Tools and Techniques in Modal Logic Marcus Kracht II ...

About this Book
This book is intended as a course in modal logic for students who have had prior
contact with modal logic and wish to study it more deeply. It presupposes training
in mathematics or logic. Very little specific knowledge is presupposed, most results
which are needed are proved in this book. Knowledge of basic logic—propositional
logic, predicate logic—as well as basic mathematics will of course be very helpful.
The book treats modal logic as a theory, with several subtheories, such as completeness
theory, correspondence theory, duality theory and transfer theory. Thus, the
emphasis is on the inner structure of the theory and the connections between the
subdisciplines and not on coverage of results. Moreover, we do not proceed by discussing
one logic after the other; rather, we shall be interested in general properties
of logics and calculi and how they interact. One will therefore not find sections devoted
to special logics, such as G, K4 or S4. We have compensated for this by a
special index of logics, by which it should be possible to collect all major results on
a specific system. Heavy use is made of algebraic techniques; moreover, rather than
starting with the intuitively simpler Kripke–frames we begin with algebraic models.
The reason is that in this way the ideas can be developed in a more direct and coherent
way. Furthermore, this book is about modal logics with any number of modal
operators. Although this may occasionally lead to cumbersome notation, it was felt
necessary not to specialize on monomodal logics. For in many applications one operator
is not enough, and so modal logic can only be really useful for other sciences
if it provides substantial results about polymodal logics.
No book can treat a subject area exhaustively, and therefore a certain selection
had to be made. The reader will probably miss a discussion of certain subjects such
as modal predicate logic, provability logic, proof theory of modal logic, admissibility
of rules, polyadic operators, intuitionistic logic, and arrow logic, to name the most
important ones. The choice of material included is guided by two principles: first, I
prefer to write about what I understand best; and second, about some subjects there
already exist good books (see [182], [43], [31], [157], [224]), and there is no need to
add another one (which might even not be as good as the existing ones).
I got acquainted with modal logic via Montague Semantics, but it was the book
[169] by Wolfgang Rautenberg that really hooked me onto this subject. It is a pity
that this book did not get much attention. Until very recently it was the only book
which treated modal logic from a mathematical point of view. (Meanwhile, however,
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