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26Revamping the restriction strategyIn Joe Salerno (ed.), New Essays on the Knowability Paradox, Oxford University Press. 2008.This study continues the anti-realist’s quest for a principled way to avoid Fitch’s paradox. It is proposed that the Cartesian restriction on the anti-realist’s knowability principle ‘ϕ, therefore 3Kϕ’ should be formulated as a consistency requirement not on the premise ϕ of an application of the rule, but rather on the set of assumptions on which the relevant occurrence of ϕ depends. It is stressed, by reference to illustrative proofs, how important it is to have proofs in normal form before ap…Read more
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18Deflationism and the Godel PhenomenaMind 111 (443): 551-582. 2002.Any consistent and sufficiently strong system of first-order formal arithmetic fails to decide some independent Gödel sentence. We examine consistent first-order extensions of such systems. Our purpose is to discover what is minimally required by way of such extension in order to be able to prove the Gödel sentence in a non-trivial fashion. The extended methods of formal proof must capture the essentials of the so-called 'semantical argument' for the truth of the Gödel sentence. We are concerned…Read more
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La barre de Scheffer dans la logique des séquents et des syllogismesLogique Et Analyse 22 (88): 505. 1979.
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88Anti-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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94Paradoxes 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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35Games some people would have all of us play: A critical study of J. Hintikka, The Principles of Mathematics Revisited (review)Philosophia Mathematica 6 (1): 226-241. 1998.
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4Written for any readers interested in better harnessing philosophy's real value, this book covers a broad range of fundamental philosophical problems and certain intellectual techniques for addressing those problems. In Introducing Philosophy: God, Mind, World, and Logic, Neil Tennant helps any student in pursuit of a 'big picture' to think independently, question received dogma, and analyse problems incisively. It also connects philosophy to other areas of study at the university, enabling all …Read more
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51Game theory and conventiontNordic Journal of Philosophical Logic 6 (1): 3-19. 2001.This paper rebuts criticisms by Hintikka of the author's account of game-theoretic semantics for classical logic. At issue are (i) the role of the axiom of choice in proving the equivalence of the game-theoretic account with the standard truth-theoretic account; (ii) the alleged need for quantification over strategies when providing a game-theoretic semantics; and (iii) the role of Tarski's Convention T. As a result of the ideas marshalled in response to Hintikka, the author puts forward a new c…Read more
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109Cut for core logicReview of Symbolic Logic 5 (3): 450-479. 2012.The motivation for Core Logic is explained. Its system of proof is set out. It is then shown that, although the system has no Cut rule, its relation of deducibility obeys Cut with epistemic gain.
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Bob Hale and Crispin Wright. The reason's proper study: Essays towards a neo-Fregean philosophy of mathematicsPhilosophia Mathematica 11 (2): 226-240. 2003.
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19The Logical Structure of Scientific Explanation and Prediction: Planetary Orbits in a Sun’s Gravitational FieldStudia Logica 95 (1-2): 207-232. 2010.We present a logically detailed case-study of explanation and prediction in Newtonian mechanics. The case in question is that of a planet's elliptical orbit in the Sun's gravitational field. Care is taken to distinguish the respective contributions of the mathematics that is being applied, and of the empirical hypotheses that receive a mathematical formulation. This enables one to appreciate how in this case the overall logical structure of scientific explanation and prediction is exactly in acc…Read more
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23The aim here is to describe how to complete the constructive logicist program, in the author’s book Anti-Realism and Logic, of deriving all the Peano-Dedekind postulates for arithmetic within a theory of natural numbers that also accounts for their applicability in counting finite collections of objects. The axioms still to be derived are those for addition and multiplication. Frege did not derive them in a fully explicit, conceptually illuminating way. Nor has any neo-Fregean done so.
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16A 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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159The Emperor’s New ConceptsNoûs 36 (s16): 345-377. 2002.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 abstracted to give the numerosity of a predicate. There are, however, logical mistakes in Peacocke
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83Logic, 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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8Review: From Logic to Philosophies (review)British Journal for the Philosophy of Science 32 (3). 1981.
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138Deflationism and the gödel phenomenaMind 111 (443): 551-582. 2002.consistent and sufficiently strong system of first-order formal arithmetic fails to decide some independent Gödel sentence. We examine consistent first-order extensions of such systems. Our purpose is to discover what is minimally required by way of such extension in order to be able to prove the Gödel sentence in a non-trivial fashion. The extended methods of formal proof must capture the essentials of the so-called ‘semantical argument’ for the truth of the Gödel sentence. We are concerned to …Read more
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1Jaakko Hintikka, The Principles of Mathematics RevisitedPhilosophia Mathematica 6 (1): 90-115. 1998.
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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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110Parts, classes and Parts of Classes : an anti-realist reading of Lewisian mereologySynthese 190 (4): 709-742. 2013.This study is in two parts. In the first part, various important principles of classical extensional mereology are derived on the basis of a nice axiomatization involving ‘part of’ and fusion. All results are proved here with full Fregean rigor. They are chosen because they are needed for the second part. In the second part, this natural-deduction framework is used in order to regiment David Lewis’s justification of his Division Thesis, which features prominently in his combination of mereology …Read more
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253Review of C. S. Jenkins, Grounding Concepts: An Empirical Basis for Arithmetical Knowledge (review)Philosophia Mathematica 18 (3): 360-367. 2010.This book is written so as to be ‘accessible to philosophers without a mathematical background’. The reviewer can assure the reader that this aim is achieved, even if only by focusing throughout on just one example of an arithmetical truth, namely ‘7+5=12’. This example’s familiarity will be reassuring; but its loneliness in this regard will not. Quantified propositions — even propositions of Goldbach type — are below the author’s radar.The author offers ‘a new kind of arithmetical epistemology’…Read more
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98Inferentialism is explained as an attempt to provide an account of meaning that is more sensitive (than the tradition of truth-conditional theorizing deriving from Tarski and Davidson) to what is learned when one masters meanings.
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