Denotation of contextual modal type theory (CMTT): Syntax and meta-programming
The modal logic S4 can be used via a Curry-Howard style correspondence to obtain a lambda-calculus. Modal (boxed) types are intuitively interpreted as 'closed syntax of the calculus'. This lambda-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. (C) 2012 Elsevier B.V. All rights reserved.
contextual modal type theory syntax meta-programming s4 modal logic curry-howard correspondence higher-order logic