Advertisement

Philosophical Studies

, Volume 158, Issue 3, pp 377–400 | Cite as

Arbitrary reference

  • Wylie Breckenridge
  • Ofra Magidor
Article

Abstract

Two fundamental rules of reasoning are Universal Generalisation and Existential Instantiation. Applications of these rules involve stipulations (even if only implicitly) such as ‘Let n be an arbitrary number’ or ‘Let John be an arbitrary Frenchman’. Yet the semantics underlying such stipulations are far from clear. What, for example, does ‘n’ refer to following the stipulation that n be an arbitrary number? In this paper, we argue that ‘n’ refers to a number—an ordinary, particular number such as 58 or 2,345,043. Which one? We do not and cannot know, because the reference of ‘n’ is fixed arbitrarily. Underlying this proposal is a more general thesis:

Arbitrary Reference (AR): It is possible to fix the reference of an expression arbitrarily. When we do so, the expression receives its ordinary kind of semantic-value, though we do not and cannot know which value in particular it receives.

Our aim in this paper is defend AR. In particular, we argue that AR can be used to provide an account of instantial reasoning (one that is better than the prominent alternatives), and we suggest that AR can also figure in offering new solutions to a range of difficult philosophical puzzles.

Keywords

Reference Arbitrary Arbitrary reference Arbitrary objects Instantial reasoning Universal generalisation Existential instantiation Natural deduction Vagueness Indiscernible Structuralism Random Indefinite 

Notes

Acknowledgments

We are grateful to audiences at Cornell University, CSMN Oslo, Macquarie University, and the University of Oxford, as well as to Ross Cameron, John Hawthorne, Stephen Kearns, Jeff King, Vann McGee, Moritz Schulz, Stewart Shapiro, Nick Smith, and Robbie Williams for helpful discussions.

References

  1. Benacerraf, P. (1965). What numbers could not be. Philosophical Review, 74, 47–73.CrossRefGoogle Scholar
  2. Cameron, R. (2010). Vagueness and naturalness. Erkenntnis, 72, 281–293.CrossRefGoogle Scholar
  3. Fara, D. (2000). Shifting sands: An interest-relative theory of vagueness. Philosophical Topics, 28, 45–81; Originally published under ‘Delia Graff’.Google Scholar
  4. Fine, K. (1983). A defence of arbitrary objects. Proceedings of the Aristotelian Society, Supplementary Volume, 57, 55–77.Google Scholar
  5. Fine, K. (1985a). Natural deduction and arbitrary objects. Journal of Philosophical Logic, 14, 57–107.CrossRefGoogle Scholar
  6. Fine, K. (1985b). Reasoning with arbitrary objects. Oxford: Basil Blackwell.Google Scholar
  7. Hawthorne, J., & Manley, D. (MS). The reference book. Unpublished manuscript.Google Scholar
  8. Kearns, S., & Magidor, O. (2008). Epistemicism about vagueness and meta-linguistic safety. Philosophical Perspectives, 22, 277–304.CrossRefGoogle Scholar
  9. Kearns, S., & Magidor, O. (forthcoming). Semantic sovereignty. Philosophy and Phenomenological Research.Google Scholar
  10. King, J. (1991). Instantial terms, anaphora and arbitrary objects. Philosophical Studies, 61, 239–265.CrossRefGoogle Scholar
  11. Mackie, J. (1958). The rules of natural deduction. Analysis, 19, 27–35.CrossRefGoogle Scholar
  12. Martino, E. (2001). Arbitrary reference in mathematical reasoning. Topoi, 20, 65–77.CrossRefGoogle Scholar
  13. Prawitz, D. (1965). Natural deduction: A proof-theoretical study. Stockholm: Almqvist & Wiksell.Google Scholar
  14. Price, R. (1962). Arbitrary individuals and natural deduction. Analysis, 22, 94–96.CrossRefGoogle Scholar
  15. Rescher, N. (1958). Can there be random individuals. Analysis, 18, 114–117.CrossRefGoogle Scholar
  16. Shapiro, S. C. (2004). A logic of arbitrary and indefinite objects. In D. Dubois, C. Welty, & M. Williams (Eds.), Principles of knowledge representation and reasoning: Proceedings of the ninth international conference (KR2004) (pp. 565–575). Menlo Park: AAAI Press.Google Scholar
  17. Tennant, N. (1983). A defence of arbitrary objects. Proceedings of the Aristotelian Society, Supplementary Volume, 57, 79–89.Google Scholar
  18. Williamson, T. (1994). Vagueness. London: Routledge.Google Scholar
  19. Williamson, T. (1997). Précis of vagueness. Philosophy and Phenomenological Research, 57, 921–928.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.Balliol CollegeUniversity of OxfordOxfordUK
  2. 2.School of Humanities and Social SciencesCharles Sturt UniversityWagga WaggaAustralia

Personalised recommendations