Logical Constants and Unrestricted Quantification

Abstract

Variants of the so-called permutation criterion have been used for distinguishing between logical and non-logical operations or expressions. Roughly, an operation is defined as logical if, and only if, it is invariant under arbitrary permutations on every domain. Thus a logical operation behaves on all objects in the same way. An expression is logical if, and only if, the operation expressed by it is logical. I consider a variant of the permutation criterion that eliminates domains: An operation is permutation-invariant if, and only if, it is invariant under arbitrary permutations of the universe. An expression is logical if, and only if, it expresses an operation that is permutation-invariant in this sense. This domain-free definition of the invariance criterion matches definitions of logical consequence without domains where first-order quantifiers are taken to range over all (first-order) objects in all interpretations. Without domains some problems of the invariance criterion disappear. In particular, an operation can behave on all objects of any domain in the same way, while still behaving very differently in each domain. On the criterion without domains, a logical operation always behaves on all objects in the same way, not only on all objects of any given domain.