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 ≡ that allows to separate denotations of sentences from their logical values. Intuitively, ≡ combines two sentences φ and ψ into a true one whenever φ and 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 LD LD, Logic of Descriptions with Suszko’s Axioms LDS LDS, Logic of Equimeaning LDE LDE) with non-Fregean versions of certain well-known paraconsistent logics (Jaśkowski’s Discussive Logic D₂ D2, Logic of Paradox LP LP, Logics of Formal Inconsistency LFI1 LFI1 and LFI2 LFI2). We prove that Grzegorczyk’s logics are either weaker than or incomparable to non-Fregean extensions of LP LP, LFI1 LFI1, LFI2 LFI2. Furthermore, we show that non-Fregean extensions of LP LP, LFI1 LFI1, LFI2 LFI2, and D₂ D2 are more expressive than their original counterparts. Our results highlight that the non-Fregean connective ≡ can serve as a tool for expressing various properties of the ontology underlying the logics under consideration.