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abstract ALONZO CHURCH alternative denial ANDRZEJ MOSTOWSKI applied argument arithmetic assertion axioms Boolean algebra Carnap closure algebra complete concept consistency construction contains decision problem deductive defined definition denote discussion elementary elements entities equivalent existence expressions finite formula free variables Frege functional calculus given Godel Hence Ibid identity II-set individual induction interpretation intuitionistic JOURNAL language Lewis system logique m-valued mathematical logic matrix MAX BLACK means method modal logic modus ponens Mostowski natural numbers negation nominalistic notation notion objects operations ordinal P-realizes pair philosophy positive integers postulates predicate primitive principle Prof proof propositional calculus provable proved quantifiers Quine real numbers recursive functions reference relation replace represented result reviewer rules of inference satisfies semantics sense sentences sequence statements symbolic logic takes the truth-value Tarski theorem theory of types tion true truth truth-value University valid W. V. Quine wffs