Modal Embedding of the Logic of Analytic Implication

Abstract The logic of analytic implication $$\textbf{PAI}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>PAI</mml:mi> </mml:math> developed by William T. Parry is part of a family of relevance logics that exhibits a strong variable-sharing property: $$\phi \rightarrow \psi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ϕ</mml:mi> <mml:mo>→</mml:mo> <mml:mi>ψ</mml:mi> </mml:mrow> </mml:math> is a theorem only if every variable occurring in $$\psi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ψ</mml:mi> </mml:math> also occurs in $$\phi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ϕ</mml:mi> </mml:math> . A demodalized version of $$\textbf{PAI}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>PAI</mml:mi> </mml:math> , known as $$\textbf{DAI}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>DAI</mml:mi> </mml:math> , was formulated in Dunn (1972). Urquhart (1973) further introduced a modal extension of $$\textbf{DAI}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>DAI</mml:mi> </mml:math> , called the logic of analytic implication with necessity, $$\textbf{AIN}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>AIN</mml:mi> </mml:math> . Urquhart conjectured that $$\textbf{PAI}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>PAI</mml:mi> </mml:math> can be embedded into $$\textbf{AIN}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>AIN</mml:mi> </mml:math> 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 $$\textbf{PAI}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>PAI</mml:mi> </mml:math> . In particular, we show that intuitionistic $$\textbf{PAI}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>PAI</mml:mi> </mml:math> can be embedded into $$\textbf{AIN}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>AIN</mml:mi> </mml:math> by the Gödel-McKinsey-Tarski translation.