Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-30T17:18:42.807Z Has data issue: false hasContentIssue false
  • English
  • Français

EXISTENTIAL-IMPORT MATHEMATICS

Published online by Cambridge University Press:  09 April 2015

JOHN CORCORAN  and
HASSAN MASOUD
JOHN CORCORAN
Affiliation:
DEPARTMENT OF PHILOSOPHY, UNIVERSITY AT BUFFALO, BUFFALO, NY 14260-4150, USAE-mail:[email protected]
HASSAN MASOUD
Affiliation:
DEPARTMENT OF PHILOSOPHY, UNIVERSITY OF ALBERTA, EDMONTON, AB T6G2E7, CANADAE-mail:[email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

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.

Keywords

first-order logicexistential importpredicateimport-carryingimport-free
Type
Articles
Information
Bulletin of Symbolic Logic , Volume 21 , Issue 1 , March 2015 , pp. 1 - 14
Copyright
Copyright © The Association for Symbolic Logic 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Boolos, G., Burgess, J., and Jeffrey, R., Computability and logic, Cambridge University Press, Cambridge, 2007.CrossRefGoogle Scholar
Church, A., Introduction to Mathematical Logic, Princeton University Press, Princeton, 1956.Google Scholar
Corcoran, J., Completeness of an ancient logic. Journal of Symbolic Logic, vol. 37 (1972), pp. 696–702.Google Scholar
Corcoran, J., Meanings of implication. Diálogos, vol. 9 (1973), pp. 59–76.Google Scholar
Corcoran, J., First-order logical form, this Bulletin, vol. 10 (2004), p. 445.Google Scholar
Corcoran, J., Schemata: the Concept of Schema in the History of Logic, this Bulletin, vol. 12 (2006), pp. 219–40.Google Scholar
Corcoran, J., Existential import, this Bulletin, vol. 13 (2007), pp. 143–144.Google Scholar
Corcoran, J., Aristotle’s many-sorted logic, this Bulletin, vol. 14 (2008), pp. 155–156.Google Scholar
Corcoran, J., and Masoud, H., Existential-import sentence schemas: classical and relativized, this Bulletin, vol. 20 (2014), pp. 402–403.Google Scholar
Corcoran, J., and Masoud, H., Existential import today: New metatheorems; historical, philosophical, and pedagogical misconceptions. History and Philosophy of Logic, vol. 36 (2015), pp. 39–61.Google Scholar
Corcoran, J., and Sagüillo, J. M., Absence of multiple universes of discourse in the 1936 Tarski consequence-definition paper. History and Philosophy of Logic, vol. 32 (2011), pp. 359–374.Google Scholar
Goldfarb, W., Deductive Logic, Hackett, Indianapolis, IN, 2003.Google Scholar
Lewis, C. I. and Langford, C. H., Symbolic Logic, Century, New York, NY, 1932.Google Scholar
Parry, W., Quantification of the Predicate and Many-sorted Logic. Philosophy and Phenomenological Research, vol. 22 (1966), pp. 342–360.Google Scholar
Quine, W., Philosophy of logic, Harvard University Press, Cambridge, 1970/1986.Google Scholar
Russell, B., Introduction to Mathematical Philosophy, Dover, New York, 1919.Google Scholar
Smiley, T., Syllogism and quantification. The Journal of Symbolic Logic, vol. 27 (1962), pp. 58–72.Google Scholar
Tarski, A., On the concept of logical consequence, Logic, Semantics, Metamathematics, Papers from 1923 to 1938 (Corcoran, John, editor), Hackett, Indianapolis, IN, 1983, pp. 409–420.Google Scholar
Tarski, A., Logic, Semantics, Metamathematics, Papers from 1923 to 1938 (Corcoran, John, editor), Hackett, Indianapolis, IN, 1983.Google Scholar
Tarski, A., and Givant, S., A Formalization of Set Theory without Variables, American Mathematical Society, Providence, RI, 1987.Google Scholar
Tennant, N., Aristotle’s syllogistic and core logic. History and Philosophy of Logic, vol. 35 (2014), pp. 120–147.Google Scholar
Wittgenstein, L., Tractatus Logico-Philosophicus, Kegan Paul, London, 1922.Google Scholar