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<p>This paper presents a comprehensive proof-theoretic analysis of Jaśkowski’s discussive (or discursive) logic, working with a set of connectives including classical negation and disjunction, as well as so-called (right-)discussive conjunction and discussive implication. By employing established techniques two labelled frameworks are introduced: sequent and natural deduction systems. The paper explores the ability of the proposed calculi to accurately represent Jaśkowski’s discussive logic, particularly in light of its paraconsistent nature, and establishes cut-admissibility and normalization theorems. Additionally, the introduced sequent calculus – shown to allow terminating proof search – is employed to prove the embedding of discussive logic within modal logic <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10992_2025_9808_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">\(\textbf{S5}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold">S</mi> <mn mathvariant="bold">5</mn> </mrow> </math></EquationSource> </InlineEquation>. Finally, it is proved that the natural deduction calculus translates into the corresponding sequent system, with soundness and completeness established for both calculi. Concluding remarks highlight the potential for expanding this study and suggest directions for future research.</p>

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Exploring Jaśkowski’s Discussive Logic

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  • 发表日期：

    2025年9月13日

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  • Fabio De Martin Polo

摘要

This paper presents a comprehensive proof-theoretic analysis of Jaśkowski’s discussive (or discursive) logic, working with a set of connectives including classical negation and disjunction, as well as so-called (right-)discussive conjunction and discussive implication. By employing established techniques two labelled frameworks are introduced: sequent and natural deduction systems. The paper explores the ability of the proposed calculi to accurately represent Jaśkowski’s discussive logic, particularly in light of its paraconsistent nature, and establishes cut-admissibility and normalization theorems. Additionally, the introduced sequent calculus – shown to allow terminating proof search – is employed to prove the embedding of discussive logic within modal logic \(\textbf{S5}\) S 5 . Finally, it is proved that the natural deduction calculus translates into the corresponding sequent system, with soundness and completeness established for both calculi. Concluding remarks highlight the potential for expanding this study and suggest directions for future research.