Shin Matsuura

The logic of analytic implication $\mathbf{PAI}$ developed by William T. Parry is part of a family of relevance logics that exhibits a strong variable-sharing property: $\phi\to\psi$ is a theorem only if every variable occurring in $\psi$ also occurs in $\phi$. A demodalized version of $\mathbf{PAI}$, known as $\mathbf{DAI}$, was formulated in \citet{dunn1972modification}. \citet{urquhart1973semantical} further introduced a modal extension of $\mathbf{DAI}$, called the logic of analytic implication with necessity, $\mathbf{AIN}$. Urquhart conjectured that $\mathbf{PAI}$ can be embedded into $\mathbf{AIN}$ by a G\"{o}del-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 $\mathbf{PAI}$. In particular, we show that intuitionistic $\mathbf{PAI}$ can be embedded into $\mathbf{AIN}$ by the G\"{o}del-McKinsey-Tarski translation.