Propositional Logics of Closed and Open Substitutions over Heyting's Arithmetic

Notre Dame Journal of Formal Logic 47 (3):299-309 (2006)
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Abstract

In this note we compare propositional logics for closed substitutions and propositional logics for open substitutions in constructive arithmetical theories. We provide a strong example where these logics diverge in an essential way. We prove that for Markov's Arithmetic, that is, Heyting's Arithmetic plus Markov's principle plus Extended Church's Thesis, the logic of closed and the logic of open substitutions are the same

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A survey of propositional realizability logic.Valery Plisko - 2009 - Bulletin of Symbolic Logic 15 (1):1-42.

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References found in this work

Incompleteness in intuitionistic metamathematics.David Charles McCarty - 1991 - Notre Dame Journal of Formal Logic 32 (3):323-358.
A semantical proof of De Jongh's theorem.Jaap van Oosten - 1991 - Archive for Mathematical Logic 31 (2):105-114.
Propositional Calculus and Realizability.Gene F. Rose - 1954 - Journal of Symbolic Logic 19 (2):126-126.

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