# On Sequential Modifications of Applied Predicate Calculi

• M. G. Rogava
Part of the Seminars in Mathematics book series (SM, volume 4)

## Abstract

In [1] Gentzen constructed a modification of the classical predicate calculus of the first degree without the cut rule (the calculus LK). This modification of predicate calculus turned out to be a convenient starting point for investigations of a number of important problems of the theory of logical inference and the construction of various deduction search algorithms. Kanger [2], Matulis [3], [4], and Curry [5] constructed sequential modifications of the classical predicate calculus of the first degree, also without the cut rule, which hence had a number of properties particularly essential for the solution of logical deduction search problems (invertibility of all deduction rules, absence of structural rules, “strict” conditions for the variation of terms in quantifier rules). In [6] Kanger mentioned a sequential modification possessing properties of the described kind, also for the classical predicate calculus of the first degree with equality. Analogous sequential modifications, in some respects, are constructed herein for axiomatically given mathematical theories, given by a finite list or a denumerable set of specific axioms, and having the logical means of classical predicate calculus of the first degree with constant functional symbols and equality as the logical deduction apparatus (any such mathematical theory is called an applied predicate calculus below). In the sequential modifications of applied predicate calculi proposed herein, the specific axioms are successfully replaced by deduction rules more adapted to the logical deduction search. When a specific axiom has the form of equality of two terms, the deduction rule replacing this specific axiom is of particularly simple form. To eliminate the cut rule from applied predicate calculi, the Gentzen method should be supplemented by a number of considerations not provided by this method.

## Keywords

Free Variable Predicate Calculus Sequential Calculus Sequential Modification Elementary Formula

## Literature Cited

1. 1.
Gentzen, G., “Untersuchungen über das logische Schliessen. I,” Math. Z., 39(2):176–210 (1934); IL Math. Z., 39(3):405–431 (1934).Google Scholar
2. 2.
Kanger, S., “Provability in logic,” Acta. Univers. Stockholm, Stockholm Studies in Philosophy. I, Stockholm, 1957.Google Scholar
3. 3.
Matulis, V. A., “Two modifications of classical predicate calculus without structural deduction rules,” Doklady Akad. Nauk SSSR, 147 (5):1029–1031 (1962).Google Scholar
4. 4.
Matulis, V. A., “On modifications of classical predicate calculus with a single deduction tree,” Doklady Akad. Nauk SSSR,148(4):768–770 (1963).Google Scholar
5. 5.
Curry, H. B., Foundations of Mathematical Logic, New York, 1963.Google Scholar
6. 6.
Kanger, S., “A simplified proof method for elementary logic,” in: Computer Programming and Formal Systems, Amsterdam, 1963, pp. 87–94.
7. 7.
Mints, G. E., “Skolem method of eliminating positive quantifiers in sequential calculi,” Doklady Akad. Nauk SSSR, 169(l):24–27 (1966).Google Scholar
8. 8.
Kleene, S. C., “Permutability of inferences in Gentzen’s calculi LK and LCI,” Mem. Am. Math. Soc., No. 10, pp. 1–26 (1952).Google Scholar
9. 9.
Kleene, S. C., Introduction to Metamathematics, Van Nostrand, Princeton, 1950.Google Scholar
10. 10.
Presburger, M., Über die Völlstandigkeit eines gewissen Systems der Arithmetik ganzer Zahlen, in welchen die Addition als einzige Operation hervortritt,” Comptes-rendus du I Congress des Mathematiciens des Pays Slaves, Warsaw, 1930, pp. 92–101, 395.Google Scholar