Finite Frames Fail: How Infinity Works Its Way into the Semantics of Admissibility
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.
KeywordsIntermediate logics Admissible rules Finite model property Projective Heyting algebras
- 3.Chagrov A., Zakharyaschev M.: Modal Logic, vol. 77 of Oxford Logic Guides. Oxford University Press, Oxford (1997)Google Scholar
- 5.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.Google Scholar
- 7.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)Google Scholar
- 12.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.Google Scholar
- 15.Iemhoff, R., A note on consequence relations, Logic Group Preprint Series 314, 2013.Google Scholar
- 16.Mints G. E.: Derivability of admissible rules. Journal of Mathematical Sciences 6, 417–421 (1976)Google Scholar
- 19.Skura T. F.: A complete syntactical characterization of the intuitionistic logic. Reports on Mathematical Logic 23, 75–80 (1989)Google Scholar
- 20.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.Google Scholar
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