NEGATION IN CONTEXT Michael De A Thesis Submitted for the Degree of PhD at the University of St Andrews 2011 Full metadata for this item is available in Research@StAndrews:FullText at: http://research-repository.st-andrews.ac.uk/ Please use this identifier to cite or link to this item: http://hdl.handle.net/10023/3031 This item is protected by original copyright This item is licensed under a Creative Commons License Negation in Context Michael De Arch´ePhilosophical Research Centre University of St Andrews Scotland [email protected] May 2011 Thesis submitted for the degree of PhD in Philosophy 1. Candidate’s declarations: I, Michael De, hereby certify that this thesis, which is approximately 60,000 words in length, has been written by me, that it is the record of work carried out by me and that it has not been submitted in any previous application for a higher degree. I was admitted as a research student in October 2007 and as a candidate for the degree of Ph.D. in March 2011; the higher study for which this is a record was carried out in the University of St Andrews between 2007 and 2011. Date 22 May 2011 signature of candidate 2. Supervisor’s declaration: I hereby certify that the candidate has fulfilled the conditions of the Resolution and Regulations appropriate for the degree of Ph.D. in the University of St Andrews and that the candidate is qualified to submit this thesis in application for that degree. Date 23/5/11 signature of supervisor 3. Permission for electronic publication: (to be signed by both candidate and supervisor) In submitting this thesis to the University of St Andrews we understand that we are giving permission for it to be made available for use in accordance with the regulations of the University Library for the time being in force, subject to any copyright vested in the work not being affected thereby. We also understand that the title and the abstract will be published, and that a copy of the work may be made and supplied to any bona fide library or research worker, that my thesis will be electronically accessible for personal or research use unless exempt by award of an embargo as requested below, and that the library has the right to migrate my thesis into new electronic forms as required to ensure continued access to the thesis. We have obtained any third-party copyright permissions that may be required in order to allow such access and migration, or have requested the appropriate embargo below. The following is an agreed request by candidate and supervisor regarding the electronic publication of this thesis: Access to all of printed copy but embargo of all of electronic publication of thesis for a period of two years on the following grounds: publication would preclude future publication. Date 23/5/11 signature of candidate signature of supervisor Contents Contents ii 1 Negation in Context 1 1.1 Introduction ............................. 1 1.2 What is negation? ......................... 5 1.3 Constructivism and empirical negation .............. 7 1.4 Consequence relations for speech acts . 25 1.5 Negation, truth and falsity ..................... 30 1.6 The definability of negation as impossibility . 33 2 What is negation? 37 2.1 Introduction ............................. 37 2.2 Syntactic characterizations of negation . 41 2.3 Semantic characterizations of negation . 46 2.4 Final remarks ............................ 60 3 Constructivism and empirical negation 63 3.1 Introduction ............................. 64 3.2 Constructivist objections to classical (and like) negations . 69 3.3 The semantics of fixed negation . 76 3.4 Validities and invalidities ..................... 80 3.5 Tableaux .............................. 84 3.6 Completeness ............................ 86 ii CONTENTS iii 3.7 Final remarks ............................ 89 4 Consequence relations for speech acts and attitudes 93 4.1 Introduction ............................. 94 4.2 Denial ................................ 97 4.3 Doubt ................................ 103 4.4 Doubt in Kleene logic . 107 4.5 Doubt in modal logic . 109 4.6 Consequence as 1-preservation . 111 4.7 Final remarks ............................ 114 5 On the nature of truth values 117 5.1 Introduction ............................. 117 5.2 Frege on truth values . 118 5.3 Every logic is gappy . 121 5.4 One-valued semantics . 124 5.5 Truthmaking ............................ 128 5.6 Maximalism ............................. 133 5.7 Positive propositions . 140 5.8 Final remarks ............................ 152 6 The definability of negation as impossibility 155 6.1 Preliminaries ............................ 157 6.2 Negation as impossibility . 159 6.3 Extended modal languages . 161 6.4 Final remarks ............................ 177 7 Final words 179 Bibliography 181 Only few wind up where they expected The rest never know where they are headed 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 JeffPelletier, 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¨orn Brodowski, Herman v vi CONTENTS 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 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. 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 Chapter 1, which puts each the chapters to follow into context, highlights a central problem negation poses to a certain area of philosophy. Chapter 2 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. Chapter 3 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. Chapter 4 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. Chapter 5 provides an account of the nature of truth values regarded as objects. It is argued that only truth vii viii CONTENTS 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. Chapter 6 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.