Abstract
Many intermediate logics, even extremely well-behaved ones such as IPC, lack the finite model property for admissible rules. We give conditions under which this failure holds. We show that frames which validate all admissible rules necessarily satisfy a certain closure condition, and we prove that this condition, in the finite case, ensures that the frame is of width 2. Finally, we indicate how this result is related to some classical results on finite, free Heyting algebras.
References
Baker K. A.: Equational axioms for classes of Heyting algebras. Algebra Universalis 6(1), 105–120 (1976)
Balbes R., Horn A.: Injective and projective Heyting algebras. Transactions of the American Mathematical Society 148(2), 549–559 (1970)
Chagrov A., Zakharyaschev M.: Modal Logic, vol. 77 of Oxford Logic Guides. Oxford University Press, Oxford (1997)
Citkin A.: On admissible rules of intuitionistic propositional logic. Mathematics of the USSR-Sbornik 31(2), 279–288 (1977)
Citkin, A., On the recognition of admissibility of some rules in intuitionistic logic, in Vth All-Union Conference in Mathematical Logic, Novosibirsk, 1979, p. 162.
de Jongh D. H. J., Troelstra A.S.: On the connection of partially ordered sets with some pseudo-Boolean algebras. Indagationes Mathematicae 28(3), 317–329 (1966)
Fedorishin B. R., Ivanov V. S.: The finite model property with respect to admissibility for superintuitionistic logics. Siberian Advances in Mathematics 13(2), 56–65 (2003)
Friedman H.: One hundred and two problems in mathematical logic. The Journal of Symbolic Logic 40(2), 113–129 (1975)
Ghilardi S.: Unification in intuitionistic logic. The Journal of Symbolic Logic 64(2), 859–880 (1999)
Ghilardi S.: Unification, finite duality and projectivity in varieties of Heyting algebras. Annals of Pure and Applied Logic 127(1–3), 99–115 (2004)
Goudsmit J. P.: Admissibility and refutation. Archive for Mathematical Logic 53(7–8), 779–808 (2014)
Grigolia, R., Free and projective Heyting and monadic Heyting algebras, in U. Höhle and E. P. Klement (eds.), Non-classical Logics and Their Applications to Fuzzy Subsets, vol. 32 of Theory and Decision Library, Springer, Dordrecht, 1995, pp. 33–52.
Iemhoff R.: On the admissible rules of intuitionistic propositional logic. The Journal of Symbolic Logic 66(1), 281–294 (2001)
Iemhoff R.: Intermediate logics and Visser’s rules. Notre Dame Journal of Formal Logic 46(1), 65–81 (2005)
Iemhoff, R., A note on consequence relations, Logic Group Preprint Series 314, 2013.
Mints G. E.: Derivability of admissible rules. Journal of Mathematical Sciences 6, 417–421 (1976)
Rybakov V. V.: A criterion for admissibility of rules in the model system s4 and the intuitionistic logic. Algebra and Logic 23, 369–384 (1984)
Rybakov V. V., Kiyatkin V. R., Oner T.: On finite model property for admissible rules. Mathematical Logic Quarterly 45(4), 505–520 (1999)
Skura T. F.: A complete syntactical characterization of the intuitionistic logic. Reports on Mathematical Logic 23, 75–80 (1989)
Troelstra, A. S., and D. van Dalen, Constructivism in Mathematics—An Introduction, vol. 121 of Studies in Logic and the Foundations of Mathematics, Elsevier, Amsterdam, 1988.
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Goudsmit, J.P. Finite Frames Fail: How Infinity Works Its Way into the Semantics of Admissibility. Stud Logica 104, 1191–1204 (2016). https://doi.org/10.1007/s11225-016-9672-1
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DOI: https://doi.org/10.1007/s11225-016-9672-1
Keywords
- Intermediate logics
- Admissible rules
- Finite model property
- Projective Heyting algebras