A tableau for set-satisfiability for extended fuzzy logic BL

Abstract

This paper presents a tableau calculus for finding a model for a set-satisfiable finite set of formulas of an extended fuzzy logic BL, a fuzzy logic BL with additional Baaz connective and the involutive negation, if such a model exists. The calculus is a generalisation of a tableau calculus for BL, which is based on the decomposition theorem for a continuous t-norm. The aforementioned tableau calculus for BL is used to prove that a formula A of the extended BL is valid with respect to all continuous t-norms or to find a continuous t-norm * and assignment V of propositional atoms to [0,1] such that *-evaluation V*(A)<1. The tableau calculus presented in this paper enables for a finite set of formulas F of the extended BL and a subset K of [0,1] to find a continuous t-norm * and assignment V of propositional atoms to [0,1] such that *-evaluation V*(A) belongs to K for all formulas A that belong to F, or alternatively to show that such a model does not exist.