Logic of Predicates with Explicit Substitutions

We present a non-commutative linear logic | the logic of predicates with equality and explicit substitutions. Thus, the position of linear logic with respect to the usual logic is given a new explanation. 1 The world according to Girard A recent introduction to linear logic, cf. 13], starts with the following explanation of the position of usual logic with respect to the linear. Linear logic is not an alternative logic ; it should rather be seen as an extension of usual logic. This paper aims at supporting the same idea. Our justiication of the claim is, however, quite diierent from the one ooered by Girard. The latter, cf. 9], translates every sequent of the usual propositional logic (classical, or intuitionistic) into a sequent of commutative linear logic. Then one shows that a sequent can be proved classically, resp., intuitionistically, ii its translation can be proved linearly. By contrast, our embedding only works on the level of predicate logic. We show that every theory of classical logic of predicates with equality lives as a theory within a non-commutative intuitionistic substructural logic: the logic of predicates with equality and explicit substitution. Also, our explanation does not require to call upon so called exponentials | the modalities introduced by Girard just to facilitate his embedding. Our construction is also diierent from other proposals to move substitutions from the level of metatheory to the theory of logic, cf. 16]. They add substitutions as modal constructions. Here, substitutions are considered new atomic formull. 1.1 Linear logic In classical or intuitionistic logics there are several equivalent ways of saying what conjunction or disjunction is. It has been argued by Avron, cf. 2], that logical connectives do not exist outside the context provided by the underlying logic, i.e., the underlying consequence relation. In other words, only after deening a consequence relation it is possible to talk about connectives associated with it. The connectives, then, are classiied according to properties they satisfy with respect to this consequence relation. It should come as no surprise that the same operation on formull may play diierent connectives for diierent consequence relations. A simple example of this phenomenon is based on the duality of the notion of the multi-conclusion consequence relation. Thus, an operation which is conjunction with respect to one consequence relation is at the same time a disjunction with respect to the dual consequence relation.