Graded consequence relations of lattice-valued propositional logic LP(X)

In this paper, graded consequence relations in lattice-valued propositional logic LP(X) are studied. First, valuation sets in LP(X) are defined and their properties are discussed. Based on these, a graded semantic consequence relation between an L-fuzzy set of formulae and a formula is specified. Accordingly, graded syntactic consequence relation is also given. It is demonstrated that these two classes of graded consequence relations are generalizations of counterparts in classical logic and even in LP(X). Furthermore, graded soundness problem, graded completeness theorem and graded deduction theorem for them are given and proven.