All Quantifiers Versus the Quantifier All

Abstract

In courses of logic for general students the general and existential quantifiers are the only ones distinguished from among all possible quantifier expressions of the natural language. One can argue that other quantifiers deserve mention, even though there are good reason for emphasizing the familiar ones: namely, they are the simplest, the universal quantifier is a counterpart of the operation of generalizing, the number of nested quantifiers is a good measure of logical complexity, and the expressive power of the general quantifier and its dual is considerable.

Yet, even in the teaching about these two simplest quantifiers it has not been resolved how to indicate the realm to which a given quantifier refers. The methods range from the Fregean assumption that they refer to the totality of objects in the world to the restricted quantifiers to many sorted logic. It turns out that these approaches are not fully equivalent, because the sorts are usually assumed to be nonempty, which results in a problem similar to the well-known issue with non-emptiness of names in syllogistics.

Logicians have studied various generalized quantifiers. It is, however, unclear how to treat the quantifier “many” and similar heavily context-dependent ones. They are not invariant under isomorphisms so no purely logical or mathematical treatment seems applicable. How else can one characterize the context-independent quantifiers among all possible quantifiers corresponding to quantifier expressions in natural language? The following thesis on quantifiers is proposed:

(Principal Thesis):

Context-independence = definability in terms of the universal quantifier.

This thesis provides an additional reason for distinguishing the universal quantifier from among all other quantifiers: it suffices for defining all context-independent ones