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 Δ1 1-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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
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.
Cocchiarella, N. B.: 1985,‘Frege's Double Correlation Thesis and Quine's Set Theories NF and ML’, Journal of Philosophical Logic 14, 1–39.
Cocchiarella, N. B.: 1986, Logical Investigations of Predication Theory and the Problem of Universals, Bibliopolis, Naples.
Cocchiarella, N. B.: 1992,‘Cantor's Power-Set Theorem versus Frege's Double-Correlation Thesis’, History and Philosophy of Logic 13, 179–201.
Frege, G.: 1893, Grundgesetze der Arithmetik I, Hermann Pohle, Jena.
Frege, G.: 1976, Wissenschaftlicher Briefwechsel, G. Gabriel, H. Hermes, F. Kambartel, C. Thiel, and A. Veraart (eds.), Felix Meiner, Hamburg.
Heck, R.: 1996,‘The Consistency of Predicative Fragments of Frege's Grundgesetze der Arithmetik’, History and Philosophy of Logic 17, 209–220.
Heck, R.: 1997a,‘Grundgesetze der Arithmetik I §§ 29–32’, Notre Dame Journal of Formal Logic 38, 437–474.
Heck, R.: 1997b,‘The Julius Caesar Objection’, in R. Heck (ed.), Language, Thought, and Logic, Oxford University Press, Oxford, pp. 273–308.
Heck, R.: 1997c,‘Frege and Semantics’, in T. Ricketts (ed.), forthcoming, The Cambridge Companion to Frege, preprint, 39 pp.
Martin, E.: 1982,‘Referentiality in Frege's Grundgesetze’, History and Philosophy of Logic 3, 151–164.
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.
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.
Schroeder-Heister, P.: 1987,‘A Model-Theoretic Reconstruction of Frege's Permutation Argument’, Notre Dame Journal of Formal Logic 28, 69–79.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Wehmeier, K.F. Consistent Fragments of Grundgesetze and the Existence of Non-Logical Objects. Synthese 121, 309–328 (1999). https://doi.org/10.1023/A:1005203526185
Issue Date:
DOI: https://doi.org/10.1023/A:1005203526185