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 \(\equiv \) $\equiv$ that allows to separate denotations of sentences from their logical values. Intuitively, \(\equiv \) $\equiv$ combines two sentences \(\varphi \) $\phi$ and \(\psi \) $\psi$ into a true one whenever \(\varphi \) $\phi$ and \(\psi \) $\psi$ 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 \(\textsf{LD}\) $\mathsf{LD}$ , Logic of Descriptions with Suszko’s Axioms \(\textsf{LDS}\) $\mathsf{LDS}$ , Logic of Equimeaning \(\textsf{LDE}\) $\mathsf{LDE}$ ) with non-Fregean versions of certain well-known paraconsistent logics (Jaśkowski’s Discussive Logic \(\textsf{D}_2\) ${\mathsf{D}}_2$ , Logic of Paradox \(\textsf{LP}\) $\mathsf{LP}$ , Logics of Formal Inconsistency \(\textsf{LFI}{1}\) $\mathsf{LFI}1$ and \(\textsf{LFI}{2}\) $\mathsf{LFI}2$ ). We prove that Grzegorczyk’s logics are either weaker than or incomparable to non-Fregean extensions of \(\textsf{LP}\) $\mathsf{LP}$ , \(\textsf{LFI}{1}\) $\mathsf{LFI}1$ , \(\textsf{LFI}{2}\) $\mathsf{LFI}2$ . Furthermore, we show that non-Fregean extensions of \(\textsf{LP}\) $\mathsf{LP}$ , \(\textsf{LFI}{1}\) $\mathsf{LFI}1$ , \(\textsf{LFI}{2}\) $\mathsf{LFI}2$ , and \(\textsf{D}_2\) ${\mathsf{D}}_2$ are more expressive than their original counterparts. Our results highlight that the non-Fregean connective \(\equiv \) $\equiv$ can serve as a tool for expressing various properties of the ontology underlying the logics under consideration.