By
Nikolaos Galatos, School of Information Science, Japan Advanced Institute of Science and Technology
Peter Jipsen, Chapman University, Orange, USA
Tomasz Kowalski, Australian National University, Canberra, Australia
Hiroakira Ono, Japan Advanced Institute of Science and Technology, Ishikawa, Japan
Description
The book is meant to serve two purposes. The first and more obvious one is to present state of the art results in algebraic research into
residuated structures related to substructural logics. The second, less obvious but equally important, is to provide a reasonably gentle
introduction to algebraic logic. At the beginning, the second objective is predominant. Thus, in the first few chapters the reader will
find a primer of universal algebra for logicians, a crash course in nonclassical logics for algebraists, an introduction to residuated
structures, an outline of Gentzen-style calculi as well as some titbits of proof theory - the celebrated Hauptsatz, or cut elimination
theorem, among them. These lead naturally to a discussion of interconnections between logic and algebra, where we try to demonstrate
how they form two sides of the same coin. We envisage that the initial chapters could be used as a textbook for a graduate course, perhaps
entitled Algebra and Substructural Logics.
As the book progresses the first objective gains predominance over the second. Although
the precise point of equilibrium would be difficult to specify, it is safe to say that we enter the technical part with the discussion
of various completions of residuated structures. These include Dedekind-McNeille completions and canonical extensions. Completions are
used later in investigating several finiteness properties such as the finite model property, generation of varieties by their finite
members, and finite embeddability. The algebraic analysis of cut elimination that follows, also takes recourse to completions. Decidability
of logics, equational and quasi-equational theories comes next, where we show how proof theoretical methods like cut elimination are
preferable for small logics/theories, but semantic tools like Rabin's theorem work better for big ones. Then we turn to Glivenko's theorem,
which says that a formula is an intuitionistic tautology if and only if its double negation is a classical one. We generalise it to the
substructural setting, identifying for each substructural logic its Glivenko equivalence class with smallest and largest element. This
is also where we begin investigating lattices of logics and varieties, rather than particular examples. We continue in this vein by presenting
a number of results concerning minimal varieties/maximal logics. A typical theorem there says that for some given well-known variety
its subvariety lattice has precisely such-and-such number of minimal members (where values for such-and-such include, but are not limited
to, continuum, countably many and two). In the last two chapters we focus on the lattice of varieties corresponding to logics without
contraction. In one we prove a negative result: that there are no nontrivial splittings in that variety. In the other, we prove a positive
one: that semisimple varieties coincide with discriminator ones.
Within the second, more technical part of the book another transition
process may be traced. Namely, we begin with logically inclined technicalities and end with algebraically inclined ones. Here, perhaps,
algebraic rendering of Glivenko theorems marks the equilibrium point, at least in the sense that finiteness properties, decidability
and Glivenko theorems are of clear interest to logicians, whereas semisimplicity and discriminator varieties are universal algebra par
exellence. It is for the reader to judge whether we succeeded in weaving these threads into a seamless fabric.
Included in series
Studies in Logic and the Foundations of Mathematics
Audience:
This book is intended for: Research mathematicians and graduate studentsand:Computer scientists
