CORCORAN-MASOUD ON EXISTENTIAL-IMPORT SENTENCE SCHEMAS

CORCORAN-MASOUD ON EXISTENTIAL-IMPORT SENTENCE SCHEMASJohn Corcoran and Hassan Masoud.  Existential-import sentence schemas: classical and relativized Bulletin of Symbolic Logic. 20 (2014) 402–3. This expository lecture was presented at a workshop while we were writing our papers for HPL and BSL. In the workshop, we emphasized the fact that understanding a proof requires understanding the theorem. It is often worthwhile to spend even more time discussing the theorem than presenting the proof.Understanding a theorem is sometimes greatly assisted by speculating about how the theorem was discovered. In this case the theorem was discovered by looking for counterexamples to the falsehood that no universalized conditional implies its corresponding existentialized conjunction and once counterexamples were accumulated, seeing a pattern of what they had in common. This procedure produced guesses that were disconfirmed until finally a true guess was stumbled on and then proved. In many cases, understanding a theorem requires suspecting that it is false: this might be such a case. Anyway, doubt is a mark of understanding. And a mark of a proof is its capacity to dissolve doubt. Creating a proof is not performing a ritual, following the rules of proving.► JOHN CORCORAN AND HASSAN MASOUD, Existential-import sentence schemas:classical and relativized.Philosophy, University at Buffalo, Buffalo, NY 14260-4150, USAE-mail: [email protected]  The variable-enhanced-English sentence schema ‘for every integer x P(x)’ translates the first-order schema ‘x P(x)’ interpreted in the integers: ‘for every integer x’ translates ‘x’ applied to integers.   The Classical Existential-Import Schema, CEIS, has as instances every conditional whose antecedent is a universal sentence and whose consequent is the corresponding existential—replacing the initial universal quantifier by the existential. If for every integer x P(x), then for some integer x P(x).[x P(x) → x P(x)]Obviously every CEIS instance is tautological [logically true].  The Relativized Existential-Import Schema, REIS, has as instances every conditional whose antecedent is a universalized conditional and whose consequent is the corresponding existentialized conjunction—replacing the initial universal quantifier by the existential and replacing the conditional connective by conjunction. If for every integer x [if A(x), then C(x)], then for some integer x [A(x) and C(x)].[x (A(x) → C(x)) → x (A(x) & C(x))]Non-tautological REIS instances are familiar. But, contrary to textbook impressions, certain instances of REIS are tautological.  A necessary and sufficient condition for REIS instances to be tautological follows.   Theorem: A REIS instance is tautological iff the existentialization x A(x) of the antecedent condition A(x) is tautological. [x (A(x) → C(x)) → x (A(x) & C(x))] is tautologicalif and only if x A(x) is tautological.  “If” is obvious. The four key ideas in our “only-if” proof are: 1) $x A(x) is tautological if ~ $x A(x) implies $x A(x).2) ~ $x A(x) implies x (A(x) → C(x)).3) x (A(x) → C(x)) implies x (A(x) & C(x)) , by the hypothesis.4) x (A(x) & C(x)) implies $x A(x).This lecture complements this BULLETIN 11(2005), p. 460; 11(2005), pp. 554-5; 12(2006) pp. 219–40 and 13(2007) pp.143–4; and 17 (2011), pp. 324–5.