Abstract
Model-theoretical (semantical) treatments of modal logic have enjoyed spectacular success ever since the pioneering work by Stig Kanger in 1957.1 Quine and others have admittedly proffered sundry philosophical objections to modal logic and its semantics but they have not impeded the overwhelming progress either of the semantical theory of intensional (modal) logics or of its applications.
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Notes
Stig Kanger, Provability in Logic. Stockholm Studies in Philosophy, Vol. 1, Stockholm, 1957. Kanger deserves much more credit for developing a viable semantics for modal logics than is given to him in the literature. For instance, the main novelty of such semantics as compared with Carnap’s old ideas is the use of the altemativeness relation. (See below for an explanation of this concept.) Kanger introduced this idea in the literature and used it in his work before anyone else, e.g., five years before Kripke.
See Saul Kripke, ‘A Completeness Theorem in Modal Logic’, J. Symbolic Logic 24 (1959), 1–4; ‘Semantical Considerations on Modal Logic’, Acta Philosophica Fennica 16 (1963), 83–94; `Semantical Analysis of Modal Logic I’, Zeitschrift für mathematische Logik und Grundlagen der Mathematik 9 (1963), 67–96; `Semantical Analysis of Modal Logic II’, in J. W. Addison, L. Henkin, and A. Tarski (eds.), The Theory of Models, North-Holland, Amsterdam, 1965, pp. 206–220; ’The Undecidability of Monadic Modal Quantification Theory’, Zeitschrift für mathematische Logik und Grundlagen der Mathematik 8 (1962), 113–116; `Semantical Analysis of Intutionistic Logic’, in J. N. Crossley and Michael Dummett (eds), Formal Systems and Recursive Functions, North-Holland, Amsterdam, 1965, pp. 92–130.
Marcel Guillaume, `Rapports entre calculs propositionels modaux et topologie impliqués par certaines extensions de la méthode des tableaux: Système de Feys—von Wright’, Comptes rendus des séances de [Academic des Science (Paris) 246 (1958), 1140–1142; `Système S4 de Lewis’, ibid.,2207–2210; ‘Système S5 de Lewis’, ibid., 247 (1958), 1282–1283; Jaakko Hintikka, `Quantifiers in Deontic Logic’, Societas Scientariarum Fennica,Commentationes humanarum litterarum, Vol. 23, 1957, No. 4; `Modality and Quantification’, Theoria 27 (1961), 119–128; `The Modes of Modality’, Acta Philosophica Fennica 16 (1963), 65–82.
See Richmond Thomason (ed.), Formal Philosophy: Selected Papers of Richard Montague, Yale University Press, New Haven, 1974, Chapters 1–2.
Op. cit., Chapters 3–8.
David Kaplan, UCLA dissertation, 1964.
Nino Cocchiarella, ‘On the Primary and Secondary Semantics of Logical Necessity’, Journal of Philosophic Logic 4 (1975), 13–27; ‘Logical Atomism and Modal Logic’, Philosophia 4 (1974), 40–66.
Alfred Tarski and Bjami Jonsson, `Boolean Algebras with Operators I—II’, American Journal of Mathematics 73 (1951), and 74 (1952).
See Leon Henkin, `Completeness in the Theory of Types’, J. Symbolic Logic 15 (1950), 81–91. (Please note that Peter Andrews has discovered a flaw in Henkin’s original argument and has shown how to repair it.)
See especially Jaakko Hintikka, `Quantifiers in Logic and Quantifiers in Natural Languages’, in S. Körner (ed.), Philosophy of Logic, Blackwell’s, Oxford, 1976, pp. 208–232; `Quantifiers vs. Quantification Theory’, Linguistic Inquiry 5 (1974), 153–177; Logic,Language-Games, and Information, Clarendon Press, Oxford, 1973. Much of the relevant literature has now been collected in Esa Saarinen (ed.), Game-Theoretical Semantics, D. Reidel, Dordrecht, 1978.
Cf. here Jaakko Hintikka and Veikko Rantala, `A New Approach to Infinitary Languages’, Annals of Mathematical Logic 10 (1976), 95–115.
For an explicit discussion of this idea see Jaakko Hintikka and Lauri Carlson, ‘Conditionals, Generic Quantifiers, and Other Applications of Subgarnes’, in A. Margalit (ed.), Meaning and Use, D. Reidel, Dordrecht, 1978.
As Dana Scott has pointed out in an unpublished note, one can in this way also obtain Gödel’s functional interpretation of first-order logic and arithmetic. See Kurt Gödel, Tine bisher noch nicht benützte Erweiterung des finiten Standpunktes’, in Logica: Studia Paul Bernays Dedicata, Editions du Griffon, Neuchatel, 1959, pp. 76–83. 14 Cf. my paper `Language Games’, in Essays on Wittgenstein in Honour of G. H. von Wright (Acta Philosophica Fennica, Vol. 28, Nos. 1–3), North-Holland, Amsterdam, 1976, pp. 105–125.
`Quantifiers vs. Quantification Theory’, Linguistic Inquiry 5 (1974), 153–177.
Of course this is not the only nor the most natural way of imposing the distinction on second-order logic.
See David Kaplan, op. cit. (note 6 above).
I have discussed these issues in the essays collected in Models for Modalities, D. Reidel, Dordrecht, 1969, and The Intentions of Intentionality and Other New Models for Modalities, D. Reidel, Dordrecht, 1975.
For backwards-looking operators, see Esa Saarinen, `Backwards, Looking Operators in Tense Logic and Natural Language’, in Jaakko Hintikka et. al (eds.), Essays on Mathematical and Philosophical Logic, D. Reidel, Dordrecht, 1978, pp. 341–367; and Esa Saarinen, `Intentional Identity Interpreted’, Linguistic and Philosophy 2 (1978), 151–223, with further references to the literature. The initiators of the diesa seem to have been Hans Kamp and David Kaplan.
`Reductions in the Theory of Types’, Acta Philosophica Fennica 8 (1955), 56–115.
See S. K. Thomason, `Semantic Analysis of Tense Logics’, J. Symbolic Logic 37 (1972), 150–158; Noncompactness in Propositional Modal Logic’, ibid., 716–720; `An Incompleteness Theorem in Modal Logic’, Theoria 40 (1974), 30–34.
Personal communcation.
Barbara Hall Partee (ed.), Montague Grammar, Academic Press, New York, 1976, and note 4 above.
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Hintikka, J. (1998). Standard vs. Nonstandard Logic: Higher-Order, Modal, and First-Order Logics. In: Language, Truth and Logic in Mathematics. Jaakko Hintikka Selected Papers, vol 3. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-2045-8_7
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