Logic, Logics, and Logicism

Abstract

The paper starts with an examination and critique of Tarski's well-known proposed explication of the notion of logical operation in the type structure over a given domain of individuals as one which is invariant with respect to arbitrary permutations of the domain. The class of such operations has been characterized by McGee as exactly those definable in the language ${\cal L}_{\infty,\infty}$. Also characterized similarly is a natural generalization of Tarski's thesis, due to Sher, in terms of bijections between domains. My main objections are that on the one hand, the Tarski-Sher thesis thus assimilates logic to mathematics, and on the other hand fails to explain the notion of same logical operation across domains of different sizes. A new notion of homomorphism invariant operation over functional type structures (with domains $M_0$ of individuals and $\{T,F\}$ at their base) is introduced to accomplish the latter. The main result is that an operation is definable from the first-order predicate calculus without equality just in case it is definable from homomorphism invariant monadic operations, where definability in both cases is taken in the sense of the $\lambda$-calculus. The paper concludes with a discussion of the significance of the results described for the views of Tarski and Boolos on logicism.