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.
