On Logicist Conceptions of Functions and Classes

Chapter
Part of the The Western Ontario Series in Philosophy of Science book series (WONS, volume 75)

Abstract

John Bell arrived in “new” London in 1989, a refugee from the academy under Margaret Thatcher. We soon became good friends, and during the early years of our friendship we collaborated on two papers (Bell and Demopoulos, 1993, 1996). The first of these collaborations was a paper on the foundational significance of results based on second-order logic and Frege’s understanding of his Begriffsschrift; the second was on various notions of independence that arise in connection with elementary propositions in the philosophy of logical atomism. I retain fond memories of both collaborations; they proceeded quickly and almost effortlessly. In this contribution to John’s Festschrift, I propose to revisit our paper on Frege. That paper was occasioned by (Hintikka and Sandu, 1992), which questioned whether Frege’s understanding of second-order logic corresponded, in his framework of functions and concepts, to what we would now regard as the standard interpretation, the interpretation that takes the domain of the function variables to be the full power-set of the domain of individuals. Hintikka and Sandu maintained that it did not on the basis of a number of arguments, all of which they took to show that Frege favored some variety of non-standard interpretation for which the domain of the function variables is something less than the characteristic functions of all subsets of the domain over which the individual variables range.

Keywords

Disjunctive Normal Form Propositional Function Transcendental Argument Combinatorial Possibility Logical Atomism 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Bibliography

  1. Bell, J. and Demopoulos, W. (1993). Frege’s theory of concepts and objects and the interpretation of second-order logic. Philosophia Mathematica, 1:139–156.CrossRefGoogle Scholar
  2. Bell, J. and Demopoulos, W. (1996). Elementary propositions and independence. Notre Dame Journal of Formal Logic, 37:112–124.CrossRefGoogle Scholar
  3. Burgess, J. (1993). Hintikka et Sandu versus Frege in re arbitrary functions. Philosophia Mathematica, 1:50–65.CrossRefGoogle Scholar
  4. Burgess, J. (2005). Fixing Frege. Princeton University Press, Princeton, NJ.Google Scholar
  5. Carnap, R. (1931). The logicist foundation of mathematics. In Putnam, H. and Benacerraf, P., editors, Philosophy of Mathematics: Selected Readings, pages 41–52. Cambridge University Press, Cambridge, second edition. Transl. E. Putnam and G. Massey.Google Scholar
  6. Dummett, M. (1981). Frege, Philosophy of Language. Harvard University Press, Cambridge, second edition.Google Scholar
  7. Dummett, M. (1991). Frege, Philosophy of Mathematics. Harvard University Press, Cambridge.Google Scholar
  8. Frege, G. (1884). The Foundations of Arithmetic: A Logico-Mathematical Enquiry into the Concept of Number. Northwestern University Press, Evanston, second revised edition. Transl. J.L. Austin. English edition 1980.Google Scholar
  9. Frege, G. (1893–1903). The Basic Laws of Arithmetic: Exposition of the System. University of California Press, Berkeley, CA. Two volumes. Partial translation by M. Furth, English version 1964.Google Scholar
  10. Heck, R. and Stanley, J. (1993). Reply to Hintikka and Sandu: Frege and second-order logic. Journal of Philosophy, 90:416–424.CrossRefGoogle Scholar
  11. Heck, R. (1995). Definition by induction in Frege’s Grundgesetze der Arithmetik. In Demopoulos, W., editor, Frege’s Philosophy of Mathematics, pages 295–333. Harvard University Press, Cambridge.Google Scholar
  12. Hintikka, J. and Sandu, G. (1992). The skeleton in Frege’s cupboard: The standard versus nonstandard distinction. Journal of Philosophy, 89:290–315.CrossRefGoogle Scholar
  13. Potter, M. (2005). Ramsey’s transcendental argument. In Lillehammer, H. and Mellor, D., editors, Ramsey’s Legacy, pages 71–82. Oxford University Press, Oxford.Google Scholar
  14. Ramsey, F. (1990). The foundations of mathematics. In Mellor, D., editor, F.P. Ramsey: Philosophical Papers, pages 164–224. Cambridge University Press, Cambridge. Ramsey’s essay was originally published in 1925.Google Scholar
  15. Russell, B. (1919). Introduction to Mathematical Philosophy. Allen and Unwin, London.Google Scholar
  16. Sandu, G. (2005). Ramsey on the notion of arbitrary function. In Frapolli, M., editor, Ramsey: Critical Reassessments, pages 237–256. Continuum Studies in British Philosophy, London.Google Scholar
  17. Wehmeier, K. F. (2008). Wittgensteinian tableaux, identity, and co-denotation. Erkenntnis, 69: 363–376.CrossRefGoogle Scholar
  18. Wehmeier, K. F. (2009). On Ramsey’s “silly delusion” regarding Tractatus 5.53. In Primiero, G. and Rahman, S., editors, Acts of Knowledge: History, Philosophy and Logic: Essays Dedicated to Goran Sundholm, tributes series. College Publications, London.Google Scholar
  19. Whitehead, A. and Russell, B. (1910). Principia Mathematica, Volume 1. Cambridge University Press, Cambridge.Google Scholar
  20. Wittgenstein, L. (1922). Tractatus Logic-Philosophicus. Routledge Classics, New York, NY, 2001 edition. Transl. D. Pears and B. McGuinness.Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Killam Research FellowUniversity of Western OntarioLondonCanada

Personalised recommendations