What question did this study set out to answer?

This paper aims to prove that the logic of analytic implication can be embedded into its modal extension.

May 11, 2026Open Access

Modal Embedding of the Logic of Analytic Implication

Key Points

This paper aims to prove that the logic of analytic implication can be embedded into its modal extension.
Provided proof of Urquhart's conjecture about the embedding of PAI into AIN.
Extended results to other logics related to PAI.
Specifically examined the embedding of intuitionistic PAI.
Successfully demonstrated the embedding of PAI into AIN under the Gödel-McKinsey-Tarski translation.
Extended findings to include intuitionistic versions of the logic.
Established broader implications for related systems of logic.

Abstract

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.

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