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Published online by Cambridge University Press: 12 March 2014
Quine has given a method for eliminating the bound variables in first-order predicate logic establishing thus a variable-free formulation called predicate-functor logic. The purpose of this paper is to give an autonomous and complete proof procedure for Quine's predicate-functor logic with identity, without presupposing axioms or inference rules for quantification theory or, for that matter, any other logic.
I acknowledge my great gratitude to Quine who has inspired and stimulated this paper. He has read a first version with proofs in extenso and made critical remarks. I am particularly thankful to him for having informed me about the homogeneous intersection functor pointed out by Dr. Steven T. Kuhn along with advice to use it in place of the original (heterogeneous) one. I am also indebted to Quine for having directed my attention to the single permutation functor. I am further indebted to my friend Ali Karatay for his detailed examination of the draft and numerous suggestions. Finally I acknowledge my gratitude to the anonymous referee of this Journal for various useful suggestions; I am especially indebted to him for the identity rules used in the last version.
2 See Quine, W.V., Algebraic logic and predicate functors, The ways of paradox and other essays, Harvard Press paperback, enlarged edition, 1976Google Scholar. Quine emphasizes in this paper the importance of finding a simple and complete proof procedure for the predicate-functor approach to logic which seems to him “somehow more basic” than the ordinary quantificational approach (pp. 305, 307). A tableau system of proof (without a formal completeness proof) was given by the present author in his Quantification without variables: Predicate-functor logic (polycopied paper in Turkish, , Middle East Technical University, Ankara, 1978)Google Scholar.
3 See Smullyan, R. M., First order logic, Springer-Verlag, New York, 1968, pp. 20 f, 52 fCrossRefGoogle Scholar.
4 See Smullyan, op. cit., pp. 15 ff. esp. p. 24, pp. 53 ff.
5 We can construct a finished systematic tableau for i 1,… in πn (see Smullyan, op. cit., p. 60) by accommodating Smullyan's systematic procedure (ibid., p. 59) for our tableau rules concerning (αβ)-type and identity.
6 Our tableau system of proof can be extended to one for the inconsistency of sets of predicates of L. We can apply the ultraproduct construction to language L* and prove Łos's Theorem. We can then prove the Compactness Theorem for L and infer the Strong Completeness Theorem for predicate-functor logic with identity (viz. that every tableau-consistent set of predicates of L is satisfiable).