CORCORAN ON EXISTENTIALLY-IMPORTANT PREDICATESThese results are explained, proved, and put into historical, philosophical, and mathematical perspective in the following.http://www.tandfonline.com/doi/full/10.1080/01445340.2014.952947 Corcoran, John. 2014. Existentially-important predicates. Bulletin of Symbolic Logic. 20 (2014) 262.JOHN CORCORAN, Existentially-important predicates.Philosophy, University at Buffalo,Buffalo, NY 14260-4150E-mail:
[email protected] Aristotle’s logic has unlimited existential-import: the universal-affirmative P belongs-to-every Simplies the corresponding existential-affirmative P belongs-to-some Sin every case. First-order logic has limited existential-import: the universalized-conditional "x [S(x) ® P(x)]implies the corresponding existentialized-conjunction $x [S(x) & P(x)]in some but not all cases. Corcoran [2, p. 144] determined which existential-import implications hold: "x [S(x) ® P(x)] implies $x [S(x) & P(x)] iff $x S(x) is tautological. The predicate (“open formula” having only x free) Q(x) is import-carrying (or “existentially-important”) iff $x Q(x) is tautological; otherwise Q(x) is import-free. The existentialized-conjunctions $x [x = t & Q(x)] used in the Gödel-Carnap Diagonal Lemma all contain import-carrying predicates [1, pp. 221f]. How widespread are import-carrying predicates? To answer, let L be any first-order language with any interpretation INT in any universe U. A subset S of U is definable [in L under INT] iff S is the extension of some predicate Q(x). S is import-carrying-definable (respectively, import-free-definable) iff S is the extension of an import-carrying (respectively, import-free) predicate. Given suitable L and INT, the set of even numbers is the extension both of the import-carrying $y [x = (y + y)] and of the import-free "y[x ≠ (y + (y + 1))]. This set is typical. Whether the existential-import implication holds is independent of the content (extension) of the antecedent S(x) if it is non-empty—just as the existential-import implication’s holding is independent of the form and content of the consequent P(x), as indicated above. Existential-importance Theorem: Let L, INT, and U be arbitrary. Every non-empty definable subset of U is both import-carrying-definable and import-free-definable. Whatever ‘widespread’ means, import-carrying predicates are quite widespread, and no less so than import-free predicates. [1] GEORGE BOOLOS, JOHN BURGESS, AND RICHARD JEFFREY, Computability and logic, Cambridge UP, 2007. [2] JOHN CORCORAN, Existential import, this BULLETIN, vol. 13 (2007) pp. 143–4.
