New Representation Theorem for Many-valued Modal Logics

We propose a new definition of Representation theorem for many-valued modal logics, based on a complete latice of algebraic truth values, and define the stronger relationship between algebraic models of a given logic L and relational structures used to define the Kripke possible-world semantics for L. Such a new framework offers clear semantics for the satisfaction algebraic relation, based on many-valued models of a logic L, avoiding the necessity to define a designated subset of logic values for a satisfaction relation, often difficult to determine for many-valued logic, especially for bilattice based logic. We define the subclass of many-valued modal logics based on distributive lattices which have compact autoreferential cannonical representation. The significant member of this subclass is the paraconsistent fuzzy logic extended by new logic values in order to deal with incomplete and inconsistent information also. The Kripke-style semantics for this subclass of modal logics have as set of possible worlds the joint-irriducible subset of the carrier set of manyvalued algebras. Such a new theory is applied for the case of autoepistemic intuitionistic many-valued logic based on Belnap’s 4-valued bilattice as minimal extension of classic logic used to manage incomplete and inconsistent information also.