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quine_1 - 258 APPENDIX Case 1 All of'zl a without end turn...

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258 A P P E N D I X .- * '*S " L g w E N H E I M * s = B E O R E M 259 Case 1: All of 'zl', 'a', ... without end turn up in sequents of S. then, by the reasoning of p. 9711 , S' will be true also under some It is evident from our general method of generating sequents that, interpretation in the full universe of positive integers. Then so will S. given any sequent Q (of S) whose first.quantifier is universal, an The notion of consistency admits of a natural extension from instantiation ofQ with respect to each of 'zl', 'z,', ... will eventually schemata to classes of schemata. A class of schemata is consistent occur. So, given the universe of positive integers and the stated if, under some interpretation in a non-empty universe, all its mem- interpretation of 'XI', 'z2', ..., it foliows that Q will count as true if hers come out true together. (If some of the schemata of the class all sequents with fewer quantifiers than Q count as true. But also contain 'F' monadically, say, and others contain 'F' dyadically, any sequent Q' whose first quantifier is existential will be true if all what sense is there in speaking of a joint interpretation? Let us sequents with fewer quantifiers are true, for these latter will inLlude settle this point by treating the monadic 'F' and the dyadic 'F' as if an instance of Q'. So, to sum up, any sequent with quantifiers will they were different letters.) Now Lowenheim's theorem admits im- count as true if all sequents with fewer quantifiers do. But 3 makes mediately of this superficial extension: If a finite class of ¶uantifica- all unquantified sequents true. Hence it also makes all singly quanti-
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