# zbMATH — the first resource for mathematics

On finite model property for admissible rules. (English) Zbl 0938.03033
The finite model property (f.m.p.) is important not only for helping to investigate the decidability problem, but also for providing a much more precise semantic description of logics under consideration. This paper is concerned with a technique for verifying the f.m.p. with respect to admissible rules. The authors present a general sufficient condition for the absence of the f.m.p. with respect to admissibility and find that no modal logic extending <span class=”textbf”>K</span>4 <span class=”textrm”>w</span>ith the co-cover property and of width $$>2$$ has the f.m.p. with respect to admissibility. A surprising number of important modal logics of width $$>2$$ are within the scope of this result: K4, S4, GL, K4.1, K4.2, S4.1, S4.2, GL.2, etc. Consequently, the situation is quite opposite to the case of the ordinary f.m.p. where the absolute majority of the important modal logics have the f.m.p. (but not with respect to admissibility). On the other hand, it is shown that all modal logics of width $$\leq 2$$ containing S4 and being not subsystems of three special tabular logics, have the f.m.p. with respect to admissibility.

##### MSC:
 03B45 Modal logic (including the logic of norms)
##### Keywords:
finite model property; admissible rules; modal logic
Full Text:
##### References:
 [1] Bull, That all extensions of S4.3 have the finite model property, Z. Math. Logik Grundlagen Math. 12 pp 341– (1966) · Zbl 0154.00407 [2] Chagrov, Modal logics (1997) [3] Dummett, Modal logics between S4 and S5, Z. Math. Logik Grundlagen Math. 5 pp 250– (1959) · Zbl 0178.30801 [4] Fine, The logics containing S4.3, Z. Math. Logik Grundlagen Math. 17 pp 371– (1971) · Zbl 0228.02011 [5] Fine, Logics Containing K4, Part I, J. Symbolic Logic 39 pp 229– (1974) · Zbl 0287.02010 · doi:10.2307/2272340 [6] Fine, Logics Containing K4, Part II, J. Symbolic Logic 50 pp 619– (1985) [7] Gabbay, A sequence of decidable finitely axiomatizable intermediate logics with the disjunction property, J. Symbolic Logic 39 pp 67– (1974) · Zbl 0289.02032 [8] Gabbay, Selective filtration in modal logics, Theoria 30 pp 323– (1970) [9] Gabbay, A general, filtration method for modal logics, J. Philosophical Logic 1 pp 29– (1972) · Zbl 0248.02025 [10] Grigolia, Investigations in Non-Classical Logics and Formal Theories pp 281– (1983) [11] Kracht, Splittings and the finite model property, J. Symbolic Logic 58 pp 139– (1993) · Zbl 0782.03006 [12] Kracht , M. Internal definability and completeness in modal logic 1991 · Zbl 0742.03004 [13] Lemmon, Algebraic semantics for modal logics I, II, J. Symbolic Logic 31 pp 46– (1966) · Zbl 0147.24805 [14] Rybakov, Decidable noncompact extension of the logic 54, Algebra i Logika 17 pp 210– (1978) · Zbl 0415.03012 · doi:10.1007/BF01670114 [15] Rybakov, Modal logics with LM-axioms, Algebra i Logika 17 pp 455– (1978) · Zbl 0427.03013 · doi:10.1007/BF01674781 [16] Rybakov, A criterion for admissibility of rules in the modal system S4 and the intuitionistic logic, Algebra i Logika 23 pp 369– (1984) · Zbl 0598.03013 [17] Rybakov, Logic Methodology and Philosophy of Science VIII pp 121– (1989) [18] Rybakov, Problems of substitution and admissibility in the modal system Grz and intuitionistic calculus, Annals Pure Appl. Logic 50 pp 71– (1990) · Zbl 0709.03009 [19] Rybakov, Rules of inference with parameters for intuitionistic logic, J. Symbolic Logic 57 pp 912– (1992) · Zbl 0788.03007 [20] Rybakov, A modal analog for Glivenko’s theorem and its applications, Notre Dame J. Formal Logic 33 pp 244– (1992) · Zbl 0788.03020 [21] Rybakov, Criteria for admissibility of inference rules, Modal and intermediate logics with the branching property. Studia Logica 53 pp 203– (1994) · Zbl 0807.03016 [22] Rybakov, Admissibility of Logical Inference Rules (1997) [23] Segerberg, Decidability of S4.1, Theoria 34 pp 7– (1968) [24] Segerberg, An Essay in Classical Modal Logic 1-3 (1971) · Zbl 0311.02028 [25] Shehtman, Riger-Nishimura’s ladders, Doklady Aacad. Nauk SSSR 241 pp 1288– (1978) [26] Wolter, The finite model property in tense logic, J. Symbolic Logic 60 pp 757– (1995) · Zbl 0836.03015
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.