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**Provability as modality.**
*(Russian)*
Zbl 0535.03008

Actual problems of logic and methodology of science, Collect. sci. Works, Kiev 1980, 193-230 (1980).

[For the entire collection see Zbl 0505.00006.]

This is a survey of the work on provability logic and related modal and superintuitionistic systems done by the first author (who died recently) and his students mainly before 1976. One of the central results is in finding what the authors call the modal and the superintuitionistic fragment of the provability logic. If the basic modality of the provability logic is denoted by \(\Delta\), then \(\square A\) stands for A & \(\Delta\) A. Then the modal (i.e. \(\square\)-) fragment of the provability logic is Grzegorczyk’s logic (this has been proved independently by several other authors). Now Gödel-Tarski translation (prefixing all subformulas by \(\square)\) allows to pass to a superintuitionistic logic corresponding to a given modal one. The former turns out to be the intuitionistic propositional calculus.

This is a survey of the work on provability logic and related modal and superintuitionistic systems done by the first author (who died recently) and his students mainly before 1976. One of the central results is in finding what the authors call the modal and the superintuitionistic fragment of the provability logic. If the basic modality of the provability logic is denoted by \(\Delta\), then \(\square A\) stands for A & \(\Delta\) A. Then the modal (i.e. \(\square\)-) fragment of the provability logic is Grzegorczyk’s logic (this has been proved independently by several other authors). Now Gödel-Tarski translation (prefixing all subformulas by \(\square)\) allows to pass to a superintuitionistic logic corresponding to a given modal one. The former turns out to be the intuitionistic propositional calculus.

Reviewer: G. E. Mints (Leningrad)