Operator fuzzy modal logic and principle of resolution

This paper generalizes operator fuzzy logic to operator fuzzy modal logic. Some properties, such as, reduction and transformation of modalities, are discussed. Based on the definitions and properties depicted above, clauses set and standard clauses set are proposed. It is proved that for every clauses set there is a standard clauses set which is logically equal to it. This paper defines λ-unsatisfiable and λ-reduced set of standard clauses set, and proves that standard clauses set is λ-unsatisfiable if and only if its λ-reduced set is unsatisfiable. Therefore, any clauses set is λ-unsatisfiable if and only if the λ-reduced set of its standard clauses set is unsatisfiable. Based on the author´s previous works, λ-reduced set is unsatisfiable if and only if the empty clause can be derived from its resolvable form. Then we complete λ-resolution on an operator fuzzy modal logic system