# Existential-Import Mathematics

Bulletin of Symbolic Logic 21 (1):1-14 (2015)

# Abstract

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.

# Other Versions

No versions found

## PhilArchive

Upload a copy of this work     Papers currently archived: 96,395

Setup an account with your affiliations in order to access resources via your University's proxy server

2015-05-12

51 (#339,802)

6 months
20 (#202,340)

# Author Profiles

Hassan Masoud
University of Alberta
John Corcoran
PhD: Johns Hopkins University; Last affiliation: University at Buffalo

# References found in this work

Tractatus logico-philosophicus.Ludwig Wittgenstein - 1922 - Filosoficky Casopis 52:336-341.
Introduction to Mathematical Philosophy.Bertrand Russell - 1919 - Revue Philosophique de la France Et de l'Etranger 89:465-466.
Philosophy of Logic (2nd Edition).W. V. Quine - 1986 - Cambridge, MA: Harvard University Press.
Symbolic Logic.C. I. Lewis & C. H. Langford - 1932 - Erkenntnis 4 (1):65-66.
Introduction to mathematical logic.Alonso Church - 1958 - Revue de Métaphysique et de Morale 63 (1):118-118.