On Sequential Modifications of Applied Predicate Calculi
In  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 , Matulis , , and Curry  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  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.
KeywordsFree Variable Predicate Calculus Sequential Calculus Sequential Modification Elementary Formula
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