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We propose a new definition of the representation theorem for many-valuedlogics, with modal operators as well, and define the stronger relationshipbetween algebraic models of a given logic and relational structures used todefine the Kripke possible-world semantics for it. Such a new framework offersa new semantics for many-valued logics based on the truth-invarianceentailment. Consequently, it is substantially different from currentdefinitions based on a matrix with a designated subset of logic values, usedfor the satisfaction relation, often difficult to fix. In the case when themany-valued modal logics are based on the set of truth-values that are completedistributive lattices we obtain a compact autoreferential Kripke-stylecanonical representation. The Kripke-style semantics for this subclass of modallogics have the joint-irreducible subset of the carrier set of many-valuedalgebras as set of possible worlds. A significant member of this subclass isthe paraconsistent fuzzy logic extended by new logic values in order to alsodeal with incomplete and inconsistent information. This new theory is appliedfor the case of autoepistemic intuitionistic many-valued logic, based onBelnap's 4-valued bilattice, as a minimal extension of classical logic used tomanage incomplete and inconsistent information as well.

A New Representation Theorem for Many-valued Modal Logics