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.
- Considers both the algebraic and logical perspective within a
common framework.
- Written by experts in the area.
- Easily accessible to graduate students and researchers from other
fields.
- Results summarized in tables and diagrams to provide an overview
of the area.
- Useful as a textbook for a course in algebraic logic, with
exercises and suggested research directions.
- Provides a concise introduction to the subject and leads directly
to research topics.
- The ideas from algebra and logic are developed hand-in-hand and
the connections are shown in every level.
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