Abstract
Kant had the right idea in his theory of mathematics, but he was misled by an antiquated philosophical dogma.1 Following his general transcendental point of view, he maintained that our ways of reasoning about existence (especially inferences from the existence or nonexistence of an individual to the existence or nonexistence of a different individual) must be grounded in the human activities through which we come to know the existence of individuals. This is a deep and intriguing idea, and Kant’s identification of the types of reasoning in question as mathematical rather than logical marks only a difference in terminology between Kant and contemporary philosophers of logic. What Kant was dealing with is unmistakably such logical reasoning as is now codified in the modern logic of quantification theory — not anything we would any longer consider distinctively mathematical reasoning.
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Notes
The interpretation summarized here of Kant’s theories of mathematical reasoning, space, time, and perception is presented in my books, Logic, Language-Games, and Information, Clarendon Press, Oxford, 1973, chapters 5–9, and in Knowledge and the Known, D. Reidel, Dordrecht, 1974, chapters 6–8. (See especially the former, chapter 5, “Logic, Language-Games and Transcendental Arguments”.)
See Logic, Language-Games, and Information chapter 3, “Language-Games for Quantifiers”. Many of my subsequent contributions to the development of game-theoretical semantics are collected in Esa Saarinen, editor, Game-Theoretical Semantics, D. Reidel, Dordrecht, 1979. See also chapter 1 above.
Several of the philosophical implications of game-theoretical semantics were spelled out in my Logic, Language-Games, and Information prior to its main applications to linguistic and logical semantics.
See my paper, “Language-Games” in Saarinen, Game-Theoretical Semantics, pp. 1–26, originally in Jaakko Hintikka, 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.
For Michael Dummett, see “What Is a Theory of Meaning? I”, in Samuel Guttenplan, editor, Mind and Language, Clarendon Press, Oxford, 1975, pp. 99–138; “What Is a Theory of Meaning? II”, in Gareth Evans and John McDowell, editors, Truth and Meaning, Clarendon Press, Oxford, 1976, pp. 67–137; The Justification of De deletion11, Proceedings of the British Academy 59 (1973), 1–34; Elements of Intuitionism, Clarendon Press, Oxford, 1977; “The Philosophical Basis of Intuitionistic Logic”, in H. E. Rose, editors, Logic Colloquium 73, North-Holland, Amsterdam1, 1975, pp. 5–40. For Dag Prawitz, see “Meaning and Proofs: On the Conflict between Classical and Intuitionistic Logic”, Theoria 43 (1977), 2–40; “On the Idea of a General Proof Theory”, Synthese 27 (1974), 63–77; “Intuitionistic Logic: A Philosophical Provocation”, in the Entretiens de DUsseldorf, Institut International de Philosophie, 1978 (unpublished); “Proofs and the Meaning and Completeness of the Logical Constants”, in Jaakko Hintikka, editors, Essays on Mathematical and Philosophical Logic, D. Reidel, Dordrecht, 1978, pp. 25–40; “Ideas and Results from Proof Theory”, in J. E. Fenstad, editor, Proceedings of the Second Scandinavian Logic Symposium, North-Holland, Amsterdam, 1975, pp. 235–50.
Dummett, “Theory of Meaning? II”, p. 115.
Dag Prawitz, “Meaning and Proofs”, p. 3.
See here my papers “Quantifiers vs. Quantification Theory” and “Quantifiers in Natural Languages: Some Logical Problems”, in Saarinen, Game-Theoretical Semantics, pp. 49–79 and 81–117, respectively.
The information set of a move M by a player P is the set of those moves whose outcome is known by Pat M. See also R. Duncan Luce and Howard Raiffa, Games and Decisions, John Wiley, New York, 1957, p. 43.
Leon Henkin, “Completeness in the Theory of Types”, Journal of Symbolic Logic 15 (1950), 81–91. Cf. Jaakko Hintikka, “Standard vs. Nonstandard Logic: Higher-Order, Modal, and First-Order Logics”, in E. Agazzi, editor. Modern Logic: A Survey, D. Reidel, Dordrecht, 1980, pp. 283–96.
See “Quantifiers vs. Quantification Theory”, in Saarinen, Game-Theoretical Semantics. (Originally published in Linguistic Inquiry 5 (1974), 153–77.)
Ibid.
P. F. Strawson, Introduction to Logical Theory, Methuen, London, 1952; “Meaning and Truth”, in Logico-Linguistic Papers, Methuen, London, 1971.
The basic writings of the Lorenzen school are conveniently available in Paul Lorenzen and Kuno Lorenz, editors, Dialogische Logik, Wissenschaftliche Buchgesellschaft, Darmstadt, 1978 (with ample further references to the literature). See also Wolfgang Stegmaller, “Remarks on the Completeness of Logical Systems Relative to the Validity-Concepts of P. Lorenzen and K. Lorenz”, Notre Dome Journal of Formal Logic 5 (1964), 81–112.
Frequently game-theoretical semantics enables us to deepen the insights obtained in the competing approaches. For instance, the verificationists have pointed out repeatedly that in their view there is no valid general reason to believe in the law of bivalence, according to which each proposition S is either true or false. In game-theoretical semantics, the truth of S means that there is a winning strategy for Myself in the correlated semantical game G(S), and the falsity of S means that Nature has a winning strategy in G(S). Not only is it immediately clear on the basis of these definitions that there is in general no reason to expect S to be either true or false, but in the game-theoretical approach we see at once what the law of bivalence amounts to. It amounts to an assumption of determinateness for semantical games, that is, to an assumption that one of the two players always has a winning strategy. Now it is known from game theory and its applications to logic that determinateness postulates are often especially strong and especially interesting assumptions. Indeed, there exists a flourishing branch of foundational studies examining the consequences of different determinateness assumptions. For example, see J. Mycielski and H. Steinhaus, “A Mathematical Axiom Contradicting the Axiom of Choice”, Bulletin de VAcademie Polonaise des Sciences (ser. II) 10 (1962), 1-3; J. Mycielski, “On the Axiom of Determinateness”, Fundamenta Mathematicae 53 (1964), 205-24; Jens Erik Fenstad, “The Axiom of Determinateness”, in Jens Erik Fenstad, editor, Proceedings of the Second Scandinavian Logic Symposium, North-Holland, Amsterdam, 1971, pp. 41–61 (with further references to the literature).
I am referring to what Lorenzen and Lorenz call material games. The introduction of these games has not led the architects of this approach to change the label “dialogical”, however.
One way of looking at the activity of attempting to prove S is as an enterprise of trying to construct a counterexample to S, ie, trying to construct a domain in which 1 don’t have a winning strategy in G(S).
Critique of Pure Reason, B xiii.
See note 18 above.
Jaakko Hintikka and Esa Saarinen, “Information-Seeking Dialogues: Some of Their Logical Properties”, Studia Logica 32 (1979), 355–63.
See Kurt Godel, “Uber eine bisher noch nicht benutzte Erweiterung des finiten Standpunktes”, in (no editor given) Logica: Studia Paul Bernays dedicata, Editions Griffon, Neuchatel, 1959, pp. 76–83. (See also section II (v) of the bibliography at the end of this volume for references to the literature on functional interpretations.) Note here how the translation into second-order logic exemplified by (2) and (5)–(6) above is of the nature of a functional interpretation. It can be varied further in different ways, eg, by limiting the values of function variables (including those of a higher type) to recursive functions.
As was first pointed out by Dana Scott [“A Game-Theoretic Interpretation of Logical Formulae”, McCarthy Seminar, Stanford University, July 1968 (unpublished)], we can, for instance, in this way obtain an eminently natural motivation for Godel’ s interpretation of elementary arithmetic (see the preceding note). For other uses of functional interpretations, cf. Jaakko Hintikka and Lauri Carlson, “Conditionals, Generic Quantifiers, and Other Applications of Subgames”, in Saarinen, Game-Theoretical Semantics, pp. 179–214. See also the next chapter. Cf. also Jean-Yves Girard, “Functional Interpretation and Kripke Models”, in Robert Butts and Jaakko Hintikka, editors, Logic, Foundations of Mathematics, and Computability Theory (Part One of the Proceedings of the Fifth International Congress of Logic, Methodology, and Philosophy of Science), D. Reidel, Dordrecht, 1977, pp. 33–57.
Cf., eg, the papers mentioned in notes 8, 18, and 23 above.
Lauri Carlson, Dialogue Games: An Approach to Discourse Analysis, D. Reidel, Dordrecht, 1983.
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Hintikka, J. (1983). Semantical Games and Transcendental Arguments. In: The Game of Language. Synthese Language Library, vol 22. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-9847-2_2
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