Conditional states in finite-valued logics. (English) Zbl 0942.06004

Dubois, Didier (ed.) et al., Fuzzy sets, logics and reasoning about knowledge. Dordrecht: Kluwer Academic Publishers. Appl. Log. Ser. 15, 161-174 (1999).
This paper deals with conditionals in finite-valued Łukasiewicz logics. Generalizing the notion of state introduced by the present reviewer in his paper: “Averaging the truth-value in Łukasiewicz logic” [Stud. Log. 55, No. 1, 113-127 (1995; Zbl 0836.03016)], the authors introduce the notion of conditional state on \(n\)-valued MV-algebras. The latter are the Lindenbaum algebras of the \(n\)-valued sentential calculus of Łukasiewicz. (For background see the monograph by R. L. O. Cignoli, I. M. L. D’Ottaviano and D. Mundici: Algebraic foundations of many-valued reasoning (Trends in Logic – Studia Logica Library 7, Kluwer Academic Publishers, Dordrecht) (2000; Zbl 0937.06009).) For any \(n\)-valued MV-algebra \(M\), the main result of the paper yields a sufficient condition for a Boolean conditional probability defined on the Boolean skeleton of \(M\) to be uniquely extendable to a conditional state on \(M\). As an application, the authors extend a representation theorem of Krauss on Boolean conditionals.
For the entire collection see [Zbl 0927.00008].
Reviewer: D.Mundici (Milano)


06D35 MV-algebras
03B50 Many-valued logic