Synthese

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Extensionality and logicality

S.I. : Intensionality in Mathematics

Abstract

Tarski characterized logical notions as invariant under permutations of the domain. The outcome, according to Tarski, is that our logic, which is commonly said to be a logic of extension rather than intension, is not even a logic of extension—it is a logic of cardinality (or, more accurately, of “isomorphism type”). In this paper, I make this idea precise. We look at a scale inspired by Ruth Barcan Marcus of various levels of meaning: extensions, intensions and hyperintensions. On this scale, the lower the level of meaning, the more coarse-grained and less “intensional” it is. I propose to extend this scale to accommodate a level of meaning appropriate for logic. Thus, below the level of extension, we will have a more coarse-grained level of form. I employ a semantic conception of form, adopted from Sher, where forms are features of things “in the world”. Each expression in the language embodies a form, and by the definition we give, forms will be invariant under permutations and thus Tarskian logical notions. I then define the logical terms of a language as those terms whose extension can be determined by their form. Logicality will be shown to be a lower level analogue of rigidity. Using Barcan Marcus’s principles of explicit and implicit extensionality, we are able to characterize purely logical languages as “sub-extensional”, namely, as concerned only with form, and we thus obtain a wider perspective on both logicality and extensionality.

Keywords

Logicality Permutation invariance Tarski Sher Barcan Marcus Extensionality Form 

References

  1. Bonnay, D. (2008). Logicality and invariance. The Bulletin of Symbolic Logic, 14(1), 29–6.CrossRefGoogle Scholar
  2. Button, T. (2006). Realistic structuralisms identity crisis: A hybrid solution. Analysis, 66(291), 216–222.CrossRefGoogle Scholar
  3. Carnap, R. (1947). Meaning and necessity. Chicago: The University of Chicago Press.Google Scholar
  4. Cresswell, M. J. (1975). Hyperintensional logic. Studia Logica, 34(1), 25–38.CrossRefGoogle Scholar
  5. Dutilh Novaes, C. (2014). The undergeneration of permutation invariance as a criterion for logicality. Erkenntnis, 79(1), 81–97.CrossRefGoogle Scholar
  6. Duží, M. (2010). The paradox of inference and the non-triviality of analytic information. Journal of Philosophical Logic, 39(5), 473–510.CrossRefGoogle Scholar
  7. Duží, M. (2012). Towards an extensional calculus of hyperintensions. Organon F, 19, 20–45.Google Scholar
  8. Feferman, S. (1999). Logic, logics and logicism. Notre Dame Journal of Formal Logic, 40(1), 31–55.CrossRefGoogle Scholar
  9. Feferman, S. (2010). Set-theoretical invariance criteria for logicality. Notre Dame Journal of Formal Logic, 51(1), 3–20.CrossRefGoogle Scholar
  10. Gómez-Torrente, M. (2015). Alfred Tarski. In E. N. Zalta (Ed.), The stanford encyclopedia of philosophy (Spring 2015 ed.). Metaphysics Research Lab, Stanford University. https://plato.stanford.edu/archives/spr2015/entries/tarski/.
  11. Klein, F. (1893). A comparative review of recent researches in geometry. Bulletin of the American Mathematical Society, 2(10), 215–249.CrossRefGoogle Scholar
  12. Levy, A. (1979). Basic set theory. Berlin: Springer.CrossRefGoogle Scholar
  13. Lindenbaum, A., & Tarski, A. (1983). On the limitations of the means of expression of deductive theories. In J. Corcoran (Ed.), Logic, semantics, metamathematics (pp. 384–392). Indianapolis: Hackett.Google Scholar
  14. Marcus, R. (1960). Extensionality. Mind, 69(273), 55–62.CrossRefGoogle Scholar
  15. Marcus, R. (1961). Modalities and intensional languages. Synthese, 13(4), 303–322.CrossRefGoogle Scholar
  16. Marquis, J. P. (2008). From a geometrical point of view: A study of the history and philosophy of category theory (Vol. 14). Berlin: Springer.Google Scholar
  17. Mautner, F. (1946). An extension of Klein’s Erlanger program: Logic as invariant-theory. American Journal of Mathematics, 68(3), 345–384.CrossRefGoogle Scholar
  18. McCarthy, T. (1981). The idea of a logical constant. The Journal of Philosophy, 78(9), 499–523.CrossRefGoogle Scholar
  19. McGee, V. (1996). Logical operations. Journal of Philosophical Logic, 25(6), 567–580.CrossRefGoogle Scholar
  20. Moschovakis, Y. N. (2006). A logical calculus of meaning and synonymy. Linguistics and Philosophy, 29(1), 27–89.CrossRefGoogle Scholar
  21. Sagi, G. (2014). Models and logical consequence. Journal of Philosophical Logic, 43(5), 943–964.CrossRefGoogle Scholar
  22. Shapiro, S. (1998). Logical consequence: Models and modality. In M. Schirn (Ed.), The philosophy of mathematics today (pp. 131–156). Oxford: Oxford Univerity Press.Google Scholar
  23. Sher, G. (1991). The bounds of logic: A generalized viewpoint. Cambridge, MA: MIT Press.Google Scholar
  24. Sher, G. (1996). Did Tarski commit ‘Tarski’s fallacy’? The Journal of Symbolic Logic, 61(2), 653–686.CrossRefGoogle Scholar
  25. Sher, G. (2013). The foundational problem of logic. Bulletin of Symbolic Logic, 19(2), 145–198.CrossRefGoogle Scholar
  26. Tarski, A. (1936). On the concept of logical consequence. In J. Corcoran (Ed.), Logic, semantics, metamathematics (pp. 409–420). Indianapolis: Hackett.Google Scholar
  27. Tarski, A. (1986). What are logical notions? History and Philosophy of Logic, 7, 143–154.CrossRefGoogle Scholar
  28. Tarski, A., & Givant, S. (1987). A formalization of set theory without variables. Providence: Colloquium publications (American Mathematical Society).CrossRefGoogle Scholar
  29. Tichý, P. (1988). The foundations of Frege’s logic. Berlin: Walter de Gruyter.CrossRefGoogle Scholar
  30. White, M., & Tarski, A. (1987). A philosophical letter of alfred tarski. The Journal of Philosophy, 84(1), 28–32.CrossRefGoogle Scholar
  31. Whitehead, A. N., & Russell, B. (1910). Principia mathematica (Vol. 1). Cambridge: Cambridge University Press.Google Scholar
  32. Woods, J. (2014). Logical indefinites. Logique et Analyse, 227, 277–307.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  1. 1.Philosophy DepartmentUniversity of HaifaHaifaIsrael

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