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26On the Adequacy of a Substructural Logic for Mathematics and SciencePhilosophical Quarterly 72 (4): 1002-1018. 2022.Williamson argues for the contention that substructural logics are ‘ill-suited to acting as background logics for science’. That contention, if true, would be very important, but it is refutable, given what is already known about certain substructural logics. Classical Core Logic is a substructural logic, for it eschews the structural rules of Thinning and Cut and has Reflexivity as its only structural rule. Yet it suffices for classical mathematics, and it furnishes all the proofs and disproofs…Read more
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26Revamping the restriction strategyIn Joe Salerno (ed.), New Essays on the Knowability Paradox, Oxford University Press. 2009.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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25On Tarski’s Axiomatization of MereologyStudia Logica 107 (6): 1089-1102. 2019.It is shown how Tarski’s 1929 axiomatization of mereology secures the reflexivity of the ‘part of’ relation. This is done with a fusion-abstraction principle that is constructively weaker than that of Tarski; and by means of constructive and relevant reasoning throughout. We place a premium on complete formal rigor of proof. Every step of reasoning is an application of a primitive rule; and the natural deductions themselves can be checked effectively for formal correctness.
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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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22Two problems for evolutionary epistemology: Psychic reality and the emergence of normsRatio 1 (1): 47-63. 1988.
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21Grandy Richard E.. Advanced logic for applications. Synthese library, vol. 110. D. Reidel Publishing Company, Dordrecht and Boston 1977, xiii + 168 pp (review)Journal of Symbolic Logic 47 (3): 714-716. 1982.
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19Foundational Adventures: essays in honour of Harvey M. Friedman (edited book)College Publications. 2014.This volume is a tribute by his peers, and by younger scholars of the next generation, to Harvey M. Friedman, perhaps the most profound foundationalist since Kurt Godel. Friedman's researches, beginning precociously in his mid-teens, have fundamentally shaped our contemporary understanding of set theory, recursion theory, model theory, proof theory and metamathematics. His achievements in concept formation and theory formulation have also renewed the standard set by Godel and Alfred Tarski for t…Read more
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19The Logic of NumberOxford University Press. 2022.This book develops Tennant's Natural Logicist account of the foundations of the natural, rational, and real numbers. Tennant uses this framework to distinguish the logical from the intuitive aspects of the basic elements of arithmetic.
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18The Logic for Mathematics without Ex Falso QuodlibetPhilosophia Mathematica. forthcoming.Informally rigorous mathematical reasoning is relevant. So too should be the premises to the conclusions of formal proofs that regiment it. The rule Ex Falso Quodlibet induces spectacular irrelevance. We therefore drop it. The resulting systems of Core Logic C and Classical Core Logic C+ can formalize all the informally rigorous reasoning in constructive and classical mathematics respectively. We effect a revised match-up between deducibility in Classical Core Logic and a new notion of relevant …Read more
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16Peacocke argues for a ‘generalized rationalism’, holding that ‘all entitlement has a fundamentally a priori component.’ (2) But his rationalism ‘differs from those of Frege and Gödel, just as theirs differ from that of Leibniz.’ He requires both substantive theories of intentional content and of understanding, and systematic formal theories of referential semantics and truth. We need an externalist theory of content: ‘Only mental states with externally individuated contents can make judgements a…Read more
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16Theories, concepts and rationality in an evolutionary account of scienceBiology and Philosophy 3 (2): 224-231. 1988.
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16Deflationism 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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14Gp’s lpIn Can Başkent & Thomas Macaulay Ferguson (eds.), Graham Priest on Dialetheism and Paraconsistency, Springer Verlag. pp. 481-506. 2019.This study takes a careful inferentialist look at Graham Priest’s Logic of Paradox. I conclude that it is sorely in need of a proof-system that could furnish formal proofs that would regiment faithfully the “naïve logical” reasoning that could be undertaken by a rational thinker within LP.
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14From Logic to Philosophies (review)British Journal for the Philosophy of Science 32 (3): 287-301. 1981.
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12Collected Papers on Epistemology, Philosophy of Science and History of Philosophy, Vols. I and IIPhilosophical Quarterly 29 (116): 270. 1979.
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12AutologicEdinburgh University Press. 1992.Shows how to program on a computer (in Prolog) the effective skills taught in introductory and intermediate logic courses. The topics include the relevance of relevance, representing formulae and proofs, avoiding loops and blind alleys, and other aspects. Of interest to computational logicians, proof-theorists, cognitive scientists, and workers in artificial intelligence. Distributed by Columbia U. Press. Annotation copyright by Book News, Inc., Portland, OR
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12The future with cloningIn James H. Fetzer (ed.), Consciousness Evolving, John Benjamins. pp. 34--223. 2002.
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11Review: Richard E. Grandy, Advanced Logic for Applications (review)Journal of Symbolic Logic 47 (3): 714-716. 1982.
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11Introduction to special issue: Carnap and Some ContemporariesPhilosophia Mathematica 16 (1): 1-3. 2008.
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