A temporal semantics for basic logic (Q1037596)

In this paper, the authors give a temporal semantics of Basic Fuzzy Logic, the many-valued logic of continuous t-norms [see \textit{P. Hájek}, Metamathematics of fuzzy logic. Dordrecht: Kluwer Academic Publishers (1998; Zbl 0937.03030)]. The algebraic counterparts of Basic Fuzzy Logic are the BL-algebras. Any linearly ordered BL-algebra can be decomposed as the ordinal sum of linearly ordered MV-algebras (that are the algebraic counterpart of Łukasiewicz logic). Using this decomposition and the completeness theorem of Basic Logic with respect to linearly ordered BL-algebras, a temporal semantics is given for Basic Logic in such a way that the logic of each instant is Łukasiewicz logic with a finite or infinite number of truth values. The main result of the paper is the soundness with respect to flows of time that do not branch in the future and completeness with respect to all finite linear flows of time.