Intensional models for first degree formulas

Abstract

In Anderson and Belnap [8] there was developed a semantics for first degree entailments (fde), i.e., entailments A → B between formulas A and B involving only truth-functions (defined in terms of “or” and “not”) and quantifiers. The key ideas were (i) the notion of a frame ⟨P, F_IP, I⟩, where P is a set of (intensional) propositions closed under negation and multiple disjunction, I is a domain of individuals, and F_IP is the set of functions from I into P; and (ii) the semantic relation of cons (consequence), as obtaining between a set of propositions taken conjunctively, and a set taken disjunctively; and (iii) the notion of an atomic frame, i.e., a frame generated by a set of propositions X closed under negation, such that for any disjoint subclasses Y and Z of X, Y does not bear cons to Z.