A new axiomatization for involutive monoidal t-norm-based logic.

*(English)*Zbl 0998.03024The paper gives an alternative axiomatization for the monoidal t-norm-based residuated logic originally introduced by F. Esteva and L. Godo [Fuzzy Sets Syst. 124, 271-288 (2001; Zbl 0994.03017)]. The presented approach stresses the geometric understanding of Girard monoids, especially the rotation invariance \(T(x,y)\leq z\) iff \(T(y,N(z))\leq N(x)\), where \(T\) is a left-continuous triangular norm and \(N\) is a strong negation. The rotation invariance is transformed into axiom schemata for logical calculi, and then serves as a substitute for the standard adjointness condition linking conjunction and implication connectives in t-norm-based residuated logics. The proposed new (and equivalent) axiomatization may give new interesting results in the recently introduced residuated fuzzy logic based on left-continuous t-norms.

Reviewer: Radko Mesiar (Bratislava)

##### MSC:

03B52 | Fuzzy logic; logic of vagueness |

##### Keywords:

axiomatization; monoidal t-norm-based residuated logic; Girard monoids; rotation invariance; triangular norm
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\textit{S. Gottwald} and \textit{S. Jenei}, Fuzzy Sets Syst. 124, No. 3, 303--307 (2001; Zbl 0998.03024)

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##### References:

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