First-order Substructural Logics: An Algebraic Approach

Substructural logics have been traditionally characterized as those logics such that, when presented by means of a (sequent) proof system, lack one or more of the usual structural rules: weakening, exchange and contraction. Nonetheless, it is well known that they can also be, very usefully, roughly described as the logics of residuated lattices. Indeed, from this point of view we have seen, specially in the last decade, a florescence of works on propositional substructural logics, mainly capitalizing on the fact that they can be given an algebraic semantics based on some class of (expansions or (sub)reducts of) residuated lattices, and hence by using the tools and techniques from (Abstract) Algebraic Logic (see e.g. [8]). The same applies, to a lesser extent, to first-order formalisms for substructural logics, inasmuch they can given a semantics which, though not purely algebraic as in the propositional case, it contains an essential algebraic part together with a domain of individuals to interpret first-order variables and terms. Prominent examples of this approach are the following: