On the algebra of relevance logics

Abstract

After recalling some prerequisites from universal algebra in Chapter 1, we recount in Chapter 2 the general theory of deductive (logical) systems. As working examples, we consider the exponential-free fragment CLL of linear logic and some of its extensions, notably the relevance logic Rt and its fragment R (which lacks a sentential `truth' constant t of Rt). In Chapter 2, we focus on what it means for two deductive systems to be equivalent (in the sense of abstract algebraic logic). To be algebraizable is to be equivalent to the equational consequence relation j=K of some class K of pure algebras. This phenomenon, rst investigated in [11], is explored in detail in Chapter 3, and nearly all of the well-known algebraization results for familiar logics can be viewed as instances of it. For example, CLL is algebraized by the variety of involutive residuated lattices. The algebraization of stronger logics is then a matter of restriction. In particular, Rt corresponds in this way to the variety DMM of De Morgan monoids, which is studied in Chapter 4. Moreover, the subvarieties of DMM algebraize the axiomatic extensions of Rt. The lattice of axiomatic extensions of Rt is naturally of logical interest, but our perspective allows us to view its structure through an entirely algebraic lens: it is interchangeable with the subvariety lattice of DMM. The latter is susceptible to the methods of universal algebra. Exploiting this fact in Chapter 5, we determine (and axiomatize) the minimal subvarieties of DMM, of which, as it happens, there are just four. It follows immediately that Rt has just four maximal consistent axiomatic extensions; they are described transparently. These results do not appear to be in the published literature of relevance logic (perhaps for philosophical reasons relating to the status of the constant t). The new ndings of Chapter 5 allow us to give, in Chapter 6, a simpler proof of a theorem of K. Swirydowicz [59], describing the upper part of the lattice of axiomatic extensions of R. Among the many potential applications of this result, we explain one that was obtained recently in [52]: the logic R has no structurally complete axiomatic consistent extension, except for classical propositional logic.