Abstract
Tradition is classical. Surely, nothing could be more pleonastic than that? The logical tradition, certainly, was squarely classical from Bolzano to Carnap, with, say, Frege, Moore, Russell and the Wittgenstein of the Tractatus as intermediaries. Propositions are construed as being in themselves true-or-false. Indeed, in this tradition, a declarative sentence S expresses a proposition (or is a proposition, depending on what version of the theory that is adopted) by being true-or-false. So the meaningfulness of a sentence consists in its being true-or-false. But S is true-orfalse, or so they say, only when S is true, or when S is false. On the classical account the presumption of bivalence is built into the very notion of meaningfulness: there is no difference between asserting that A is a proposition and asserting that A is true-or-false. The matter came to the fore in the foundations of set theory. In his first attempt at giving an application criterion for sets Cantor noted:
Eine Mannigfaltigkeit (ein Inbegriff, eine Menge) von Elementen, die irgendwelcher Begriffssphäre angehören, nenne ich wohldefiniert, wenn auf Grund ihrer Definition und infolge des logischen Prinzips vom ausgeschlossenen Dritten es als intern bestimmt angesehen werden muss, sowohl ob irgendein derselben Begriffssphäre angehöriges Objekt zu der gedachten Mannigfaltigkeit gehört oder nicht, wie auch, ob zwei zur Mannigfaltigkeit gehörige Objekte trotz formaler Unterschiede in der Art des Gegebenseins einander gleich sind oder nicht.1
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References
Paul Bernays, “Sur le platonisme dans les mathématiques”, in: L’enseignement mathématique, Vol. 34, 1935, pp. 52–69; English translation by Charles D. Parsons, in: Paul Benacerraf and Hilary Putnam, Philosophy of Mathematics, Oxford: Blackwell 1964, pp. 274–286.
Bernard Bolzano, Wissenschaftslehre, Vols I - IV, Sulzbach: J. von Seidel 1837.
Luitzen Egbertus Jan Brouwer, Over de Grondslagen van de Wiskunde (1907), Nordhoff, Groningen, second edition with additional material (ed. D. van Dalen ), Amsterdam: Mathematisch Centrum 1981.
Luitzen Egbertus Jan Brouwer, “De onbetrouwbaarheid der logische principes”, Tijdschrii t voor wijsbegeerte 2, 1908, pp. 152–158.
Luitzen Egbertus Jan Brouwer, “Mathematik, Wissenschaft und Sprache”, in: Monatshefte ftir Mathematik und Physik, Vol. 36, pp. 153–64. English translation by Walter van Stigt in: P. Mancosu (ed.), From Brouwer to Hilbert, New York: Oxford University Press 1986, pp. 45–53.
Georg Cantor, “Über unedliche lineare Punktmannigfaltigkeiten”, Mathematische Annalen Nr. 3, 20, 1882, pp. 113–121.
Georg Cantor, Letters to Richard Dedekind, in: Georg Cantor, Gesammelte Abhandlungen, (Ernst Zermelo, editor), Anhang, 1899, pp. 443–451.
Rudolf Carnap, Abriss der Logistik,Wien: Springer 1929
Rudolf Carnap, Logische Syntax der Sprache (1934), quoted after: The Logical Syntax of Language, English translation by Amethe Smeaton, Countess von Zeppelin, London: Routledge and Kegan Paul 1937.
Alonzo Church, “A set of postulates for the foundation of logic, Annals of Mathematics,Vol. 33, 1932, pp. 346–366; Vol 34, 1933, p. 839–864.
Gottlob Frege, Grundgesetze der Arithmetik, Band I, Band II, Jena: H. Pohle 1893, 1903.
G.H. Hardy, “Mathematical proof”, Mind, Vol. 38, 1929, pp. 1–25.
Jean van Heijenoort, “Logic as calculus versus logic as language”, Synthese, Vol. 17, 1967, pp. 324–330.
Jaakko Hintikka, “On the development of the model-theoretic viewpoint in logical theory”, Synthese, Vol. 77, 1988, pp. 1–36.
Jaakko Hintikka, Lingua Universalis vs. Calculus Ratiocinator: An Ultimate Presupposition of Twentieth-Century Philosophy, Dordrecht: Kluwer 1996.
L. Kronecker, 1882. Martin Kusch, “Husserl and Heidegger on meaning”, Synthese, Vol. 77, 1988, pp. 99–127.
Per Martin-Ulf, “On the meaning of the logical constants and the justification of the logical laws” (1985), in: Nordic Journal of Philosophical Logic, Vol.1 (1996), pp. 11–60. Originally distributed in 1985.
Per Martin-Löf, “Truth of a proposition, evidence of a judgment, validity of a proof”, Synthese,Vol. 73, 1987, pp. 407420.
Karl Menger, “Bemerkungen zu Grundlagenfragen, I. Über Verzweigungsmengen”, Jahresbericht der Deutschen Mathematiker-Vereinigung, Vol.37, 1928, pp. 213–226. Cited from the English version in Menger [ 1979a ].
Karl Menger, “Der Intuitionismus”, in: Blätter far Deutsche Philosophie, Vol. 4, 1930, pp. 311–235. Cited from the English version in Menger [ 1979a ].
Karl Menger, “My memories of L. E. J. Brouwer” (1979), in Menger [1979a], pp. 23–255.
Karl Menger, Selected Papers in Logic and Foundations, Didactics, Economics, (Vienna Circle Collection, Vol. 10 ), Dordrecht: Reidel 1979a.
Henri Poincaré, “Les mathématiques et la logique”, in: Revue de Métaphysique et de Morale,Vol. 14, 1906, pp.17–34, 294–317, 866–868.
Henri Poincaré, “La logique de l’infini”, in: Revue de Métaphysique et de Morale,Vol. 17, 1909, pp. 451–482.
Willard Van Orman Quine, A System of Logistic, Cambridge, Mass.: Harvard University Press 1934.
Aame Ranta, Type-theoretical Grammar, Oxford: Clarendon Press 1994.
Bertrand Russell, Principles of Mathematics, Cambridge: Cambridge University Press 1903.
Thoralf Skolem, “Einige Bemerkungen zur axiomatischen Begründung der Mengenlehre”, in: Den feinte skandinaviska matematikerkongressen, Redogörelse, Helsingfors: Akademiska Bokhandeln 1923, pp. 217–232.
B.G. Sundholm, “Constructions, proofs and the meaning of the logical constants”, Journal of Philosophical Logic, 12, 1983, pp. 151–72.
B.G. Sundholm, “Sätze der Logik: An Alternative Conception”, in: Rudolf Haller/Johannes Brandl (eds.), Wittgenstein–Eine Neubewertung, Akten des 14. Internationalen Wittgenstein-Symposiums, Wien: Hölder-Pichler-Tempsky 1990, pp. 51–61.
B.G. Sundholm, “Questions of proof”, in: Manuscrito (Campinas), Vol. 16, 1993, pp. 47–70.
B.G. Sundholm, “Existence, proof and truth-making: a perspective on the intuitionistic conception of truth”, TOPOI, Vol 13, 1994, pp. 117–26
B.G. Sundhohn, “Vestiges of realism”, in: Brian McGuinness and Gianluigi Oliven, The Philosophy of Michael Dummett, Dordrecht: Kluwer 1994a, pp. 137–166.
Alfred Tarski, “Der Wahrheitsbegriff in den formalisierten Sprachen”, in: Studia Philosophica (Lemberg), Vol. 1, 1935, pp. 261–405. (German translation, with a novel Nachwort,of a Polish original published in 1933.)
Hao Wang, Reflections on Kurt Gödel, Cambridge: MIT Press 1987.
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Sundholm, G. (1999). Intuitionism and Logical Tolerance. In: Woleński, J., Köhler, E. (eds) Alfred Tarski and the Vienna Circle. Vienna Circle Institute Yearbook [1998], vol 6. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-0689-6_12
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