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Core Logic: A Conspectus

  • Neil TennantEmail author
Chapter
Part of the Palgrave Innovations in Philosophy book series (PIIP)

Abstract

This chapter presents an ‘absolutist’ view about logic—Core Logic. Core Logic is relevant, in a sense heretofore not satisfactorily explicated. The so-called loss of unrestricted transitivity of deduction in Core Logic brings with it epistemic gain. Core Logic suffices for Intuitionistic Mathematics, Classical Mathematics, the hypothetico-deductive testing of scientific theories against empirical evidence, and the reasoning involved in the logical and semantic paradoxes. Core Logic is the minimal inviolable core of logic that is needed, and suffices, for rational belief revision. Core Logic is the logic of ‘conceptual constitution’: it suffices for neo-logicist derivation of number-theoretic axioms from deeper logical principles governing the operator #xΦ(x) (the number of Φs). Core Logic has the nice property that its natural deductions and sequent proofs are structurally isomorphic, and it is obtained by smoothly generalizing an inferential reading of the truth-tables. There is an automated proof-finder for (propositional) Core Logic, whose decision problem is PSPACE-complete (like that of Intuitionistic Logic). Any classical core proof provides a means of ‘quasi-’effectively transforming any verifications of its premises, relative to a model M, into a verification of its conclusion, relative to M.

Keywords

Logical monism Logical pluralism Logic Classical logic Intuitionistic logic Relevant logics Core logic Classical mathematics Constructive mathematics Proof Natural deduction Sequent calculus 

References

  1. Anderson, A. R., and N. D. Belnap, Jr. 1975. Entailment: The Logic of Relevance and Necessity I. Princeton: Princeton University Press.Google Scholar
  2. Friedman, H., and R.K. Meyer. 1992. Whither Relevant Arithmetic? Journal of Symbolic Logic 7: 824–831.CrossRefGoogle Scholar
  3. Gentzen, G. 1934. Untersuchungen über das logische schliessen. Mathematische Zeitschrift I:176–210. In The American Philosophical Quarterly 1, Translated by M. E. Szabo as ‘Investigations into Logical Deduction’, 228–306, 1964.Google Scholar
  4. Tennant, N. 1996. The Law of Excluded Middle Is Synthetic a Priori, If Valid. Philosophical Topics 24: 205–229.CrossRefGoogle Scholar
  5. ———. 1997. The Taming of the True. Oxford: Oxford University Press.Google Scholar
  6. ———. 2010. Inferential Semantics. In The Force of Argument: Essays in Honor of Timothy Smiley, ed. J. Lear and A. Oliver, 223–257. London: Routledge.Google Scholar
  7. ———. 2012a. Changes in Mind: An Essay on Rational Belief Revision. Oxford: Oxford University Press.CrossRefGoogle Scholar
  8. ———. 2012b. Cut for Core Logic. Review of Symbolic Logic 5 (3): 450–479.CrossRefGoogle Scholar
  9. ———. 2015a. Cut for Classical Core Logic. Review of Symbolic Logic 8 (4): 236–256.CrossRefGoogle Scholar
  10. ———. 2015b. The Relevance of Premises to Conclusions of Core Proofs. Review of Symbolic Logic 8 (4): 743–784.CrossRefGoogle Scholar
  11. ———. 2017. Core Logic. Oxford: Oxford University Press.Google Scholar
  12. ———. 2018. A Logical Theory of Truthmakers and Falsitymakers. In The Oxford Handbook of Truth, ed. M. Glanzberg, 355–393. Oxford: Oxford University Press.Google Scholar
  13. Urquhart, A. 1984. The Undecidability of Entailment and Relevant Implication. Journal of Symbolic Logic 49: 1059–1073.CrossRefGoogle Scholar

Copyright information

© The Author(s) 2018

Authors and Affiliations

  1. 1.Department of PhilosophyOhio State UniversityColumbusUSA

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