First-order logic has limited existential import: the universalized conditional ∀x [S(x) → P(x)] implies its corresponding existentialized conjunction ∃x [S(x) & P(x)] in some but not all cases. We prove the Existential-Import Equivalence:
∀x [S(x) → P(x)] implies ∃x [S(x) & P(x)] iff ∃x S(x) is logically true.
The antecedent S(x) 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(x) is one whose existentialization, ∃x Q(x), is logically true; otherwise, Q(x) 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(x) in L, S is the truth-set of Q(x) 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.