Denotation of contextual modal type theory (CMTT): Syntax and meta-programming

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Abstract

The modal logic S4 can be used via a Curry–Howard style correspondence to obtain a λ-calculus. Modal (boxed) types are intuitively interpreted as ‘closed syntax of the calculus’. This λ-calculus is called modal type theory—this is the basic case of a more general contextual modal type theory, or CMTT.

CMTT has never been given a denotational semantics in which modal types are given denotation as closed syntax. We show how this can indeed be done, with a twist. We also use the denotation to prove some properties of the system.

Keywords

S4 modal logic
Curry–Howard correspondence
Contextual modal type theory
Meta-programming
Higher-order logic
Syntax

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