Abstract
We prove two completeness results, one for the extension of dependence logic by a monotone generalized quantifier Q with weak interpretation, weak in the sense that the interpretation of Q varies with the structures. The second result considers the extension of dependence logic where Q is interpreted as “there exist uncountably many.” Both of the axiomatizations are shown to be sound and complete for FO(Q) consequences.
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References
Engström, F.: Generalized quantifiers in dependence logic. Journal of Logic, Language and Information 21, 299–324 (2012)
Engström, F., Kontinen, J.: Characterizing quantifier extensions of dependence logic. Journal of Symbolic Logic 78(1), 307–316 (2013)
Mostowski, A.: On a generalization of quantifiers. Fund. Math. 44, 12–36 (1957)
Keisler, H.: Logic with the quantifier “there exist uncountably many”. Annals of Mathematical Logic 1(1), 1–93 (1970)
Peters, S., Westerståhl, D.: Quantifiers in Language and Logic. Clarendon Press (2006)
Kolaitis, P.G., Väänänen, J.A.: Generalized quantifiers and pebble games on finite structures. Ann. Pure Appl. Logic 74(1), 23–75 (1995)
Hella, L., Väänänen, J., Westerståhl, D.: Definability of polyadic lifts of generalized quantifiers. J. Logic Lang. Inform. 6(3), 305–335 (1997)
Barwise, J.: On branching quantifiers in English. J. Philos. Logic 8(1), 47–80 (1979)
Immerman, N.: Languages that capture complexity classes. SIAM J. Comput. 16(4), 760–778 (1987)
Dahlhaus, E.: Skolem normal forms concerning the least fixpoint. In: Börger, E. (ed.) Computation Theory and Logic. LNCS, vol. 270, pp. 101–106. Springer, Heidelberg (1987)
Väänänen, J.: Dependence Logic - A New Approach to Independence Friendly Logic. London Mathematical Society Student Texts, vol. 70. Cambridge University Press, Cambridge (2007)
Henkin, L.: Some remarks on infinitely long formulas. In: Infinitistic Methods (Proc. Sympos. Foundations of Math., Warsaw, 1959, pp. 167–183. Pergamon, Oxford (1961)
Kontinen, J., Väänänen, J.A.: On definability in dependence logic. Journal of Logic, Language and Information 18(3), 317–332 (2009)
Lindström, P.: First order predicate logic with generalized quantifiers. Theoria 32, 186–195 (1966)
Kaufmann, M.: The quantifier “there exist uncountably many”, and some of its relatives. In: Barwise, J., Feferman, S. (eds.) Perspectives in Mathematical Logic. Model Theoretic Logics, pp. 123–176. Springer (1985)
Makowsky, J., Tulipani, S.: Some model theory for monotone quantifiers. Archive for Mathematical Logic 18(1), 115–134 (1977)
Kontinen, J., Väänänen, J.: Axiomatizing first order consequences in dependence logic. Annals of Pure and Applied Logic (June 6, 2013)
Engström, F., Kontinen, J., Väänänen, J.: Dependence logic with generalized quantifiers: Axiomatizations. arxiv:1304.0611 (2013)
Barwise, J.: Some applications of henkin quantifiers. Israel Journal of Mathematics 25(1), 47–63 (1976)
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Engström, F., Kontinen, J., Väänänen, J. (2013). Dependence Logic with Generalized Quantifiers: Axiomatizations. In: Libkin, L., Kohlenbach, U., de Queiroz, R. (eds) Logic, Language, Information, and Computation. WoLLIC 2013. Lecture Notes in Computer Science, vol 8071. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39992-3_14
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DOI: https://doi.org/10.1007/978-3-642-39992-3_14
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