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<p>In this paper, we propose a relational semantics of propositional language, which unifies the relational semantics of intuitionistic logic, Visser’s Basic Propositional Logic and orthologic. Working in language <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11225_2025_10186_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="75" /> </InlineMediaObject> <EquationSource Format="TEX">\(\{\bot ,\wedge ,\lnot \}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">{</mo> <mi>⊥</mi> <mo>,</mo> <mo>∧</mo> <mo>,</mo> <mo>¬</mo> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11225_2025_10186_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="80" /> </InlineMediaObject> <EquationSource Format="TEX">\(\{\bot ,\wedge ,\rightarrow \}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">{</mo> <mi>⊥</mi> <mo>,</mo> <mo>∧</mo> <mo>,</mo> <mo stretchy="false">→</mo> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> respectively, we axiomatize this basic logic as well as stronger ones corresponding to different combinations of frame conditions: reflexivity, symmetry, and transitivity. We also provide translations from these propositional logics into modal logics.</p>

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A Unified Relational Semantics for BPL, IPL and OL: Axiomatization Without Disjunction

  • 期刊论文

  • 发表日期：

    2025年5月26日

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  • Zhicheng Chen

摘要

In this paper, we propose a relational semantics of propositional language, which unifies the relational semantics of intuitionistic logic, Visser’s Basic Propositional Logic and orthologic. Working in language \(\{\bot ,\wedge ,\lnot \}\) { ⊥ , ∧ , ¬ } and \(\{\bot ,\wedge ,\rightarrow \}\) { ⊥ , ∧ , → } respectively, we axiomatize this basic logic as well as stronger ones corresponding to different combinations of frame conditions: reflexivity, symmetry, and transitivity. We also provide translations from these propositional logics into modal logics.