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Resolution in type theory

Extract

In [8] J. A. Robinson introduced a complete refutation procedure called resolution for first order predicate calculus. Resolution is based on ideas in Herbrand's Theorem, and provides a very convenient framework in which to search for a proof of a wff believed to be a theorem. Moreover, it has proved possible to formulate many refinements of resolution which are still complete but are more efficient, at least in many contexts. However, when efficiency is a prime consideration, the restriction to first order logic is unfortunate, since many statements of mathematics (and other disciplines) can be expressed more simply and naturally in higher order logic than in first order logic. Also, the fact that in higher order logic (as in many-sorted first order logic) there is an explicit syntactic distinction between expressions which denote different types of intuitive objects is of great value where matching is involved, since one is automatically prevented from trying to make certain inappropriate matches. (One may contrast this with the situation in which mathematical statements are expressed in the symbolism of axiomatic set theory.).

References

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[1]Andrews, Peter B., A transfinite type theory with type variables, North-Holland Publishing Company, Amsterdam, 1965.
[2]Church, Alonzo, A formulation of the simple theory of types, this Journal, vol. 5 (1940), pp. 5668.
[3]Gould, William Eben, A matching procedure for ω-order logic, Ph.D. thesis, Princeton University, 1966; reprinted as Sci. Rep. No. 4 AFCRL 66–781, Oct. 15, 1966 (Contract No. AF 19(628)-3250), AD 646 560.
[4]Henkin, Leon, Completeness in the theory of types, this Journal, vol. 15 (1950), pp. 8191.
[5]Hindley, Roger, An abstract form of the Church-Rosser theorem. I, this Journal, vol. 34 (1969), pp. 545560.
[6]Kolodner, Ignace I., Fixed points, American Mathematical Monthly, vol. 71 (1964), p. 906.
[7]Prawitz, Dag, Hauptsatz for higher order logic, this Journal, vol. 33 (1968), pp. 452457.
[8]Robinson, J. A., A machine-oriented logic based on the resolution principle, Journal of the Association for Computing Machinery, vol. 12 (1965), pp. 2341.
[9]Schütte, Kurt, Syntactical and semantical properties of simple type theory, this Journal, vol. 25 (1960), pp. 305326.
[10]Smullyan, Raymond M., A unifying principle in quantification theory, Proceedings of the National Academy of Sciences, vol. 49 (1963), pp. 828832.
[11]Smullyan, Raymond M., First-order logic, Springer-Verlag, New York Inc., 1968.
[12]Moto-o-Takahashi, , A proof of cut-elimination in simple type theory, Journal of the Mathematical Society of Japan, vol. 19 (1967), pp. 399410.

Resolution in type theory

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