Cartesian Isomorphisms Are Symmetric Monoidal: A Justification of Linear Logic

It is proved that all the isomorphisms in the Cartesian category freely generated by a set of objects (i.e., a graph without arrows) can be written in terms of arrows from the symmetric monoidal category freely generated by the same set of objects. This proof yields an algorithm for deciding whether an arrow in this free Cartesian category is an isomorphism. Introduction. We believe that a logic should be completely determined by its assumptions concerning structural rules. Assumptions concerning logical constants are secondary, because they should be invariable when one passes from one logic to another. This point of view, which one may also reach on a priori grounds (see [3]), is corroborated by what one finds in the area of substructural logics-namely, logics with restricted structural rules, like intuitionistic, relevant and linear logic (see [6]). To justify a substructural logic one should first of all find a good reason for the choice of structural rules. Assumptions concerning logical constants will not differ from those one could make in classical logic. What may happen only is that classical connectives split into several nonequivalent variants. One finds this already in intuitionistic logic, where A -* B and -,(A A -'B) cease to be equivalent. All the assumptions for connectives and quantifiers of linear logic may be made in classical logic, too. Only the notorious split between so-called "multiplicative" and "additive" connectives will disappear. The assumptions for the modal operators of linear logic (called "exponentials") are also common: when they are added to a classical base of structural rules, they produce just the modal logic S4. From this point of view, a justification of linear logic should consist primarily in finding reasons for the rejection of the structural rules of contraction and thinning. The literature on linear logic looks usually for these reasons in applications envisaged in computer science. Our ambition is more modest. We find that there may be other reasons, of a technical nature and internal to logic, which may also serve to justify the rejection of contraction and thinning. We want to show that if among structural rules one wants to keep only those that replace collections of premises by isomorphic collections of premises, one should reject exactly contraction and thinning. Received October 15, 1996; revised July 1, 1997. ? 1999. Association for Symbolic Logic 0022-4812/99/6401-001 6/$2.60