Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-17T15:03:55.843Z Has data issue: false hasContentIssue false
  • English
  • Français

The Road to Modern Logic—An Interpretation

Published online by Cambridge University Press:  15 January 2014

José Ferreirós
José Ferreirós*
Affiliation:
Departamento de Filosofía Y Lógica, Universidad De Sevilla, Camilo José Cela s/n, E-41018 Sevilla, Spain, E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper aims to outline an analysis and interpretation of the process that led to First-Order Logic and its consolidation as a core system of modern logic. We begin with an historical overview of landmarks along the road to modern logic, and proceed to a philosophical discussion casting doubt on the possibility of a purely rational justification of the actual delimitation of First-Order Logic. On this basis, we advance the thesis that a certain historical tradition was essential to the emergence of modern logic; this traditional context is analyzed as consisting in some guiding principles and, particularly, a set of exemplars (i.e., paradigmatic instances). Then, we proceed to interpret the historical course of development reviewed in section 1, which can broadly be described as a two-phased movement of expansion and then restriction of the scope of logical theory. We shall try to pinpoint ambivalencies in the process, and the main motives for subsequent changes. Among the latter, one may emphasize the spirit of modern axiomatics, the situation of foundational insecurity in the 1920s, the resulting desire to find systems well-behaved from a proof-theoretical point of view, and the metatheoretical results of the 1930s. Not surprisingly, the mathematical and, more specifically, the foundational context in which First-Order Logic matured will be seen to have played a primary role in its shaping.

Mathematical logic is what logic, through twenty-five centuries and a few transformations, has become today. (Jean van Heijenoort)

Type
Research Article
Information
Bulletin of Symbolic Logic , Volume 7 , Issue 4 , December 2001 , pp. 441 - 484
Copyright
Copyright © Association for Symbolic Logic 2001

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.)

References

REFERENCES

[1] Arnauld, Antoine and Nicole, Pierre, La logique ou l'art de penser, Flammarion, Paris, 1970 (original edition 1662).Google Scholar
[2] Bernays, Paul, A system of axiomatic set theory, The Journal of Symbolic Logic, vol. 2 (1937), pp. 65–77.CrossRefGoogle Scholar
[3] Birkhoff, Garrett, A source book in classical analysis, Harvard University Press, 1973.Google Scholar
[4] Bocheński, Joseph M., Formale Logik, Alber, München, 1956.Google Scholar
[5] Boole, George, The mathematical analysis of logic, Macmillan, Cambridge, 1847, references to the reprint Basil Blackwell, Oxford, 1951.Google Scholar
[6] Carnap, Rudolf, Abriss der Logistik, mit besonderer Berücksichtigung der Relationstheorie und ihrer Anwendungen, Springer, Wien, 1929.Google Scholar
[7] Carnap, Rudolf, Die logizistische Grundlegung der Mathematik, Erkenntnis, vol. 2 (1931), pp. 91–105, references to the English translation in P. Benacerraf and H. Putnam, Philosophy of Mathematics : selected readings, Cambridge University Press, 1983, 41–52.Google Scholar
[8] Church, Alonzo, Review of Chwistek, L., Überwindung des Begriffsrelativismus, The Journal of Symbolic Logic, vol. 2 (1937), pp. 169–170.Google Scholar
[9] Church, Alonzo, The present situation in the foundations of mathematics, Philosophie mathématique (Gonseth, F., editor), Hermann, Paris, 1939.Google Scholar
[10] Church, Alonzo, A formulation of the simple theory of types, The Journal of Symbolic Logic, vol. 5 (1940), pp. 56–68.Google Scholar
[11] Church, Alonzo, Introduction to mathematical logic, Princeton University Press, 1956.Google Scholar
[12] Corry, Leo, Modern algebra and the rise of mathematical structures, Birkhäuser, Basel, 1996.Google Scholar
[13] Creath, Richard (editor), Dear Carnap, Dear Van: The Quine-Carnap correspondence and related work, University of California Press, 1990.Google Scholar
[14] Dawson, John W. Jr., Completing the GOdel-Zermelo correspondence, Historia Mathematica, vol. 12 (1985), pp. 66–70.CrossRefGoogle Scholar
[15] de Morgan, Augustus, On the syllogism: III and on logic in general, Transactions of the Cambridge Philosophical Society, 1858, references to (Heart, P., editor), On the Syllogism and other logical writings , pp. 74–146, Routledge & Kegan Paul, London, 1966.Google Scholar
[16] de Morgan, Augustus, On the syllogism: IV and on the logic of relations, Transactions of the Cambridge Philosophical Society (1860), references to (Heart, P., editor), On the Syllogism and other logical writings , pp. 208–246, Routledge & Kegan Paul, London, 1966.Google Scholar
[17] Dedekind, Richard, Stetigkeit und irrationale Zahlen, Gesammelte mathematische Werke, vol. 3, Chelsea, New York, 1969, reprintof original edition (1872). English translation in W. B. Ewald (editor), From Kant to Hilbert , Oxford University Press, vol. 2.Google Scholar
[18] Dedekind, Richard, Was sind und was sollen die Zahlen?, Gesammelte mathematische Werke, vol. 3, Chelsea, New York, 1969, reprint of original edition (1888). English translation in W. B. Ewald (editor), From Kant to Hilbert , Oxford Univ. Press, vol. 2.Google Scholar
[19] Dreben, Burton and van Heuenoort, J., Introductory note, Collected works of Gödel, K., vol. 1, Oxford University Press, 1986, pp. 44–59.Google Scholar
[20] Drobisch, Moritz W., Neue Darstellung der Logik nach ihren einfachen Verhältnissen, Voss, Leipzig, 1836 (four editions up to 1875).Google Scholar
[21] Ferreirós, José, Traditional logic and the early history of sets, 1854-1908, Archive for History of Exact Sciences, vol. 50 (1996), pp. 5–71.Google Scholar
[22] Ferreiros, Jose, Notes on types, sets and logicism, 1930–1950, Theoria, vol. 12 (1997), pp. 91–124.Google Scholar
[23] Ferreiros, Jose, Labyrinth of thought. A history ofset theory and its role in modern mathematics, Birkhäuser, Basel, 1999.Google Scholar
[24] Frege, Gottlob, Begriffsschrift, Nebert, Halle, 1879, reprint as Begriffsschrift und andere Aufsatze (I. Angelelli, editor) in Olms, Hildesheim, 1964.Google Scholar
[25] Frege, Gottlob, Grundgesetze der Arithmetik, vol. 1, Pohl, Jena, 1893, reprint Olms, Hildesheim, 1966.Google Scholar
[26] Frege, Gottlob, Grundgesetze der Arithmetik, vol. 2, Pohl, Jena, 1903, reprint Olms, Hildesheim, 1966.Google Scholar
[27] Frege, Gottlob, Anwendungen der Begriffsschrift, Jenaischer Zeitschrift für Naturwiss, 1879. References to the reprint in Begriffsschrift und andere Aufsätze (I. Angelelli, editor), Olms, Hildesheim, 1964.Google Scholar
[28] Frege, Gottlob, Der Gedanke. Eine logische Untersuchung, Beiträge zur Philosophie des deutschen Idealismus, vol. 1 (1918), references to the English translation in B. McGuinness, editor. Collected Papers on Mathematics, Logic, and Philosophy , pp. 351–372, Basil Blackwell, Oxford, 1984.Google Scholar
[29] Gödel, Kurt, Über formal unentscheidbare Satze der Principia Mathematica und verwandter Systeme, Monatshefte für Mathematik und Physik, vol. 38 (1931), pp. 173–198, references to the reprint in Collected works , vol. 1, Oxford University Press, 1986.CrossRefGoogle Scholar
[30] Godel, Kurt, Consistency proof for the generalized continuum hypothesis, Proceedings of the National Academy of Sciences of the United States of America, vol. 24 (1939), pp. 220–224, references to the reprint in Collected works , vol. 2, Oxford University Press, 1990.Google Scholar
[31] Godel, Kurt, The consistency of the axiom of choice and of the generalized continuum hypothesis with the axioms of set theory, Princeton University Press, 1940, references to the reprint in Collected works , vol. 2, Oxford University Press, 1990.Google Scholar
[32] Goldfarb, Warren, Logic in the twenties: The nature of the quantifier, The Journal of Symbolic Logic, vol. 44 (1979), pp. 351–368.Google Scholar
[33] Grattan-Guinness, Ivor, In memoriam Kurt Godel: His 1931 correspondence with Zermelo, Historia Mathematica, vol. 6 (1979), pp. 294–304.Google Scholar
[34] Grattan-Guinness, Ivor, Living together and living apart. On the interactions between mathematics and logics from the French Revolution to the First World War, South African Journal of Philosophy, vol. 7 (1988), pp. 73–82.Google Scholar
[35] Hllbert, David, Axiomatisches Denken, Mathematische Annalen, vol. 78 (1918), pp. 405–415, references to the reprint in Gesammelte Abhandlungen , vol. 3, Springer, Berlin, 1935, 146–156.Google Scholar
[36] Hllbert, David, Die logischen Grundlagen der Mathematik, Mathematische Annalen, vol. 88 (1923), pp. 151–165, references to the reprint in Gesammelte Abhandlungen , vol. 3, Springer, Berlin, 1935, 178–191.Google Scholar
[37] Hllbert, David and Ackermann, Wilhelm, Grundzüge der theoretischen Logik, Springer, Berlin, 1928.Google Scholar
[38] Klein, Felix, Vorlesungen über die Entwicklung der Mathematik, Springer, Berlin, 1926, 2 vols. (reprinted in 1979).Google Scholar
[39] Kneale, William and Kneale, Martha, The development of logic, Clarendon, Oxford, 1962.Google Scholar
[40] Kuhn, Thomas, The structure of scientific revolutions, 2nd ed., Chicago University Press, 1970.Google Scholar
[41] Lewis, Clarence I., A survey of symbolic logic, University of California Press, Berkeley, 1918, reprint Dover, New York.Google Scholar
[42] Łukasiewicz, Jan, Aristotle's syllogistics, Clarendon, Oxford, 1957.Google Scholar
[43] Moore, Gregory H., Beyond first-order logic. The historical interplay between mathematical logic and axiomatic set theory, History and Philosophy of Logic, vol. 1 (1980), pp. 95–137.CrossRefGoogle Scholar
[44] Moore, Gregory H., Zermelo's axiom of choice, Springer, Berlin, 1982.Google Scholar
[45] Moore, Gregory H., The emergence offirst-order logic, History and philosophy of modern mathematics (Aspray, W. and Kitcher, P., editors), University of Minnesota Press, 1988.Google Scholar
[46] Moore, Gregory H., Hilbert and the emergence of modern mathematical logic, Theoria, vol. 12 (1997), pp. 65–90.Google Scholar
[47] Moore, Gregory H., Logic, early twentieth century, Routledge encyclopedia of philosophy (Craig, E., editor), Routledge, London, 1998.Google Scholar
[48] Nagel, Ernest, The formation of modern conceptions of formal logic in the development of geometry (original edition 1939), Teleology revisited, Columbia University Press, 1979.Google Scholar
[49] Pasch, Moritz, Vorlesungen über neuere Geometrie, Springer, Berlin, 1926 (first edition 1882).Google Scholar
[50] Peano, Giuseppe, Arithmeticesprincipia, nova methodo exposita, Bocca, Torino, 1889, partial English translation in [72].Google Scholar
[51] Peckhaus, Volker, The way of logic into mathematics, Theoria, vol. 12 (l997), pp. 39–64.Google Scholar
[52] Peckhaus, Volker, Mathesis universalis. Leibniz und die Entdeckung der formalen Logik im 19. Jahrhundert, Akademie-Verlag, Berlin, 1998.Google Scholar
[53] Peckhaus, Volker, 19th century logic between philosophy and mathematics, this Bulletin, vol. 5 (1999), pp. 433–450.Google Scholar
[54] Quine, Willard van O., Set-theoretic foundations for logic, The Journal of Symbolic Logic, vol. 1 (1936), pp. 45–57.Google Scholar
[55] Peckhaus, Volker, Mathematical logic, Norton, New York, 1940.Google Scholar
[56] Peckhaus, Volker, From a logical point of view, Harvard University Press, 1953.Google Scholar
[57] Peckhaus, Volker, Philosophy of logic, Prentice-Hall, Englewood Cliffs, NJ, 1970.Google Scholar
[58] Peckhaus, Volker, Autobiographical notes, The philosophy of W. V. Quine (Hahn, L. E. and Schilpp, P. A., editors), Open Court, La Salle, IL, 1986, pp. 1–46.Google Scholar
[59] Ramsey, Frank, The foundations of mathematics, Proceedings of the London Mathematical Society, vol. 25, 1926, reprinted in Foundations , London, Routledge & Kegan Paul, 1978.Google Scholar
[60] Russell, Bertrand, The principles of mathematics, Cambridge University Press, 1903, (2nd edition 1937). Reprint London, Allen & Unwin, 1948.Google Scholar
[61] Russell, Bertrand, Mathematical logic as based on the theory of types, American Journal of Mathematics, vol. 30 (1908), pp. 222–262, references to the reprint in [72].Google Scholar
[62] Scanlan, Michael, Who were the American postulate theorists?, The Journal of Symbolic Logic, vol. 56 (1991), pp. 981–1002.Google Scholar
[63] Shapiro, Stewart, Foundations without foundationalism. A case for second-order logic, Oxford University Press, 1990.Google Scholar
[64] Skolem, Thoralf, Einige Bemerkungen zur axiomatischen Begründung der Mengenlehre, Dem femte skandinaviska mathematikerkongressen (Helsinki), Akademiska Bokhan-deln, 1923, also in Selected Works in Logic , Universitetsforlaget, Oslo, 1970. References to the English translation in [72], 290–301.Google Scholar
[65] Skolem, Thoralf, Über die mathematische Logik, Norsk matematisk tidsskrift, vol. 10 (1928), pp. 125–142, references to the English translation in [72], 512–524. Also in Selected Works in Logic .Google Scholar
[66] Skolem, Thoralf, Über einige Grundlagenfragen der Mathematik, Videnskaps-selskapets Skrifter, (1929), no. 4, pp. 1–49, references to the reprint in Selected Works in Logic , Universitetsforlaget, Oslo, 1970.Google Scholar
[67] Tarski, Alfred, Der Wahrheitsbegrifff in den formalisierten Sprachen, Studia philosophica, vol. 1 (1935), (Polish original, without postscript, in 1933). References to the English translation in Logic, Semantics, Metamathematics , Oxford University Press, 1956.Google Scholar
[68] Tarski, Alfred, Introduction to logic and the methodology of deductive sciences, Harvard University Press, 1941.Google Scholar
[69] Tarski, Alfred, Über den Begriff der logischen Folgerung, Actes du congrès international de philosophie scientifique, (1936), references to the English translation in Logic, semantics, metamathematics , Oxford University Press, 1956.Google Scholar
[70] Ueberweg, Friedrich, System der Logik und Geschichte der logischen Lehren, 5th ed., Marcus, A., Bonn, 1882, English translation of the 3rd edition, London, Longmans, Green & Co., 1871.Google Scholar
[71] van der Waerden, Bartel L., Moderne Algebra, Springer, Berlin, 1930.CrossRefGoogle Scholar
[72] van Heiuenoort, Jean, From Frege to Godel, Harvard University Press, 1967.Google Scholar
[73] von Neumann, John, Eine Axiomatisierung der Mengenlehre, Journal für die reine und angewandte Mathematik, vol. 154 (1925), pp. 219–240, reprint in Collected Works , vol. 1, Oxford, Pergamon, 1961. English trans. in [72], 394–413.CrossRefGoogle Scholar
[74] von Neumann, John, Zur Hilbert sehen Beweistheorie, Mathematische Zeitschrift, vol. 26 (1927), pp. 1–46, references to Collected Works , vol. 1, Oxford, Pergamon, 1961, 256–300.Google Scholar
[75] von Neumann, John, Die formalistische Grundlegung der Mathematik, Erkenntnis, vol. 2 (1931), references to the English translation in P. Benacerraf and H. Putnam, Philosophy of Mathe-matics: selected readings , Cambridge University Press, 1983.Google Scholar
[76] Weyl, Hermann, Über die Definitionen der mathematischen Grundbegriffe, Mathematisch-naturwissenschaftliche Blatter 7, (1910), references to the reprint in Gesammelte Abhandlungen , Springer, Berlin, 1968, vol. 1, 298–304.Google Scholar
[77] Weyl, Hermann, Das Kontinuum: Kritische Untersuchungen über die Grundlagen der Analysis, Veit, Leipzig, 1918, references to the reprint Chelsea, New York.Google Scholar
[78] Whately, Richard, Elements of logic, J. Mawman, London, 1827.Google Scholar
[79] Whitehead, Alfred N. and Russell, Bertrand, Principia mathematica, Cambridge University Press, 1910–1913 (2nd edition 1925–1927). References to the 1978 reprint.Google Scholar
[80] Zermelo, Ernst, Untersuchungen über die Grundlagen der Mengenlehre, Mathematische Annalen, vol. 65 (1908), pp. 261–281, English translation in [72], 199–215.Google Scholar
[81] Zermelo, Ernst, Über Stufen der Quantification und die Logik des Unendlichen, Jahresbericht der Deutschen Mathematiker-Vereinigung, vol. 41 (1932), pp. 85–88.Google Scholar