Natural Logicism via the Logic of Orderly Pairing

  • Neil Tennant
Part of the Synthese Library book series (SYLI, volume 341)


The 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. These outstanding axioms need to be derived in a way fully in keeping with the spirit and the letter of Frege’s logicism and his doctrine of definition. To that end this study develops a logic, in the Gentzen-Prawitz style of natural deduction, for the operation of orderly pairing. The logic is an extension of free first-order logic with identity. Orderly pairing is treated as a primitive. No notion of set is presupposed, nor any set-theoretic notion of membership. The formation of ordered pairs, and the two projection operations yielding their left and right coordinates, form a coeval family of logical notions. The challenge is to furnish them with introduction and elimination rules that capture their exact meanings, and no more. Orderly pairing as a logical primitive is then used in order to introduce addition and multiplication in a conceptually satisfying way within a constructive logicist theory of the natural numbers. Because of its reliance, throughout, on sense-constituting rules of natural deduction, the completed account can be described as ‘natural logicism’.


Cardinal Number Natural Deduction Elimination Rule Introduction Rule Free Logic 
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  1. 1.
    John P. Burgess. Fixing Frege. Princeton University Press, Princeton and Oxford, 2005.zbMATHGoogle Scholar
  2. 2.
    Georg Cantor. Grundlagen einer allgemeinen Mannigfaltigkeitslehre. Ein mathematisch-philosophischer Versuch in der Lehre des Unendlichen. B. G. Teubner, Leipzig, 1883.Google Scholar
  3. 3.
    Georg Cantor. Beiträge zur Begründung der transfiniten Mengenlehre. Mathematische Annalen, 46:481–512, 1895. English translation by Philip E. B. Jourdain in Contributions to the Founding of the Theory of Transfinite Numbers, 1952, Dover publications (originally Open Court, 1915), pp. 85–136.CrossRefGoogle Scholar
  4. 4.
    Michael Dummett. Frege: Philosophy of Mathematics. Harvard University Press, Cambridge, Massachusetts, 1991.Google Scholar
  5. 5.
    Gottlob Frege. Grundgesetze der Arithmetik. I. Band. Georg Olms Verlagsbuchhandlung, Hildesheim, 1893; reprinted 1962.Google Scholar
  6. 6.
    Gottlob Frege. Grundgesetze der Arithmetik. II. Band. Georg Olms Verlagsbuchhandlung, Hildesheim, 1903; reprinted 1962.Google Scholar
  7. 7.
    Gottlob Frege. The Basic Laws of Arithmetic: Exposition of the System. Translated and edited, with an Introduction, by Montgomery Furth. University of California Press, Berkeley, Los Angeles, London, 1964.Google Scholar
  8. 8.
    Gerhard Gentzen. Untersuchungen über das logische Schliessen. Mathematische Zeitschrift, I, II:176–210, 405–431, 1934, 1935. Translated as ‘Investigations into Logical Deduction’, in The Collected Papers of Gerhard Gentzen, edited by M. E. Szabo, North-Holland, Amsterdam, 1969, pp. 68–131.Google Scholar
  9. 9.
    Richard Heck, Jr. The Development of Arithmetic in Frege’s Grundgesetze der Arithmetik. Journal of Symbolic Logic, 58:579–601, 1993.zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Richard Heck, Jr. Definition by Induction in Frege’s Grundgesetze. In William Demopoulos, editor, Frege’s Philosophy of Mathematics, Harvard University Press, Cambridge, MA, 1995, pp. 295–333.Google Scholar
  11. 11.
    Akihiro Kanamori. The empty set, the singleton, and the ordered pair. Bulletin of Symbolic Logic, 9:273–288, 2003.zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Casimir Kuratowski. Sur la notion de l’ordre dans la théorie des ensembles. Fundamenta Mathematicae, 2:161–171, 1921.Google Scholar
  13. 13.
    Michael Potter. Reason’s Nearest Kin: Philosophies of Arithmetic from Kant to Carnap. Oxford University Press, Oxford, 2000.zbMATHGoogle Scholar
  14. 14.
    Dag Prawitz. Natural Deduction: A Proof-Theoretical Study. Almqvist & Wiksell, Stockholm, 1965.Google Scholar
  15. 15.
    Ian Rumfitt. Frege’s logicism. Proceedings of the Aristotelian Society, Supplementary Volume, 73:151–180, 1999.Google Scholar
  16. 16.
    Neil Tennant. Natural Logic. Edinburgh University Press, Edinburgh, 1978.zbMATHGoogle Scholar
  17. 17.
    Neil Tennant. Anti-Realism and Logic: Truth as Eternal. Clarendon Library of Logic and Philosophy, Oxford University Press, USA, 1987.Google Scholar
  18. 18.
    Neil Tennant. The Taming of the True. Oxford University Press, Oxford, 1997.zbMATHGoogle Scholar
  19. 19.
    Neil Tennant. A general theory of abstraction operators. The Philosophical Quarterly, 54(214):105–133, 2004.CrossRefMathSciNetGoogle Scholar
  20. 20.
    Crispin Wright. Frege’s Conception of Numbers as Objects. Aberdeen University Press, Aberdeen, 1983.zbMATHGoogle Scholar

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© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  • Neil Tennant
    • 1
  1. 1.Department of PhilosophyThe Ohio State UniversityColumbus

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