Hassan Masoud

First-order logic haslimitedexistential import: the universalized conditional ∀x[S(x) → P(x)] implies its corresponding existentialized conjunction ∃x[S(x) & P(x)] insome but not allcases. We prove theExistential-Import Equivalence:∀x[S(x) → P(x)] implies ∃x[S(x) & P(x)] iff ∃xS(x) is logically true.The antecedent S(x) of the universalized conditional alone determines whether the universalized conditionalhas existential import: implies its corresponding existentialized conjunction.Apredicateis a formula having onlyxfree. Anexistential-importpredicate Q(x) is one whose existentialization, ∃xQ(x), is logically true; otherwise, Q(x) isexistential-import-freeor simplyimport-free. Existential-import predicates are also said to beimport-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 isdefinable inLunderINT iff for some predicate Q(x) in L, S is the truth-set of Q(x) under INT. S isimport-carrying definableiff S is the truth-set of an import-carrying predicate. S isimport-free definableiff 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 isbothimport-carrying definableandimport-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.

Contrary to common misconceptions, today's logic is not devoid of existential import: the universalized conditional ∀ x [S→ P] implies its corresponding existentialized conjunction ∃ x [S & P], not in all cases, but in some. We characterize the proexamples by proving the Existential-Import Equivalence: The antecedent S of the universalized conditional alone determines whether the universalized conditional has existential import, i.e. whether it implies its corresponding existentialized conjunction.A predicate is an open 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.How abundant or widespread is existential import? How abundant or widespread are existential-import predicates in themselves or in comparison to import-free predicates? We show that existential-import predicates are quite abundant, and no less so than import-free predicates. Existential..