Proof theory for fuzzy logics.

*(English)*Zbl 1168.03002
Applied Logic Series 36. Dordrecht: Springer (ISBN 978-1-4020-9408-8/hbk; 978-1-4020-9409-5/ebook). 285 p. (2009).

The class of mathematical fuzzy logics is a natural extension of the class of t-norm-based \([0,1]\)-valued logics. The investigation of these logics started with P. Hájek’s seminal monograph [Metamathematics of fuzzy logic. Dordrecht: Kluwer Academic Publishers (1998; Zbl 0937.03030)], which developed the logic of all continuous t-norms. The core basic examples are the infinite-valued Gödel and Łukasiewicz logics, and the product logic. All these fuzzy logics usually are determined by algebraic semantics, and have adequate and rather natural axiomatizations by Hilbert-type calculi, sometimes, however, with infinitary deduction rules.

After a careful explanation of these basic facts, the present monograph offers a study of proof-theoretically more interesting Gentzen-type calculi for such logics. However, one has to use hypersequents instead of ordinary sequents, to get suitable, adequate axiomatizations. Nevertheless, only for some of these mathematical fuzzy logics, Gentzen-type axiomatizations are known, particularly for the core ones.

Main emphasis here is upon the propositional logics. But there is also a chapter on Gentzen-type calculi for first-order fuzzy logics focussing on analogues of the usual Herbrand theorem and Skolemization.

This monograph is a well readable and up-to-date presentation of its topic, which clearly indicates which interesting results have been proved, but which also shows how much remains to be done. It is excellently written by some of the leading experts in the field.

After a careful explanation of these basic facts, the present monograph offers a study of proof-theoretically more interesting Gentzen-type calculi for such logics. However, one has to use hypersequents instead of ordinary sequents, to get suitable, adequate axiomatizations. Nevertheless, only for some of these mathematical fuzzy logics, Gentzen-type axiomatizations are known, particularly for the core ones.

Main emphasis here is upon the propositional logics. But there is also a chapter on Gentzen-type calculi for first-order fuzzy logics focussing on analogues of the usual Herbrand theorem and Skolemization.

This monograph is a well readable and up-to-date presentation of its topic, which clearly indicates which interesting results have been proved, but which also shows how much remains to be done. It is excellently written by some of the leading experts in the field.

Reviewer: Siegfried J. Gottwald (Leipzig)

##### MSC:

03-02 | Research exposition (monographs, survey articles) pertaining to mathematical logic and foundations |

03B50 | Many-valued logic |

03B52 | Fuzzy logic; logic of vagueness |

03F03 | Proof theory in general (including proof-theoretic semantics) |