Abstract
We consider arrow logics (i.e., propositional multi-modal logics having three -- a dyadic, a monadic, and a constant -- modal operators) augmented with various kinds of infinite counting modalities, such as 'much more', 'of good quantity', 'many times'. It is shown that the addition of these modal operators to weakly associative arrow logic results in finitely axiomatizable and decidable logics, which fail to have the finite base property.
Similar content being viewed by others
References
AndrÉka, H., I. Hodkinson and I. NÉmeti, ‘Finite algebras of relations are representable on finite sets', Journal of Symbolic Logic 64 (1999), 243-267.
Burris, S., and H. Sankappanavar, A course in universal algebra, Springer Verlag, Berlin, 1981.
Gargov, G., S. Passy and T. Tinchev, ‘Modal environment for Boolean speculations’, in: Math. Logic and Applications, Plenum Press, New York, 1987, p. 253-263.
Henkin, L., J. D. Monk and A. Tarski, Cylindric Algebras. Part I, North Holland, 1971.
Henkin, L., J. D. Monk and A. Tarski, Cylindric Algebras. Part II, North Holland, 1985.
van der Hoek, W., ‘Modalities for reasoning about knowledge and quantities’, Ph.D. Dissertation, Free Univ. of Amsterdam, 1992.
van der Hoek, W., M. de Rijke, ‘Counting objects’, Journal of Logic and Computation 5 (1995), 325-345.
Kurucz, Á., ‘Decision problems in algebraic logic’, Ph.D. Dissertation, Hungarian Academy of Sci., 1997.
Maddux, R., ‘Some varieties containing relation algebras’, Trans. Amer. Math. Soc. 272 (1982), 501-526.
Marx, M., ‘Algebraic relativization and arrow logic’, Ph.D. Dissertation, Univ. of Amsterdam, 1995.
Marx, M., ‘Complexity of modal logics of relations’, ILLC Research Report ML-97-02, 1997.
Marx, M., and Sz. MikulÁs, ‘Undecidable relativizations of algebras of relations’, Journal of Symbolic Logic 64 (1999), 747-760.
Marx, M., L. PÓlos and M. Masuch (eds.), Arrow Logic and Multi-Modal Logic, Studies in Logic, Language and Information, CSLI Publications, Stanford, 1996.
Marx, M., and Y. Venema, Multi-dimensional modal logic, Applied Logic Series 4, Kluwer Academic Publishers, 1997.
MikulÁs, Sz., ‘Taming logics’, Ph.D. Dissertation, Univ. of Amsterdam, 1995.
NÉmeti, I., ‘Decidability of relation algebras with weakened associativity’, Procs. Amer. Math. Soc. 100 (1987), 340-344.
Ohlbach, H. J., ‘Translation methods for non-classical logics: an overview’, Bulletin of the IGPL 1,1 (1993), 69-90.
Stebletsova, V., ‘Weakly associative relation algebras with polyadic composition operations’, Logic Group Preprint Series no. 169, Dept. of Philosophy, Utrecht Univ., 1996 (to appear in Studia Logica).
Tarski, A., and S. Givant, A Formalization of Set Theory without Variables, AMS Colloquium Publications vol. 41, 1988.
Venema, Y., ‘A crash course in arrow logic’, in [13], p. 3-34.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Kurucz, Á. Arrow Logic and Infinite Counting. Studia Logica 65, 199–222 (2000). https://doi.org/10.1023/A:1005215730377
Issue Date:
DOI: https://doi.org/10.1023/A:1005215730377