Modal Embedding of the Logic of Analytic Implication

Abstract The logic of analytic implication PAI PAI developed by William T. Parry is part of a family of relevance logics that exhibits a strong variable-sharing property: ϕ → ψ is a theorem only if every variable occurring in ψ also occurs in ϕ. A demodalized version of PAI PAI, known as DAI DAI, was formulated in Dunn (1972). Urquhart (1973) further introduced a modal extension of DAI DAI, called the logic of analytic implication with necessity, AIN AIN. Urquhart conjectured that PAI PAI can be embedded into 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 PAI PAI. In particular, we show that intuitionistic PAI PAI can be embedded into AIN AIN by the Gödel-McKinsey-Tarski translation.