The Undergeneration of Permutation Invariance as a Criterion for Logicality
- 321 Downloads
Permutation invariance is often presented as the correct criterion for logicality. The basic idea is that one can demarcate the realm of logic by isolating specific entities—logical notions or constants—and that permutation invariance would provide a philosophically motivated and technically sophisticated criterion for what counts as a logical notion. The thesis of permutation invariance as a criterion for logicality has received considerable attention in the literature in recent decades, and much of the debate is developed against the background of ideas put forth by Tarski in a 1966 lecture (Tarski 1966/1986). But as noted by Tarski himself in the lecture, the permutation invariance criterion yields a class of putative ‘logical constants’ that are essentially only sensitive to the number of elements in classes of individuals. Thus, to hold the permutation invariance thesis essentially amounts to limiting the scope of logic to quantificational phenomena, which is controversial at best and possibly simply wrong. In this paper, I argue that permutation invariance is a misguided approach to the nature of logic because it is not an adequate formal explanans for the informal notion of the generality of logic. In particular, I discuss some cases of undergeneration of the criterion, i.e. the fact that it excludes from the realm of logic operators that we have good reason to regard as logical, especially some modal operators.
KeywordsModal Logic Accessibility Relation Quantificational Phenomenon Logical Constant Numerical Identity
- Dutilh Novaes, C. (2012). Reassessing logical hylomorphism and the demarcation of logical constants. Synthese, 185(3), 387–410.Google Scholar
- Lindenbaum, A., & Tarski, A. (1934–1935). “Über die Beschränktheit der Ausdrucksmittel deduktiver Theorien” (“On the Limitations of the Means of Expression of Deductive Theories”). Ergebnisse eines mathematischen Kolloquiums, 7, 15–22 (English translation in Tarski, 1983. Logic, Semantics, Metamathematics. J. Corcoran (ed.). Indianapolis: Hackett).Google Scholar
- Lindström, P. (1966). First order predicate logic with generalized quantifiers. Theoria, 32, 186–195Google Scholar
- MacFarlane, J. (2000). What does it mean to say that logic is formal? PhD dissertation, Pittsburgh University.Google Scholar
- MacFarlane, J. (2009). Logical constants. In E. Zalta (Ed.), Stanford encyclopedia of philosophy. Available at http://plato.stanford.edu/entries/logical-constants/.
- Mostowski, A. (1957). On a generalization of quantifiers. Fundamenta Mathematicae, 44, 12–35.Google Scholar
- Sher, G. (1991). The bounds of logic: A generalized viewpoint. Cambridge, MA: MIT Press.Google Scholar
- Tarski, A. (1936/2002). On the concept of following logically. History and Philosophy of Logic, 23, 155–196.Google Scholar