Applications of the Elementary Theory

Abstract

It will be important to have definite descriptions (which involve the primitive logical notion the) to facilitate the following applications. Let us stipulate that where Ø is any formula with one free x-variable, (ix) Ø (“the object x such that Ø”) is to be a complex object term of our language. Some examples might be: (iy)(E! y & Typ) (“the object which exists and taught Plato”) and (ix)(A!x amf; (F)(xF = F = R v F = S)) (“the object which encodes just roundness and squareness”). Semantically, we interpret descriptions like (ix) Ø as denoting the unique object which satisfies Ø, if there is one, and as not denoting anything if there is not one. To guarantee that descriptions work in our system just as we would expect them to a priori, we add a proper axiom schema which asserts that atomic formulas or defined identity formulas Ψ in which there occurs a description (ix) Ø are true if there is a unique object satisfying Ø and there is something which satisfies both Ø and Ψ¹.