## An extension principle for fuzzy logics.(English)Zbl 0811.03019

In this paper “an extension principle” for closure operators and, in particular, for deduction systems is proposed and examined. Namely any closure operator $$J$$ defined on the class of all subsets of a set $$S$$ is extended into a fuzzy closure operator $$J^*$$ defined on the class of all fuzzy subsets of $$S$$. Thus, the notion of a canonical extension of a deduction system and interesting examples of fuzzy logics are given. In particular, the canonical extension of classical propositional calculus is defined. Further, the canonical extension of first-order logic makes it possible to define the fuzzy Herbrand models of fuzzy programs. Finally, it is shown that the extension principle makes it possible to obtain fuzzy logics related to fuzzy subalgebra theory and graded consequence relation theory.

### MSC:

 03B52 Fuzzy logic; logic of vagueness 03B70 Logic in computer science
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### References:

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