Advertisement

What’s So Bad About Second-Order Logic?

  • Jason TurnerEmail author
Part of the Synthese Library book series (SYLI, volume 373)

Abstract

Second-order logic is generally thought problematic by the philosophical populace. Philosophers of mathematics and logic may have sophisticated reasons for rejecting second-order logic, but ask the average philosopher-on-the-street what’s wrong with second-order logic and they will probably mumble something about Quine, ontological commitment, and set theory in sheep’s clothing. In this paper, I try to get more precise about exactly what might be behind these mumblings. I offer four potential arguments against second-order logic and consider several lines of response to each. Two arguments target the coherence of second-order quantification generally, and stem from concerns about ontological commitment. The other two target the expressive power of ‘full’ (as opposed to ‘Henkin’) second-order logic, and give content to the concern that second-order logic is in fact “set theory in sheep’s clothing”. My aim is to understand the dialectic, not take sides; still, second-order logic comes through looking more promising than we might have initially thought.

Keywords

Ontological Commitment Continuum Hypothesis Genuine Consequence Epistemic Obligation Topic Neutrality 
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.

References

  1. 1.
    Barcan Marcus, R. 1972. Quantification and ontology. Nous 6(3): 240–250.CrossRefGoogle Scholar
  2. 2.
    Beall, J.C., and G. Restall. 2006. Logical pluralism. Oxford: Oxford University Press.Google Scholar
  3. 3.
    Boolos, G. 1975. On second-order logic. The Journal of Philosophy 72(16): 509–527.CrossRefGoogle Scholar
  4. 4.
    Dorr, C. 2003. Vagueness without ignorance. Philosophical Perspectives 17(1): 83–113.CrossRefGoogle Scholar
  5. 5.
    Field, H. 1991. Metalogic and modality. Philosophical Studies 62(1): 1–22.CrossRefGoogle Scholar
  6. 6.
    Field, H. 2008. Saving truth from paradox. Oxford: Oxford University Press.CrossRefGoogle Scholar
  7. 7.
    Haack, S. 1978. Philosophy of logics. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  8. 8.
    Henkin, L. 1950. Completeness in the theory of types. The Journal of Symbolic Logic 15(2): 81–91.CrossRefGoogle Scholar
  9. 9.
    Kaplan, D. 1989. Demonstratives. In Themes from Kaplan, ed. J. Almog, J. Perry, and H. Wettstein, 481–563. New York: Oxford University Press.Google Scholar
  10. 10.
    Kreisel, G. 1967. Informal rigor and completeness proofs. In Problems in the philosophy of mathematics, ed. I. Lakatos, 138–171. Amsterdam: North-Holland.CrossRefGoogle Scholar
  11. 11.
    Lewis, D. 1970. How to define theoretical terms. The Journal of Philosophy 67: 427–446. Reprinted in Lewis, D. 1983. Philosophical papers, vol. 1, 78–95. Oxford: Oxford University Press.Google Scholar
  12. 12.
    Prior, A.N. 1971. Objects of thought. Oxford: Clarendon.CrossRefGoogle Scholar
  13. 13.
    Quine, W. 1940. Mathematical logic. Cambridge, MA: Harvard University Press.Google Scholar
  14. 14.
    Quine, W. 1960. Word and object. Cambridge: MIT.Google Scholar
  15. 15.
    Quine, W. 1970. Philosophy of logic. Englewood Cliffs: Prentice-Hall.Google Scholar
  16. 16.
    Rayo, A. (2008). On specifying truth-conditions. The Philosophical Review 117: 385–443.CrossRefGoogle Scholar
  17. 17.
    Rayo, A., and T. Williamson. 2003. A completeness theorem for unrestricted first-order languages. In Liars and heaps: New essays on paradox, ed. J.C. Beall, chapter 15, 331–356. Oxford: Oxford University Press.Google Scholar
  18. 18.
    Rayo, A., and S. Yablo. 2002. Nominalism through de-nominalization. Noûs 35(1): 74–92.CrossRefGoogle Scholar
  19. 19.
    Rossberg, M. Forthcoming. Somehow things do not relate: On the interpretation of polyadic second-order logic. The Journal of Philosophical Logic.Google Scholar
  20. 20.
    Russell, G. 2008. One true logic? The Journal of Philosophical Logic 37(8): 593–611.CrossRefGoogle Scholar
  21. 21.
    Shapiro, S. 1991. Foundations without foundationalism. Oxford: Oxford University Press.Google Scholar
  22. 22.
    Shapiro, S. 2005. Higher-order logic. In The Oxford handbook of philosophy of mathematics and logic, ed. S. Shapiro, 751–780. Oxford: Oxford University Press.CrossRefGoogle Scholar
  23. 23.
    van Inwagen, P. 1981. Why I don’t understand substitutional quantification. Philosophical Studies 39: 281–285. Reprinted in van Inwagen, P. 2001. Ontololgy, identity, and modality, 32–36. Cambridge: Cambridge University Press.Google Scholar
  24. 24.
    van Inwagen, P. 1998. Meta-ontology. Erkenntnis 38: 223–250. Reprinted in van Inwagen, P. 2001. Ontololgy, identity, and modality, 13–31. Cambridge: Cambridge University Press.Google Scholar
  25. 25.
    van Inwagen, P. 2004. A theory of properties. In Oxford studies in metaphysics, vol. 1, ed. D.W. Zimmerman, 107–138. Oxford: Oxford University Press.Google Scholar
  26. 26.
    Williams, J. 2010. Fundamental and derivative truths. Mind 119(473): 103–141.CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Department of PhilosophySaint Louis UniversitySt. LouisUSA

Personalised recommendations