## Notre Dame Journal of Formal Logic

### Classical Negation and Game-Theoretical Semantics

Tero Tulenheimo

#### Abstract

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 $\Sigma^{2}_{1}$-fragment of third-order logic. Consequently the classical negation remains inexpressible in $L_{2}$.

#### Article information

Source
Notre Dame J. Formal Logic, Volume 55, Number 4 (2014), 469-498.

Dates
First available in Project Euclid: 7 November 2014

https://projecteuclid.org/euclid.ndjfl/1415382952

Digital Object Identifier
doi:10.1215/00294527-2798709

Mathematical Reviews number (MathSciNet)
MR3276408

Zentralblatt MATH identifier
1342.03030

#### Citation

Tulenheimo, Tero. Classical Negation and Game-Theoretical Semantics. Notre Dame J. Formal Logic 55 (2014), no. 4, 469--498. doi:10.1215/00294527-2798709. https://projecteuclid.org/euclid.ndjfl/1415382952

#### 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.
• [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.
• [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.
• [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.
• [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.
• [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.
• [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.
• [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.
• [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.
• [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.
• [17] Kontinen, J., and V. Nurmi, “Team logic and second-order logic,” Fundamenta Informaticae, vol. 106 (2011), pp. 259–72.
• [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.
• [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.