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The Road to Modern Logic—An Interpretation


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)

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[2] Paul Bernays , A system of axiomatic set theory, The Journal of Symbolic Logic, vol. 2 (1937), pp. 6577.

[7] Rudolf Carnap , Die logizistische Grundlegung der Mathematik, Erkenntnis, vol. 2 (1931), pp. 91105, references to the English translation in P. Benacerraf and H. Putnam, Philosophy of Mathematics: selected readings, Cambridge University Press, 1983, 41–52.

[10] Alonzo Church , A formulation of the simple theory of types, The Journal of Symbolic Logic, vol. 5 (1940), pp. 5668.

[14] John W. Dawson Jr., Completing the GOdel-Zermelo correspondence, Historia Mathematica, vol. 12 (1985), pp. 6670.

[21] José Ferreirós , Traditional logic and the early history of sets, 1854-1908, Archive for History of Exact Sciences, vol. 50 (1996), pp. 571.

[32] Warren Goldfarb , Logic in the twenties: The nature of the quantifier, The Journal of Symbolic Logic, vol. 44 (1979), pp. 351368.

[33] Ivor Grattan-Guinness , In memoriam Kurt Godel: His 1931 correspondence with Zermelo, Historia Mathematica, vol. 6 (1979), pp. 294304.

[35] David Hllbert , Axiomatisches Denken, Mathematische Annalen, vol. 78 (1918), pp. 405415, references to the reprint in Gesammelte Abhandlungen, vol. 3, Springer, Berlin, 1935, 146156.

[36] David Hllbert , Die logischen Grundlagen der Mathematik, Mathematische Annalen, vol. 88 (1923), pp. 151165, references to the reprint in Gesammelte Abhandlungen, vol. 3, Springer, Berlin, 1935, 178–191.

[43] Gregory H. Moore , Beyond first-order logic. The historical interplay between mathematical logic and axiomatic set theory, History and Philosophy of Logic, vol. 1 (1980), pp. 95137.

[44] Gregory H. Moore , Zermelo's axiom of choice, Springer, Berlin, 1982.

[49] Moritz Pasch , Vorlesungen über neuere Geometrie, Springer, Berlin, 1926 (first edition 1882).

[54] Willard van O. Quine , Set-theoretic foundations for logic, The Journal of Symbolic Logic, vol. 1 (1936), pp. 4557.

[61] Bertrand Russell , Mathematical logic as based on the theory of types, American Journal of Mathematics, vol. 30 (1908), pp. 222262, references to the reprint in [72].

[62] Michael Scanlan , Who were the American postulate theorists?, The Journal of Symbolic Logic, vol. 56 (1991), pp. 9811002.

[71] Bartel L. van der Waerden , Moderne Algebra, Springer, Berlin, 1930.

[74] John von Neumann , Zur Hilbert sehen Beweistheorie, Mathematische Zeitschrift, vol. 26 (1927), pp. 146, references to Collected Works, vol. 1, Oxford, Pergamon, 1961, 256–300.

[80] Ernst Zermelo , Untersuchungen über die Grundlagen der Mengenlehre, Mathematische Annalen, vol. 65 (1908), pp. 261281, English translation in [72], 199215.

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Bulletin of Symbolic Logic
  • ISSN: 1079-8986
  • EISSN: 1943-5894
  • URL: /core/journals/bulletin-of-symbolic-logic
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