## 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.

### MSC:

 03B45 Modal logic (including the logic of norms)

### Citations:

Zbl 0774.03005; Zbl 0574.03008
Full Text:

### References:

  On the complexity of countermodels for intuitionistic calculus (1980)  Proceedings of the third scandinavian logic symposium pp 15– (1975)  IGPL Newsletter 1 pp 7– (1992)  Canonical formulas for K4. part 1: Basic results 57 pp 1377– (1992)  DOI: 10.1007/BF01982017 · Zbl 0708.03011  Mathematical Sbornik 180 pp 1415– (1989)  Soviet Mathematics Doklady 27 pp 274– (1983)  DOI: 10.1007/BF01053062 · Zbl 0772.03008  DOI: 10.1002/malq.19710170141 · Zbl 0228.02011  Modal logic (1965)  An undecidable problem in correspondence theory 56 pp 1261– (1991) · Zbl 0737.03018  Proceedings of the Xth USSR conference for mathematical logic pp 163– (1990)  Logical methods for constructing effective algorithms pp 135– (1986)  The undecidability of the disjunction property of propositional logics and other related problems 58 pp 967– (1993) · Zbl 0799.03009  DOI: 10.1093/logcom/5.3.287 · Zbl 0856.03017  Mathematical logic, mathematical linguistics and algorithm theory pp 75– (1983)  DOI: 10.1002/malq.19660120129 · Zbl 0154.00407  Philosophical Studies 13 (1971)  DOI: 10.1007/BF00713542 · Zbl 0466.03008  The decidability of certain intermediate logics 33 pp 258– (1968) · Zbl 0175.27103  Algebra and Logic 13 pp 188– (1974)  DOI: 10.1007/BF01668576 · Zbl 0319.02019  Logic notebook (1986)  Logics containing K4, Part 11 50 pp 619– (1985)  Modal logic audits neighbours (1995)
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