Independent bases for rules admissible in pretabular logics.

*(English)*Zbl 0956.03024Study of independence of admissible rules in modal and intermediate logics (or, equivalently, of independent axiomatization for quasivarieties of modal and Heyting algebras) is an interesting problem, of which little is known yet. The problem is non-trivial since there exist tabular modal logics without independent bases for admissibility , whereas many well-known logics do have such bases [see V. V. Rybakov, Admissibility of logical inference rules (Stud. Logic Found. Math. 136, Elsevier Publ., Amsterdam, New York) (1997; Zbl 0872.03002)]. The paper considers this problem for those intermediate and modal logics above S4 which are pretabular (recall that a logic is tabular iff it is determined by a single finite algebra, and pretabular iff it is maximal among non-tabular logics). By the results of Maksimova, Esakia and Meskhi [cf. A. V. Chagrov and M. V. Zakharyaschev, Modal logic (Oxford University Press) (1997; Zbl 0871.03007)], it is known that there exist exactly three pretabular intermediate logics and exactly five pretabular S4-logics, and Rybakov [loc. cit.] showed that among them only one intermediate and three modal logics have finite bases for admissibility. The paper constructs infinite independent bases of admissible rules for the remaining four logics.

Reviewer: V.Shekhtman (Moskva)

##### MSC:

03B45 | Modal logic (including the logic of norms) |

03B55 | Intermediate logics |

03C15 | Model theory of denumerable and separable structures |