<p>Since Meyer and Dunn showed that the rule <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11225_2025_10198_Article_IEq1.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">\(\gamma \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>γ</mi> </math></EquationSource> </InlineEquation> is admissible in <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11225_2025_10198_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">\({\textbf{E}}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="bold">E</mi> </math></EquationSource> </InlineEquation>, relevantists have produced new proofs of the admissibility of <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11225_2025_10198_Article_IEq1.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">\(\gamma \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>γ</mi> </math></EquationSource> </InlineEquation> for an ever more expansive list of relevant logics. We show in this paper that this is not cause to think that this is the norm; rather <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11225_2025_10198_Article_IEq1.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">\(\gamma \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>γ</mi> </math></EquationSource> </InlineEquation> fails to be admissible in a wide variety of relevant logics. As an upshot, we suggest that the proper view of <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11225_2025_10198_Article_IEq1.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">\(\gamma \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>γ</mi> </math></EquationSource> </InlineEquation>-admissibility is as a coherence criterion, and thus as a selection criterion for logical theory choice.</p>

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Failures of \(\gamma \)

  • Tore Fjetland Øgaard,
  • Shawn Standefer

摘要

Since Meyer and Dunn showed that the rule \(\gamma \) γ is admissible in \({\textbf{E}}\) E , relevantists have produced new proofs of the admissibility of \(\gamma \) γ for an ever more expansive list of relevant logics. We show in this paper that this is not cause to think that this is the norm; rather \(\gamma \) γ fails to be admissible in a wide variety of relevant logics. As an upshot, we suggest that the proper view of \(\gamma \) γ -admissibility is as a coherence criterion, and thus as a selection criterion for logical theory choice.