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

Logical systems with classical negation and means for sentential or propositional self-reference involve, in some way, paradoxical statements such as the liar. However, the paradox disappears if one replaces classical by an appropriate non-classical negation such as a paraconsistent one (no paradox arises if the liar is both true and false). We consider a non-Fregean logic which is a revised and extended version (Lewitzka 2012) of Epsilon-T-Logic originally introduced by (Straeter 1992) as a logic with a total truth predicate and propositional quantifiers. Self-reference is achieved by means of equations between formulas which are interpreted over a model-theoretic universe of propositions. Paradoxical statements, such as the liar, can be asserted only by unsatisfiable equations and do not correlate with propositions. In this paper, we generalize Epsilon-T-Logic to a four-valued logic related to Dunn/Belnap logic B4. We also define three-valued versions related to Kleene's logic K3 and Priest's Logic of Paradox P3, 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 abstract parameter logics such that parameter logic and extension are of the same logical type. For this purpose, we define and study abstract logics of type B4, K3 and P3. Using semantic methods we show compactness of the consequence relation of abstract logics of type B_4, give a representation as minimally generated logics and establish a connection to the approach of (Font 1997). Finally, we present a complete sequent calculus for the Epsilon-T-style extension of classical abstract logics simplifying constructions originally developed by (Straeter 1992, Zeitz 2000, Lewitzka 1998).