Branching Time and the Semantics of Future Contingents
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Branching Time and the Semantics of Future Contingents Jacek Wawer PhD dissertation prepared under the supervision of Prof. dr hab. Tomasz Placek Institute of Philosophy Jagiellonian University Kraków, Poland Acknowledgments First and foremost, I would like to express my gratitude to Professor Tomasz Placek, the supervisor of my dissertation. Over a decade ago, he introduced me to the world of branching and has helped me to travel it ever since. The assistance he provided during my work cannot be overestimated. The philosophical community was impoverished due to his dedication, but I benefited enormously. Also, thanks to his continuous sup- port, I had the best possible environment to develop my thought. I have never been an obedient student and often challenged, rather than developed his philosophical views. Nonetheless, he only doubled his support and helped me to express my views better than I could have done it myself. I count myself to be immensely lucky to be his student. I would like to thank Professors Fabrice Correia, John MacFarlane, and Thomas Müller who mentored me during my research visits in Geneva, Berkeley, and Konstanz respectively. Many of their helpful suggestions influenced my views. I am indebted to Alex Malpass, Leszek Wronski,´ Juliusz Doboszewski, Antje Rumberg, and Michał Marczyk for their feedback on various fragments of this work and many inspiring dis- cussions which helped me clarify my ideas and recognize new paths. I would have not accomplished this work, if not for the continuous support and encouragement of my wife, Karolina. My research was possible due to the financial, administrative, and scientific sup- port of many institutions. Most importantly, the Jagiellonian University, and also (in alphabetic order) of the Foundation for Polish Science, Ministry of Science and Higher Education of the Republic of Poland, the Polish National Science Centre, the Polish- U.S. Fulbright Commission, and the Rector’s Conference of the Swiss Universities supplemented my research. Last, but not least, I would like to thank Professor Nuel Belnap and Professor Peter Øhrstrøm. I had a chance to personally discuss philosophy with them only a few times. Nonetheless, their influence and inspiration will be visible at virtually every page of this work. I disagree with some of their views (always with considerable discomfort), but the problems I undertake are their problems, expressed in their vocabulary, and approached using their methods. If I was able to see anything at all, it is only because I stood on the shoulders of the giants. Contents 1 Introduction1 2 Branching Realism5 2.1 Naïve Branching Realism........................7 2.1.1 Futures in abundance......................8 2.1.2 Problem with the trousers universe............... 11 2.2 Genuine Branching Realism...................... 13 2.2.1 Two perspectives........................ 16 2.2.2 Two languages......................... 18 2.2.3 Which perspective is basic?.................. 20 2.2.4 Are possible histories possible?................ 23 2.2.5 Indexical actuality....................... 25 2.2.6 Towards Branching Actualism................. 27 3 Ockhamist semantics 31 3.1 Branching structure........................... 31 3.2 Ockhamist truth............................. 32 3.3 A few remarks on the logic of Ockhamism............... 37 3.4 Future tense operator.......................... 39 3.5 Modal operator............................. 41 3.6 Sentences and propositions....................... 44 3.7 From semantics to postsemantics.................... 45 4 Semantics of Branching Realism 47 4.1 Metaphysical constraint of semantics.................. 47 4.2 Extremism................................ 49 4.3 Modalism................................ 59 4.4 Many-valued semantics......................... 62 4.5 Supervaluations............................. 76 4.6 Assessment relativism.......................... 83 4.7 History relativism............................ 93 4.8 Local relativism............................. 99 4.8.1 Recognized possibilities.................... 99 4.8.2 Counterfactual branches.................... 105 iii CONTENTS 4.8.3 Continuations.......................... 110 4.8.4 Sets of transitions........................ 119 4.8.5 A problem with local relativism................ 126 5 Thin Red Line 129 5.1 Metaphysics of the Thin Red Line................... 129 5.2 Semantic impact............................. 133 5.3 Objections to the Thin Red Line.................... 137 5.3.1 Metaphysics.......................... 137 5.3.2 Epistemology.......................... 137 5.3.3 Actuality............................ 138 5.3.4 Semantics............................ 139 5.3.5 Postsemantics.......................... 146 6 Branching Actualism 166 6.1 Metaphysical background........................ 166 6.2 Semantic impact............................. 173 6.3 Response to objections......................... 179 6.3.1 Metaphysics.......................... 179 6.3.2 Epistemology.......................... 180 6.3.3 Actuality............................ 181 6.3.4 Semantics............................ 186 6.3.5 Postsemantics.......................... 188 6.3.6 Possible predictions...................... 192 6.4 Branching possibilities......................... 198 6.4.1 Are actualist possibilities sufficiently real?........... 198 6.4.2 Are genuine possibilities sufficiently real?........... 203 6.4.3 The nature of branching possibilities.............. 206 6.4.4 Divergence or branching?................... 213 6.5 Localism and trans-localism...................... 217 Summary 230 APPENDIX 232 iv Chapter 1 Introduction Aristotle has stated the famous problem of the so-called future contingents in section 9 of De Interpretatione. Future contingents are statements concerning possible future eventualities. Aristotle describes them as “affirmations regarding particulars that are going to be” with respect to which “both possibilities are open, both being and not being, and consequently, both coming to be and not coming to be” (18a28–18a32 and 19a10-19a12). When Aristotle reflected on such affirmations, he noticed that we are torn by two conflicting ideas. On the one hand, if something might happen and it might not happen, then we can- not truly say that it will happen, nor can we truly say that it will not happen. If the affirmation were true, then the affirmed state would necessarily have to happen. If the negation were true, then the state would necessarily have to not to happen. Therefore, neither the affirmation, nor the negation of a contingent event is true (De Interpreta- tione, 18a34–18b16). On the other hand, the contingent occurrence either will happen, or will not happen. If it will happen, then the affirmation of the future occurrence is true and if it will not happen, then the negation of the future occurrence is true. It is thus absurd to say that neither the affirmation, nor the negation is true because, “Take a sea-battle: it would have neither to happen nor not to happen” (De Interpretatione, 18b17–18b25).1 Aristotle identified the tension between our notions of truth, possibility, and time. Every solution generates some substantial conceptual costs (otherwise, the issue would not have been so fervently disputed for over almost two and a half millennia in both European and Arab philosophy). In fact, it is not even entirely clear what was Aris- totle’s own reaction to the dialectical situation he outlined, since his comments are vague enough to allow mutually exclusive interpretations. This work addresses Aristo- tle’s problem yet again, this time in the form it assumed in the modern temporal logic. Specifically, in the context of the model of branching time. To get a better taste of Aristotle’s puzzle, let us use a simple example. The Greeks triremes were berthed at the harbor of Salamis. The captains were awaiting the orders, ready to fight the approaching Persian fleet. As Herodotus re- 1Throughout my dissertation, I use the translations of Aristotle’s work contained in (Aristotle, 1991). 1 CHAPTER 1. INTRODUCTION counts, Queen Artemisia of Caria advised emperor Xerxes against attacking the Greeks at Salamis. As she had provided very good reasons, she could have swayed him in fa- vor of another tactics. At that point, the faith of the naval campaign was unsettled. Had Xerxes listened to Artemisia, there would not have been a sea battle, had he rejected her advice, there would have been one. At the same time, the Greek leaders—Themistocles and Eurybiades—had observed the maneuvers of the Persian fleet from a hill nearby Salamis and Themistocles said to Eurybiadeds, “There will be a sea battle tomorrow.” Let me depict the story with a very simple, branching model: no sea battle m3 m2 sea battle There will be a sea m0 battle tomorrow m1 Xerxes and Artemisia Confer Point m0 represents the moment at which the future sea battle is being decided. The right “branch” of the treelike model represents the continuation of this moment which holds the sea battle, the left branch represents the opposite continuation. At moment m1, Themistocles says, “There will be a sea battle tomorrow.” The Aristotelian question is: is the sentence true? Even a quick glance on the model suffices to recognize that the issue is highly prob- lematic. All that we have at our disposal is two alternative continuations of moment m1, which seem to be of little help if we want to answer the Aristotelian question. Based on the picture alone, one can easily argue that the sentence might be true and it might be false, but there is