<p>The logic of analytic implication <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\textbf{PAI}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="bold">PAI</mi> </math></EquationSource> </InlineEquation> developed by William T. Parry is part of a family of relevance logics that exhibits a strong variable-sharing property: <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\phi \rightarrow \psi $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ϕ</mi> <mo stretchy="false">→</mo> <mi>ψ</mi> </mrow> </math></EquationSource> </InlineEquation> is a theorem only if every variable occurring in <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\psi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ψ</mi> </math></EquationSource> </InlineEquation> also occurs in <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\phi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ϕ</mi> </math></EquationSource> </InlineEquation>. A demodalized version of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\textbf{PAI}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="bold">PAI</mi> </math></EquationSource> </InlineEquation>, known as <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\textbf{DAI}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="bold">DAI</mi> </math></EquationSource> </InlineEquation>, was formulated in Dunn (<CitationRef CitationID="CR2">1972</CitationRef>). Urquhart (<CitationRef CitationID="CR16">1973</CitationRef>) further introduced a modal extension of <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\textbf{DAI}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="bold">DAI</mi> </math></EquationSource> </InlineEquation>, called the logic of analytic implication with necessity, <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\textbf{AIN}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="bold">AIN</mi> </math></EquationSource> </InlineEquation>. Urquhart conjectured that <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\textbf{PAI}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="bold">PAI</mi> </math></EquationSource> </InlineEquation> can be embedded into <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\textbf{AIN}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="bold">AIN</mi> </math></EquationSource> </InlineEquation> by a Gödel-McKinsey-Tarski style translation. This paper provides a proof of this previously unverified conjecture and extends the result to other logics in the neighbourhood of <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\textbf{PAI}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="bold">PAI</mi> </math></EquationSource> </InlineEquation>. In particular, we show that intuitionistic <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\textbf{PAI}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="bold">PAI</mi> </math></EquationSource> </InlineEquation> can be embedded into <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$\textbf{AIN}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="bold">AIN</mi> </math></EquationSource> </InlineEquation> by the Gödel-McKinsey-Tarski translation.</p>

Modal Embedding of the Logic of Analytic Implication

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2026年5月9日

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Shin Matsuura

摘要

The logic of analytic implication $\textbf{PAI}$ $PAI$ developed by William T. Parry is part of a family of relevance logics that exhibits a strong variable-sharing property: $\phi \rightarrow \psi $ $ϕ \to ψ$ is a theorem only if every variable occurring in $\psi $ $ψ$ also occurs in $\phi $ $ϕ$ . A demodalized version of $\textbf{PAI}$ $PAI$ , known as $\textbf{DAI}$ $DAI$ , was formulated in Dunn (1972). Urquhart (1973) further introduced a modal extension of $\textbf{DAI}$ $DAI$ , called the logic of analytic implication with necessity, $\textbf{AIN}$ $AIN$ . Urquhart conjectured that $\textbf{PAI}$ $PAI$ can be embedded into $\textbf{AIN}$ $AIN$ by a Gödel-McKinsey-Tarski style translation. This paper provides a proof of this previously unverified conjecture and extends the result to other logics in the neighbourhood of $\textbf{PAI}$ $PAI$ . In particular, we show that intuitionistic $\textbf{PAI}$ $PAI$ can be embedded into $\textbf{AIN}$ $AIN$ by the Gödel-McKinsey-Tarski translation.