Paraconsistency in Non-Fregean Framework

Abstract

A non-Fregean framework aims to provide a formal tool for reasoning about semantic denotations of sentences and their interactions. Extending a logic to its non-Fregean version involves introducing a new connective $$\equiv $$ ≡ that allows to separate denotations of sentences from their logical values. Intuitively, $$\equiv $$ ≡ combines two sentences $$\varphi $$ φ and $$\psi $$ ψ into a true one whenever $$\varphi $$ φ and $$\psi $$ ψ have the same semantic correlates, describe the same situations, or have the same content or meaning. The paper aims to compare non-Fregean paraconsistent Grzegorczyk’s logics (Logic of Descriptions $$\textsf{LD}$$ LD , Logic of Descriptions with Suszko’s Axioms $$\textsf{LDS}$$ LDS , Logic of Equimeaning $$\textsf{LDE}$$ LDE ) with non-Fregean versions of certain well-known paraconsistent logics (Jaśkowski’s Discussive Logic $$\textsf{D}_2$$ D 2 , Logic of Paradox $$\textsf{LP}$$ LP , Logics of Formal Inconsistency $$\textsf{LFI}{1}$$ LFI 1 and $$\textsf{LFI}{2}$$ LFI 2 ). We prove that Grzegorczyk’s logics are either weaker than or incomparable to non-Fregean extensions of $$\textsf{LP}$$ LP , $$\textsf{LFI}{1}$$ LFI 1 , $$\textsf{LFI}{2}$$ LFI 2 . Furthermore, we show that non-Fregean extensions of $$\textsf{LP}$$ LP , $$\textsf{LFI}{1}$$ LFI 1 , $$\textsf{LFI}{2}$$ LFI 2 , and $$\textsf{D}_2$$ D 2 are more expressive than their original counterparts. Our results highlight that the non-Fregean connective $$\equiv $$ ≡ can serve as a tool for expressing various properties of the ontology underlying the logics under consideration.