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Logical systems with classical negation and means for sentential orpropositional self-reference involve, in some way, paradoxical statements suchas the liar. However, the paradox disappears if one replaces classical by anappropriate non-classical negation such as a paraconsistent one (no paradoxarises if the liar is both true and false). We consider a non-Fregean logicwhich is a revised and extended version (Lewitzka 2012) of Epsilon-T-Logicoriginally introduced by (Straeter 1992) as a logic with a total truthpredicate and propositional quantifiers. Self-reference is achieved by means ofequations between formulas which are interpreted over a model-theoreticuniverse of propositions. Paradoxical statements, such as the liar, can beasserted only by unsatisfiable equations and do not correlate withpropositions. In this paper, we generalize Epsilon-T-Logic to a four-valuedlogic related to Dunn/Belnap logic B_4. We also define three-valued versionsrelated to Kleene's logic K_3 and Priest's Logic of Paradox P_3, respectively.In this many-valued setting, models may contain liars and other "paradoxical"propositions which are ruled out by the more restrictive classical semantics.We introduce these many-valued non-Fregean logics as extensions of abstractparameter logics such that parameter logic and extension are of the samelogical type. For this purpose, we define and study abstract logics of typeB_4, K_3 and P_3. Using semantic methods we show compactness of the consequencerelation of abstract logics of type B_4, give a representation as minimallygenerated logics and establish a connection to the approach of (Font 1997).Finally, we present a complete sequent calculus for the Epsilon-T-styleextension of classical abstract logics simplifying constructions originallydeveloped by (Straeter 1992, Zeitz 2000, Lewitzka 1998).

On some many-valued abstract logics and their Epsilon-T-style extensions