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SOME OBSERVATIONS ABOUT GENERALIZED QUANTIFIERS IN LOGICS OF IMPERFECT INFORMATION

Published online by Cambridge University Press:  12 April 2019

FAUSTO BARBERO
FAUSTO BARBERO*
Affiliation:
Department of Philosophy, History, Culture and Art Studies, University of Helsinki
*
*DEPARTMENT OF PHILOSOPHY, HISTORY, CULTURE AND ART STUDIES UNIVERSITY OF HELSINKI, P.O. BOX 24 (UNIONINKATU 40 A) FIN - 00014, FINLAND E-mail: [email protected]
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Abstract

We analyse the two definitions of generalized quantifiers for logics of dependence and independence that have been proposed by F. Engström, comparing them with a more general, higher order definition of team quantifier. We show that Engström’s definitions (and other quantifiers from the literature) can be identified, by means of appropriate lifts, with special classes of team quantifiers. We point out that the new team quantifiers express a quantitative and a qualitative component, while Engström’s quantifiers only range over the latter. We further argue that Engström’s definitions are just embeddings of the first-order generalized quantifiers into team semantics, and fail to capture an adequate notion of team-theoretical generalized quantifier, save for the special cases in which the quantifiers are applied to flat formulas. We also raise several doubts concerning the meaningfulness of the monotone/nonmonotone distinction in this context. In the appendix we develop some proof theory for Engström’s quantifiers.

Keywords

03B6003C80IF logiclogics of dependence and independencegeneralized quantifiersteam semanticshigher order quantifiers
Type
Research Article
Information
The Review of Symbolic Logic , Volume 12 , Issue 3 , September 2019 , pp. 456 - 486
Copyright
Copyright © Association for Symbolic Logic 2019 

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References

BIBLIOGRAPHY

Aczel, P. (1975). Quantifiers, games and inductive definitions. In Kanger, S., editor. Proceedings of the Third Scandinavian Logic Symposium. Studies in Logic and the Foundations of Mathematics, Vol. 82. New York: North-Holland, Amsterdam and Oxford, and American Elsevier Publishing Company, pp. 1–14.CrossRefGoogle Scholar
Andersson, A. (2002). On second-order generalized quantifiers and finite structures. Annals of Pure and Applied Logic, 115 (1), 1–32.CrossRefGoogle Scholar
Barbero, F. & Sandu, G. (2019). Interventionist counterfactuals on causal teams. In Finkbeiner, B. and Kleinberg, S., editors. Proceedings 3rd Workshop on Formal Reasoning about Causation, Responsibility, and Explanations in Science and Technology, pp. 16–30.Google Scholar
Burtschick, H.-J. & Vollmer, H. (1998). Lindström quantifiers and leaf language definability. International Journal of Foundations of Computer Science, 9(3), 277–294.CrossRefGoogle Scholar
Caicedo, X., Dechesne, F., & Janssen, T. M. V. (2009). Equivalence and quantifier rules for logic with imperfect information. Logic Journal of the IGPL, 17, 91–129.CrossRefGoogle Scholar
Dechesne, F. (2005). Game, Set, Maths: Formal Investigations into Logic with Imperfect Information. Ph.D. Thesis, Tilburg University, Tilburg.Google Scholar
Van der Does, J. (1993). Sums and quantifiers. Linguistics and Philosophy, 16(5), 509–550.CrossRefGoogle Scholar
Durand, A., Ebbing, J., Kontinen, J., & Vollmer, H. (2011). Dependence logic with a majority quantifier. In Chakraborty, S. and Kumar, A., editors. IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011). Leibniz International Proceedings in Informatics (LIPIcs), Vol. 13. Dagstuhl, Germany: Schloss Dagstuhl – Leibniz-Zentrum für Informatik, pp. 252–263.Google Scholar
Durand, A., Ebbing, J., Kontinen, J., & Vollmer, H. (2015). Dependence logic with a majority quantifier. Journal of Logic, Language and Information, 24, 289–305.CrossRefGoogle Scholar
Durand, A., Hannula, M., Kontinen, J., Meier, A., & Virtema, J. (2016). Approximation and dependence via multiteam semantics. Proceedings of the 9th International Symposium on Foundations of Information and Knowledge Systems. Lecture Notes in Computer Science, Vol. 9616. Heidelberg: Springer, pp. 271–291.CrossRefGoogle Scholar
Engström, F. (2012). Generalized quantifiers in Dependence logic. Journal of Logic, Language and Information, 21, 299–324.CrossRefGoogle Scholar
Engström, F. & Kontinen, J. (2013). Characterizing quantifier extensions of dependence logic. Journal of Symbolic Logic, 78(1), 307–316.CrossRefGoogle Scholar
Engström, F., Kontinen, J., & Väänänen, J. (2013). Dependence logic with generalized quantifiers: Axiomatizations. In Libkin, L., Kohlenbach, U., and de Queiroz, R., editors. Logic, Language, Information, and Computation 20th International Workshop, WoLLIC 2013, Darmstadt, Germany, Vol. 8071. Berlin: Springer, pp. 138–152.Google Scholar
Galliani, P. (2012). Inclusion and exclusion dependencies in team semantics-on some logics of imperfect information. Annals of Pure and Applied Logic, 163(1), 68–84.CrossRefGoogle Scholar
Galliani, P. (2013). Epistemic operators in Dependence logic. Studia Logica, 101, 367–397.CrossRefGoogle Scholar
Grädel, E. & Väänänen, J. (2013). Dependence and independence. Studia Logica, 101, 399–410.CrossRefGoogle Scholar
Henkin, L. (1961). Some remarks on infinitely long formulas. Infinitistic methods. In Proceedings of the Symposium on Foundations of Mathematics. Warsaw and Oxford: Państwowe Wydawnictwo Naukowe and Pergamon Press, pp. 167–183.Google Scholar
Hintikka, J. & Sandu, G. (1989). Informational independence as a semantical phenomenon. In Fenstad, J. E., Frolov, I. T., and Hilpinen, R., editors. Logic, Methodology and Philosophy of Science VIII. Amsterdam: Elsevier Science Publishers B.V., pp. 571–589.Google Scholar
Hodges, W. (1997a). Compositional semantics for a language of imperfect information. Logic Journal of the IGPL, 5, 539–563.CrossRefGoogle Scholar
Hodges, W. (1997b). Some strange quantifiers. In Mycielski, J., Rozenberg, G., and Salomaa, A., editors. Structures in Logic and Computer Science. Lecture Notes in Computer Science , Vol. 1261. London, UK: Springer-Verlag, pp. 51–65.CrossRefGoogle Scholar
Hyttinen, T., Paolini, G., & Väänänen, J. (2015). Quantum team logic and bell’s inequalities. Review of Symbolic Logic, 08(04), 722–742.CrossRefGoogle Scholar
Kontinen, J. (2010). Definability of second order generalized quantifiers. Archive for Mathematical Logic, 49(3), 379–398.CrossRefGoogle Scholar
Kontinen, J. & Väänänen, J. (2009). On definability in dependence logic. Journal of Logic, Language and Information, 18(3), 317–332.CrossRefGoogle Scholar
Lindström, P. (1966). First-order predicate logic with generalized quantifiers. Theoria, 32, 186–195.Google Scholar
Mann, A. L., Sandu, G., & Sevenster, M. (2011). Independence-Friendly Logic - A Game-Theoretic Approach. London Mathematical Society Lecture Note Series, Vol. 386. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Mostowski, A. (1957). On a generalization of quantifiers. Fundamenta Mathematicae, 44(2), 12–36.CrossRefGoogle Scholar
Peters, S. & Westerståhl, D. (2006). Quantifiers in Language and Logic. Oxford: Oxford University Press.Google Scholar
Sandu, G. (1993). On the logic of informational independence and its applications. Journal of Philosophical Logic, 22, 29–60.CrossRefGoogle Scholar
Scha, R. (1981). Distributive, collective and cumulative quantification. In Groenendijk, J. A. G., Janssen, T. M. V., and Stokhof, M. B. J., editors. Formal Methods in the Study of Language, Part 2. Amsterdam: Mathematisch Centrum, pp. 483–512.Google Scholar
Sevenster, M. (2014). Dichotomy result for independence-friendly prefixes of generalized quantifiers. The Journal of Symbolic Logic, 79(04), 1224–1246.CrossRefGoogle Scholar
Väänänen, J. (2007). Dependence Logic: A New Approach to Independence Friendly Logic. London Mathematical Society Student Texts, Vol. 70. Cambridge: Cambridge University Press.CrossRefGoogle Scholar