Notre Dame Journal of Formal Logic

Frege's Proof of Referentiality

Øystein Linnebo

Abstract

I present a novel interpretation of Frege's attempt at Grundgesetze I §§29--31 to prove that every expression of his language has a unique reference. I argue that Frege's proof is based on a contextual account of reference, similar to but more sophisticated than that enshrined in his famous Context Principle. Although Frege's proof is incorrect, I argue that the account of reference on which it is based is of potential philosophical value, and I analyze the class of cases to which it may successfully be applied.

Article information

Source
Notre Dame J. Formal Logic, Volume 45, Number 2 (2004), 73-98.

Dates
First available in Project Euclid: 16 September 2004

Permanent link to this document
https://projecteuclid.org/euclid.ndjfl/1095386645

Digital Object Identifier
doi:10.1305/ndjfl/1095386645

Mathematical Reviews number (MathSciNet)
MR2130801

Zentralblatt MATH identifier
1104.03005

Subjects
Primary: 03A05: Philosophical and critical {For philosophy of mathematics, see also 00A30}

Keywords
Frege reference context principle abstraction principles

Citation

Linnebo, Øystein. Frege's Proof of Referentiality. Notre Dame J. Formal Logic 45 (2004), no. 2, 73--98. doi:10.1305/ndjfl/1095386645. https://projecteuclid.org/euclid.ndjfl/1095386645


Export citation

References

  • [1] Boolos, G., "The standard of equality of numbers. Reprinted from Meaning and Method: Essays in Honor of Hilary Putnam, pp. 261--77, Cambridge University Press, Cambridge, 1990", pp. 234--54 in Frege's Philosophy of Mathematics, edited by W. Demopoulos, Harvard University Press, Cambridge, 1995. Reprinted in [?].
  • [2] Boolos, G., Logic, Logic, and Logic, edited by R. Jeffrey, Harvard University Press, Cambridge, 1998.
  • [3] Burgess, J. P., "Frege and arbitrary functions", pp. 89--107 in Frege's Philosophy of Mathematics, edited by W. Demopoulos, Harvard University Press, Cambridge, 1995. Revised version of ``Hintikka et Sandu versus Frege in re arbitrary functions,'' Philosophia Mathematica (3), vol. 1(1993), pp. 50--65.
  • [4] Dummett, M., Frege: Philosophy of Language, 2d edition, Harvard University Press, Cambridge, 1981.
  • [5] Dummett, M., The Interpretation of Frege's Philosophy, Harvard University Press, Cambridge, 1981.
  • [6] Dummett, M., Frege: Philosophy of Mathematics, Harvard University Press, Cambridge, 1991.
  • [7] Dummett, M., ``Neo-Fregeans: In bad company?'' pp. 369--405 in The Philosophy of Mathematics Today (Munich, 1993), edited by M. Schirn, Oxford University Press, New York, 1998. With a reply by Crispin Wright.
  • [8] Frege, G., Grundgesetze der Arithmetik. Band I, II, Georg Olms Verlagsbuchhandlung, Hildesheim, 1962.
  • [9] Frege, G., Foundations of Arithmetic, Blackwell, Oxford, 1950. Translated by J. L. Austin.
  • [10] Frege, G., The Basic Laws of Arithmetic. Exposition of the System, edited by M. Furth, University of California Press, Berkeley, 1964.
  • [11] Frege, G., Philosophical and Mathematical Correspondence, edited by G. Gabriel et al., University of Chicago Press, Chicago, 1980. Translated from the German by H. Kaal.
  • [12] Hale, B., "Grundlagen \S"64, Proceedings of the Aristotelian Society, vol. 97 (1997), pp. 243--61. Reprinted with a postscript in [?].
  • [13] Hale, B., and C. Wright, The Reason's Proper Study. Essays Towards a Neo-Fregean Philosophy of Mathematics, The Clarendon Press, Oxford, 2001.
  • [14] Heck, R. G., Jnr., "Grundgesetze der Arithmetik, I, \S\S"29--32, Notre Dame Journal of Formal Logic, vol. 38 (1997), pp. 437--74.
  • [15] Heck, R. G., Jr., "Frege and semantics", in The Cambridge Companion to Frege, edited by T. Ricketts, Cambridge University Press, Cambridge, forthcoming.
  • [16] Heck, R. G., Jr., and J. Stanley, "Reply to J. Hintikka and G. Sandu. Frege and second-order logic: `The skeleton in Frege's cupboard: The standard versus nonstandard distinction', Journal of Philosophy, vol. 89 (1992), pp. 290--315", The Journal of Philosophy, vol. 90 (1993), pp. 416--24.
  • [17] Hintikka, J., and G. Sandu, "The skeleton in Frege's cupboard: The standard versus nonstandard distinction", The Journal of Philosophy, vol. 89 (1992), pp. 290--315.
  • [18] Linnebo, Ø., "Predicative fragments of Frege arithmetic", Bulletin of Symbolic Logic, vol. 10 (2004), pp. 153--74.
  • [19] Linnebo, Ø., "To be is to be an \emphF". forthcoming in Dialectica.
  • [20] Martin, E., Jr., "Referentiality in Frege's Grundgesetze", History and Philosophy of Logic, vol. 3 (1982), pp. 151--64.
  • [21] Parsons, C., "Frege's theory of number", p. 365 in Mathematics in Philosophy. Selected Essays, Cornell University, Ithaca, 1983.
  • [22] Potter, M., and T. Smiley, "Abstraction by recarving", Proceedings of the Aristotelian Society, vol. 101 (2001), pp. 327--38.
  • [23] Potter, M., and T. Smiley, "Recarving content: Hale's final proposal", Proceedings of the Aristotelian Society, vol. 102 (2002), pp. 301--403.
  • [24] Resnik, M., "Frege's proof of referentiality", pp. 177--95 in Frege Synthesized, edited by L. Haaparanta and J. Hintikka, Dordrecht Reidel, 1986.
  • [25] Ricketts, T., "Truth-values and courses-of-value in Frege's Grundgesetze", pp. 187--211 in Early Analytic Philosophy: Frege, Russell, Wittgenstein, edited by W. W. Tait, Open Court, Peru, 1997.
  • [26] Weiner, J., "Section 31 revisited", pp. 149--82 in From Frege to Wittgenstein: Perspectives on Early Analytic Philosophy, edited by E. Reck, Oxford University Press, Oxford, 2001.
  • [27] Wright, C., Frege's Conception of Numbers as Objects, vol. 2 of Scots Philosophical Monograph Series, Aberdeen University Press, Aberdeen, 1983.