MICHAEL DE Negation in Context Thesis submitted for the degree of PhD in Philosophy University of St Andrews, Scotland, May 2011 Only few wind up where they expected The rest never know where they are headed Abstract The present essay includes six thematically connected papers on negation in the areas of the philosophy of logic, philosophical logic and metaphysics. Each of the chapters besides Chapter1 , which puts each the chapters to follow into context, highlights a central problem negation poses to a certain area of philosophy. Chapter2 discusses the problem of logical revisionism and whether there is any room for genuine disagreement, and hence shared mean- ing, between the classicist and deviant's respective uses of `not'. If there is not, revision is impossible. I argue that revision is indeed possible and pro- vide an account of negation as contradictoriness according to which a number of alleged negations are declared genuine. Among them are the negations of FDE (First-Degree Entailment) and a wide family of other relevant logics, LP (Priest's dialetheic \Logic of Paradox"), Kleene weak and strong 3-valued logics with either \exclusion" or \choice" negation, and intuitionistic logic. Chapter3 discusses the problem of furnishing intuitionistic logic with an em- pirical negation for adequately expressing claims of the form `A is undecided at present' or `A may never be decided' the latter of which has been argued to be intuitionistically inconsistent. Chapter4 highlights the importance of various notions of consequence-as-s-preservation where s may be falsity (ver- sus untruth), indeterminacy or some other semantic (or \algebraic") value, in formulating rationality constraints on speech acts and propositional attitudes such as rejection, denial and dubitability. Chapter5 provides an account of the nature of truth values regarded as objects. It is argued that only truth exists as the maximal truthmaker. The consequences this has for semantics representationally construed are considered and it is argued that every logic, from classical to non-classical, is gappy. Moreover, a truthmaker theory is developed whereby only positive truths, an account of which is also developed therein, have truthmakers. Chapter6 investigates the definability of negation as \absolute" impossibility, i.e. where the notion of necessity or possibility in question corresponds to the global modality. The modality is not readily de- finable in the usual Kripkean languages and so neither is impossibility taken in the broadest sense. The languages considered here include one with counterfac- tual operators and propositional quantification and another bimodal language with a modality and its complementary. Among the definability results we give some preservation and translation results as well. iii Acknowledgements I remember vibrantly my first years as an undergraduate in philosophy. Since philosophy was not offered as part of my high school curriculum, this was my first taste of the real thing. How sweet it was. During that time I was influenced by two people who were responsible for steering me in the direction I lie today. Those two incisive philosophers are David DeVidi and Tim Kenyon. David enthusiastically introduced me to the wonderful world of logic. Tim acted as my Honors thesis advisor and we met weekly, early, and even sometimes on weekends. I recall the innumerable hours I spent in the library preparing for those meetings. At the time I thought I was doing serious research. How I've learned! There are two other incisive philosophers to whom I would like to express my deepest gratitude. The first of those is Jeff Pelletier, my Masters advisor. His continued support and guidance through the beginnings of my research career had a significant impact on me as both a philosopher and a person. The second is Stephen Read, my doctoral thesis advisor, who served to both inspire and guide. Our weekly meetings in the Arch´eLogic Group, The Foundations of Logical Consequence and Medieval Logic Group seminars were highly stim- ulating and enjoyable. Much of his influence can be seen in nearly every page of this essay. During my time at the Arch´ePhilosophical Research Centre, I had the plea- sure of countless enjoyable exchanges with some very bright minds. Among those, I would like to thank Ralf Bader, Derek Ball, Bj¨ornBrodowski, Herman Cappelen, Colin Caret, Yuri Cath, Paul Dimmock, Dylan Dodd, Roy Dyck- hoff, Philip Ebert, Noah Friedman-Biglin, Patrick Greenough, Katherine Haw- ley, Andri Hj´almarsson,Thomas Hodgson, Torfinn Huvenes, Ole Hjortland, Jonathan Ichikawa, Frederique Janssen-Lauret, Dirk Kindermann, Ira Kiourti, Toby Meadows, Julien Murzi, Dilip Ninan, Andrea Onofri, Walter Pedriali, Laura Porro, Graham Priest, David Ripley, Marcus Rossberg, Gil Sagi, An- ders Schoubye, Daniele Sgaravatti, Stewart Shapiro, Martin Smith, Andreas Stokke, Margot Strohminger, Crispin Wright, and Elia Zardini. I would like to express special gratitude to Julia Langkau who, among endless other generosities, proof read a draft of the thesis and quarreled with me uncompromisingly over points of style and presentation. Thanks is owed to the Institute for Logic, Language and Computation at the University of Amsterdam for their warm hospitality during the winter months of 2009 and, in particular, to Johan van Benthem, Catarina Dutilh Novaes, Dick de Jongh, Alexandru Morcoi, Sarah Uckelman, Jouko V¨a¨an¨nen,and Yde Venema for making the visit most highly enjoyable. Thanks also to the St An- drews/Stirling Philosophy Graduate Program and the Philosophical Quarterly iv v who, in the first place, made this visit possible. Many others have had a positive impact on me in other ways, and a final thanks is owed to them as well. Contents Contents vi 1 Negation in Context1 1.1 Introduction............................. 1 1.2 What is negation? ......................... 4 1.3 Constructivism and empirical negation.............. 5 1.4 Consequence relations for speech acts............... 16 1.5 Negation, truth and falsity..................... 19 1.6 The definability of negation as impossibility........... 21 2 What is negation? 24 2.1 Introduction............................. 24 2.2 Syntactic characterizations of negation.............. 26 2.3 Semantic characterizations of negation.............. 30 2.4 Final remarks............................ 39 3 Constructivism and empirical negation 41 3.1 Introduction............................. 41 3.2 Constructivist objections to classical (and like) negations . 45 3.3 The semantics of fixed negation.................. 49 3.4 Validities and invalidities ..................... 52 3.5 Tableaux .............................. 54 3.6 Completeness............................ 56 3.7 Final remarks............................ 58 4 Consequence relations for speech acts and attitudes 60 4.1 Introduction............................. 60 4.2 Denial................................ 62 4.3 Doubt................................ 66 4.4 Doubt in Kleene logic ....................... 69 4.5 Doubt in modal logic........................ 70 4.6 Consequence as 1-preservation................... 71 4.7 Final remarks............................ 74 5 On the nature of truth values 75 5.1 Introduction............................. 75 5.2 Frege on truth values........................ 76 5.3 Every logic is gappy ........................ 78 5.4 One-valued semantics ....................... 80 vi CONTENTS vii 5.5 Truthmaking ............................ 82 5.6 Maximalism............................. 86 5.7 Positive propositions........................ 90 5.8 Final remarks............................ 98 6 The definability of negation as impossibility 99 6.1 Preliminaries ............................ 100 6.2 Negation as impossibility ..................... 101 6.3 Extended modal languages..................... 103 6.4 Final remarks............................ 113 7 Final words 114 Bibliography 116 Chapter 1 Negation in Context Abstract This chapter sets up the background for the chapters to follow. It includes historical references, puts the problems discussed in the forthcoming chapters into context, and ties together a number of the main themes and problems. 1.1 Introduction Plato's beard and contemporary issues When one takes a close look at \new" problems in philosophy one may often find that they are really old problems in modern guise. This is not a bad thing. The old problems that are truly central to philosophy take on new form and become amenable to solutions involving a host of new tools and insights. There may be a lot to learn by viewing the same problem in different contexts. It is important, however, that one sees the connection between the old and the new so that an appreciation of the insights garnered centuries, indeed thousands, of years ago may likewise be appreciated in modern light. Some of the problems that I have in mind concern negation and falsity. For instance, Plato's discussion (in the Sophist, Theaetetus, and Parmenides) of the problem of speaking about that which is not, a problem labeled Plato's beard by Quine in [W. 61]1, recurred in numerous forms throughout history. The problem was arguably the primary motivation of Russell's theory of de- scriptions, a theory which provided a more ontologically amenable solution to 1It is claimed, e.g. by Read [Rea94, ch. 5], that Quine actually used the label `Plato's beard' to denote what he thought was Plato's solution to what I've called the problem. But it's not clear whether Quine meant the problem or a particular solution.