CORCORAN 2007 ON EXISTENTIAL IMPORT

CORCORAN 2007 ON EXISTENTIAL IMPORT2007. Existential import. Bulletin of Symbolic Logic. 13, 143–4. This abstract makes some important points about existential import. The first point, often omitted in texts that should include it, is that in standard first-order logic absolutely every universal implies the corresponding existential—got by changing the universal quantifier to an existential. The abstract goes on to treat—not arbitrary universals—but universalized-conditionals, where it is widely known that some of them do not imply the corresponding existentialized-conjunction. Of course even in these cases the universalized-conditional does imply its corresponding existential—which is not an existentialized-conjunction but an existentialized-conditional. Students are often not clear whether ‘universal’ is used in the narrow sense or the broad sense, or even whether the difference is important.To the best of my knowledge this abstract contains the first occurrence in print of what was later named THE EXISTENTIAL IMPORT EQUIVALENCE, a proposition later succinctly and conveniently expressed as follows.In order for a given universalized-conditional to imply its corresponding existentialized-conjunction it is necessary and sufficient for the existentialization of the universalized-conditional’s antecedent to be tautological.Of course to understand that sentence it is necessary to understand ‘universalized-conditional’, ‘existentialized-conjunction’, ‘universalized-conditional’s corresponding existentialized-conjunction’, and ‘the existentialization of the universalized-conditional’s antecedent’. To the best of my knowledge, it was I who coined all four of these expressions: I needed them to express my theorem.This 2007 abstract was written before the four expressions were coined and thus should be understandable without them.I could not be sure that someone else previously published substantially the same theorem—regardless of whether they proved it. If someone knows anything to the contrary, I will gladly congratulate them and withdraw my tentative claim of priority.I could not be sure that someone else published expressions for substantially the same four concepts. If someone knows anything to the contrary, I will gladly congratulate them and withdraw my tentative claim of priority.