Logica Universalis

, Volume 6, Issue 3–4, pp 597–613 | Cite as

Logic as a Science and Logic as a Theory: Remarks on Frege, Russell and the Logocentric Predicament

Article

Abstract

Since its publication in 1967, van Heijenoort’s paper, “Logic as Calculus and Logic as Language” has become a classic in the historiography of modern logic. According to van Heijenoort, the contrast between the two conceptions of logic provides the key to many philosophical issues underlying the entire classical period of modern logic, the period from Frege’s Begriffsschrift (1879) to the work of Herbrand, Gödel and Tarski in the late 1920s and early 1930s. The present paper is a critical reflection on some aspects of van Heijenoort’s thesis. I concentrate on the case of Frege and Russell and the claim that their philosophies of logic are marked through and through by acceptance of the universalist conception of logic, which is an integral part of the view of logic as language. Using the so-called “Logocentric Predicament” (Henry M. Sheffer) as an illustration, I shall argue that the universalist conception does not have the consequences drawn from it by the van Heijenoort tradition. The crucial element here is that we draw a distinction between logic as a universal science and logic as a theory. According to both Frege and Russell, logic is first and foremost a universal science, which is concerned with the principles governing inferential transitions between propositions; but this in no way excludes the possibility of studying logic also as a theory, i.e., as an explicit formulation of (some) of these principles. Some aspects of this distinction will be discussed.

Mathematics Subject Classification

03A05 

Keyword

Philosophy of logic 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Dummett M.: The Logical Basis of Metaphysics. Harvard University Press, Cambridge (1991)Google Scholar
  2. 2.
    Frege G.: Conceptual notation: a formula language of pure thought, modelled upon the formula language of arithmetic. In: Bynum, T.W. (trans. and ed.) Conceptual Notation and Related Articles., pp. 101–208. Clarendon Press, Oxford (1879)Google Scholar
  3. 3.
    Frege, G.: On the aim of conceptual notation. In: Bynum, T.W. (trans. and ed.) Conceptual Notation and Related Articles, pp. 90–100. Clarendon Press, Oxford (1882)Google Scholar
  4. 4.
    Frege G.: On Mr. Peano’s conceptual notation (1897). In: McGuinness, B.F. (ed.) Collected Papers on Mathematics, Logic and Philosophy, pp. 234–248. Basil Blackwell, Oxford (1984)Google Scholar
  5. 5.
    Frege, G.: The basic laws of arithmetic. In: Furth, M. (trans. & ed.) Exposition of the System. University of California Press, Berkeley (1964)Google Scholar
  6. 6.
    Frege, G.: Conceptual notation and related articles. In: Bynum, T.W. (trans. & ed.) Clarendon Press, Oxford (1972)Google Scholar
  7. 7.
    Goldfarb W.D.: Logic in the twenties: the nature of the quantifier. J. Symb. Log. 44, 352–368 (1979)MathSciNetGoogle Scholar
  8. 8.
    Goldfarb W.D.: Logicism and logical truth. J. Philos. 79, 692–695 (1982)CrossRefGoogle Scholar
  9. 9.
    Grattan-Guinness, I.: Dear Russell—Dear Jourdain: a commentary on Russell’s logic. In: Based on his Correspondence with Philip Jourdain. Duckworth, London (1977)Google Scholar
  10. 10.
    Haaparanta L.: Frege’s doctrine of being. Acta Philos. Fennica 39, 1–182 (1985)Google Scholar
  11. 11.
    Heck R.G. Jr.: Frege and semantics. In: Potter, M., Ricketts, T. (ed.) The Cambridge Companion to Frege., pp. 342–378. Cambridge University Press, Cambridge (2010)CrossRefGoogle Scholar
  12. 12.
    Hintikka, J.: Lingua Universalis vs. Calculus Ratiocinator: an Ultimate Presupposition of Twentieth-century Philosophy. Kluwer, Dordrecht (1997)Google Scholar
  13. 13.
    Hintikka M.B., Hintikka J.: Investigating Wittgenstein. Basil Blackwell, Oxford (1986)Google Scholar
  14. 14.
    Hylton P.: Russell’s substitutional theory. Synthese 45, 1–31 (1980)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Hylton P.: Logic in Russell’s logicism. In: Bell, D., Cooper, N. (eds) The Analytic Tradition:Meaning,Thought and Knowledge, pp. 137–172. Basil Blackwell, xford (1990)Google Scholar
  16. 16.
    Kemp G.: Propositions and reasoning in Frege and Russell. Pac. Philos. Q. 79, 218–235 (1998)CrossRefGoogle Scholar
  17. 17.
    Kusch, M.: Language as Calculus vs. language as universal medium: a study in Husserl, Heidegger and Gadamer. Kluwer, Dordrecht (1989)Google Scholar
  18. 18.
    Landini G.: Russell’s hidden substitutional theory. Oxford University Press, New York (1998)MATHGoogle Scholar
  19. 19.
    Peckhaus, V.: Logik, Mathesis universalis und allgemeine Wissenschaft. Leibniz und die Wiederentdeckung der formalen Logik im 19. Jahrhundert. Akademie Verlag, Berlin (1997)Google Scholar
  20. 20.
    Prawitz D.: On the idea of a general proof theory. Synthese 27, 63–77 (1974)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Ricketts T.: Logic and truth in Frege. Proc. Aristot. Soc., Suppl. 70, 121–140 (1996)Google Scholar
  22. 22.
    Ricketts T.: Frege’s 1906 foray into metalogic. Philos. Top. 25, 169–187 (1997)Google Scholar
  23. 23.
    Russell, B.: Recent Italian work on the foundations of mathematics (1901). In: Moore, G.H. (ed.) The Collected Papers of Bertrand Russell, vol. 3. The Principles of Mathematics, 1900–02, pp. 350–362. Routledge, London (1993)Google Scholar
  24. 24.
    Russell B.: The Principles of Mathematics. Cambridge University Press, Cambridge (1903)MATHGoogle Scholar
  25. 25.
    Russell B.: The Principles of Mathematics. 2nd edn. Allen and Unwin, London (1937)MATHGoogle Scholar
  26. 26.
    Russell B.: Theory of implication. Am. J. Math. 28, 159–202 (1906)MATHCrossRefGoogle Scholar
  27. 27.
    Russell B.: The Problems of Philosophy. Home University Library of Modern Knowledge, No. 35.. Williams and Norgate, London (1912)Google Scholar
  28. 28.
    Russell B.: The Problems of Philosophy, 9th impression, with Appendix. Oxford University Press, Oxford and New York (1980)Google Scholar
  29. 29.
    Russell B.: Introduction to Mathematical Philosophy. Allen and Unwin, London (1919)MATHGoogle Scholar
  30. 30.
    Sheffer H.M.: Review of A N. Whitehead and B. Russell. Principia Math. vol. 1, 2nd edn. Isis 226– (1926)Google Scholar
  31. 31.
    Tappenden J.: Metatheory and mathematical practice in Frege. Philos. Top. 25, 213–264 (1997)Google Scholar
  32. 32.
    Van Heijenoort J.: Logic as language and logic as calculus. Synthese 17, 324–330 (1967)MATHCrossRefGoogle Scholar
  33. 33.
    Van Heijenoort, J.: Systéme et métasystème chez Russell. In: The Paris Logic Group (ed.) Logic Colloquium’85, pp. 111–122. North-Holland, Amsterdam (1987)Google Scholar
  34. 34.
    Whitehead A.N., Russell B.: Principia Mathematica, vol. 1. Cambridge University Press, Cambridge (1910)Google Scholar
  35. 35.
    Wittgenstein, L.: Tractatus logico-philosophicus. English translation by C.K. Ogden. With an Introduction by B. Russell. Kegan Paul, London. (1922) (revised edn. 1933)Google Scholar

Copyright information

© Springer Basel AG 2012

Authors and Affiliations

  1. 1.Department of Philosophy, History, Culture and Art StudiesUniversity of HelsinkiUniversity of HelsinkiFinland

Personalised recommendations