Functional completeness for subsystems of intuitionistic propositional logic

The problem of functional completeness for a given logic 5r is the problem of finding a set F of logical operations of Lf such that every logical operation of'Se is explicitly definable by a finite number of compositions from the elements of F. This note presents a generalization of von Kutschera's (1968) approach to the problem of functional completeness for intuitionistic propositional logic (IPL). Besides Lorenz's (1968) analysis wrt a game-theoretical semantics for IPL, yon Kutschera's proof seems to be the earliest published result on functional completeness for IPL. It makes use of a proof-theoretic interpretation specifying general rule schemata in a higher-level Gentzen-style sequent calculus and shows the set of intuitionistic connectives F1 = {-7, A, v, ~ } to be functionally complete for IPL. A proof of functional completeness of FI and F2 = {_1_, A, v, D} wrt Kripke's semantics for IPL can be found in McCullough (1971). Inspired by von Kutschera (1968), Schroeder-Heister (1984) has proved functional completeness of F 2 for IPL wrt an extended natural deduction framework that allows for assumptions of arbitrary finite level. Functional completeness of F2 for IPL wrt natural deduction has also been shown by Prawitz (1979). Zuker & Tragesser (1978) consider a so-called 'inferential' interpretation of Gentzen's natural deduction according to which the meaning of each logical operation is given by its set of introduction rules. They show that in a natural deduction framework for every connective F one can find a finite combination of connectives from r 2 with the same set of introduction rules and thus with the same meaning as F. It is not clear, however, whether the presence of shared introduction rules implies exchangeability in all deductive contexts, which follows by explicit definability. Up to now the probIem of functional completeness for substructural subsystems of IPL, i.e. subsystems of IPL with a restricted set