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

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, i.e. 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.