A tableau for set-satisfiability for extended fuzzy logic BL

This paper presents a tableau calculus for finding a model for a set-satisfiable finite set of formulas of a 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 ψ of 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⋆(ψ) < 1. The tableau calculus presented in this paper enables for a finite set of formulas Ψ of BL△∼ and K ⊆ [0, 1] to find a continuous t-norm ⋆ and assignment V of propositional atoms to [0,1] such that ⋆-evaluation V⋆(ψ) ∈ K for all ψ ∈ Ψ , or alternatively to show that such a model does not exist.