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Structural completeness of Medvedev’s propositional calculus. (English) Zbl 0358.02024
The paper shows that the intermediate logic of Medvedev satisfies conditions referred to in a conjecture of Friedman, namely, it is a calculus, with constant \(0\) and connectives \(\land ,\lor \) and \(\rightarrow\) such that (i) \(0\notin T\), (ii) \(\alpha \land \beta \in T\) if and only if \(\alpha, \beta \in T\), (iii) \(\alpha \lor \beta \in T\) if and only if at least one of \(\alpha\), \(\beta\) is in \(T\), (iv) \(\alpha \rightarrow \beta \in T\) if and only if for every substitution, if the substituted form of \(\alpha\) is in \(T\), so is the substituted form of \(\beta\). Condition (iv) is equivalent to the structural completeness concept of Pogorzelski. The intuitionistic calculus itself was known to satisfy the first three of the conditions but not the fourth. The connective \(\sim\) of negation is introduced as required abbreviationally by the definition ‘\(\sim\gamma\)’ for ‘\(\gamma \rightarrow 0\)’. Proofs in the paper are given in a sketched form.
Reviewer: R. Harrop

03B55 Intermediate logics