# Book

Petr Hajek (hajek@uivt.cas.cz)
Wed, 3 Mar 1999 05:23:48 +0100 (MET)

METAMATHEMATICS OF FUZZY LOGIC
Petr Hajek

Trends in Logic vol. 4
Kluwer 1998, 308 pp.
http:/www.wkap.nl/book.htm/0-7923-5238-6
Hardbound, USD 130.00

This book presents a systematic treatment of deductive aspects and
structures of fuzzy logic understood as many valued logic {\em sui
generis}. Some important systems of real-valued propositional and
predicate calculus are defined and investigated. The aim is to
show that fuzzy logic as a logic of imprecise (vague) propositions
does have well developed formal foundations and that most things
usually named fuzzy inference" can be naturally understood as
logical deduction.

There are two main groups of intended readers. First, logicians:
they can see that fuzzy logic is indeed a branch of logic and may
find several very interesting open problems. Second, equally
important, researchers involved in fuzzy logic applications and
soft computing. As a matter of fact, most of these are not
professional logicians so that it can easily happen that an
application, clever and succcessful as it may be, is presented in a
way which is logically not entirely correct or may appear
simple-minded. (Standard presentations of the logical aspects of
fuzzy controllers are the most typical example.) This fact would
not be very important if only the {\em bon ton} of logicians were
harmed; but it is the opinion of the author (who is a mathematical
logician) that a better understanding of the strictly logical basis
of fuzzy logic (in the usual broad sense) is very useful for fuzzy
logic appliers since if they know better what they are doing, they
may hope to do it better. Still more than that: a better mutual
understanding between (classical) logicians and researchers in
fuzzy logic promises to lead to deeper cooperation and new results.

Contents:
Ch. 1 Preliminaries
Ch. 2 Many-valued propositional calculi
Ch. 3 Lukasiewicz propositional logic
Ch. 4 Product logic, Godel logic
Ch. 5 Many-valued predicate logic
Ch. 6 Complexity and undecidability
Ch. 7 On approximate inference
Ch. 8 Generalized quantifiers and modalities
Ch. 9 Miscellanea
Ch. 10 Historical remarks.

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