Simply-typed constant-domain modal lambda calculus I: distanced beta reduction and combinatory logic

Abstract

A system $\boldsymbolλ_θ$ is developed that combines modal logic and simply-typed lambda calculus, and that generalizes the system studied by Montague and Gallin. Whereas Montague and Gallin worked with Church's simple theory of types, the system $\boldsymbolλ_θ$ is developed in the typed base theory most commonly used today, namely the simply-typed lambda calculus. Further, the system $\boldsymbolλ_θ$ is controlled by a parameter $θ$ which allows more options for state types and state variables than is present in Montague and Gallin. A main goal of the paper is to establish some basic metatheory of $\boldsymbolλ_θ$: (i) an Andrews-like characterization of its models in terms of combinatory logic is given, and this combinatory logic involves a $\mathsf{BCKW}$-like basis rather than an $\mathsf{SKI}$-like basis and (ii) semantic conservation and expressibility results relating $\boldsymbolλ_θ$ to the maximal system $\boldsymbolλ_ω$ are proven. Similar results are proven for the relation between $\boldsymbolλ_ω$ and$\boldsymbolλ$, the corresponding ordinary simply-typed lambda calculus. This answers a question of Zimmermann in the semantics of the simply typed setting. In a companion paper this is extended to Church's simple theory of types. We further develop a partial correspondence between a pure combinatory logic centered on the $\mathsf{BCKW}$-like basis and the weak deductive system for $\boldsymbolλ_ω$ wherein $β$-reduction is not allowed under a lambda abstract, and we use this to show partial deductive conservation between the maximal system $\boldsymbolλ_ω$ and the intermediary systems $\boldsymbolλ_θ$.