Logic of predicates versus linear logic

The purpose of this note is to suggest an alternative explanation of the relationship between predicate logics with equality and linear logic. The explanation given by Girard resorts to so called exponentials, which are believed to extract from a linear formula A its classical content !A and ?A. Thus, Girard gives a top-down explanation of the relationship between linear and classical logics: the latter lives within the banged fragment of the former. His explanation is given in terms of propositional logic, and works for predicate logic as well. Here, a bottom-up approach is put forward: instead of extracting the classical fragment of a linear logic one looks for a linear extension of a given logic. One such extension is arises very naturally when one starts form predicate logic with equality (without quantiiers). Our attempt to extend such a logic with explicit substitutions leads to a non-commutative intuitionistic linear conservative extension. 1 The world according to Girard In this section the relationship between the usual logic and linear logic as described by Girard is presented and criticized. A recent introduction to linear logic by Girard, cf. 15], starts with the following explanation of the position of usual logics 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 envisaged by Girard. The latter, cf. 11], is proof-theoretic in nature. Firstly, every sequent of classical, resp., intuitionistic, logic is translated into a sequent of commutative linear logic with exponentials. Then one shows that the former can be proved classically, resp., intuitionistically, ii its translation can be proved linearly. Here it is shown that every theory of classical logic of predicates with equality lives in a suuciently rich theory built over a non-commutiative intuitionistic substructural logic: the logic of predicates with explicit substitution. This perspective does not require to call upon exponentials introduced by Girard just to facilitate his embedding.