Erkenntnis

, Volume 74, Issue 3, pp 321–349 | Cite as

The Metamathematics of Putnam’s Model-Theoretic Arguments

Original Article

Abstract

Putnam famously attempted to use model theory to draw metaphysical conclusions. His Skolemisation argument sought to show metaphysical realists that their favourite theories have countable models. His permutation argument sought to show that they have permuted models. His constructivisation argument sought to show that any empirical evidence is compatible with the Axiom of Constructibility. Here, I examine the metamathematics of all three model-theoretic arguments, and I argue against Bays (2001, 2007) that Putnam is largely immune to metamathematical challenges.

References

  1. Bays, T. (2001). On Putnam and his models. The Journal of Philosophy, 98, 331–350.CrossRefGoogle Scholar
  2. Bays, T. (2007). More on Putnam’s models: A reply to Bellotti. Erkenntnis, 67, 119–135.CrossRefGoogle Scholar
  3. Bays, T. (2009). Skolem’s paradox. In E. N. Zalta (Ed.), Stanford encyclopedia of philosophy. [http://plato.stanford.edu/entries/paradox-skolem/]—visited on 14.vi.2010.
  4. Bellotti, L. (2005). Putnam and constructibility. Erkenntnis, 62, 395–409.CrossRefGoogle Scholar
  5. Benacerraf, P. (1985). Skolem and the Skeptic. Proceedings of the Aristotelian Society, 59, 85–115.Google Scholar
  6. Boolos, G. S., & Jeffrey, R. C. (1989). Computability and Logic, 3rd ed. Cambridge: Cambridge University Press.Google Scholar
  7. Franzen, T. (2004). Inexhaustibility: A non-exhaustive treatment. Wellesley, MA: Association for Symbolic Logic, Lecture Notes in Logic 16.Google Scholar
  8. Hodges, W. (1993). Model theory. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  9. Kunen, K. (1980). Set theory: An introduction to independence proofs. Volume 102, Studies in logic and the foundations of mathematics. Oxford: North-Holland.Google Scholar
  10. Lewis, D. (1984). Putnam’s paradox. Australasian Journal of Philosophy, 62, 221–236.CrossRefGoogle Scholar
  11. McIntosh, C. (1979). Skolem’s criticisms of set theory. Noûs, 13, 313–334.CrossRefGoogle Scholar
  12. Mostowski, A. (1969). Constructible sets with applications. Amsterdam: North-Holland, Studies in Logic and the Foundations of Mathematics.Google Scholar
  13. Potter, M. (2004). Set theory and its philosophy. Oxford: Oxford University Press.Google Scholar
  14. Putnam, H. (1978). Meaning and the moral sciences. London: Routledge and Kegan Paul.Google Scholar
  15. Putnam, H. (1980). Models and reality. Journal of Symbolic Logic, 45, 464–482.CrossRefGoogle Scholar
  16. Putnam, H. (1981). Reason, truth and history. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  17. Putnam, H. (1983). Realism and reason: Philosophical papers (Vol. 3). Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  18. Putnam, H. (1989). Model theory and the ‘factuality’ of semantics. In A. George (Ed.), Reflections on chomsky (pp. 213–232). Oxford: Basil Blackwell.Google Scholar
  19. Putnam, H. (1992). Replies. Philosophical Topics, 20, 347–408.Google Scholar
  20. Simpson, S. G. (1999). Subsystems of second-order arithmetic. Berlin: Springer-VerlagGoogle Scholar
  21. Skolem, T. (1922). Some remarks on axiomatised set theory. In: J. van Heijenoort (Ed.), From Frege to Gödel: A source book in mathematical logic, 1879–1931 (1967) (pp. 290–301). Cambridge, MA: Harvard University Press.Google Scholar
  22. Skolem, T. (1958). Une relativisation des notions mathématiques fondamentales. In: E. J. Fenstad (Ed.), Selected works in logic (1970) (pp. 633–638). Oslo: Universitetsforlaget.Google Scholar
  23. Velleman, D. J. (1998). ‘Review of Levin’s ‘Putnam on reference and constructible sets’ (1997)’. Mathematical Reviews, 98c:03015, 1364Google Scholar
  24. Wright, C. (1985). Skolem and the Skeptic. Proceedings of the Aristotelian Society, 59, 117–137.Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Darwin College, Cambridge UniversityCambridgeUK

Personalised recommendations