Extensionality and logicality

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.

Invariance Criteria as Meta-Constraints

“…26 For such a line of argument, see (Sher, 2016, p. 304). 27 Indeed, it is customary to interpret Tarski's later work as completing the explication of logical consequence he set out on in his earlier paper (Tarski, 1936), but see (Sagi, 2021) for an alternative interpretation.…”

“…Ultimately, the appropriateness of a system of constraints, as any system of logic, depends on the use it is intended for. Specifically, a system of semantic constraints can be used in empirical semantics to model logical consequence in natural language, or it can be used as a framework of commitments made by a reasoner [28]. It may be that these two uses (not to exclude other possible uses) will pull us in completely different directions, but it may also be that they do not differ categorically in the conditions they require on constraints, rather only in emphasis and degree: while the former aims at empirical adequacy, the latter might entail a preference for coherence and robustness.…”