A reduction of classical propositional logic to the conjunction-negation fragment of an intuitionistic relevant logic

It is well known that the conjunction-negation fragment of the classical propositional calculus coincides with the conjunction-negation fragment of the Heyting propositional calculus (Godel, 1933; Kleene, 1952, $81). Since conjunction and negation form a sufficient basis for classical propositional logic, this reduces, in a certain sense, classical propositional logic to a fragment of Heyting propositional logic. It seems to be less well known that the same relation obtains between the classical propositional calculus and the Kolmogorov-Johansson propositional calculus usually called “minimal” (Prior, 1962, p. 259; Curry, 1963, p. 279). As a matter of fact, Kohnogorov (1925) introduced this calculus, which is properly included in Heyting’s, with the idea of obtaining some such reduction. It can be asked whether there are naturally motivated propositional calculi properly included in Kolmogorov-Johansson’s for which the same relation with the classical propositional calculus obtains. In this paper we shall present an intuitionistic relevant propositional calculus for which this is the case. Except for some trivial notational modifications, the theoretical framework of this logic is provided by Anderson and Belnap (1975).