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 B_4. We also define three-valued versions related 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 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 B_4, K_3 and P_3. 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).

💡 Research Summary

The paper tackles the well‑known liar paradox that arises in classical logics equipped with a total truth predicate and propositional self‑reference. The authors observe that the paradox disappears when the classical negation is replaced by a non‑classical, paraconsistent negation that allows a sentence to be both true and false simultaneously. Building on Lewitzka’s revised non‑Fregean Epsilon‑T‑Logic (originally introduced by Straeter), which uses equations between formulas to achieve self‑reference, the authors first recall how unsatisfiable equations isolate paradoxical statements such as the liar, preventing them from denoting any genuine proposition in the model.

The core contribution is a systematic many‑valued generalisation of Epsilon‑T‑Logic. Three families of logics are defined: a four‑valued system of type B₄ (the Dunn/Belnap logic), a three‑valued system of type K₃ (Kleene’s strong three‑valued logic), and a three‑valued system of type P₃ (Priest’s Logic of Paradox). In the B₄‑extension the truth values are {True, False, Both, Neither}; the liar receives the value “Both”. In K₃ the values are {True, False, Undefined}, disallowing true‑false contradictions, while in P₃ the values are {True, False, Both}, allowing contradictions but not undefinedness. Each extension is obtained by enriching the original non‑Fregean semantics with a valuation function that respects the corresponding truth tables and by adapting the interpretation of equations accordingly.

To place these logics in a broader meta‑logical framework, the authors introduce the notion of an “abstract logic of type X”. An abstract logic consists of a set of formulas, a satisfaction relation, and a collection of parameterised operators (quantifiers, propositional connectives, and the equation‑introduction operator). The crucial requirement is that a parameter logic and its Epsilon‑T‑style extension belong to the same type (B₄, K₃ or P₃). Within this setting the paper proves several meta‑theoretical results.

First, compactness of the consequence relation for type B₄ abstract logics is established. The proof adapts the classic filter‑extension argument to the four‑valued setting, showing that if every finite subset of a set of formulas is satisfiable, then the whole set is satisfiable. Second, each abstract logic of type X is shown to be minimally generated: it can be reconstructed from a small set of primitive valuation clauses and the equation‑introduction rule. This representation mirrors Font’s (1997) approach to abstract logics, thereby linking the present work to an established algebraic tradition.

Finally, the authors present a complete sequent calculus for the Epsilon‑T‑style extension of classical abstract logics. Compared with earlier calculi by Straeter (1992), Zeitz (2000) and Lewitzka (1998), the new system eliminates several auxiliary rules and relies on a streamlined set of structural rules together with a single “equation‑introduction” rule that captures self‑reference. Soundness and completeness are proved by constructing canonical models that respect the many‑valued semantics. The calculus works uniformly for all three types, demonstrating that the paraconsistent treatment of negation does not jeopardise proof‑theoretic properties.

In summary, the paper delivers a unified many‑valued treatment of non‑Fregean self‑referential logics, establishes compactness and minimal generation for the corresponding abstract logics, connects these results to earlier algebraic frameworks, and supplies a clean, complete proof system. The work opens the way for applying such logics to areas where paradoxical self‑reference is unavoidable, such as formal semantics of natural language, reflective programming languages, and inconsistent knowledge bases.