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 \(\varphi \) and \(\psi \) into a true one whenever \(\varphi \) and \(\psi \) 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 \(\textsf{LD}\), Logic of Descriptions with Suszko’s Axioms \(\textsf{LDS}\), Logic of Equimeaning \(\textsf{LDE}\)) with non-Fregean versions of certain well-known paraconsistent logics (Jaśkowski’s Discussive Logic \(\textsf{D}_2\), Logic of Paradox \(\textsf{LP}\), Logics of Formal Inconsistency \(\textsf{LFI}{1}\) and \(\textsf{LFI}{2}\)). We prove that Grzegorczyk’s logics are either weaker than or incomparable to non-Fregean extensions of \(\textsf{LP}\), \(\textsf{LFI}{1}\), \(\textsf{LFI}{2}\). Furthermore, we show that non-Fregean extensions of \(\textsf{LP}\), \(\textsf{LFI}{1}\), \(\textsf{LFI}{2}\), and \(\textsf{D}_2\) 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.