Canonical formulas for K4. II: Cofinal subframe logics. (English) Zbl 0884.03014

This paper continues Part I by the same author [J. Symb. Log. 57, 1377-1402 (1992; Zbl 0774.03005)]. It studies properties of cofinal subframe (csf) and subframe (sf) \({\mathbf K4}\)-modal and intermediate logics. These classes are defined in terms of canonical formulas introduced in Part I. An equivalent definition is given by Theorems 5.1, 5.7: a logic is csf (respectively, sf) iff the class of its frames is closed under cofinal subframes (respectively, under subframes). For the intermediate case the syntactic definition has the following form: a logic is csf (respectively, sf) iff it is axiomatizable by disjunction free formulas (respectively, by implicative formulas). It is proved (both in the modal and the intermediate cases) that there exists a continuum of sf-logics, and a continuum of csf-logics, which are not sf. Every csf-logic has the f.m.p. (Theorem 4.1); this stregthens the corresponding result for sf-logics [K. Fine, J. Symb. Log. 50, 619-651 (1985; Zbl 0574.03008)]. Moreover, every csf-logic has the exponential size model property (Theorem 4.3). Another result of K. Fine also transfers to csf-logics: within this family elementarity, canonicity and compactness are equivalent (Theorem 6.1), and for the intermediate case every csf-logic enjoys these properties (Theorem 6.8). Also, unlike the general case, these properties are decidable for finitely axiomatizable csf-logics (Theorem 6.4), and every csf-logic is elementary on the class of finite frames (Corollary 6.5). It is also proved that canonicity transfers from every \({\mathbf S4}\)-logic to its intermediate fragment, and from every intermediate logic to its minimal modal companion (Theorem 6.6). The last section considers quasinormal csf- and sf- logics.


03B45 Modal logic (including the logic of norms)
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