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Superintuitionistic logics as fragments of extensions of the provability logic. (Russian) Zbl 0642.03016
The authors search for relations among different propositional logics used for axiomatization of the notion of provability. They have studied [Actual problems of logic and methodology of science, Collect. Sci. Works, Kiev 1980, 193-230 (1980; Zbl 0535.03008); Vsesayuz. Konf. Mat. Logik., Kishinev (1976)] the logic, which appeared to be an extension of that one considered by R. Solovay [Israel J. Math. 25, 287-304 (1976; Zbl 0352.02019)], defined as \[ G = Cl\circ \Delta + (\Delta (p\supset q)\supset (\Delta p\supset \Delta q)) + (\Delta (\Delta p\supset p)\supset \Delta p) + (A/\Delta A). \] They also considered the logics \[ \text{Grz} = S4 + (\square(\square(p\supset \square p)\supset p)\supset p), \]
\[ I^{\Delta} = I\circ \Delta + (p\supset \Delta p) + ((\Delta p\supset p)\supset p) + (((p\supset q)\supset p)\supset (\Delta q\supset p)) \] and I (intuitionistic porpositional logic.
The present paper is devoted to the proof of the commutativity of the following diagram: \[ \begin{matrix} {\mathcal L}I^\Delta &{\begin{matrix} @>\kappa>> \\ @<<\lambda< \end{matrix}} & {\mathcal L}G \\ \nabla\downarrow && \downarrow\mu \\ {\mathcal L}I &{\begin{matrix} @<{\sigma^{-1}}<< \\ @>>\sigma> \end{matrix}} &{\mathcal L}\text{Grz} \end{matrix} \] Moreover it is shown that contrary to the functions \(\sigma\),\(\kappa\),the functions \(\nabla\) and \(\mu\) are not homeomorphisms of the considered structures.
Reviewer: O.Štěpánková
03B55 Intermediate logics
03B45 Modal logic (including the logic of norms)
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