Paraconsistency in Non-Fregean Framework

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 φ and ψ ψ into a true one whenever φ φ and ψ 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.