<p>Prawitz proved that the basic logical connectives <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11225_2025_10215_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="81" /> </InlineMediaObject> <EquationSource Format="TEX">\(\wedge ,\vee ,\rightarrow ,\bot \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>∧</mo> <mo>,</mo> <mo>∨</mo> <mo>,</mo> <mo stretchy="false">→</mo> <mo>,</mo> <mi>⊥</mi> </mrow> </math></EquationSource> </InlineEquation> are not interdefinable in intuitionistic logic. His proof relied essentially on treating negation as a defined connective, and a key lemma fails when <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11225_2025_10215_Article_IEq2.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">\(\lnot \)</EquationSource> <EquationSource Format="MATHML"><math> <mo>¬</mo> </math></EquationSource> </InlineEquation> is included as a primitive connective. I provide a new proof of this result that applies when negation is a primitive connective and which also shows that <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11225_2025_10215_Article_IEq3.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="78" /> </InlineMediaObject> <EquationSource Format="TEX">\(\wedge ,\vee ,\rightarrow ,\lnot \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>∧</mo> <mo>,</mo> <mo>∨</mo> <mo>,</mo> <mo stretchy="false">→</mo> <mo>,</mo> <mo>¬</mo> </mrow> </math></EquationSource> </InlineEquation> are not interdefinable.</p>

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A Proof-Theoretic Note on the Independence of Intuitionistic Connectives

  • Ethan Brauer

摘要

Prawitz proved that the basic logical connectives \(\wedge ,\vee ,\rightarrow ,\bot \) , , , are not interdefinable in intuitionistic logic. His proof relied essentially on treating negation as a defined connective, and a key lemma fails when \(\lnot \) ¬ is included as a primitive connective. I provide a new proof of this result that applies when negation is a primitive connective and which also shows that \(\wedge ,\vee ,\rightarrow ,\lnot \) , , , ¬ are not interdefinable.