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Fuzzy logics based on $$[0,1)$$-continuous uninorms. (English) Zbl 1128.03015
Preceding studies of fuzzy logics were based on triangular norms. The authors turn attention to fuzzy logics where uninorms play the same role. Uninorms are common generalizations of triangular norms and conorms; the neutral element can be an arbitrary number from the interval $$[0,1]$$.
The only continuous residuated uninorms are triangular norms; thus the authors assume continuity only on the halfopen interval $$[0,1)$$ and require so-called conjunctive uninorms, i.e., attaining $$0$$ at $$(0,1)$$. This class includes a lot of interesting examples, including the cross ratio uninorm.
Following the work of Hájek, the authors first introduce an analogue of the basic (fuzzy) logic, a logic where the conjunction (resp. implication) is evaluated by the above uninorms (resp. their residua). Then they study extensions whose semantics is based on specific uninorms. Completeness theorems are proved and the results are put into the context of previous fuzzy logics. A Gentzen-style hypersequent calculus is provided and used to establish co-NP completeness results.

##### MSC:
 03B52 Fuzzy logic; logic of vagueness 03B50 Many-valued logic
##### Keywords:
uninorm; t-norm; fuzzy logic; cross ratio; hypersequent calculus
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