Consistency degrees of finite theories in Łukasiewicz propositional fuzzy logic (Q703371)

The authors introduce a measure of consistency degree in Łukasiewicz propositional logic. As is well known [see \textit{R. L. O. Cignoli}, \textit{I. M. L. D'Ottaviano} and \textit{D. Mundici}, Algebraic foundations of many-valued reasoning. Kluwer, Dordrecht (2000; Zbl 0937.06009)], every \(n\)-variable formula determines a McNaughton function \(f\). The (average) truth degree of \(f\) is defined here to be the integral of \(f\) over the unit \(n\)-cube. Building on a previous paper by \textit{G.-J. Wang} and \textit{Y. Leung} [Fuzzy Sets Syst. 136, No. 1, 71--91 (2003; Zbl 1024.03027)], the authors introduce a pseudo-metric on the set of formulas, and a ``divergence degree'' on deductively closed sets of formulas. Inconsistent theories are then shown to have maximum divergence degree -- but not vice versa. Then they exhibit a sequence of consistent theories whose divergence degrees descend to zero. In the final part, a notion of ``consistency degree'' is introduced for finitely axiomatizable theories, and a necessary and sufficient condition for inconsistency is obtained.