Valuation sets in lattice-valued propositional logic LP(X)

In this paper, model theory properties of lattice-valued propositional logic LP(X) are studied. We allows to transfer the results of classical model theory to those of LP(X) in a natural way. First, valuation sets of L-fuzzy subsets of formulae and formulae in LP(X) are defined and their properties are discussed. Based on these, a new model relation between an L-fuzzy subset of formulae and a formula is defined and this is the generalization of counterparts in LP(X) and even classical two-valued logic. based on this model relation, a new class of semantic consequence operators are defined and the relation between the model relation and them are discussed. Furthermore, the relationship of logical consequences between L-fuzzy subsets of formulae is investigated. These results will lay a logical foundation for approximate reasoning at every level.