Superintuitionistic logics as fragments of extensions of the provability logic. (Russian) Zbl 0642.03016

Kuznetsov, A. V.; Muravitskij, A. Yu.

Mat. Issled. 98, 15-39 (1987).

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

MSC:

03B55	Intermediate logics
03B45	Modal logic (including the logic of norms)

Keywords:

axiomatization of provability; propositional logics