Closure operators in lattice-valued propositional logics

In this paper, first the relation between two lattice-valued logic systems based on lattice implication algebras, LP(X) and Lvpl, is discussed. And it shows that LP(X) is not a special case of Lvp, since the generalized modus ponens rule in LP(X) is not a rule in Lvpl. On the other hand, in most of non-classical logics, closure operators are defined by classical inclusion relation between sets, only implicative closure operators, L_K-closure operators and the semantic closure operator C^Xξ in Lvpl are all defined by graded inclusion, which accords with the spirit of fuzzy set theory. In the last section of this paper, it is proved that C^Xξ is both an implicative closure operator and an L_K-closure operator. In addition, we illustrate that the semantic closure operator Con in LP(X) is neither an implicative closure operator nor an L_K-closure operator.