A Short Note on Intuitionistic Propositional Logic with Multiple Conclusions

It is a common misconception among logicians to think that intuitionism is necessarily tied-up with single conclusion (sequent or Natural Deduction) calculi. Single conclusion calculi can be used and are convenient, but they are by no means necessary, as shown by such influential authors as Kleene, Takeuti and Dummett, to cite only three. If single conclusions are not necessary, how do we guarantee that only intuitionistic derivations are allowed? Traditionally one insists on restrictions on particular rules: implication right, negation right and universal quantification right are required to be single conclusion rules. In this note we show that instead of a cardinality restrictionm such as one conclusion only, we can use a notion of dependency between formulae to enforce the constructive character of derivations. Since Gentzen’s pioneering work it has been traditional to associate intuitionism with a single-conclusion sequent calculus or Natural Deduction system. Gentzen’s own sequent calculus presentation of intuitionistic logic, the famous system LJ, is obtained from his classical system LK by means of a cardinality restriction imposed on the succedent of every sequent. It is well-known that Gentzen’s formulation of classical logic, the system LK, uses sequents, expressions of the form Γ ⇒ ∆, where both Γ and ∆ may contain several formula occurrences. The intuition is that the conjunction of the formulae in Γ entails the disjunction of the formulae in ∆. In Gentzen’s calculus for intuitionistic logic LJ sequents are restricted to succedents with at most one formula occurrence. This is convenient, but by no means necessary. Since at least Maehara’s work in the fifties (see [Mae54]) it has been known that intuitionistic logic can be presented via a multiple-conclusion system. Maehara’s system is described in Takeuti’s influential book ([Tak75]), which calls the system LJ’. Also Kleene in his monograph ([Kl52]) presents systems which constitute multiple-conclusion versions of intuitionistic logic. But while both of these (classes of) systems stick to the idea that sequents can have multiple conclusions, they still keep some form of (local) cardinality restriction on succedents: the rules for implication right, negation right and universal quantification right must be modified in that they can only be performed if there is a single formula in the succedent of the premise to which these rules are applied. If we