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Classical negation and game-theoretical semantics. (English) Zbl 1342.03030
Summary: Typical applications of Hintikka’s game-theoretical semantics (GTS) give rise to semantic attributes – truth, falsity – expressible in the $$\Sigma^1_1$$-fragment of second-order logic. Actually a much more general notion of semantic attribute is motivated by strategic considerations. When identifying such a generalization, the notion of classical negation plays a crucial role. We study two languages, $$L_1$$ and $$L_2$$, in both of which two negation signs are available: $$\rightharpoondown$$ and $$\sim$$. The latter is the usual GTS negation which transposes the players’ roles, while the former will be interpreted via the notion of mode. Logic $$L_1$$ extends independence-friendly (IF) logic; $$\rightharpoondown$$ behaves as classical negation in $$L_1$$. Logic $$L_2$$ extends $$L_1$$, and it is shown to capture the $$\Sigma_1^2$$-fragment of third-order logic. Consequently the classical negation remains inexpressible in $$L_2$$.

##### MSC:
 03B60 Other nonclassical logic 03C80 Logic with extra quantifiers and operators 03B15 Higher-order logic; type theory (MSC2010)
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##### References:
 [1] Burgess, J. P., “A remark on Henkin sentences and their contraries,” Notre Dame Journal of Formal Logic , vol. 44 (2003), pp. 185-88. · Zbl 1071.03023 · doi:10.1305/ndjfl/1091030856 [2] Caicedo, X., F. Dechesne, and T. M. V. Janssen, “Equivalence and quantifier rules for logic with imperfect information,” Logic Journal of the IGPL , vol. 17 (2009), pp. 91-129. · Zbl 1160.03007 · doi:10.1093/jigpal/jzn030 [3] Carlson, L. and J. Hintikka, “Conditionals, generic quantifiers, and other applications of subgames” pp. 179-214 in Game-Theoretical Semantics , edited by E. Saarinen, vol. 5 of Synthese Language Library , Reidel, Dordrecht, Netherlands, 1978. [4] Enderton, H. B., “Finite partially-ordered quantifiers,” Zeitschrift für mathematische Logik und Grundlagen der Mathematik , vol. 16 (1970), pp. 393-97. · Zbl 0193.29405 · doi:10.1002/malq.19700160802 [5] Figueira, S., D. Gorín, and R. Grimson, “On the formal semantics of IF-like logics,” Journal of Computer and System Sciences , vol. 76 (2010), pp. 333-46. · Zbl 1197.03028 · doi:10.1016/j.jcss.2009.10.006 [6] Figueira, S., D. Gorín, and R. Grimson, “On the expressive power of IF-logic with classical negation,” pp. 135-45 in Logic, Language, Information and Computation , edited by L. Beklemishev and R. de Queiroz, Springer, Berlin, 2011. · Zbl 1328.03029 · doi:10.1007/978-3-642-20920-8_16 [7] Hintikka, J., “Language-games for quantifiers” pp. 46-72 in Studies in Logical Theory , edited by N. Rescher, Basil Blackwell, Oxford, 1968. [8] Hintikka, J., The Principles of Mathematics Revisited , with an appendix by G. Sandu, Cambridge Univ. Press, Cambridge, 1996. · Zbl 0869.03003 [9] Hintikka, J., “Negation in logic and in natural language,” Linguistics and Philosophy , vol. 25 (2002), pp. 585-600. [10] Hintikka, J., “Truth, negation and other basic notions of logic” pp. 195-219 in The Age of Alternative Logics: Assessing Philosophy of Logic and Mathematics Today , edited by J. van Benthem, et al., Springer, Berlin, 2006. [11] Hintikka, J., and J. Kulas, The Game of Language: Studies in Game-Theoretical Semantics and Its Applications , vol. 22 of Synthese Language Library , D. Reidel, Dordrecht, Netherlands, 1983. [12] Hodges, W., “Compositional semantics for a language of imperfect information,” Logic Journal of the IGPL , vol. 5 (1997), pp. 539-63. · Zbl 0945.03034 · doi:10.1093/jigpal/5.4.539 [13] Hodges, W., “Some strange quantifiers,” pp. 51-65 in Structures in Logic and Computer Science , edited by J. Mycielski, G. Rozenberg, and A. Salomaa, vol. 1261 of Lecture Notes in Computer Science , Springer, Berlin, 1997. · doi:10.1007/3-540-63246-8_4 [14] Hodges, W., “Elementary predicate logic” pp. 1-129 in Handbook of Philosophical Logic, Vol. 1 , edited by D. M. Gabbay and F. Guenthner, Kluwer, Dordrecht, Netherlands, 2001. · doi:10.1007/978-94-015-9833-0_1 [15] Hodges, W., “Logics of imperfect information: Why sets of assignments?” pp. 117-33 in Interactive Logic , edited by J. van Benthem, B. Löwe, and D. Gabbay, vol. 1 of Texts in Logic and Games , Amsterdam Univ. Press, Amsterdam, 2007. [16] Janssen, T. M. V., “Independent choices and the interpretation of IF logic: Logic and games,” Journal of Logic, Language and Information , vol. 11 (2002), pp. 367-87. · Zbl 1003.03025 · doi:10.1023/A:1015542413718 [17] Kontinen, J., and V. Nurmi, “Team logic and second-order logic,” Fundamenta Informaticae , vol. 106 (2011), pp. 259-72. · Zbl 1250.03048 · doi:10.3233/FI-2011-386 [18] Kontinen, J., and J. Väänänen, “A remark on negation in dependence logic,” Notre Dame Journal of Formal Logic , vol. 52 (2011), pp. 55-65. · Zbl 1216.03048 · doi:10.1215/00294527-2010-036 [19] Krynicki, M., “Henkin quantifiers” pp. 193-262 in Quantifiers: Logics, Models and Computation, Vol. 1 , edited by M. Krynicki, M. Mostowski, and L. Szczerba, Kluwer, Dordrecht, Netherlands, 1995. [20] Mostowski, M., “Arithmetic with the Henkin Quantifier and its Generalizations,” pp. 1-25 in Séminaire du Laboratoire Logique, Algorithmique et Informatique Clermontois, Vol. 2 , edited by F. Gaillard and D. Richard, Institut universitaire de technologie de Clermont-Ferrand, Aubière, 1991. [21] Väänänen, J., Dependence Logic: A New Approach to Independence Friendly Logic , vol. 70 of London Mathematical Society Student Texts , Cambridge Univ. Press, Cambridge, 2007. [22] Walkoe, W., “Finite partially-ordered quantification,” Journal of Symbolic Logic , vol. 35 (1970), pp. 535-55. · Zbl 0219.02008 · doi:10.2307/2271440
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