Implicational F-Structures and Implicational Relevance Logics

We describe a method for obtaining classical logic from intuitionistic logic which does not depend on any proof system, and show that by applying it to the most important implicational relevance logics we get relevance logics with nice semantical and proof-theoretical properties. Semantically all these logics are sound and strongly complete relative to classes of structures in which all elements except one are designated. Proof-theoretically they correspond to cut-free hypersequential Gentzen-type calculi. Another major property of all these logic is that the classical implication can faithfully be translated into them. The intuitionistic implicational logic is, as is well-known, the minimal logic for which the standard deduction theorem obtains. The classical implicational calculus, in turn, is a sort of a completion of the intuitionistic one, since the set of theorems of any non-trivial implicational logic which extends intuitionistic implicational logic should be a subset of the set of the classical tautologies. Now each of the various standard substructural implicational logics is also usually characterized as the minimal system which satisses a certain deduction theorem. Since minimality does not necessarily mean optimality, it should be interesting to try to apply to implicational relevance logics the process of completion that leads from the intuitionistic implicational logic to the classical one. But how exactly do we get classical logic from intuitionistic logic? The usual answer is that this is done by a passage from a single-conclusion sequential calculus to a multiple-conclusion one, in which the logical rules remain the same, but applications of the oocial structural rules are allowed on both sides of the sequents. However, in logics like the basic implicational relevance logic R ! such a passage yields no new provable sequents, since it is still impossible to deduce there sequents which are not single-conclusion. So this method of 1