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Residuated fuzzy logics with an involutive negation. (English) Zbl 0965.03035
The paper gives an extension of BL-logic (basic logic), which is a logic of continuous t-norms interpreting conjunction and their residua interpreting implication. It has been proved that such a logic is complete. BL-logic can be extended to Gödel (G), product (\(\Pi\)) or Łukasiewicz (Ł) logics where conjunction is interpreted by minimum, product, or Łukasiewicz conjunction, respectively. The negation in all logics is interpreted as a derived operation \(\neg a= a\Rightarrow 0\). Unlike Łukasiewicz logic, where negation is involutive, i.e. \(\neg\neg a= a\), the negation in the other two logics is Gödel, i.e. \(\neg a=1\) for \(a=0\) and \(\neg a=0\) otherwise. The paper introduces the concept of strict BL-logic (SBL), for which the linearly ordered BL-algebras satisfying its axioms have Gödel negation. This logic is complete with respect to the class of linearly ordered SBL-algebras. Furthermore, it analyzes the possibility to extend SBL by involutive negation \(\sim\). The axioms for the logic SBL\(_\sim\) and its extensions G\(_\sim\) and \(\Pi_\sim\) are introduced and completeness for all of them is proved. The same is also proved for the predicate versions of these logics. The paper also discusses their further extensions to fuzzy logics with evaluated syntax (in the paper called Rational Pavelka extensions), which when adding infinitary deduction rules become again complete in the sense that the provability and truth degrees coincide.

MSC:
03B52 Fuzzy logic; logic of vagueness
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