Abstract
Publisher Summary The necessity of simplifying derivations in formal systems has led to the study of the class of all rules of inference such that the use of these rules in derivations does not change the set of provable formulas. This class has been called “the class of admissible rules of inference.” Investigations of the class of admissible rules have, for the most part, dealt with the intuitionistic propositional calculus H of Heyting. The substitution problem for propositional logics λ consists in the recognition, given an arbitrary formula A(xi, pj), whether there exist formulas Bi such that A(Bi, pj) ∈ A. The chapter describes the solution of the above stated problems. The chapter adopts an algebraic approach, using properties of free algebra. A solution is obtained for H by a reduction to the analogous problems for the systems S4 and Grz.
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