Synthese

, Volume 121, Issue 3, pp 309–328 | Cite as

Consistent Fragments of Grundgesetze and the Existence of Non-Logical Objects

  • Kai F. Wehmeier

Abstract

In this paper, I consider two curious subsystems ofFrege's Grundgesetze der Arithmetik: Richard Heck's predicative fragment H, consisting of schema V together with predicative second-order comprehension (in a language containing a syntactical abstraction operator), and a theory TΔ in monadic second-order logic, consisting of axiom V and Δ11-comprehension (in a language containing anabstraction function). I provide a consistency proof for the latter theory, thereby refuting a version of a conjecture by Heck. It is shown that both Heck and TΔ prove the existence of infinitely many non-logical objects (TΔ deriving,moreover, the nonexistence of the value-range concept). Some implications concerning the interpretation of Frege's proof of referentiality and the possibility of classifying any of these subsystems as logicist are discussed. Finally, I explore the relation of TΔ toCantor's theorem which is somewhat surprising.

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REFERENCES

  1. Boolos, G.: 1987,‘The Consistency of Frege's Foundations of Arithmetic’, in J. J. Thomson (ed.), On Being and Saying: Essays for Richard Cartwright,MIT Press, Cambridge,MA, pp. 3–20.Google Scholar
  2. Cocchiarella, N. B.: 1985,‘Frege's Double Correlation Thesis and Quine's Set Theories NF and ML’, Journal of Philosophical Logic 14, 1–39.Google Scholar
  3. Cocchiarella, N. B.: 1986, Logical Investigations of Predication Theory and the Problem of Universals, Bibliopolis, Naples.Google Scholar
  4. Cocchiarella, N. B.: 1992,‘Cantor's Power-Set Theorem versus Frege's Double-Correlation Thesis’, History and Philosophy of Logic 13, 179–201.Google Scholar
  5. Frege, G.: 1893, Grundgesetze der Arithmetik I, Hermann Pohle, Jena.Google Scholar
  6. Frege, G.: 1976, Wissenschaftlicher Briefwechsel, G. Gabriel, H. Hermes, F. Kambartel, C. Thiel, and A. Veraart (eds.), Felix Meiner, Hamburg.Google Scholar
  7. Heck, R.: 1996,‘The Consistency of Predicative Fragments of Frege's Grundgesetze der Arithmetik’, History and Philosophy of Logic 17, 209–220.Google Scholar
  8. Heck, R.: 1997a,‘Grundgesetze der Arithmetik I §§ 29–32’, Notre Dame Journal of Formal Logic 38, 437–474.Google Scholar
  9. Heck, R.: 1997b,‘The Julius Caesar Objection’, in R. Heck (ed.), Language, Thought, and Logic, Oxford University Press, Oxford, pp. 273–308.Google Scholar
  10. Heck, R.: 1997c,‘Frege and Semantics’, in T. Ricketts (ed.), forthcoming, The Cambridge Companion to Frege, preprint, 39 pp.Google Scholar
  11. Martin, E.: 1982,‘Referentiality in Frege's Grundgesetze’, History and Philosophy of Logic 3, 151–164.Google Scholar
  12. Parsons, T.: 1987,‘On the Consistency of the First-Order Portion of Frege's Logical System’, Notre Dame Journal of Formal Logic 28, 161–168.Google Scholar
  13. Schirn,M.: 1996,‘Introduction: Frege on the Foundations of Arithmetic and Geometry’, in M. Schirn (ed.), Frege: Importance and Legacy, de Gruyter, Berlin/New York, pp. 1–42.Google Scholar
  14. Schroeder-Heister, P.: 1987,‘A Model-Theoretic Reconstruction of Frege's Permutation Argument’, Notre Dame Journal of Formal Logic 28, 69–79.Google Scholar

Copyright information

© Kluwer Academic Publishers 1999

Authors and Affiliations

  • Kai F. Wehmeier
    • 1
  1. 1.Philosophical InstituteRijksuniversiteit LeidenLeidenThe Netherlands

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