Non-commutative Łukasiewicz propositional logic (Q818518)

The system \(\mathcal{PL}\) is introduced as the non-commutative counterpart of Łukasiewicz propositional logic. The models are the non-commutative generalizations of MV-algebras as presented by \textit{G. Georgescu} and \textit{A. Iorgulescu} in ``Pseudo-MV algebras'' [Mult.-Valued Log. 6, No.~1--2, 95--135 (2001; Zbl 1014.06008)]. Since any psMV-algebra is a pseudo BL-algebra with some additional properties, this paper follows the ideas presented by \textit{P. Hájek} in ``Observations on non-commutative fuzzy logic'' [Soft Comput. 8, No.~1, 38--43 (2003; Zbl 1075.03009)]. After analyzing the system \(\mathcal{PL}\), some further axiomatic extensions \(\mathcal{PL}_{r}\) and \(\mathcal{L}\) are considered. Adding one unary logical connective, which satisfies the modal axiom (K), leads to the propositional calculus \(\mathcal {PL}_{vt}\) that can be interpreted as an axiomatization for the fuzzy truth value ``very true'' in the non-commutative Łukasiewicz logic. \(\mathcal{PL}_{vt}\) is proved to be a conservative extension of \(\mathcal{PL}_{r}\).