii) get rid of an explicit notation for the aboutness relation. Since I shall no longer assume that thoughts are objects, I revert to the use of third-order predicates for classes of thoughts. Let M be one such class. Russell (1903) called a thought of the form ‘ p Mp p ’ the “logical product” of M.40 For the intuitive justication of the premises,∀ it( matters→ (a)) that logical products be about a certain class, e.g. that the thought that p Mp p is about M, and (b) that for every class there be a logical product for that∀ class.( Together,→ ) (a) and (b) correspond to Premise 1. Intuitively, (a) can be justied by reecting that, as Frege was the rst to realize, quantication is a higher-order analog of predication. Hence just like the thought that 2 is self-identical is, intuitively, about the number 2, so the thought that every M is true is about M.

We need a single fourth-order predicate that applies to predicates of thoughts— this is analogous to Ψ above. For example, theQ formula ‘ X’ might be ‘ p Xp p ’, following Russell. We can then rely on the subject/predicateQ distinction∀ in( the syntax→ )

40Russell’s language here suggests that he might have thought of thoughts about classes as (big) conjunctions—in conformity with his “Russellian” theory of propositions. Indeed, this is how Kripke (2011) discusses the Russell-Myhill, but the use of innitary logic is unnecessary, and so I avoid it.

120 of e.g. ‘ M ’, and dispense with explicit aboutness relations. Derivations of Russell-

Myhill fromQ( these) assumptions are given by Goodman (2017) and Hodes (2015), in dierent frameworks. The cost of a formulation closer to Russell is that the role of aboutness in motivating the premises is hidden inside syntactic structure—and this may lead to the wrong diagnosis, as we shall see.

In the derivation I gave earlier, in which function application of a higher-order predicate to a class term is replaced by the one of relation and plural terms, all propositional variablesQ occur in name position. This is signicant, because one inuential approach to the Russell-Myhill is to dismiss it, alleging that Russell blurred the distinction between sense and reference. Indeed, this was Frege’s reaction to the Russell-Myhill.41 More recently, Harold Hodes (2015) has also expressed misgivings concerning the use of bound variables in sentence position, a topic reminescent of Frege’s worries. The legitimacy of such binding, however, is orthogonal to the question about the source of absurdity in the

Russell-Myhill. The paradox is shown to be not the result of Russell’s confusion, if the notion of aboutness is properly isolated from syntactic structure.

Another mistaken diagnosis is given by Jeremy Goodman, who puts the blame on Rus- sell’s Premise 2, which I formalized as ID-Plur, and which is concerned with the granularity of thoughts about classes. In eect, Goodman gives a direct proof of no structure:

X Y Z Z X Z Y p Xp Y p

∃ ∃ (∀ ( Q ↔ Q ) ∧ ∃ (¬ ∧ )) There are classes X and Y such that the thoughts X and Y are higher-order indis- cernible only if they are coextensional. no structureQ is the denialQ of a regimentation of

41Russell gave Frege communication of the Russell-Myhill in a letter dated September 29, 1902 (Frege, 1980, 147). See the discussion in Klement (2002), especially chapter 6. The paradox was independently rediscovered by Myhill (1958) in Church’s (1951) LSD, whence its name.

121 Premise 2 in Goodman’s notation—a claim Goodman calls structure:

X Y Z Z X Z Y p Xp Y p

∀ ∀ (∀ ( Q ↔ Q ) → ∀ ( ↔ )) Following my preferred conventions, structure can be equivalently stated as follows:

(STR) X Y X Y p Xp Y p

∀ ∀ ((Q ⇔ Q ) → ∀ ( ↔ )) For any classes of thoughts X and Y , if the thought that X is is hyperintensionally equivalent to the thought that Y is , then X is Y , i.e. X andQY are coextensional. I shall present below a derivation of Russell-MyhillQ in my preferred framework, from (STR).

Goodman comments on the signicance of no structure: Our conclusion ... implies that reality is not structured in the manner of the sentence [sic] we use to talk about it. (Goodman, 2017, 46)

In a similar vein, according to Uzquiano the Russell-Myhill paradox “is best regarded as a limitative result on propositional granularity” (Uzquiano, 2015a, 328). Finally, the same lesson about the Russell-Myhill is endorsed by Cian Dorr (2016) in the course of an argument against the thesis that thoughts are individuated as nely as sentences. So this interpretation is quite popular.42

I do not wish to dispute what seems to be a robust consensus in Russell scholarship, on why Russell himself took Premise 2 to be plausible, namely his commitment to an account of thoughts on which thoughts are structured in something like the way sentences are. Russell seems to have assumed that “identical propositions must have identical constituents” (Linsky, 1999, 155). But it is controversial to assume that “propositions”, or

42Dorr’s interpretation of the Russell-Myhill is part of an argument in favor of an approach to the logic of identication, in a higher-order language with a λ abstraction operator, that validates β-conversion principles. Dorr is careful to restrict his conclusions to extensional and modal contexts, because β-conversion is notoriously in tension with certain natural assumptions about the logic of attitudes (Salmon, 2010). The picture of the logic of identication emerging from my remarks is an intermediate one, which rejects both the structured view for the reasons given in chapter 5, and β-conversion, in order to account for the logic of attitudes. I have some tentative remarks about this in chapter 7.

122 thoughts, have constituents. The more general justication for the truth of Premise 2, and so of its formal counterpart ID-Plur, goes back to a more abstract notion of aboutness.

Contemporary accounts of structured propositions provide a metaphysical model of aboutness, as mereological constituency in a structured object. Furthermore, they identify thoughts with such structured objects (Salmon, 1986; Soames, 2015), resulting in very ne- grained accounts of thoughts. However, while the source of aboutness considerations may be sentential structure, the constituency model is not the only proposal on the market: the relation with sentential structure may be less direct. Indeed, Armstrong and Stanley (2011) question the explanatory value of structured theories on aboutness, and recent work by

Yablo and Fine, mentioned earlier, presents alternative accounts of this notion.

Uzquiano, Goodman, and Dorr, are correct, it seems to me, in thinking that the Russell-

Myhill is a limitative result on propositional identity, but not in taking it to be an acceptable one.43 For the paradox follows not from assuming that thoughts are structured, which is a controversial philosophical thesis, but that thoughts’ identity conditions are sensitive to aboutness constraints. This is considerably less controversial. For if ID-Plur is false, the notion of aboutness is too weak to deliver an hyperintensional account of thoughts. For example, at least one thought asserted by Locke was not rejected by Hume. So Locke’s assertions are not Hume’s rejections (the two sets of thoughts don’t have the same members). However, if Uzquiano, Goodman, and Dorr, are correct, the thought that Locke asserted only truths and the thought that Hume rejected only truths may be the same thought. If so, a hard-core Lockean, who believes that Locke asserted only truths, but is

43At present time Gabriel Uzquiano “does not disagree” with this conclusion, at least relative to the particular formulation of the Russell-Myhill I am working with (personal communication).

123 somewhat friendly to Hume, and does not believe that Hume rejected only truths, would turn out to be conceptually confused, and self-refuting. This is very hard to fathom.

There are of course models of thoughts on which ID-Plur is false. For example, the thought that 2 is self-identical and the thought that 3 is self-identical, being both necessarily true, turn out to be identical if thoughts are intensions—as according to the theory of

Lewis (1986) and Stalnaker (1984). But this is a failure of coarse-grained theories. Theories of aboutness have been developed in the attempt to capture signicant hyperintensional distinctions among thoughts that may be necessarily equivalent. Coarse-grained theories fail as plausible models for thoughts, and if we want to study hyperintensionality with the tools of higher-order logic, the logic must make room for it.

6.2 Predicative logic

I’ve argued so far that the Russell-Myhill is neither the result of confusion on Russell’s part, nor of semantic ascent, nor of naïve set theory, nor, nally, of controversial metaphysical claims about structured propositions. In this section I shall explain what, in my view, is the source of trouble. A predicativist response will then be forthcoming, and well motivated by a dynamic understanding of abstraction. The proposal is cashed out in modal terminology. As we shall see in the next section, it comes with the extra benet of a nice account of Prior’s paradox, and of reducing Prior’s theorem to a harmless triviality.

It is noteworthy that Russell discusses the Russell-Myhill at the very end of his Prin- ciples of Mathematics (Russell, 1903), recognizing that this paradox cannot be solved by enforcing the simple type-theoretic distinctions of a broadly Fregean logic. Indeed, Rus- sell was consequently led to ramify his intensional logic, a move that has been widely regarded as unacceptable—at least since Ramsey’s spurious distinction of the paradoxes

124 in two classes hid the Russell-Myhill from plain sight. Ramication was Russell’s version of a predicative conception of higher-order quantication.44 There are perhaps better versions of predicativism, but predicativism does seem to be the most promising option.

We have an impredicative plural comprehension principle Comp-Plur in the background logic, or alternatively an impredicative class comprehension principle. Their status, as I said, does not seem what’s at issue with the Russell-Myhill. But we also have the two principles EX-Plur and ID-Plur, and it is important to recognize that together they almost constitute a theory of abstraction. For illustration, following Burgess (2005), consider Frege’s (inconsistent) theory of abstraction for extensions. Let ζ be Frege’s extension of operator, so that ‘ζxX’ reads ‘x is the extension of X’. Frege’s theory of abstraction for extensions is thus:

(EXζ) X x.ζxX

(IDζ*) ∀X,Y∃ x, y ζxX ζxY x y z z X z Y

∀ ∀ ( ∧ → ( = ↔ ∀ ( ∈ ↔ ∈ ))) EXζ and IDζ∗ are almost notational variants of EX-Plur and ID-Plur, the dierence being a biconditional in the consequent of IDζ∗ where ID-Plur has a conditional. As is well known, EXζ and IDζ∗ are equivalent to (a formulation of) Frege’s infamous Basic Law V.

For EXζ and IDζ∗ together entail that for every class X there is exactly one extension of that class: X !xζxX.45 This result legitimates the use of a function ε from classes to extensions.∀ Thus∃ ‘εX’ is read ‘the extension of X’. With this denition, we can derive the

44Ramication has been endorsed by Whitehead and Russell (1927), Church (1993), and Anderson (1980). A number of commentators agree that ramication is a powerful tool against the intensional paradoxes, including Church (1976, fn. 25), Kripke (2011, 376-377), Williamson (2016), and nally David Kaplan (1995), whose remarks are developed further by Anderson (2009) and Lindström (2009). General presentations of ramied type theory may be found in Church (1956, 1976), and Hazen (1983), and see Hylton (1990) and Linsky (1999) for its metaphysical underpinnings. 45IDζ∗ entails ζxX ∧ ζyX → (x = y ↔ ∀z(z ∈ X ↔ z ∈ X)). The rst conjunct in the antecedent is given by EXζ, and the second is assumed. It follows that x = y. Hence ∀y(ζyX → x = y), which is the uniqueness claim to be added to EXζ to establish ∀X∃!xζxX.

125 following formulation of blv from EXζ and IDζ∗:

(blv) X,Y x, y εX εY z z X z Y

∀ ∀ ( = ↔ ∀ ( ∈ ↔ ∈ )) So the theory of abstraction for extensions consisting in EXζ and IDζ∗ can be repackaged in an abstraction principle for extensions blv. As is well known, only the left-to-right direction in the main biconditional in blv is necessary for the derivation of Russell’s paradox in Grundgesetze. When we unpackage blv in two principles, keeping only the direction indispensable for the paradox, we obtain:

(EXζ) X x.ζxX

(IDζ) ∀X,Y∃ x, y ζxX ζxY x y z z X z Y

∀ ∀ ( ∧ → ( = → ∀ ( ∈ ↔ ∈ ))) EXζ and IDζ are just notational variants of EX-Plur and ID-Plur. So EX-Plur and ID-Plur do not constitute a full theory of abstraction, but they are in all relevant respects analogous to the inconsistent fragment of Frege’s theory of extensions. Behind the Russell-Myhill paradox, therefore, lies the machinery responsible for inconsistency in abstraction principles. Although hitherto unrecognized, the Russell-Myhill paradox is a paradox of the same formal nature as Russell’s set theoretic paradox—at least in its original formulation, as it appears in Grundgesetze.

It is clear that the abstraction of thoughts was not Russell’s goal, for in Principles of

Mathematics he wrote:

of the three kinds of denition admitted by Peano—the nominal denition, the denition by postulates, and the denition by abstraction [footnote omitted]— I recognize only the nominal ... (Russell, 1903, 112)

Russell did not consider abstraction principles indispensable for mathematics, for reasons we do not need to get into. This being the case, his views about denitions might well

126 explain why he would not state the full abstraction principle for thoughts underlying the

Russell-Myhill contradiction. After all, there is no need to subscribe to a full theory of

abstraction, that one regards as independently controversial, if weaker premises lead to

contradiction and already enjoy intuitive motivation.

Russell’s two principles for the Russell-Myhill are notational variants of the principles

Frege assumed in his theory of extensions in Grundgesetze, and so we can locate the source

of inconsistency in the same kind of impredicativity. It is crucial to recognize that impred-

icativity can take many forms. Even if we accept unrestricted comprehension principles

as Comp-Plur for the sake of absolute generality, we don’t have to accept an unrestricted

theory of abstraction—especially since it is well known that some such theories lead to

contradiction.

A lot of discussion in the neologicist literature has been dedicated to attempts to sort

out “good” abstraction principles (like Hume’s principle) from the “bad” ones (like blv):

a problem known as Bad Company (Tennant, 1987; Boolos, 1987). A very rich discussion about criteria of acceptability for abstraction principles ensued. It is not clear, however, that such discussion has led to an organic understanding of the underlying diculty, and some authors have consequently explored other approaches. For example, Linnebo

(2018) develops a dynamic conception of abstraction, according to which the abstraction of higher-order (intensional) entities from a given domain should be understood as a pro- ductive process: the initial resources expand as abstraction proceeds. With Linnebo, Studd

(2016) argues that dynamic approaches to abstraction accommodate all forms of abstraction, “good” and “bad” alike, undercutting Bad Company.

There is more than one way to formally develop a dynamic conception of abstraction, but all accounts rely on a certain predicative perspective on higher-order quantication.

127 Following these developments, I shall present predicativism in modal clothing, to illustrate

how the account works, and its strengths.

6.2.1 Dynamic abstraction

We shall proceed as follows: rst, we move to a free-logical setting in which we can control

for existence assumptions, following Tennant (2017). This is independently motivated,

given the discussion in chapter 4. Secondly, we let the theory of abstraction introduce

such assumptions under a modal layer, following Linnebo (2018). (See also Studd (2016)

for a closely related proposal.)

In chapters 4 and 5, I motivated the adoption of an abstraction principle for thoughts,

called Thought Abstraction, given an hyperintensional connective of propositional iden-

tity ‘ ’ and a function ‘ ’ that takes a sentence to the thought it expresses: J K (TA)⇔ ⋅ α β α β J K J K The thought expressed by ‘α’ is the thought= ⇔ expressed( ⇔ ) by ‘β’ if and only if α is hyperintensionally equivalent to β. According to TA, thoughts are the abstracts of sentences.

Since hyperintensional equivalence suces for material equivalence, TA entails EX and

ID:

(EX) p p α J K (ID) ∃p,( q =p α) q β p q α β J K J K EX is the claim that∀ there( is= a thought∧ = expressed→ ( = by↔ ‘α(’, and⇔ ID))) is the claim that any thoughts expressed by ‘α’ and ‘β’ are the same just in case α is hyperintensionally equivalent to β.

As one would expect, TA suces for the derivation of a version of Russell-Myhill. The derivation is not immediately obvious, because TA is an explicitly metalinguistic principle

128 (the ‘ ’ operator is a quotation device). Following Tennant (2017), a straightforward J K proposal⋅ to integrate the theory of abstraction with the ambient higher-order logic is to

suppose that we are working in a free logic. For the considerations made in chapter 4, this

proposal is independently motivated in the case of reference to thoughts. In a free logical

setting, inferences licensed by the laws of quantication are conditional upon existence

claims. In particular, Universal Instantiation,

(UI) pψ ψ t p

∀ → [ ~ ] where ‘ψ t p ’ is the result of replacing every occurrence of p in ψ for a singular term t, is restricted[ only~ ] to parameters and free variables. Substitution of singular terms takes the form of a law of Free Universal Instantiation:

(FUI∗) pψ q q t ψ t p

∀ → (∃ ( = ) → [ ~ ])) I presented in chapter 4 a theory of canonical Descriptive Singular Terms for thoughts, i.e. singular terms denoting thoughts that have the form ‘ α ’, for some sentence ‘α’ that J K species the thought thereby referred to. These are the singular terms on the right hand side of TA. Thus the proper form of FUI∗, in a Russellian intensional logic, is a schema

FUI:46

(FUI) pψ q q α ψ α p J K 46Strictly speaking, the stipulation∀ about→ (dst∃ (s results= in:) →∀pψ[→ ~(∃q])(q = α ) → ψ( α ~p])). However, as noted in the Introduction, in a Russellian intensional logic sentences workJ K doubleJ shift:K they express thoughts, and they denote them as well. This is no blunder: the sense/reference distinction disappears, as this is a logic of sense only. From this perspective, FUI seems thus in good order, at least “in a Russellian intensional logic”. There are intensional logics that retain the sense/reference distinction: most prominently, Church’s (1951) LSD. It was Myhill (1958) who noted that the Russell-Myhill does not disappear in this setting. For a predicative account of Russell-Myhill in Church’s LSD, see Walsh (2015). Walsh fails to recognize that the source of trouble in Russell-Myhill is related to impredicative abstraction, and consequently he fails to appreciate the broader philosophical picture in the background. Other than that, Walsh’s discussion is quite close to mine.

129 It is now the job of the theory of abstraction, EX in particular, to supply the existential

assumptions required by FUI.

For the Russell-Myhill paradox in my preferred framework, we need a single one-place

predicate ‘ ’ of fourth-order, taking classes of thoughts as arguments (analogous to ‘Ψ’, in the pluralQ version of the paradox). Consider the class dened by an impredicative

instance of the comprehension principle for classes of thoughts:R

(Comp) X p Xp ϕp

∃ ∀ ( ↔ ) p p X p X Xp i.e. any p is just in case there∀ (R is a↔ class∃ (X ⇔suchQ that∧ ¬p is hyperintensionally)) equivalent with the thoughtR that X, and p is not X. Furthermore, we assume, in line with Russell, the equivalent of Goodman’sQ structure under current assumptions:

(STR) X,Y X Y p Xp Y p

∀ (Q ⇔ Q → ∀ ( ↔ )) For the reasons noted above, something like STR ought to obtain if the logic of ‘ ’ is compatible with a plausible logic of aboutness: contraposing on STR, if classes X and⇔ Y are distinct, the thoughts that they are -ed are distinct.47 Together with ID, STR entails all instances of the following version ofQ Russell’s Premise 2:

p, q p X q Y p q r Xr Y r J K J K ∀ ( = Q ∧ = Q → ( = → ∀ ( ↔ ))) i.e. the thought expressed by ‘ X’ is the thought expressed by ‘ Y ’ only if X is Y .

For the derivation of the paradox,Q however, I shall not use this formalizationQ of Russell’s

47What I really would need to do here is to explain how ‘⇔’ interacts with λ abstraction, because presumably structural claims such as STR should fall out of the logic of the higher-order identity connective, given the type of ‘X’ and ‘Y ’ (which are extensional entities). Identity statements among thoughts are formed by anking ‘⇔’ with closed formulas, and the question is how identity statements among propositional functions may be derived. This generalization is obviously called for but not at all without diculties: see chapter 7 for some tentative remarks, and Dorr (2016) for some of the issues involved.

130 Premise 2, but rather the two principles from which it follows, namely ID and STR. The

reasoning is pretty much as above, with the necessary adjustments:

1. p p EX 2. p pJ XK p X Xp Comp 3. ∃ ( = QRX ) X X 1, 2, FUI 4. ∀ (R ↔ ∃ ( ⇔ Q ∧ ¬ )) Reductio assumption 5. RQRX ↔ ∃ (QRX ⇔XQ ∧ ¬ QR) 3, 4 6. RQR X X elim 7. ∃p (QRp ⇔XpQ ∧ ¬ QR) 6, STR 8. QR ⇔ Q ∧ ¬ QR 4, 6,∃ 7, FUI 9. ∀ (R ↔ ) 8 10. ¬XRQR X X 9 11. QR ⇔ QR ∧ ¬RQR 2, 10 ∃ (QR ⇔ Q ∧ ¬ QR) Contradiction atRQR lines 8 and 11. Notice that for the step to line 9, I assume the reexivity of equivalence. As I mentioned above, a promising reaction is to look at abstraction as a dynamic process. This seems well motivated, for the same kind of impredicativity that generates absurdity in Frege’s Grundgesetze theory of extensions can be seen at work in

the Russell-Myhill paradox. For this purpose, I add a symbol ‘ ’ to the language, intu-

itively read ‘Potentially’, with the syntax of a monadic sentential⟐ operator. Its dual ‘ ’, which we may read ‘Absolutely’,48 is dened as ‘ ’. Following Linnebo (2013), potentiality and absoluteness satisfy the necessitation¬ rule ⟐ ¬ Nec and the S4.2 axioms:

(Nec) ϕ only if ϕ (K) ⊢ ϕ ψ ⊢ ϕ ψ (M) ϕ( →ϕ ) → ( → ) (4) ϕ → ϕ (G) ϕ→ ϕ 48Of course, the Aristotelian dual of⟐ ‘Potentially’→ is⟐ ‘Actually’, but ‘Actually’ in contemporary Philosophy has acquired a completely dierent meaning.

131 The trade-o of ideology for ontology that the modal approach recommends is a sound

deal only if the new piece of ideology to be adopted is in good standing. So what kind of

modal operator is ?

Contingency and⟐ necessity are relative notions: they are relative to a body of laws,

which can be metaphysical, logical, moral, and so forth. Modal operators determine the

truth of a formula across variations in a modal space subject to the constraints imposed

only by such laws (Kment, 2006). Thus metaphysical laws determine metaphysical possi-

bility and necessity, ‘ ’ and ‘ ’, which are read ‘Possibly’ and ‘Necessarily’ respectively.

A proposal about potentialityn ◻ is Fine’s (2006) “interpretational” modality: potentiality and

absoluteness might correspond to contingency and necessity relative to the semantic laws.

Approximately along these lines, an informal reading of ‘ ϕ’ may be: there is an inter-

pretation of the initial domain such that ϕ. This modality,⟐ though articial (it does not

exist in natural language), may oer an independent understanding of . (Care is needed, for quantication over interpretations should not be taken at face value,⟐ if paradoxes are to be avoided (Williamson, 2003).)

If one thinks of the initial domain as containing absolutely all objects (as it is not ruled out so far), the combination ‘ ’ is understood as introducing more things to the domain of absolutely every objects.⟐∃ This might seem puzzling. However, these extra things are, of course, not objects, i.e. they are not in the range of rst-order quantiers.

Indeed, they result from abstraction. For example, the domain of absolutely everything includes sentence tokens. But there is an interpretation of statements about sentence tokens on which they are about their types: abstract objects resulting from abstraction. In my opinion, the dierence between statements about tokens and statements about types is best understood not as an ontological dierence, but rather as a dierence in what

132 identity statements depend on. We start by assuming that identity on abstract objects is undened. Iterated applications of the abstraction principle yield progressive renements of the identity relation. The potentiality operator varies over renements of the identity relation among terms for abstract objects. Thus we can say: given these sentence tokens, potentially there are their types.49

The dynamic nature of abstraction is captured by modalizing the theory of abstraction— or, equivalently, the abstraction principle TA. Consider rst the formalization in plural logic with an ‘about’ predicate. We shall now no longer hold EX-Plur and ID-Plur but rather:

(EX-Plur●) xx p.about p, xx (ID-Plur●) ∀ ⟐ ∃ ( ) xx, yy p, q about p, xx about q, yy p q r r xx r yy ∀ ∀ ( ( ) ∧ ( ) → ( = → ∀ ( ≺ ↔ ≺ ))) EX-Plur● asserts that absolutely for all things there potentially is a thought about them, and ID-Plur● asserts that absolutely for all of these things and all of those things, it is absolutely the case that thoughts about them are identical only if these things are identical to those things.

On this approach, it is Russell’s innocent-looking Premise 1 that should be rejected. It is not the case that for all things there is a thought about them. However, it is absolutely the case that for all things there potentially is a thought about them. Russell-Myhill is no longer trouble because a modalized version of plural comprehension is both unmotivated

49Formal details are given in the Addendum. There are other ways to understand ⟐. One might suppose that it is just n, and that thoughts exist contingently. Stalnaker (2010) argues that some do, and Yu (2017) presents a contingentist view of propositions in response to the paradoxes. Some results on propositional contingentism are discussed by Fritz (2016). If we think that bearing properties is not modally robust, and furthermore that existence is a kind of predication (from a higher-order perspective), then perhaps existence comes and goes in the modal space quite easily. Existence is not special, so to speak. On the other hand, according to some philosophers, modality does not mark distinctions in matters of existence (Williamson, 2016). After all, existence is not a property of objects, and modality is well understood as variation in the bearing of properties.

133 and provably false:

(Comp-Plur●) xx p p xx ϕp ⟐ ∃ ∀ ( ≺ ↔ ) That is, potentially there are things such that absolutely everything that satises ϕ is one of them. Comp-Plur● is unmotivated because it does not follow from a dynamically understood abstraction principle for pluralities (there isn’t one, since pluralities are extensional); moreover, it is not the case that potentially there are things such that absolutely everything that satises the condition dening the Russell plurality is one of them, as shown by the Russell-Myhill derivation. So Comp-Plur● is false.

A modalized abstraction principle for thoughts, TA●, and its consequences, look as follows:

(TA●) α β α β J K J K (EX●) ( p p= α ⇔ ( ⇔ )) J K (ID●) ⟐ ∃p,( q =p α) q β p q α β J K J K ∀ ( = ∧ = → ( = ↔ ( ⇔ ))) TA● states that it is absolutely the case that the thoughts expressed by ‘α’ and ‘β’ are identical just in case α is hyperintensionally equivalent to β. EX● states that potentially there is a thought expressed by ‘α’, and nally ID● states that it is absolutely the case that the thoughts expressed by ‘α’ and ‘β’ are identical just in case α β. With EX● and ID●, the Russell-Myhill reasoning no longer implies absurdity for essentially⇔ the reasons just given: a modalized comprehension principles for classes of thoughts is both unmotivated and provably false.

The Russell-Myhill paradox, far from being a quirk of Russell’s formalism, is a powerful challenge to hyperintensional theories of thoughts: theories that are sophisticated

134 enough to capture a signicant notion of aboutness. The Russell-Myhill is neither truth- theoretic nor a set-theoretic paradox, contrary to Ramsey’s classication. Instead, it is a paradox of aboutness. Absurdity comes from essentially the same source as in Rus- sell’s set-theoretic paradox as it appears in Frege’s Grundgesetze: impredicative abstraction. There is an hitherto unrecognized theory of abstraction behind the premises that generate the Russell-Myhill contradiction. Some predicative approaches to higher-order intensional quantication are designed to handle these diculties, developing a dynamic conception of abstraction. On this interpretation, identity among terms for abstracta is built progressively, and a potentiality operator varies on the stages of this progression.

The approach I presented leads to a consistent theory of thoughts that is compatible with a plausible conception of aboutness. Other accounts of the Russell-Myhill, advocated by

Goodman, Uzquiano, and Dorr, fail to deliver this result: they sacrice genuine hyperintensionality to save consistency in a thoroughly impredicative setting.

6.2.2 Revised Prior’s theorem

So the logic of thoughts should be predicative. This stance is further supported by recognizing that predicativism makes Prior’s speculative metaphysics quite harmless. Prior’s

(1961) paradox is the intensional version of the Liar paradox. We can show that a predicative understanding of quantication over thoughts leads to a very natural solution to

Prior’s paradox. So the predicativist’s unique vantage point on the realm of thought en- joys a pleasant look on the landscape. No other consistent account of the intensional paradoxes has this kind of generality. (Graham Priest’s (1991) inconsistent account of the

135 intensional paradoxes is also very general and unied; but it seems less motivated inso-

far as Priest’s view is about the nature of truth, and the intensional paradoxes are not

truth-theoretic.)

For Prior’s paradox, let us revert to the unmodalized EX and ID. We need a single third-

order predicate of thoughts ‘Q’, and an instance of a Kaplanian principle of uniqueness,

asserting intuitively that the thought that p Qp p , abbreviated as ‘η’, can be Q-ed uniquely up to material equivalence: ∃ ( ∧ ¬ )

q Qq η q

n∀ ( ↔ ( ↔ )) The paradox may be derived from EX, in free intuitionistic relevant logic, as follows:

1. p p η EX 2. q Qq J K η q Assumption 3. η∃ ( = ) Reductio assumption 4.Q∀ p( p↔ ( ↔ )) elim 5.Qp η p 2, UI 6. η ∧ ¬ 3, 4,∃ 5, UI 7. p Qp↔ ( ↔ p) 6 8. Qη¬ η 1, 7, FUI 9. ∀ (η → ¬¬ ) 2, 8 → ¬¬ Contradiction at lines 6¬¬ and 9. If instead we rely on the modalized version of EX, namely

EX●, contradiction is avoided for the following reason: the assumption in line 2 holds in the metaphysical modal context introduced by ‘ ’ by Kaplanian uniqueness. It is plausible to assume that EX● holds necessarily, and son ‘ p p η ’ is true in those possible J K circumstances. However, it still does not follow⟐∃ that( line= 2 holds) in the modal context introduced by . Thus there are circumstances in which η is the case, any Q-ed thought is materially equivalent⟐ to η, and potentially there is a thought expressed by η, but it does not follow that η is Q-ed. Intuitively: Epimenides believes a falsehood, anything believed by Epimenides is materially equivalent to the thought that Epimenides believes

136 a falsehood, but it is not the case that Epimenides believes that he believes a falsehood because that falsehood is merely potential. A contradiction would follow, if we assumed that absolutely every Q-ed thought is materially equivalent to η, i.e. q Qq η q . However, that is clearly false. ∀ ( ↔ ( ↔ ))

Something more can be understood about the situation by looking at Prior’s theorem. Recall that Prior’s theorem follows in (impredicative) higher-order logic from the assumption that the thought that p Qp p is Q-ed:

(PT*) Q p Qp ∀p ( →p ¬Qp) p p Qp p from which PT follows∀ by( necessitation→ ¬ ) → (∃ and( existential∧ ¬ ) ∧ ∃ generalization.( ∧ )) Intuitively, PT* states that Epimenides believes that everything he believes is false only if he believes a truth as well as a falsehood. For a proof, set ‘ϑ’ as short for ‘ p Qp p ’: 1. p p ϑ EX ∀ ( → ¬ ) 2. Qϑ J K Assumption 3. ϑ∃ ( = ) Reductio assumption 4. Qϑ ϑ 1, 3, FUI 5. ϑ 2, 3, 4 6. Qϑ → ¬ϑ 2, 5 7. ¬p Qp p 1, 6, FUI 8. p Qp∧ ¬ p 5 9. Qϑ∃ ( ∧p ¬ Qp) p p Qp p 2, 7, 8 ∃ ( ∧ ) Consider now an analogous proof from EX●. It is consistent to assume (i) that potentially → (∃ ( ∧ ) ∧ ∃ ( ∧ ¬ )) there is a thought expressed by ‘ϑ’, (ii) that ϑ is Q-ed, and (iii) that ϑ is the case. These are circumstances in which everything Epimenides believes is false, Epimenides believes so, and potentially there exists a thought that he believes. That is, roughly, there is a way of approximating the identity relation among terms referring to thoughts such that there exists a thought he believes. More simply and less precisely: if I say that potentially there exists a thought believed by Epimenides, there is an interpretation of what I just said on which there exists a thought believed by Epimenides.

137 Suppose now that it is absolutely the case that Epimenides believes that everything

he believes is false, i.e. Q p Qp p : 1. p p ϑ ∀ ( → ¬ ) EX● 2. Qϑ J K Assumption 3. ϑ⟐∃ ( = ) Reductio assumption 4. Qϑ ϑ 1, 3, FUI 5. Qϑ 2, M 6. ϑ → ¬ 3, 4, 5 7. Qϑ ϑ 5, 6 8. ¬p Qp p 1, 7, FUI 9. p Qp∧ ¬ p 6 10. ∃ (p Qp∧ ¬ )p p Qp p 1, 8, 9, M 11. ∃ Qϑ( ∧ ) p Qp p p Qp p 2, 10 ⟐∃ ( ∧ ¬ ) ∧ ⟐∃ ( ∧ ) So if it is absolutely the→ ⟐∃ case( that∧ Epimenides ¬ ) ∧ ⟐∃ ( believes∧ ) that everything he believes is false, then potentially he believes something true, and potentially he believes something false.

This result is valid, but trivial. There is clearly no sense of inevitability, or necessity,

“interpretational” or otherwise, attached to the supposition that Epimenides holds a belief.

It is not absolutely the case that Epimenides believes that everything he believes is false. So the antecedent of the conclusion is plainly false. Prior’s real theorem, therefore, that I have just derived, far from being a mysterious metaphysical truth handed over by logic alone, is a quite harmless triviality of the material conditional. The modal approach to abstraction has the advantage over other proposals on the topic of the paradoxes of thoughts, that it leads to a unied resolution of many diculties in this domain.

138 6.3 Addendum

This section summarizes the formal system, including a model for which I sketch a proof

of soundness. By a stipulation, the consistency of the model can be established quickly.

A sketch of Henkin completeness is laid out at the end.

Type theory

The only basic type is e, the type of individuals. There is one formation rule for higher

types:

for 0 i j, if νi, ..., νj are types, νi, ..., νj is a type,

≤ ≤ ( ) In the special case where 0 i j, we have the type of propositions, or thoughts, .A

functional type νi, ..., νj is= the= type of functions from νi, ..., νj to , which are called()

propositional functions( , after) Whitehead and Russell (1927). ()

Syntax

The language L includes, besides left and right parenthesis, the following (countable)

sets of expressions as its primitive nonlogical categories:

individual constants a, b, ..., of type e;

n-ary predicates of individuals F, G, ..., of type e1, ..., ei for i 1;

monadic propositional attitude predicates P, Q,( ..., of type) ;≥

propositional attitude predicate variables X, Y, ..., of type (());

propositional variables p, q, ..., of type ; (())

a single monadic predicate of type () .

Q ((())) The last item on this list comes up only for the discussion of the Russell-Myhill paradox.

139 There are also two sets of metavariables: α, β, ..., for expressions of type , and

ϕ, ψ, ..., for expressions of the types of any propositional functions, which will be() helpful to state some abbreviations and generalizations in the following. Notice that in the o- cial language L there are no individual variables, nor predicate variables for the type e , i.e. there are no ordinary rst or second-order variables. This is for simplicity, given( the) limited scope of the present work.

Moreover, L includes:

a term-forming operator of type e .50 J K canonical descriptive singular⋅ terms formed( ) by the following rule: if ‘α’ is a sentence,

‘ α ’ is a canonical dst. J K and logical expressions:

a binary identity relation of type , ;

primitive connectives: negation= of(() type()) , implication , and hyperintensional

equivalence , both of type ¬, . Other(()) binary connectives→ are dened as usual:

conjunction ⇔ϕ ψ ϕ (()ψ ()), disjunction ϕ ψ ϕ ψ, and material equiv-

alence ϕ ψ ∧ ⇔ϕ ¬(ψ → ¬ψ ) ϕ . ∨ ⇔ ¬ →

universal↔ quantiers⇔ ( binding→ ) ∧ ( the→ two) kinds of variables introduced above: proposi-

tional variables, and variables over predicates of thoughts. An existential quantier

is dened as usual in both cases.

primitive modal operators, of type : possibly , and potentially . Two more

50I assume for simplicity that sentences (to be dened(()) below) aren individual objects, hence⟐ of type e.‘ ⋅ ’ is undened on objects that are not sentences. The assumption is, strictly speaking, false: while utterancesJ K are individual objects, the arguments of ‘ ⋅ ’ are not utterances but sentence-types. However, since there is no context-sensitivity in the langauge, makingJ K this stipulation simplies the theory at a point where more precision is simply pedantry.

140 modal operators are dened: necessarily ϕ ϕ, and absolutely ϕ ϕ. ◻ ⇔ ¬ ◻ ¬ ⇔

Notice¬ that ⟐ ¬ the quantiers ‘ ’ are not generalized: they express generality, but they also bound free variables in their∀ scope. It is more perspicuous, and more correct, to have a variable-binding operator λ for the latter task. We would then, for instance, write

‘ λp P,Q ’ instead of ‘ p P p Qp ’. However, in this dissertation I decided to avoid the∀ use( of λ) abstraction,∀ that( introduces→ ) some complications I preferred to defer to a later point, as I mention in chapter 7.

The set Σ of formulas is dened inductively:

Propositional variables are formulas;

if t1, ..., tn are individual constants and A an n-ary predicate of individuals, At1, ..., tn

is a formula;

if p and q are propositional variables, p q is a formula;

if α and β are dsts, α β is a formula,= # J K J K J K J K if ϕ is a formula and ψ is a propositional= # attitude predicate or predicate variable,

ψϕ is a formula;

if ψ #is a propositional attitude predicate or predicate variable, ψ is a formula;

if ϕ is a formula, so are ϕ , ϕ , and ϕ ; Q #

if ϕ and ψ are formulas, so¬ are# nϕ # ψ and⟐ ϕ# ψ ;

if ϕp is a formula in which ‘p’ occurs → free,# pϕp⇔ is# a formula;

if ψX is a formula in which ‘X’ occurs free,∀ XψX# is a formula.

Formulas that do not have free occurrences of variables∀ are#sentences.

The deductive system includes any standard axioms and rules for classical S5 propositional logic for , , and . I add S4.2 axioms and rules for , including a rule of

¬ → n 141 ⟐ necessitation, following other work on potentiality (Linnebo, 2013). The notion of hyper-

intensional equivalence is characterized by (at least) these axioms and axiom schemas: p p p (Re) ⇔ (Iden) p, q p q p q p, q p q p q (Subs) ⊢ ∀p,( q ⇔p q) X Xp Xq (RBV) ⊢ ∀pϕp(( ⇔qϕq) ⇔ = ) ⊢ ∀XϕX(( ⇔ Y) ϕY→ = ) (Neg) ⊢ ∀p, q( p= → q∀ (p →q )) pϕp XϕX (STR) ⊢ ∀X,Y⇔ ∀X Y p Xp Y p ⊢ ∀ ⇔ ∀ (Idem) ⊢ ∀p p(¬ ⇔p¬ →p ⇔ ) ⊢ ¬(p∀p ⇔p ∀ p ) ⊢ ∀ ((Q ⇔ Q ) → ∀ ( ↔ )) Reexivity (Re),⊢ the∀ ( identity⇔ ¬( axioms→ ¬ )) (Iden), and the Negation⊢ axiom∀ ( ⇔ schemas¬ → (Neg),) are taken from Church’s (1984) axioms for strict equivalence. The antisymmetry and transitivity of hyperintensional equivalence follow from Iden together with the axioms for quantication theory below. Notice that it is an easy consequence of Iden and the substitutivity of identity (Subs), that hyperintensional equivalence is a congruence on the third-order domain, as desired:

(Cong) p, q p q X Xp Xq

∀ ( ⇔ → ∀ ( → )) Replacement of Bound Variables (RBV) is the only criterion of propositional identity for syntactically distinct sentences accepted as an axiom in Church’s Alternative (0) (Church,

1973a, 1974).51 Structure (STR) is an aboutness axiom: it states that for any classes of thoughts X and Y , the thoughts that X and that Y are hyperintensionally equivalent only if X is Y . Identity among classesQ is implicitlyQ taken to be coextensionality. Had I allowed (at the cost of some complications) rst-order variables and quantiers, it would have been straightforward to state a rst-order analog of STR: x, y ϕx ϕy x y . Axioms of this kind capture the intuitive idea that thoughts∀ do not(( blur⇔ distinctions) → = among) the things they are about. The idempotency axioms (Idem) for conjunction and

51Replacement of independently stipulated equivalences is also an accepted principle. But the project of stipulating equivalences (such as the philosophers’ favorite: x is a bachelor and x is an unmarried man) is very unattractive.

142 disjunction have been motivated in chapter 5. No doubt more axioms about hyperinten-

sional equivalence are desirable, but I haven’t argued for them.

There are also axiom schemas and schematic rules for the quantiers: (Distr) p ϕ ψ pϕ pψ X ϕ ψ Xϕ Xψ (Gen) ⊢ ϕ∀ ( ψp→only) → if(∀ ϕ → ∀pψp) ⊢ ϕ∀ (ψX→only) → if(∀ ϕ → ∀XψX) ⊢ → ⊢ → ∀ For both generalization⊢ → rules, the variable⊢ → ∀ must not occur free in ϕ nor in any premise of the deduction.

(FUI) pϕ q q α ϕ α p J K (UI) ∀Xψ→ (ψ∃ (Y =X ) → [ ~ ])

(Comp) ∀X p→Xp[ ~ψp]

∃ ∀ ( ↔ ) In UI, Y is a predicate variable free for X in ψ. In Comp, a comprehension axiom scheme for second order propositional functions, X must not occur free in ψ. The system is complete with an axiom schema for propositional identity TA●, or Predicative Thought

Abstraction:

(TA●) α β α β J K J K ( = ⇔ ( ⇔ )) I have not dened identity on propositional functions, but rather eliminated any of its occurrences everywhere, exploiting the fact that propositional functions are (plausibly) extensional. All hyperintensionality in the system comes from thoughts.

Notice that the use of a term-forming operator on sentences implies that some of J K these principles, in particular TA●, are inevitably schematic.⋅ One cannot generalize in the position of ‘α’ and ‘β’ in TA● because it would mean to quantify into a quotation environment (Quine, 1970). The main consequence for the theory presented here is that TA●

143 generalizes only over thoughts expressible by sentences of L . This is usually regarded as a limitation: for example, schematic rst-order principles generalize only over objects that have a name in the language, which seems arbitrary. However as I argued in chapter

4, this does not seem to be a problem here, given that it is a feature of the present account that thoughts have canonical “names”, and these are the sentences that express them.52

Semantics

A sketch of the semantics described here is contained in a passage from David Kaplan

(who assumes that thoughts are modeled as intensions): At level 0, there were only the possibilities of, say, dierent distributions of earth, air, re, and water. Now there are dierent distributions of earth, air, re, and water with it being queried whether all is earth, and there are the same distributions with it not being queried whether all is earth. These new possibilities provide new propositions to query—for example, whether it is queried whether all is earth. The question whether this new proposition is queried is not already answered by the initial assignment of a family of sets of models of level 0 to the operator [It is queried that]. (Kaplan, 1995, 45)

At the basic level, there are only thoughts about extensional matters, e.g. the thought that all is earth. Thoughts are divided into a hierarchy of levels, the higher-level thoughts depending, in some sense, on the lower-level thoughts. This dependence relation is captured by , which intuitively allows us to “jump” to a higher level. Levels are represented here as models.⟐ In practice, I shall dene an ω-sequence of models for the interpretation of L .

The dynamic aspect of potentiality is thus in eect parasitic on the indenite extensibility of the set Ord of ordinal numbers (Shapiro and Wright, 2006). Kaplan’s sketchy remarks do not address the question what forces the levels to go up. The answer I give here is inspired by recent work on grounded abstraction (Leitgeb, 2016; Horsten and Linnebo, 2016).

52Another consequence is that schematic generality is universal, not existential. This entails that the 1 strength of the second-order quantiers is at most Π1. I am not sure if and why this should occasion concern.

144 Intuitively, we go up the sequence of models by rening the extension of the identity relation. The following is perhaps not philosophically satisfactory in some respects (since I’ll help myself to quantication over the ordinals), but it should be at least formally precise.

Sketches of proofs of soundness and completeness are at the end.

σ,w Each model Mn in the sequence M M0, M1, ... is a tuple E,Tn,Fn, W, R ,I , with E an arbitrary nonempty set of individuals,= ⟨ Tn⟩and Fn are⟨ two disjoint{ sets, }W a set⟩ of possible worlds ordered by an accessibility relation R, and I an interpretation function relative to a variable assignment σ and a possible world w. E and W are always constant.

Intuitively, Tn and Fn are the sets of true and false thoughts of each model.

Building on Church (1973b), each type is assigned a domain in each model as follows:

n the domain De of e is E, the domain Dp of is Tn Fn, and the domain of each remaining

n nDa n functional type Dap Dp f Da ()Dp .⋃ The interpretation of the basic lexical categories is dened as= follows:= { ∶ ↦ }

σ,w If t is a rst-order term, IMn t E;

σ,w n if A is an n-ary predicate of individuals,( ) ∈ n 1, IMn A f E 0 ... n−1 E Dp ; Iσ,w Dn Mn (pp)p; ≥ ( ) ∈ { ∶ × × ↦ }

σ,w σ,w IM (Qα) ∈ IM α ; n J K n σ,w n if p is( a propositional) = ( ) variable, IMn p σ p Dp ;

σ,w n if ϕ is an attitude predicate, IMn ϕ( )D=pp;( ) ∈

σ,w n if X is a predicate variable, IMn (X) ∈ σ X Dpp;

σ,w n if s is a sentence or a dst, IMn s( )D=p ;( ) ∈

if ψ is a formula and ϕ a propositional( ) ∈ attitude predicate or predicate variable,

σ,w σ,w σ,w IMn ϕψ IMn ϕ IMn ψ ;

( ) = [ ( )]( ( ))

145 σ,w and nally, if ϕ is a propositional attitude predicate or predicate variable, IMn ϕ

σ,w σ,w IMn IMn ϕ ; (Q ) =

[ (Q)]( ( )) Thoughts correspond to formulas in the right way. In particular, to each equivalence class of formulas under in every model, there corresponds at least one and at most one thought in the domain of⇔ the model: that’s the thought expressed by those formulas. For each model in M , consider the set Σ ⇔ of all equivalence classes of formulas under , and we require that there is a thought~ corresponding to each of its members. I dene⇔ a

n “meaning” function m, relative to each Mn, m Σ ⇔ Dp , as follows:

n σ,w Meaning: For all U Σ ⇔ and all p ∶Dp~, m↦U p i for all α U, IMn α p n Quotient: For all p Dp , there is a unique U Σ ⇔ such that m U p ∈ ~ ∈ ( ) = ∈ ( ) = Thus in every model every sentence∈ denotes the same∈ thought~ as all of its( ) =-equals.

M is ordered by a relation , on which several conditions should be imposed.⇔ First, a no-shrinking condition on how propositional≤ domains are populated. The intuitive idea is that progress in the sequence of models never makes thoughts disappear:

n m No-Shrink: Mn Mm only if Dp Dp

≤ ⊆ Next, recall that identity among terms is dened only for thoughts. The idea of the sequence of models is that identity gets dened as we go. Intuitively, in M0 no identities

are true and none are false: no identications have been “declared”, so to speak. As we

go up the sequence, thoughts are identied or distinguished by successive rounds of the

abstraction process.

+ For this purpose, following Horsten and Linnebo (2016), consider two disjoint sets Hn

− n + n n and Hn of members of Dp . In particular, let Hn be an equivalence relation on Dp Dp . An

+ − approximation (of the identity relation) is a pair Hn Hn ,Hn . As we shall see,× this pair coincides in the limit with the extension and the anti-extension= ⟨ ⟩ of the identity relation.

146 + − + + An approximation Km Km,Km is an extension of another Hn just in case Hn Km

− − and Hn Km. Let Km be= ⟨ an admissible⟩ extension of Hn i: ⊆

⊆ Km is an equivalence relation;

+ − Km is conservative: Hn Km and Hn Km

′ ′ ′ ′ Km is closed under substitution:⊆ if s,⋂ s , t, t =are∅ dsts such that s s t t , then

′ ′ t, t Km only if s, s Km. = [ ~ ]

⟨ ⟩ ∈ ⟨ ⟩ ∈ A renement Ref takes us from an approximation to its renement, by adding to an approximation Hn pairs of thoughts identied by for any admissible extension of Hn.

+ − Let Ref Hn Ref Hn , Ref Hn such that⇔

( + ) = ⟨ ( ) ( )⟩σ,w σ,w Ref Hn is the set of all pairs IM α ,IM β such that for every admissi- n J K n J K ble extensions( ) Km of Hn, Mm, σ ⟨ α ( β;) ( )⟩

− σ,w σ,w Ref Hn is the set of all pairs ⊧IM ⇔α ,IM β such that for every admissi- n J K n J K ble extensions( ) Km of Hn, Mm, σ ⟨ α( )β . ( )⟩

⊧ ¬( ⇔ ) Horsten and Linnebo show that the image under Ref of an approximation is itself an

approximation, and other relevant results for this method. The renement operation may

be iterated through the ordinals:

+ − H0 H0 ,H0 ,

= ⟨ + ⟩ = ⟨∅ −∅⟩ Hn+1 Ref Hn , Ref Hn

= ⟨ ( +) − ( )⟩ Hλ Hγ , Hγ γ<λ γ<λ = ⟨ ⟩ Let Hn Hm just in case Hm can be obtained from Hn by successive renements. Say

that a model⊏ Mn is faithful to an approximation Hn, Mn,Hn , just in case for all mem-

σ,w σ,w bers of Hn, IM α ,IM β Hn i Mn, σ Fα( β .) Finally, if Mn,Hn and n J K n J K J K J K ⟨ ( ) ( )⟩ ∈ 147 ⊧ = F( ) Mm,Hm , we set Mn Mm just in case Hn Hm. So models are ordered by accord-

Fing( to renements) of the≤ identity relation. Intuitively,⊏ a model follows another≤ in the

sequence M only if all identities that the latter “sees”, the former can “see” too. With this machinery, I can now state what identity statements correspond to in the model:

σ,w σ,w σ,w If ϕ has the form t1 t2 , IMn ϕ IMn t1 ,IMn t2 Hn for an approximation

Hn such that Mn,H= n #. ( ) = ⟨ ( ) ( )⟩ ∈

F( ) So identity is interpreted as the claim that the interpretations of terms anking it belong

to a faithful approximation of the model.

Finally, I add further axioms for . In particular, I shall assume the following:

Reexivity: Mg Mg ≤ Transitivity: Mg Mh Mh Mi Mg Mi Antisymmetry: Mg≤ Mh Mh Mg Mg Mh Convergence: (Mg ≤ Mh ∧ Mg ≤ Mi ) → j ≤Mh Mj Mi Mj ( ≤ ∧ ≤ ) → = This is to make sure that( the logic≤ of∧ is≤ sound) → ∃ in( the sequence≤ ∧ of≤ models,) given that the S4.2 axioms are satised: the semantic⟐ clause for , relative to a model, makes reference

to what happens elsewhere in the sequence of models.⟐

Interpretation can now be extended to all formulas, together with their allocation to

one of the sets Tn or Fn for each model:

σ,w σ,w σ,w If ϕ has the form β , IMn ϕ Tn i IMn β Fn, otherwise IMn ϕ Fn;

σ,w σ,w σ,w if ϕ has the form ¬β1 # β2 ,(IM)n ∈ ϕ Tn i(IM)n∈ β1 Fn or IMn β(2 ) ∈Tn, other-

σ,w wise IMn ϕ Fn; → # ( ) ∈ ( ) ∈ ( ) ∈

σ,w σ,w σ,w if ϕ has the( ) form∈ β1 β2 , IMn ϕ Tn i IMn β1 IMn β2 , otherwise

σ,w IMn ϕ Fn; ⇔ # ( ) ∈ ( ) = ( )

σ,w ′ ′ if ϕ(has) ∈ the form β , IMn ϕ Tn i there is a w W such that wRw and

σ,w′ σ,w IMn β Tn, otherwisen # IMn (ϕ ) ∈Fn; ∈

( ) ∈ ( ) ∈ 148 σ,w ′ if ϕ has the form pψp , IMn ϕ Tn i for every assignment σ that agrees with

σ′,w σ,w σ except possibly on∀ p, I#Mn ψp( ) ∈Tn, otherwise IMn ϕ Fn;

σ,w ′ if ϕ has the form XψX ,(IMn) ∈ϕ Tn i for every( assignment) ∈ σ that agrees

σ′,w σ,w with σ except possibly∀ on X# , IMn ( ψX) ∈ Tn, otherwise IMn ϕ Fn;

σ,w if ϕ has the form β , IMn ϕ T(n i) there∈ is an m Ord (and) ∈ a model Mm M

σ,w σ,w such that Mn Mm⟐and#IMm (β ) ∈Tm, otherwise IMn ϕ∈ Fn; ∈

σ,w Finally, if ϕ has≤ the form t(1 ) ∈t2 , IMn ϕ Tn i( there) ∈ is an approximation Hn

such that Mn,Hn , and for= all its# renements( ) ∈ Hm such that Hn Hm, it is the

σ,w σ,w σ,w case that FIM(n t1 ,I)Mn t2 Hm, otherwise IMn ϕ Fn. ⊏

⟨ ( ) ( )⟩ ∈ ( ) ∈ This completes the notion of truth relative to a model. A formula ϕ is satisable in a

σ,w model Mn relative to a variable assignment σ just in case IMn ϕ Tn, in which case I

σ,w write Mn, σ ϕ. A formula ϕ is valid in a model Mn just in case IM(n )ϕ∈ Tn for all variable assignments,⊧ in which case I write Mn ϕ. A formula ϕ is a tautology( ) ∈ in a sequence of models M if and only if Mn ϕ for all M⊧n M .

Soundness (sketch) ⊧ ∈

With the exception of the clauses for , , and identity, each model is an ordinary model for (a fragment of) fourth-order predicate⇔ ⟐ logic. The axioms for do not pose any particular challenge, and the S4.2 axioms for are valid by design,⇔ given the structure of the sequence of models. The validity of TA⟐● can be established as follows. Suppose for reductio that, for some instances of ‘α’ and ‘β’, for some model Mn M and some variable assignment σ, it is not the case that Mn, σ α β ∈α β . It follows J K J K σ,w that IM α β α β Tn. So⊧ for( some= m and⇔ ( model⇔ M))m M such n J K J K σ,w σ,w σ,w that Mn (⟐¬Mm(, IM = α ⇔β( ⇔α )) ∈β Fm. So IM α β IM ∈ α β . m J K J K m J K J K m ≤ ( = ⇔ ( ⇔ )) ∈ ( = )= ~ ( ⇔ ) 149 σ,w σ,w Interpretation of the latter is the condition on the model that IMm α ,IMm β belongs to the eld of the identity relation. Interpretation of the former⟨ is( the) condition( )⟩ on the

σ,w σ,w model that there is a faithful approximation Hm to which IM α ,IM β belongs. m J K m J K Since identity in the model is a faithful approximation (of⟨ itself),( absurdity) ( follows.)⟩

Henkin completeness (sketch)

The general question of completeness is moot because of Gödel’s results. However, Henkin completeness can be established. We need to show that if a set of formulas Γ is consistent, then it is satisable in a sequence of models M . For this, we build the Henkin expansion of Γ as usual, namely a consistent set Γ+ that is (i) saturated and (ii) complete, i.e. (i) for each formula ϕ in which p occurs free, there is a sentence α such that pϕ ϕα Γ+, and (ii) for every sentence β, either β Γ+ or β Γ+. The expansion¬∀¬ is→ carried# ∈ out relative to an enumeration of formulas of∈ L that¬ contain# ∈ a free variable, making sure that at each step a consistent set is being dened. To prove (i), I assume an innite supply of

“new” sentences for each formula. (Sentences are in eect just like constants in a Henkin proof of completeness for ordinary predicate logic.)

From Γ+ a sequence of models M can be built, and it can be shown that it satises

Γ+. The construction of M is roughly analogous to the standard construction of a term model for rst-order languages, in which one takes the set of constants in the language as domain of the model. By an argument of essentially the following kind, one can establish the relative consistency of the models described here.

Take the set Σ ⇔ of equivalence classes of formulas modulo as propositional domain, in eect identifying~ thoughts with equivalence classes of formulas,⇔ and individual

+ constants for the domain E. First I dene a model Mn Γ relative to each ordinal n as

+ n follows: sentences of Γ are assigned elements of Dp (Σ ⇔), predicates of any functional

150 = ~ n + type are assigned functions into Dp , so that all atomic formulas of Γ come out satised

+ + by Mn Γ . For each sentence of the form t1 t2 in Γ with t1, t2 dsts, I dene a term-

˜ + ˜ + approximation( ) Hn such that t1, t2 Hn , subject = # to the same restrictions laid out above

˜ + for approximations; in particular,⟨ if⟩ ∈t1 α and t2 β , then t1, t2 Hn if and only J K J K + + if Mn Γ α β. For each sentence= of the form= β in Γ ⟨, I dene⟩ ∈ another model

+ ˜ ˜ Mm Γ( relative) ⊧ ⇔ to an ordinal m such that Hn Hm in⟐ which# β comes out satised, and the( rest) is as before. It can be shown by induction⊏ on the complexity of formulas that for

+ + + each formula ϕ Γ , the resulting sequence M0 Γ , M1 Γ , ... satises ϕ. Thus each

ϕ Γ is satisable,∈ and so is Γ. ⟨ ( ) ( ) ⟩

∈

151 Chapter 7: Conclusions and further work

Part of my discussion has been guided by the problem of propositional identity: to give an account of the identity of thoughts. The two most popular alternatives, the intensional view on which thoughts are sets of possible worlds and the structured view on which thoughts are complex objects with proper parts, fall short of very plausible desiderata.

The intensional view entails that substitution of necessarily equivalent sentences in the scope of propositional attitude operators is truth-preserving, and the best attempts to accommodate the evidence to the contrary lead to the two-dimensional paradox. The structured view entails that a sentence is never synonymous with a logical complication of itself, even as simple as its self-conjunction, undermining the standard Gricean explanation of some ordinary facts about language use. These are the main negative conclusions

I have reached in the present work.

The discussion of propositional identity can be organized around an abstraction principle, for whatever exactly is the nature of the equivalence relation on the right hand side, some notion of identity for thoughts may be dened on the left hand side. The debate then moves on to pinning down the exact features of hyperintensional equivalence, but any resolution on this front is straightforwardly compatible with the outline of the abstractionist account. A few existing proposals (e.g. Fine’s (2017) truthmaker logic, inquisitive logic

(Ciardelli et al., 2017), and Leitgeb’s (2018) hyperintensional logic) are candidate notions

152 for hyperintensional equivalence that I have not ruled out so far. These should be care- fully assessed, particularly with regard to additional topics regarding the logic of thoughts that I have set aside in the Introduction, such as Frege’s puzzles and the puzzles of logical omniscience.

Any (interesting) formal language denes a notion of synonymy, in terms of which the relevant abstraction principle would deliver thoughts of the right kind relative to the logical properties of that language. The real question is, of course, about synonymy in natural language. One could be a pluralist about this, and, certainly, until we know more, let a thousand owers bloom. However, I am inclined to think that, even if we don’t know it yet and perhaps we never will, we should think that there is just one right notion of synonymy for natural language—at least until proven otherwise. Consequently, there is just one abstraction principle for “natural” thoughts. Plausibly, context-sensitivity will have to be factored in, and the account will have to rely on an independent understanding of those parts of the meaning of sentences that do not belong to semantics (i.e., presuppo- sitions). Furthermore, the account would presumably extend to illocutionary acts other than declaratives, and deliver identity conditions for the semantic objects expressed by interrogatives and imperatives (i.e., questions and commands). Some of the logics indicated above may be better equipped for these developments. Progress in this respect is also, in part, a matter of empirical research.

The second line of investigation I have followed is concerned with the paradoxes. An important benet of abstracting thoughts is a well-motivated resolution to many intensional paradoxes, provided abstraction is understood as a dynamic process. Dynamic abstraction supports a predicative solution to the Russell-Myhill antinomy, which allows us to maintain an important component of the hyperintensionality of the theory of thoughts:

153 that thoughts are individuated at least as nely as the things they are about. Moreover, the awkward metaphysical consequences of Prior’s account disappear under predicative quantiers: this shows that, like impredicativity more generally in other domains, an impredicative logic of thoughts gives answers to questions that should not be answered by logic. There are some paradoxes I have not discussed, but that it would be desirable to account for under the same premises—Kripke’s (2011) Time and Thought paradox in particular, for which Kripke himself airs the hypothesis of ramication.

The project of ramifying the type theory has a very bad reputation, though in part undeservedly so. The variety of predicativism I presented here is indeed closely related to Russell’s original approach to the paradoxes, but the nature of this relation ought to be understood better. Intuitively, a simple way to think about a ramied logic for the framework I have presented is to let the syntax explicitly mark the domain in which a formula is interpreted with exponents for ordinal numbers, given that the semantics already interprets a formula relative to a model in an ordered set of models. Thus, every formula gets relativized to a stage in the abstraction process, and we are in all relevant respects in a ramicationist regime. It remains to be determined how the the resulting ramied language relates to Russell’s original approach. More ambitiously, there are other varieties of predicativism in the vicinity, in particular in the work of Feferman (2007), and it would be desirable to understand how a suitable generalization of the present project relates to these.

However, one would rst have to extend the approach I laid out here to other types, in order to appreciate its value all things considered. Metaphysically, the abstraction of thoughts invites the denition of further semantic material. For it is very natural to think of properties as obtained by “plugging holes” in thoughts by means of a λ operator. For a

154 philosophers’ favorite example, this would allow one to state identications such as that

λx.Vixen x λx.Female x Fox x , assuming of course that the thought that all vixens are vixens( ) ⇔ is equivalent( to) the∧ thought( ) that all vixens are female foxes—an assumption that some are willing to make. However, how the λ-calculus should be integrated with higher-order intensional logic is a matter of considerable controversy. In particular, while many authors accept that η-conversion preserves hyperintensional equivalence,

η-conversion: ϕ ϕ′, where ϕ′ is derived from ϕ by replacing some constituent of

the form λxi, ...x⇔j.F xi, ..., xj , where xi, ..., xj are not free in F , with F .

( ) some do not accept that β-conversion does,

β-conversion: ϕ ϕ′, where ϕ′ is derived from ϕ by replacing some constituent of

the form λxi, ...x⇔j.F Gi, ..., Gj with F Gn xn .

( ) [ ~ ] Intuitively, η-conversion converts an expression with its λ-expansion, whereas β-conversion is function application. Both Salmon (2010) and Fine (2012) argue that substitution of β- converts fails in the scope of propositional attitude operators, on the basis of arguments as this: if Descartes thinks that a is F , he might fail to think that a is such that it is

F . (The construction “is such that it is so-and-so” is how application of a λ-term to its argument is sometimes glossed informally.) If β-conversion preserves hyperintensional equivalence, Descartes might not fail to think so in those circumstances. Dorr (2016) defends both η- and β-conversion principles in an abductive argument for a general logic of identications, but his considerations in case the language includes propositional attitude operators are at best tentative. If Fine and Salmon are correct, and Dorr is correct

155 about extensional languages, it seems to follow that β-conversion has the eect of “de- hyperintensionalizing” the language, but then one wonders how the logic of properties could be derived from the logic of thoughts.

The project described here, of which this dissertation is perhaps a rst contribution, encourages ambitious applications of the technology of abstraction principles beyond the area of inquiry for which it was initially developed. The theory of abstraction with regard to the natural numbers is the result of the common eort of a number of scholars over the course of the last four decades, and what they have produced is a robust conception of abstract objects that can be applied elsewhere in Metaphysics. Showing convincingly that this is so, with regard to the foundations of semantics, is perhaps the most important ambition of the present work. There are a number of topics related to abstraction that I have not addressed, but that would have to be discussed in future work: to begin with, well-known issues for abstractionism such as the Caesar Problem. In part, problems of this kind are topic-independent, for they seem to be about the general strategy.

The analogy with the abstraction of natural numbers is helpful, but it is only an analogy: some problems arise specically for semantics. For example, some philosophers think that there are thoughts that cannot be expressed by any sentence (Wrigley, 2006), but it is just wildly implausible to imagine numbers that do not count the members of any set. These worries would have to be addressed elsewhere.

Perhaps the most important dierence between the abstraction of thoughts and the abstraction of numbers is that the nature of the equivalence relation on sentences that denes thoughts is not easily describable, whereas equinumerosity is quite uncontrover- sially the correct equivalence relation for the nite cardinals. This asymmetry points to further issues, that I have not investigated here. Traditionally, many philosophers have

156 wanted to explain synonymy (e.g. between ‘Snow is white’ and ‘Der Schnee ist weiss’) in terms of sameness of thought. I explain sameness of thought in terms of synonymy.

This turn has a lot in favor of it, as I argued, but should one want an explanatory account of synonymy across languages, one would have to look elsewhere. I am not sure this is too bad: after all, translations do not play a theoretical role anywhere in contemporary semantic theory (and perhaps one could nd an account of cross-linguistic synonymy elsewhere).

However, the issue of synonymy across languages is really, more broadly, an issue of sameness of thought across dierent representational systems. This raises the question how an abstraction principle for thoughts relates to a representational system such as the human mind. Here I have presented abstraction as a rational reconstruction of our (apparently successful) referential practices concerning thoughts. A parallel argument could be given to account for our (apparently successful) epistemological practices concerning them, e.g. reliable belief formation. A dierent set of questions is whether synonymy might explain some features of our actual cognitive lives, i.e. not just as a matter of rational reconstruction. At least in the “Scottish” variety of Neologicism, an understanding of Hume’s Principle as something like a conceptual truth leads directly to an explanation of the intentionality of beliefs about the natural numbers and of arithmetical language

(Hale and Wright, 2001). There is a parallel move to be made in the case of thoughts, which would amount to say, roughly, that a grasp of synonymy as a condition of identity of thoughts is implicit in our cognitive lives. After all, ordinary speakers do volunteer judgements about synonymy, at least in simple cases, and plausibly they may do so on the basis of their linguistic competence. The possibility of extending this ambitious account to our grasp of meaning remains to be seen.

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