Hostname: page-component-7dd5485656-zlgnt Total loading time: 0 Render date: 2025-10-31T00:37:13.087Z Has data issue: false hasContentIssue false
  • English
  • Français

GENERALITY AND EXISTENCE: QUANTIFICATIONAL LOGIC IN HISTORICAL PERSPECTIVE

Published online by Cambridge University Press:  30 December 2014

JAN VON PLATO
JAN VON PLATO*
Affiliation:
UNIVERSITY OF HELSINKI, FINLAND, 00014 HELSINKI, FINLANDE-mail: jan.vonplato@helsinki.fi
Get access Rights & Permissions [Opens in a new window]

Abstract

Frege explained the notion of generality by stating that each its instance is a fact, and added only later the crucial observation that a generality can be inferred from an arbitrary instance. The reception of Frege’s quantifiers was a fifty-year struggle over a conceptual priority: truth or provability. With the former as the basic notion, generality had to be faced as an infinite collection of facts, whereas with the latter, generality was based on a uniformity with a finitary sense: the provability of an arbitrary instance.

Keywords

quantifierspredicate logicFregeRussellSkolemHilbertBernaysBrouwerHeytingGentzenWittgensteinGoodstein

Information

Type
Articles
Information
Bulletin of Symbolic Logic , Volume 20 , Issue 4 , December 2014 , pp. 417 - 448
Copyright
Copyright © The Association for Symbolic Logic 2014 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

REFERENCES

Acerbi, F. (2010) Il silenzio delle sirene: la matematica greca antica. Carocci editore, Rome.Google Scholar
Ackermann, W. (1924) Begründung des “tertium non datur” mittels der Hilbertschen Theorie der Widerspruchsfreiheit. Mathematische Annalen, vol. 93, pp. 1–36.Google Scholar
van Atten, M. (2005) The correspondence between Oskar Becker and Arend Heyting. In Peckhaus, V., ed, Oskar Becker und die Philosophie der Mathematik, pp. 119–142, Fink Verlag, Munich.Google Scholar
Bernays, P. (1918) Beiträge zur axiomatischen Behandlung des Logik-Kalküls. Manuscript Hs. 973:193, Bernays collection, ETH-Zurich.Google Scholar
Bernays, P. (1926) Axiomatische Untersuchung des Aussagen-Kalkuls der “Principia mathematica.” Mathematische Zeitschrift, vol. 25, pp. 305–320.Google Scholar
Brouwer, L. (1924) Beweis, dass jede volle Funktion gleichmässig stetig ist. Koninklijke Akademie van Wetenschappen te Amsterdam, Proceedings of the Section of Sciences, vol. 27, pp. 189–193.Google Scholar
Brouwer, L. (1924a) Bemerkungen zum Beweise der gleichmässigen Stetigkeit voller Funktionen. Ibid., pp. 644–646.Google Scholar
Brouwer, L. (1925) Zur Begründung der intuitionistischen Mathematik. I.Mathematische Annalen, vol. 93, pp. 244–257.Google Scholar
Brouwer, L. (1927) Über Definitionsbereiche von Funktionen. Mathematische Annalen, vol. 97, pp. 60–75.CrossRefGoogle Scholar
Brouwer, L. (1928) Intuitionistische Betrachtungen über den Formalismus. Sitzungsberichte der Preussischen Akademie der Wissenschaften, pp. 48–52.Google Scholar
Burris, S. (1995) Polynomial time uniform word problems, Mathematical Logic Quarterly, vol. 41, pp. 173–182.Google Scholar
Coquand, T. (2004) About Brouwer’s fan theorem. Revue internationale de philosophie, no. 240, pp. 483–489.Google Scholar
Cosmadakis, S. (1988) The word and generator problem for lattices, Information and Computation, vol. 77, pp. 192–217.CrossRefGoogle Scholar
Curry, H. (1941) A formalization of recursive arithmetic. American Journal of Mathematics, vol. 63, pp. 263–282.Google Scholar
Frege, G. (1879) Begriffsschrift, eine nach der arithmetischen nachgebildete Formelsprache des reinen Denkens. Nebert, Halle. English translation in Van Heijenoort.Google Scholar
Frege, G. (1893) Grundgesetze der Arithmetik, begriffschriftlich abgeleitet, vol. 1, Pohle, Jena.Google Scholar
Gentzen, G. (1934-35) Untersuchungen über das logische Schliessen. Mathematische Zeitschrift, vol. 39, pp. 176–210 and 405–431.Google Scholar
Gentzen, G. (2008) The normalization of derivations. The Bulletin of Symbolic Logic, vol. 14, pp. 245–257.Google Scholar
Goldfarb, W. (1979) Logic in the twenties: the nature of the quantifier. The Journal of Symbolic Logic, vol. 44, pp. 351–368.Google Scholar
Goodstein, R. (1939) Mathematical systems. Mind, vol. 48, pp. 58–73.Google Scholar
Goodstein, R. (1945) Function theory in an axiom-free equation calculus. Proceedings of the London Mathematical Society, vol. 48, pp. 58–73.Google Scholar
Goodstein, R. (1951) Constructive Formalism. Leicester U.P.Google Scholar
Goodstein, R. (1957) Recursive Number Theory. North-Holland.Google Scholar
Goodstein, R. (1958) On the nature of mathematical systems. Dialectica, vol. 12, pp. 296–316.CrossRefGoogle Scholar
Goodstein, R. (1972) Wittgenstein’s philosophy of mathematics. In Ambrose, A. and Lazerowitz, M., eds, Ludwig Wittgenstein: Philosophy and Language, pp. 271–286, Allen and Unwin, London.Google Scholar
van Heijenoort, J., ed, (1967) From Frege to Gödel, A Source Book in Mathematical Logic, 1879–1931, Harvard University Press.Google Scholar
Heyting, A. (1930a) Die formalen Regeln der intuitionistischen Logik. Sitzungsberichte der Preussischen Akademie von Wissenschaften, Physikalisch-mathematische Klasse, pp. 42–56.Google Scholar
Heyting, A. (1930b) Die formalen Regeln der intuitionistischen Mathematik. Sitzungsberichte der Preussischen Akademie von Wissenschaften, Physikalisch-mathematische Klasse, pp. 57–71.Google Scholar
Heyting, A. (1935) Intuitionistische wiskunde. Mathematica B, vol. 4, pp. 72–83.Google Scholar
Hilbert, D. (1922) Neubegründung der Mathematik. Erste Mitteilung. Abhandlungen aus dem Mathematischen Seminar der Hamburgischen Universität, vol. 1, pp. 157–177. English translation in Mancosu 1998.Google Scholar
Hilbert, D. (1923) Die logischen Grundlagen der Mathematik. Mathematische Annalen, vol. 88, pp. 151–165.Google Scholar
Hilbert, D. (1927) Die Grundagen der Mathematik. Abhandlungen aus dem mathematischen Seminar der Hamburgischen Universität, vol. 6, pp. 65–85. Page references to the English translation in Mancosu (1998).Google Scholar
Hilbert, D. and Ackermann, W. (1928) Grundzüge der theoretischen Logik. Springer.Google Scholar
Hibert, D. (1931) Beweis des Tertium non datur. Nachrichten von der Gesellschaft für Wissenschaften zu Göttingen, Mathematisch-physikalische Klasse, pp. 120–125.Google Scholar
Hilbert, D. and Bernays, P. (1934, 1939) Grundlagen der Mathematik I–II. Springer.Google Scholar
Kahle, R. (2013) David Hilbert and the Principia Mathematica. In Griffin, N. and Linsky, B., eds, The Palgrave Centenary Companion to Principia Mathematica, pp. 21–34.Google Scholar
Mancosu, P. (1998) From Brouwer to Hilbert: The Debate on the Foundations of Mathematics in the 1920s. Oxford.Google Scholar
Mancosu, P. (2002) On the constructivity of proofs: a debate among Behmann, Bernays, Gödel, and Kaufmann. In Reflections of the Foundations of Mathematics, eds. Sieg, W. et al. ., pp. 349–371. ASL Lecture Notes in Logic.Google Scholar
Mancosu, P. and Marion, M. (2003) Wittgensteins constructivization of Eulers proof of the infinity of primes. In The Vienna Circle and Logical Empiricism, ed Stadler, F., pp. 171–188, Kluwer.Google Scholar
Marion, M. and Okada, M. (2012) Wittgenstein and Goodstein on the equation calculus and the uniqueness rule. Presented at the Goodstein centenary symposium, Leicester 2012.Google Scholar
Negri, S. and von Plato, J. (2001) Structural Proof Theory. Cambridge.CrossRefGoogle Scholar
von Plato, J. (2007) In the shadows of the Löwenheim-Skolem theorem: early combinatorial analyses of mathematical proofs. The Bulletin of Symbolic Logic, vol. 13, pp. 189–225.Google Scholar
von Plato, J. (2008) Gentzen’s proof of normalization for natural deduction. The Bulletin of Symbolic Logic, vol. 14, pp. 240–244.Google Scholar
von Plato, J. (2010) Combinatorial analysis of proofs in projective and affine geometry. Annals of Pure and Applied Logic, vol. 162, pp. 144–161.Google Scholar
von Plato, J. (2012) Gentzen’s proof systems: byproducts in a program of genius. The Bulletin of Symbolic Logic, vol. 18, pp. 313–367.Google Scholar
von Plato, J. (2013) Elements of Logical Reasoning. Cambridge.Google Scholar
von Plato, J. (2013a) Greek mathematical texts as a literary genre. Essay review of Fabio Acerbi, Il silenzio delle sirene: la matematica greca antica. History and Philosophy of Logic, vol. 34, pp. 381–392.Google Scholar
von Plato, J. (2015) Saved from the Cellar: Gerhard Gentzen’s Shorthand Notes on Logic and Foundations of Mathematics. To appear.Google Scholar
Prawitz, D. (1965) Natural Deduction: A Proof-Theoretical Study. Almqvist & Wicksell.Google Scholar
Russell, B. (1903) The Principles of Mathematics. Cambridge.Google Scholar
Russell, B. (1906) The theory of implication. American Journal of Mathematics, vol. 28, pp. 159–202.Google Scholar
Russell, B. (1908) Mathematical logic as based on the theory of types. American Journal of Mathematics, vol. 30, pp. 222–262. Page references to the reprint in Van Heijenoort.Google Scholar
Skolem, T. (1919) Untersuchungen über die Axiome des Klassenkalküls und über Produktations- und Summationsprobleme, welche gewisse Klassen von Aussagen betreffen. As reprinted in Skolem 1970, pp. 67–101.Google Scholar
Skolem, T. (1920) Logisch-kombinatorische Untersuchungen über die Erfüllbareit oder Beweisbarkeit mathematischer Sätze, nebst einem Theoreme über dichte Mengen. As reprinted in Skolem 1970, pp. 103–136. Section 1 translated into English in Van Heijenoort.Google Scholar
Skolem, T. (1922) Einige Bemerkungen zur axiomatischen Begründung der Mengenlehre. As reprinted in Skolem 1970, pp. 137–152. English translation in Van Heijenoort.Google Scholar
Skolem, T. (1923) Begrüngung der elementaren Arithmetik durch die rekurrierende Denkweise ohne Anwendung scheinbarer Veränderliche mit unendlichem Ausdehnungsbereich. As reprinted in Skolem 1970, pp. 153–188. English translation in Van Heijenoort.Google Scholar
Skolem, T. (1928) Über die mathematische Logik. As reprinted in Skolem 1970, pp. 189–206.Google Scholar
Skolem, T. (1970) Selected Works in Logic. ed. Fenstad, J. E., Universitetsforlaget, Oslo.Google Scholar
Troelstra, A. (1990) On the early history of intuitionistic logic. In Petkov, P., ed, Mathematical Logic, pp. 3–17, Plenum Press.Google Scholar
Whitehead, A. and Russell, B. (1910) Principia Mathematica, vol. I. Cambridge. Second edition 1927.Google Scholar
Wittgenstein, L. (1922) Tractatus Logico-Philosophicus. Routledge, London.Google Scholar
Wittgenstein, L. (1964) Philosophische Bemerkungen. Blackwell, London.Google Scholar
Wittgenstein, L. (1969) Philosophische Grammatik. Blackwell, London.Google Scholar