Invertible Sequential Variant of Constructive Predicate Calculus

• S. Yu. Maslov
Part of the Seminars in Mathematics book series (SM, volume 4)

Abstract

It is known that from the viewpoint of the practical search for a logical deduction, sequential variants of the calculus in which the so-called “corollary formula property” is satisfied, possess enormous advantages. The invertible calculuses, i.e., those for which the sequent-conclusion is de-ducible for any rule of deduction if and only if the sequent-premises are deducible, are particularly advantageous. In attempting a deduction search in such calculuses, by transferring from the deducible sequent Σ, whose deduction we wish to obtain, to the sequents (or sequent) from which Σ. is deducible by any of the rules of calculus, we can be assured that deducibilities are not lost, and we do not knowingly deal with a hopeless situation; we seek a deduction of the sequent Σ by using nondeducible sequents (sequent). Such invertible sequential variants possessing the corollary formula property are widely known for classical predicate calculus (see, for example, [1 Ch. XV], [2], [3]). The sequent concept is generalized below, a calculus θ is constructed which is an invertible sequential variant of constructive (intuitionistic) predicate calculus with functional signs and equality, without structural rules, and some properties of calculus θ are also studied. Basically, the calculus θ is similar to the intuitionistic G3 [1, p. 425] and has common features with Eo [2], and to a still greater extent, with Jo [4].

Keywords

Predicate Calculus Logical Deduction Deduction Rule Invertible Version Principal Formula
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Literature Cited

1. 1.
Kleene, S. C., Introduction to Metamathematics. Van Nostrand, Princeton, 1950.Google Scholar
2. 2.
Matulis, V. A., “Two versions of classical predicate calculus without structural deduction rules,” Doklady Akad. Nauk SSSR, 147(5):1029 (1962).Google Scholar
3. 3.
Kanger, S., “A simplified proof method for elementary logic,” in: Computer Programming and Formal Systems, Amsterdam, 1963, p. 87.
4. 4.
Plyushkevichus, R. A., “On a version of constructive predicate calculus without structural deduction rules,” Doklady Akad. Nauk SSSR, 161(2):292 (1965).Google Scholar
5. 5.
Shanin, N. A., Davydov, G. V., Maslov, S. Yu., Mints, G. E., Orevkov, V. P., and Slisenko, A. O., Algorithm of Machine Search for a Natural Logical Deduction in Propositional Calculus (in Russian), Moscow, 1965.Google Scholar
6. 6.
Curry, H. B., Foundations of Mathematical Logic, New York, 1963.Google Scholar
7. 7.
Matulis, V. A., “On versions of classical predicate calculus with a single deduction tree,” Doklady Akad. Nauk SSSR, 148(4):768 (1963).Google Scholar
8. 8.
Kleene, S. C., Per mutability of inferences in Gentzen’s calculi LK and LJ/ Mem. Am. Math. Soc., No. 10, pp. 1–26 (1952).Google Scholar
9. 9.
Lifshits, V. A., “Normal form for deductions in predicate calculus with equality and functional symbols,” this volume, p. 21.Google Scholar