MAT. SBORNIK MATH. USSR SBORNIK TOM 102(144)(1977), NO. 2 Vol. 31 (1977), No. 2 ON ADMISSIBLE RULES OF INTUITIONISTIC PROPOSITIONAL LOGIC UDC 517.12 A. I. CITKIN ABSTRACT. This paper studies modus rules of deduction admissible in intuitionistic propositional logic (a rule is called a modus rule if it corresponds to some sequence and allows passage from the results of any substitution in the formulas in its antecedent to the result of the same substitution in its succedent). Examples of such rules are considered, as well as the derivability of certain rules from others by means of the intuitionistic propositional calculus. An infinite independent system of admissible modus rules is constructed. It is proved that a finite Godel pseudo-Boolean algebra in which all modus rules are valid (i.e. the quasi-identities corresponding to them are valid) is isomorphic to a sequential union of Boolean algebras of power not greater than 4. Bibliography: 17 titles. Figures: 3. In the foundations of mathematics an important role is played by the intuitionistic propositional logic which is given by the well-known calculus [2]-a list of ten axiom schemes and the deductive rule of modus ponens. This calculus yields not only the logic, understood extensionally as the set of formulas true or valid in it, i.e. deducible (provable) in the calculus, but also a relation of deducibility (by means of this calculus) of certain formulas from others. It is natural to raise the question of the possibility of altering the relation of deducibility by postulating other rules of deduction without altering the logic itself. It is known (see [9], for example) that for intuitionistic proposi- tional logic there exist admissible rules of deduction (i.e. rules relative to which the logi© is closed) that are not derivative^) (i.e. the conclusion of the rule cannot be deduced from the premises of the rule with the aid of only the axioms and the rule of modus ponens). The application of admissible rules does not extend the set of formulas valid in intuitionistic logic, but broadens the possibilities for constructing proofs. The problem arises of studying the relation of deducibility of certain formulas from others specified by the collection of all deductive rules (including the axiom schemes) admissible in intuitionistic logic; in particular, the problem of whether this relation can be given by a finite number of rules of an easily discernible form.( 2 ) AMS (MOS) subject classifications (1970). Primary 02B05, 02C15, 02J05, 02D99, 02E05, 06A35; Secondary 02B99, 02E99, 06A25, 02H10, 06A40, 08A15. (')Sometimes, in comparing admissible with derivative rules, the latter are called deducible, for example in [16]. The term "derivative" is taken from [9]. ( 2 )Made appropriately precise (see [9] and below), the latter question is equivalent to the following: can the quasi-variety of pseudo-Boolean algebras, generated by the free algebras, be specified by a finite number of quasi-identities? Cf. Problem 3.30 in [4], posed by D. M. Smirnov, concerning the analogous quasi-variety of groups, and also Problem 7 in [ 3 ]. © American Mathematical Society 1978 279