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Analytic calculi for logics of ordinal multiples of standard t-norms. (English) Zbl 1139.03018
Among several extensions of BL-logics, introduced by P. Hájek [Metamathematics of fuzzy logic. Dordrecht: Kluwer Academic Publishers (1998; Zbl 0937.03030)], this paper opens a new direction. Based on ordinal multiples of the product and of the Łukasiewicz t-norm, respectively, M$$\Pi$$ and MŁ logics are introduced and analytic proof systems for these logics are formulated. These systems are based on $$r$$-hypersequents and fulfil a weakened form of the subformula property, such that the stepwise decomposition of a proposition leads to $$r$$-hypersequents containing no binary connectives. An effective search of a proof of a proposition is done in three steps described in the paper. Interesting results of this paper bring a new task: is it possible to develop an analytic proof system for BL-logics based on standard $$r$$-hypersequents?

##### MSC:
 03B52 Fuzzy logic; logic of vagueness
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