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The Game of Language pp 1–31Cite as

Game-Theoretical Semantics: Insights and Prospects

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Part of the book series: Synthese Language Library ((SLAP,volume 22))

Abstract

The paradigm problem for game-theoretical semantics (GTS) is the treatment of quantifiers, primarily logicians’ existential and universal quantifiers. As far as the uses of quantifiers in logic and mathematics are concerned, the basic ideas codified in GTS have long been an integral part of logicians’ and mathematicians’ folklore. Everybody who has taken a serious course in calculus remembers the definition of what it means for a function y = f(x) to be continuous at x0. It means that, given a number w, however small, we can find ɛ such that |f(x) - F(x0) | < δ given any x such that | x – x0| < ɛ.1 The most natural way of making this jargon explicit is to envision each choice of the value of an existentially bound variable to be my own move in a game, and each choice of the value of a universally bound variable a move in the same game by an imaginary opponent. The former is what is covered by such locutions as “we can find”, whereas the latter is what is intended by references to what is “given” to us. This is indeed what is involved in the continuity example.

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Notes

  1. Cf., eg, G. H. Hardy, A Course of Pure Mathematics, seventh edition, Cambridge University Press, Cambridge, 1938, p. 186.

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  2. See section II (i) of the bibliography at the end of this volume.

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  3. See section II (iv) of the bibliography.

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  4. See section II (v) of the bibliography.

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  5. In Dana Scott, “A Game-Theoretic Interpretation of Logical Formulae”, McCarthy seminar, Stanford University, July 1968 (unpublished).

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  6. See section II (vi) of the bibliography.

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  7. Most of the early work on GTS has been collected in Esa Saarinen, editor, Game-Theoretical Semantics, D. Reidel, Dordrecht, 1979. The second essay reprinted there (Jaakko Hintikka, “Quantifiers in Logic and Quantifiers in Natural Languages”) comes perhaps closest to an introductory discussion.

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  8. As has been explained, semantical games are played on the domain of individuals that our interpreted language can be used to convey information about. (For instance, moves connected with quantifiers are choices of members of the domain D.) Semantical games are hence essentially different from so-called dialogical games, whose moves are utterances or other kinds of propoundings of sentences. This distinguishes the semantical games of GTS from, eg, the dialogical games of Paul Lorenzen. (For them, see section IV (i) of the bibliography.) I will comment on the relation of Lorenzen’s games to mine briefly in chapter 2 below.

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  9. Even though the game-theoretical concepts used here are almost self-explanatory, a couple of brief explanations may be in order. A “strategy” of a player is a rule that tells the player in question what to do in each conceivable situation that can come up in the game. In this paper, only pure (nonprobabilistic) strategies are considered. By means of the concept of strategy, the whole game can always be reduced to a choice of strategy by each player. This is known as the “normal form” of a game. If a strategy of a player in a two-person zero-sum game wins against any strategy of one’s opponent, it is said to be a “winning” one.

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  10. Cf. here the classic paper by David Gale and F. M. Stewart, “Infinite Games with Perfect Information”, in H. W. Kuhn and A. W. Tucker, editors. Contributions to the Theory of Gomes, vol. 2 (Annals of Mathematics Studies, vol. 28), Princeton University Press, Princeton, 1953, pp. 245–66, and cf. Morton David, “Infinite Games of Perfect Information”, in M. Dresher, L. S. Shapley, and A. W. Tucker, editors, Advances in Game Theory (Annals of Mathematics Studies, vol. 52 ), Princeton University Press, Princeton, 1964, pp. 85–101.

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  11. See here section II (viii) of the bibliography.

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  12. For the distinction, see, eg, John von Neumann and Oskar Morgenstern, Theory of Games and Economic Behavior, Princeton University Press, Princeton, 1953, pp. 76–84.

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  13. See here chapter 3 below.

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  14. See section II (v) of the bibliography.

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  15. This observation was first made in the following papers: A. Mostowski, “On a System of Axioms Which Has No Recursively Enumerable Arithmetic Model”, Fundamenta Mathematicae 40 (1953), 56–61; A. Mostowski, “A Formula with No Recursively Enumerable Model”, Fundamenta Mathematicae 42 (1955), 125–40; G. Kreisel, “A Note on Arithmetic Models for Consistent Formulae of the Predicate Calculus”, Proceedings of the Xlth International Congress of Philosophy, vol. 14, Amsterdam & Louvain, 1953, pp. 39–49.

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  16. See Leon Henkin, “Completeness in the Theory of Types”, Journal of Symbolic Logic 15 (1950), 81–91 (but cf. Peter Andrews, “General Models and Extensionality”, Journal of Symbolic Logic 37 (1972), 395–97); Jaakko Hintikka, “Standard vs. Nonstandard Logic: Higher-Order, Modal, and First-Order Logics”, in Evandro Agazzi, editor, Modern Logic: A Survey, D. Reidel, Dordrecht, 1980, pp. 283–96; Jaakko Hintikka, “Is Alethic Modal Logic Possible?” Acta Philosophica Fennica 35 (1982), 89-105; and the further literature referred to in the last two papers.

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  17. See section II (v) of the bibliography and chapter 3 below.

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  18. For their logical theory, and for applications to natural languages, see section II (i) of the bibliography.

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  19. See Jaakko Hintikka, “Quantifiers vs. Quantification Theory”, Linguistic Inquiry 5 (1979), 153–77, and cf. M. Krynicki and A. H. Lachlan, “On the Semantics of the Henkin Quantifier”, Journal of Symbolic Logic 44 (1979), 184–200.

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  20. See Jon Barwise, “On Branching Quantifiers in English”, Journal of Philosophical Logic 8 (1979), 47–80.

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  21. For a discussion of such reduction problems, see W. J. Walkoe, Jr., “Finite Partially Ordered Quantification”, Journal of Symbolic Logic 35 (1970), 535–55.

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  22. See section ll (iii) of the bibliography.

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  23. See Jaakko Hintikka, “Leibniz on Plenitude, Relations, and the ‘Reign of Law’”, in Simo Knuuttila, editor, Reforging the Great Chain of Being, D. Reidel, Dordrecht, 1980, pp. 259–86, especially p. 272.

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  24. See section II (ix) of the bibliography.

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  25. See, respectively, Veikko Rantala, Aspects of Definability (Acta Philosophica Fennica, vol. 29, nos. 2–3), North-Holland, Amsterdam, 1977; and Jaakko Hintikka, “Impossible Possible Worlds Vindicated”, Journal of Philosophical Logic 4 (1975), 475–84, reprinted (and expanded) in Saarinen, Game-Theoretical Semantics, pp. 367–79.

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  26. Cf. Lauri Carlson and Alice ter Meulen, “Informational Independence in Intensional Contexts”, in Esa Saarinen et al, editors, Essays in Honour of Jaakko Hintikka, D. Reidel, Dordrecht, 1979, pp. 61–72.

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  27. See Jaakko Hintikka, “Questions with Outside Quantifiers”, in Robinson Schneider, editors, Papers from the Parasession on Nondedaratives, CLS, April 17, 1982, Chicago Linguistic Society, Chicago, 1982, pp. 83–92; “On Games, Questions, and Strange Quantifiers”, in Tom Pauli, editor, Philosophical Essays Dedicated to Lennart Aqvist on His Fiftieth Birthday, Philosophical Society and Department of Philosophy, Uppsala University, Uppsala, 1982, pp. 159–69.

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  28. Various further explanations are needed here. For instance, more has to be said about how anaphoric relations are determined in the output sentence of (G. some). Even more obviously, something ought to be said about what happens when the restrictions just mentioned are removed. A detailed discussion of these matters will not be attempted here, however.

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  29. See chapter 9 below; and see sections 10, 12, and 13 of chapter 4.

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  30. In other formal and natural languages negation can nevertheless be treated in a more informative way by giving explicit rules as to how the (semantical) negation (contradictory) of a given sentence can be formed syntactically. Such a treatment would be more informative, but is has not yet been attempted in print. See here chapter 4 below for problems with syntactical rules for negation-forming.

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  31. See Jaakko Hintikka and Jack Kulas, “Towards a Semantical Theory of Pronominal Anaphora” (in preparation).

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  32. See Richmond Thomason, editor, Formal Philosophy: Selected Papers of Richard Montague, Yale University Press, New Haven, 1974; Noam Chomsky, Aspects of the Theory of Syntax, The MIT Press, Cambridge, Mass., 1965 (see especially pp. 224–25, note 9).

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  33. These ordering conventions clearly are closely related to various linguistic principles, such as the cyclic principle of transformational grammarians and George Lakoffs “derivational constraints” in his generative semantics. Unlike the claims made for these principles, however, my ordering conventions admit exceptions.

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  34. See Tanya Reinhart, “Syntactic Domains for Semantic Rules”, in F. Guenthner and S. J. Schmidt, editors. Formal Semantics and Pragmatics for Natural Languages, D. Reidel, Dordrecht, 1979, pp. 107-30, For a more extensive discussion of syntactic domains, see her dissertation, The Syntactic Domain of Anaphora, 1976, unpublished, M.I.T. Guy Carden has pointed out problems with her view, in his “Blocked Forward Coreference”, presented at the 56th Annual Meeting of the Linguistic Society of America, New York, 1981. See also Susumu Kuno, “Reflexivization in English”, Communication and Cognition 16 (1983), 65–80.

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  35. This means that the “logical form” of (12) is (x)F(x) → G while that of (13) is (x)(F(x) → G), which can be rewritten as (Ex)F(x) → G.

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  36. See chapter 7 below.

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  37. See chapter 8 below.

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  38. This approach is represented in Joan Bresnan, editor, The Mental Representation of Grammatical Relations, The MIT Press, Cambridge, Mass., 1982.

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  39. See here Jaakko Hintikka, “Quantifiers in Natural Languages: Some Logical Problems”, in Saarinen, Game-Theoretical Semantics, and see chapter 10 below. See also Jaakko Hintikka, “On the Limitations of Generative Grammar”, in (no editor given) Proceedings of the Scan-dinavian Seminar on Philosophy of Language, Vol. 1, (Filosofiska Studier, Vol. 26 ) Philosophical Society and Department of Philosophy, Uppsala University, Uppsala, Sweden, 1975, pp. 1–92.

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  40. See Noam Chomsky, Rules and Representations, Columbia University Press, New York, 1980, pp. 122–28.

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  41. See chapter 10 below.

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  42. Cf. here Jaakko Hintikka, “A Counterexample to Tarski-Type Truth-Definitions As Applied to Natural Languages”, Philosophia 5 (1975), 204–12.

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  43. Cf. here chapter 8 below.

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  44. John Stuart Mill, A System of Logic, London, 1843 (8th edition, New York, 1881 ), Book I, Chapter IV, §1.

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  45. Augustus De Morgan, Formal Logic, Taylor and Walton, London, 1847 (reprinted by The Open Court Co., London, 1926), pp. 49–50.

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  46. On this subject, see chapter 7 below, as well as Jaakko Hintikka and Simo Knuuttila, editors, The Logic of Being: Historical Studies (forthcoming).

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  47. Cf. here chapter 8 below.

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  48. Aristotle uses different question words (and phrases) in ancient Greek as names for his categories; he introduces categories as semantically determined classes of simple predicates; he treats categories as the widest genera of entities we can meaningfully consider together; and he frequently says that the several categories go together with different senses or uses of τò εἶυα∫,, the Greek verb for being.

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  49. See Jaakko Hintikka, The Semantics of Questions and the Questions of Semantics (Acta Philosophica Fennica, vol. 28, no. 4), North-Holland, Amsterdam, 1976; and cf. idem, “New Foundations for a Theory of Questions and Answers”, forthcoming.

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  50. See here chapter 5 below.

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  51. See here chapter 3 below, and Jaakko Hintikka and Lauri Carlson, “Conditionals, Generic Quantifiers, and Other Applications of Subgames”, in Saarinen, Game-Theoretical Semantics, pp. 179–214.

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  52. See Jaakko Hintikka and Lauri Carlson, “Pronouns of Laziness in Game-Theoretical Semantics”, Theoretical Linguistics 4 (1977), 1–29.

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  53. See Jaakko Hintikka and Jack Kulas, “Towards a Semantical Theory of Pronominal Anaphora” (in preparation).

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  54. Important work in this direction has been done by Lauri Carlson. See his book, Dialogue Games: An Approach to Discourse Analysis, D. Reidel, Dordrecht, 1983.

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  55. See chapter 4 below.

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  56. See Lauri Carlson, Dialogue Games.

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  57. See Jaakko Hintikka, “Language-Games”, in Jaakko Hintikka et al., editors, Essays on Wittgenstein in Honour of G. H. von Wright (Acta Philosophica Fennica, vol. 28, nos. 1–3), North-Holland, Amsterdam, 1976, pp. 105–25, (reprinted in Saarinen, Game-Theoretical Semantics, pp. 1-26); and cf. “Language-Games for Quantifiers”, in Jaakko Hintikka, Logic, Language-Games and Information, Clarendon Press, Oxford, 1973, pp. 53–82.

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  58. See Jaakko Hintikka, “Transcendental Arguments Revived”, in A. Mercier and M. Svilar, editors, Philosophers on Their Own Work, vol. 9, Peter Lang, Bern, 1982, pp. 119–33.

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  1. Department of Philosophy, Florida State University, Tallahassee, USA

    Jaakko Hintikka

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© 1983 D. Reidel Publishing Company, Dordrecht, Holland

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Hintikka, J. (1983). Game-Theoretical Semantics: Insights and Prospects. In: The Game of Language. Synthese Language Library, vol 22. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-9847-2_1

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  • DOI: https://doi.org/10.1007/978-94-010-9847-2_1

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