Abstract

First-order logic has limited existential import: the universalized conditional ∀x [S → P] implies its corresponding existentialized conjunction ∃x [S & P] in some but not all cases. We prove the Existential-Import Equivalence:∀x [S → P] implies ∃x [S & P] iff ∃x S is logically true.The antecedent S of the universalized conditional alone determines whether the universalized conditional has existential import: implies its corresponding existentialized conjunction.A predicate is a formula having only x free. An existential-import predicate Q is one whose existentialization, ∃x Q, is logically true; otherwise, Q is existential-import-free or simply import-free. Existential-import predicates are also said to be import-carrying.How widespread is existential import? How widespread are import-carrying predicates in themselves or in comparison to import-free predicates? To answer, let L be any first-order language with any interpretation INT in any [sc. nonempty] universe U. A subset S of U is definable in L under INT iff for some predicate Q in L, S is the truth-set of Q under INT. S is import-carrying definable iff S is the truth-set of an import-carrying predicate. S is import-free definable iff S is the truth-set of an import-free predicate.Existential-Importance Theorem: Let L, INT, and U be arbitrary. Every nonempty definable subset of U is both import-carrying definable and import-free definable.Import-carrying predicates are quite abundant, and no less so than import-free predicates. Existential-import implications hold as widely as they fail.A particular conclusion cannot be validly drawn from a universal premise, or from any number of universal premises.—Lewis-Langford, 1932, p. 62.