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93Paradoxes of pure curiosityTheory and Decision 38 (3): 321-330. 1995.We consider how a rational decision theorist would justify committing resources to an investigation designed to satisfy pure curiosity. We derive a strange result about the need to be completely open-minded about the outcome
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91A general theory of abstraction operatorsPhilosophical Quarterly 54 (214): 105-133. 2004.I present a general theory of abstraction operators which treats them as variable-binding term- forming operators, and provides a reasonably uniform treatment for definite descriptions, set abstracts, natural number abstraction, and real number abstraction. This minimizing, extensional and relational theory reveals a striking similarity between definite descriptions and set abstracts, and provides a clear rationale for the claim that there is a logic of sets (which is ontologically non- committa…Read more
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87Anti-realist aporiasMind 109 (436): 825--854. 2000.Using a quantified propositional logic involving the operators it is known that and it is possible to know that, we formalize various interesting philosophical claims involved in the realism debate. We set out inferential rules for the epistemic modalities, ranging from ones that are obviously analytic, to ones that are epistemologically more substantive or even controversial. Then we investigate various aporias for the realism debate. These are constructively inconsistent triads of claims from …Read more
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86A note on the irrelevance of probabilistic irrelevanceAnalysis 66 (1): 32-35. 2006.In his book Bayes or Bust?, John Earman (1992: 63–65) seeks to set out the Bayesian reasoning that would vindicate the pre-theoretic intuition that a theory receives confirmation from having its observational predictions borne out by experience.
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82Intuitionistic mathematics does not needex falso quodlibetTopoi 13 (2): 127-133. 1994.We define a system IR of first-order intuitionistic relevant logic. We show that intuitionistic mathematics (on the assumption that it is consistent) can be relevantized, by virtue of the following metatheorem: any intuitionistic proof of A from a setX of premisses can be converted into a proof in IR of eitherA or absurdity from some subset ofX. Thus IR establishes the same inconsistencies and theorems as intuitionistic logic, and allows one to prove every intuitionistic consequence of any consi…Read more
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81Deductive versus Expressive Power: A Pre-Godelian PredicamentJournal of Philosophy 97 (5): 257. 2000.
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77On Turing machines knowing their own gödel-sentencesPhilosophia Mathematica 9 (1): 72-79. 2001.Storrs McCall appeals to a particular true but improvable sentence of formal arithmetic to argue, by appeal to its irrefutability, that human minds transcend Turing machines. Metamathematical oversights in McCall's discussion of the Godel phenomena, however, render invalid his philosophical argument for this transcendentalist conclusion
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75Existence and Identity in Free Logic: A Problem for Inferentialism?Mind 116 (464): 1055-1078. 2007.Peter Milne (2007) poses two challenges to the inferential theorist of meaning. This study responds to both. First, it argues that the method of natural deduction idealizes the essential details of correct informal deductive reasoning. Secondly, it explains how rules of inference in free logic can determine unique senses for the existential quantifier and the identity predicate. The final part of the investigation brings out an underlying order in a basic family of free logics
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71Logic, Mathematics, and the A Priori, Part I: A Problem for RealismPhilosophia Mathematica 22 (3): 308-320. 2014.This is Part I of a two-part study of the foundations of mathematics through the lenses of (i) apriority and analyticity, and (ii) the resources supplied by Core Logic. Here we explain what is meant by apriority, as the notion applies to knowledge and possibly also to truths in general. We distinguish grounds for knowledge from grounds of truth, in light of our recent work on truthmakers. We then examine the role of apriority in the realism/anti-realism debate. We raise a hitherto unnoticed prob…Read more
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71Normalizability, cut eliminability and paradoxSynthese 199 (Suppl 3): 597-616. 2016.This is a reply to the considerations advanced by Schroeder-Heister and Tranchini as prima facie problematic for the proof-theoretic criterion of paradoxicality, as originally presented in Tennant and subsequently amended in Tennant. Countering these considerations lends new importance to the parallelized forms of elimination rules in natural deduction.
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71Discussion. Changing the theory of theory change: reply to my criticsBritish Journal for the Philosophy of Science 48 (4): 569-586. 1997.‘Changing the Theory of Theory Change: Towards a Computational Approach’ (Tennant [1994]; henceforth CTTC) claimed that the AGM postulate of recovery is false, and that AGM contractions of theories can be more than minimally mutilating. It also described an alternative, computational method for contracting theories, called the Staining Algorithm. Makinson [1995] and Hansson and Rott [1995] criticized CTTC's arguments against AGM-theory, and its specific proposals for an alternative, computationa…Read more
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70Is every truth knowable? Reply to WilliamsonRatio 14 (3). 2001.This paper addresses an objection raised by Timothy Williamson to the ‘restriction strategy’ that I proposed, in The Taming of The True, in order to deal with the Fitch paradox. Williamson provides a new version of a Fitch-style argument that purports to show that even the restricted principle of knowability suffers the same fate as the unrestricted one. I show here that the new argument is fallacious. The source of the fallacy is a misunderstanding of the condition used in stating the restricte…Read more
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69Deflationism and the Godel Phenomena: Reply to CieslinskiMind 119 (474): 437-450. 2010.I clarify how the requirement of conservative extension features in the thinking of various deflationists, and how this relates to another litmus claim, that the truth-predicate stands for a real, substantial property. I discuss how the deflationist can accommodate the result, to which Cieslinski draws attention, that non-conservativeness attends even the generalization that all logical theorems in the language of arithmetic are true. Finally I provide a four-fold categorization of various forms…Read more
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68Frege's content-principle and relevant deducibilityJournal of Philosophical Logic 32 (3): 245-258. 2003.Given the harmony principle for logical operators, compositionality ought to ensure that harmony should obtain at the level of whole contents. That is, the role of a content qua premise ought to be balanced exactly by its role as a conclusion. Frege's contextual definition of propositional content happens to exploit this balance, and one appeals to the Cut rule to show that the definition is adequate. We show here that Frege's definition remains adequate even when one relevantizes logic by aband…Read more
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68Logic, Mathematics, and the A Priori, Part II: Core Logic as Analytic, and as the Basis for Natural LogicismPhilosophia Mathematica 22 (3): 321-344. 2014.We examine the sense in which logic is a priori, and explain how mathematical theories can be dichotomized non-trivially into analytic and synthetic portions. We argue that Core Logic contains exactly the a-priori-because-analytically-valid deductive principles. We introduce the reader to Core Logic by explaining its relationship to other logical systems, and stating its rules of inference. Important metatheorems about Core Logic are reported, and its important features noted. Core Logic can ser…Read more
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67The Law of Excluded Middle Is Synthetic A Priori, If ValidPhilosophical Topics 24 (1): 205-229. 1996.
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63New Foundations for a Relational Theory of Theory-revisionJournal of Philosophical Logic 35 (5): 489-528. 2006.AGM-theory, named after its founders Carlos Alchourrón, Peter Gärdenfors and David Makinson, is the leading contemporary paradigm in the theory of belief-revision. The theory is reformulated here so as to deal with the central relational notions 'J is a contraction of K with respect to A' and 'J is a revision of K with respect to A'. The new theory is based on a principal-case analysis of the domains of definition of the three main kinds of theory-change (expansion, contraction and revision). Th…Read more
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62IntroductionPhilosophia Mathematica 16 (1): 1-3. 2008.Christopher Peacocke, in A Study of Concepts, motivates his account of possession conditions for concepts by means of an alleged parallel with the conditions under which numbers are abshacted to give the numerosity of a predicate. There are, however, logical mistakes in Peacocke's treatment of numbers, which undermine his intended analogy. Nevertheless Peacocke's account of possession conditions for concepts is not rendered inadequate simply by virtue of being deprived of the intended analogy an…Read more
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61The logical structure of evolutionary explanation and prediction: Darwinism’s fundamental schemaBiology and Philosophy 29 (5): 611-655. 2014.We present a logically detailed case-study of Darwinian evolutionary explanation. Special features of Darwin’s explanatory schema made it an unusual theoretical breakthrough, from the point of view of the philosophy of science. The schema employs no theoretical terms, and puts forward no theoretical hypotheses. Instead, it uses three observational generalizations—Variability, Heritability and Differential Reproduction—along with an innocuous assumption of Causal Efficacy, to derive Adaptive Evol…Read more
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59Recursive Semantics For Knowledge and BeliefThe Monist 60 (3): 419-430. 1977.1. This paper is an informal exposition of a model-theoretic semantics for knowledge and belief set out in full detail else where. Considerations of space and simplicity prevent any recapitulation of tracts of formal definitions. My aim is simply to inform the reader of the alleged existence of one “new direction” in semantics, and to direct him to the original source for its detailed development. I shall explain certain self-imposed limitations on the scope and adequacy conditions of this treat…Read more
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58A New Unified Account of Truth and ParadoxMind 124 (494): 571-605. 2015.I propose an anti-realist account of truth and paradox according to which the logico-semantic paradoxes are not genuine inconsistencies. The ‘global’ proofs of absurdity associated with these paradoxes cannot be brought into normal form. The account combines epistemicism about truth with a proof-theoretic diagnosis of paradoxicality. The aim is to combine a substantive philosophical account of truth with a more rigorous and technical diagnosis of the source of paradox for further consideration b…Read more
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57Ultimate Normal Forms for Parallelized Natural DeductionsLogic Journal of the IGPL 10 (3): 299-337. 2002.The system of natural deduction that originated with Gentzen , and for which Prawitz proved a normalization theorem, is re-cast so that all elimination rules are in parallel form. This enables one to prove a very exigent normalization theorem. The normal forms that it provides have all disjunction-eliminations as low as possible, and have no major premisses for eliminations standing as conclusions of any rules. Normal natural deductions are isomorphic to cut-free, weakening-free sequent proofs. …Read more
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57On Some Mistaken Beliefs About Core Logic and Some Mistaken Core Beliefs About LogicNotre Dame Journal of Formal Logic 59 (4): 559-578. 2018.This is in part a reply to a recent work of Vidal-Rosset, which expresses various mistaken beliefs about Core Logic. Rebutting these leads us further to identify, and argue against, some mistaken core beliefs about logic.
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53Critical Studies / Book Reviews (review)Philosophia Mathematica 6 (1): 90-90. 1998.The over-arching theme is that we can redeem Frege's key philosophical insights concerning (natural and real) numbers and our knowledge of them, despite Russell's famous discovery of paradox in Frege's own theory of classes. That paradox notwithstanding, numbers are still logical objects, in some sense created or generated by methods or principles of abstractionââ¬â which of course cannot be as ambitious as Frege's Basic Law U. These principles not only bring numbers into existence, as it we…Read more
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53Is every truth knowable? Reply to hand and KvanvigAustralasian Journal of Philosophy 79 (1). 2001.This Article does not have an abstract
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50Truth table logic, with a survey of embeddability resultsNotre Dame Journal of Formal Logic 30 (3): 459-484. 1989.Kalrnaric. We set out a system T, consisting of normal proofs constructed by means of elegantly symmetrical introduction and elimination rules. In the system T there are two requirements, called ( ) and ()), on applications of discharge rules. T is sound and complete for Kalmaric arguments. ( ) requires nonvacuous discharge of assumptions; ()) requires that the assumption discharged be the sole one available of highest degree. We then consider a 'Duhemian' extension T*, obtained simply by droppi…Read more
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