Advertisement

The “Triumph” of First-Order Languages

  • Stewart Shapiro
Part of the Synthese Library book series (SYLI, volume 305)

Abstract

Is second-order logic the answer? As the saying goes, it depends on the question. There are several important metaphysical and epistemic issues in the vicinity: the nature and status of classes, and their place in logic, and the relationship between logic and mathematics. We also have to address significant details of logic: variables, connectives, and quantifiers.

Keywords

Propositional Function Implicit Definition Logic Colloquium Theoretical Syntax Restricted Functional Calculus 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Ackermann, W. 1924 Begründung des ‘Tertium non datur’ mittels der Hilbertschen Theorie der Widerspruchsfreiheit, Mathematische Annalen, vol. 93, pp. 1–36.CrossRefGoogle Scholar
  2. Barwise, J., and S. Feferman 1985 Model-theoretic logics, Springer-Verlag, New York.Google Scholar
  3. Benacerraf, P. 1985 Skolem and the skeptic, Proceedings of the Aristotelian Society, Supplementary Volume 59, pp. 85–115.Google Scholar
  4. Bernays, P. 1918 Beiträge zur axiomatischen Behandlung des Logik-Kalküls, Habilitationsschrift, Göttingen.Google Scholar
  5. Bernays, P. 1928 Die Philosophie der Mathematik und die Hilbertsche Beweistheorie, Blätter für Deutsche Philosophie, vol. 4, pp. 326–367.Google Scholar
  6. Bernays, P. 1935 Platonism in mathematics, Philosophy of mathematics (P. Benacerraf and H. Putnam, editors), second edition, Cambridge University Press, Cambridge, England, 1983, pp. 258–271.Google Scholar
  7. Bernays, P., and M. Schönfinkel 1928 Zum Entscheidungsproblem der mathematischen Logic, Mathematische Annalen, vol. 99, pp. 342–372.CrossRefGoogle Scholar
  8. Boole, G. 1847 The mathematical analysis of logic, being an essay toward a calculus of reasoning, London.Google Scholar
  9. Boole, G. 1854 An investigation of the laws of thought on which are founded the mathematical theories of logic and probabilities, London.Google Scholar
  10. Boolos, G. 1975 On second-order logic, The Journal of Philosophy, vol. 72, pp. 509–527.CrossRefGoogle Scholar
  11. Boolos, G. 1984 To be is to be a value of a variable (or to be some values of some variables), The Journal of Philosophy, vol. 81, pp. 430–449.CrossRefGoogle Scholar
  12. Boolos, G. 1985 Nominalist Platonism, The Philosophical Review, vol. 94, pp. 327–344.CrossRefGoogle Scholar
  13. Boolos, G., and R. Jeffrey 1980 Computability and logic, second edition, Cambridge University Press, Cambridge, England.Google Scholar
  14. Carnap, R. 1943 Formalization of logic, Harvard University Press, Cambridge, Massachusetts.Google Scholar
  15. Carnap, R., and F. Bachmann 1936 Über Extremalaxiome, Erkenntnis, vol. 6, pp. 166–188; translated by H. G. Bohnert, History and Philosophy of Logic, vol. 2, 1981, pp. 67–85.CrossRefGoogle Scholar
  16. Church, A. 1951 A formulation of the logic of sense and denotation, Structure, method and meaning (P. Henle et al., editors), Liberal Arts Press, New York, pp. 3–24.Google Scholar
  17. Church, A. 1956 Introduction to mathematical logic, vol. 1, Princeton University Press, Princeton, New Jersey.Google Scholar
  18. Corcoran, J. 1973 Gaps between logical theory and mathematical practice, The methodological unity of science (M. Bunge, editor), Reidel, Dordrecht, pp. 23–50.CrossRefGoogle Scholar
  19. Corcoran, J. 1980 Categoricity, History and Philosophy of Logic, vol. 1, pp. 187–207.CrossRefGoogle Scholar
  20. Dawson, J. 1985 Completing the Gödel-Zermelo correspondence, Historia Mathematica, vol. 12, pp. 66–70.CrossRefGoogle Scholar
  21. Dedekind, R. 1888 The nature and meaning of numbers, Essays on the theory of numbers (W. W. Berman, editor), Dover, New York, 1963, pp. 31–115.Google Scholar
  22. Etchemendy, J. 1988 Tarski on truth and logical consequence, The Journal of Symbolic Logic, vol. 53, pp. 51–79.CrossRefGoogle Scholar
  23. Field, H. 1984 Is mathematical knowledge just logical knowledge?, The Philosophical Review, vol. 93, pp. 509–552.CrossRefGoogle Scholar
  24. Fraenkel, A. 1921 Über die Zermelosche Begründung der Mengenlehre, Jahresbericht der Deutschen Mathematiker-Vereinigung, vol. 30, second section, pp. 97–98.Google Scholar
  25. Fraenkel, A. 1922 Zu den Grundlagen der Mengenlehre, Jahresbericht der Deutschen Mathematiker-Vereinigung, vol. 31, second section, pp. 101–102.Google Scholar
  26. Fraenkel, A. 1922a Der Begriff ‘definit’ und die Unabhängigkeit des Auswahlaxioms, Sitzungsberichte der Preussichen Akademie der Wissenschaften, Physikalisch-mathematische Klasse, pp. 253–257; translated in van Heijenoort 1967, pp. 284–289.Google Scholar
  27. Fraenkel, A. 1925 Untersuchungen Über die Grundlagen der Mengenlehre, Mathematische Zeitschrift, vol. 22, pp. 250–273.CrossRefGoogle Scholar
  28. Fraenkel, A., Y. Bar-Hillel, and A. Levy 1973 Foundations of set theory, second revised edition, North Holland, Amsterdam.Google Scholar
  29. Frege, G. 1879 Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens, Louis Nebert, Halle; translated in van Heijenoort 1967, pp. 1–82.Google Scholar
  30. Frege, G. 1883 Über den Zweck der Begriffsschrift, Sitzungsberichte der Jenaischen Gesellschaft für Medicin und Naturwissenschaft, vol. 16, pp. 1–10.Google Scholar
  31. Frege, G. 1884 Die Grundlagen der Arithmetik, Koebner, Breslau; translated by J. Austin, second edition, Harper, New York, 1960.Google Scholar
  32. Frege, G. 1893 Grundgesetze der Arithmetik 1, Olms, Hildescheim.Google Scholar
  33. Frege, G. 1895 Review of Schröder’s Vorlesungen Über die Algebra der Logik, Archiv für systematische Philosophie, vol. 1, pp. 433–456; translated in Translations from the philosophical writings of Gottlob Frege (P. Geach and M. Black, editors), Blackwell, Oxford, pp. 86–106.Google Scholar
  34. Gandy, R. 1988 The confluence of ideas in 1936, The universal Turing machine (R. Herken, editor), Oxford University Press, New York, pp. 55–111.Google Scholar
  35. Gilmore, P. 1957 The monadic theory of types in the lower-predicate calculus, Summaries of talks presented at the Summer Institute of Symbolic Logic at Cornell, Institute for Defense Analysis.Google Scholar
  36. Gödel, K. 1930 Die Vollständigkeit der Axiome des logischen Funktionenkalkuls, Montatshefte für Mathematik und Physik, vol. 37, pp. 349–360; translated in van Heijenoort 1967, pp. 582–591.CrossRefGoogle Scholar
  37. Gödel, K. 1931 Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I, Montatshefte für Mathematik und Physik, vol. 38, pp. 173–198; translated in van Heijenoort 1967, pp. 596–616.CrossRefGoogle Scholar
  38. Gödel, K. 1944 Russell’s mathematical logic, Philosophy of mathematics (P. Benacerraf and H. Putnam, editors), second edition, Cambridge University Press, Cambridge, England, 1983, pp. 447–469.Google Scholar
  39. Goldfarb, W. 1979 Logic in the twenties: The nature of the quantifier, The Journal of Symbolic Logic, vol. 44, pp. 351–368.CrossRefGoogle Scholar
  40. Goldfarb, W. 1988 Poincaré against the logicists, History and philosophy of modern mathematics (W. Aspray and P. Kitcher, editors), Minnesota Studies in the Philosophy of Science, vol. 11, University of Minnesota Press, pp. 61–81.Google Scholar
  41. Gonseth, F. 1941 Les entretiens de Zurich, 6-9 décembre 1938, Leeman, Zurich. Grattan-Guinness, I.Google Scholar
  42. Gonseth, F. 1977 Dear Russell — Dear Jourdain, Columbia University Press, New York.Google Scholar
  43. Gonseth, F. 1979 In memoriam Kurt Gödel: His 1931 correspondence with Zermelo on his incompletability theorem, Historia Mathematica, vol. 6, pp. 294–304.CrossRefGoogle Scholar
  44. Hazen, A. 1983 Predicative logics, Handbook of philosophical logic 1 (D. Gabbay and F. Guenthner, editors), Reidel, Dordrecht, pp. 331–407.CrossRefGoogle Scholar
  45. Henkin, L. 1950 Completeness in the theory of types, The Journal of Symbolic Logic, vol. 15, pp. 81–91.CrossRefGoogle Scholar
  46. Hilbert, D. 1899 Grundlagen der Geometrie, Teubner, Leipzig; Foundations of geometry, (E. Townsend, translator), Open Court, La Salle, Illinois, 1959.Google Scholar
  47. Hilbert, D. 1900 Über den Zahlbegriff, Jahresbericht der Deutschen Mathematiker-Vereinigung, vol. 8, pp. 180–194.Google Scholar
  48. Hilbert, D. 1900a Mathematische Problem, Bulletin of the American Mathematical Society, vol. 8 (1902), pp. 437–479.CrossRefGoogle Scholar
  49. Hilbert, D. 1902 Les principes fondamentaux de la géometrie, Gauthier-Villars, Paris; French translation of Hilbert 1899.Google Scholar
  50. Hilbert, D. 1905 Über der Grundlagen der Logik und der Arithmetik, Verhandlungen des dritten internationalen Mathematiker-Kongresses in Heidelberg vom 8. bis 13 August 1904, Teubner, Leipzig; translated in van Heijenoort 1967, pp. 129–138.Google Scholar
  51. Hilbert, D. 1923 Die logischen Grundlagen der Mathematik, Mathematische Annalen, vol. 88, pp. 151–165.CrossRefGoogle Scholar
  52. Hilbert, D. 1929 Probleme der Grundlegung der Mathematik, Mathematische Annalen, vol. 102, pp. 1–9.CrossRefGoogle Scholar
  53. Hilbert, D., and W. Ackermann. 1928 Grundzüge der theoretischen Logik, Springer, Berlin.Google Scholar
  54. Huntington, E. 1902 A complete set of postulates for the theory of absolute continuous magnitude, Transactions of the American Mathematical Society, vol. 3, pp. 264–279.CrossRefGoogle Scholar
  55. Huntington, E. 1905 A complete set of postulates for ordinary complex algebra, Transactions of the American Mathematical Society, vol. 6, pp. 209–229.CrossRefGoogle Scholar
  56. Kreisel, G. 1967 Informal rigour and completeness proofs, Problems in the philosophy of mathematics (I. Lakatos, editor), North-Holland, Amsterdam, pp. 138–186.CrossRefGoogle Scholar
  57. Lewis, C. I. 1918 A survey of symbolic logic, University of California Press, Berkeley.Google Scholar
  58. Lindström, P. 1969 On extensions of elementary logic, Theoria, vol. 35, pp. 1–11.CrossRefGoogle Scholar
  59. Löwenheim, L. 1915 Über Möglichkeiten im Relativkalkül, Mathematische Annalen, vol. 76, pp. 447–479; translated in van Heijenoort 1961, pp. 228–251.CrossRefGoogle Scholar
  60. Montague, R. 1965 Set theory and higher-order logic, Formal systems and recursive functions (J. Crossley and M. Dummett, editors), North-Holland, Amsterdam, pp. 131–148.CrossRefGoogle Scholar
  61. Moore, G. 1978 The origins of Zermelo’s axiomatization of set theory, Journal of Philosophical Logic, vol. 7, pp. 307–329.CrossRefGoogle Scholar
  62. Moore, G. 1980 Beyond first-order logic, the historical interplay between logic and set theory, History and Philosophy of Logic, vol. 1, pp. 95–137.CrossRefGoogle Scholar
  63. Moore, G. 1982 Zermelo’ s axiom of choice: Its origins, development, and influence, Springer-Verlag, New York.CrossRefGoogle Scholar
  64. Moore, G. 1988 The emergence of first-order logic, History and philosophy of modern mathematics (W. Aspray and P. Kitcher, editors), Minnesota Studies in the Philosophy of Science, vol. 11, University of Minnesota Press, pp. 95–135.Google Scholar
  65. Myhill, J. 1951 On the ontological significance of the Löwenheim-Skolem theorem, Academic freedom, logic and religion (M. White editor), American Philosophical Society, Philadelphia, pp. 57–70; also in Contemporary readings in logical theory (I. Copi and J. Gould, editors), Macmillan, New York, 1967, pp. 40–54.Google Scholar
  66. Peano, G. 1889 Arithmetices principia, Nova methodo exposita, Turin; translated in van Heijenoort 1967, pp. 85–97.Google Scholar
  67. Poincaré, H. 1909 La Logique de l’infini, Revue de métaphysique et morale, vol. 17, pp. 461–482.Google Scholar
  68. Quine, W. V. O. 1941 Whitehead and the rise of modem logic, The philosophy of Alfred North Whitehead (P. A. Schilpp, editor), Tudor Publishing Company, New York, pp. 127–163.Google Scholar
  69. Quine, W. V. O. 1960 Word and object, The MIT Press.Google Scholar
  70. Quine, W. V. O. 1986 Philosophy of logic, second edition, Prentice-Hall, Englewood Cliffs.Google Scholar
  71. Ramsey, F. 1925 The foundations of mathematics, Proceedings of the London mathematical society, series 2, 25, pp. 338–384.Google Scholar
  72. Rang, B., and W. Thomas 1981 Zermelo’s discovery of the ‘Russell paradox’, Historia Mathematica, vol. 8, pp. 15–22.CrossRefGoogle Scholar
  73. Resnik, M. 1980 Frege and the philosophy of mathematics, Cornell University Press, Ithaca.Google Scholar
  74. Schoenflies, A. 1911 Über die Stellung der Definition in der Axiomatik, Jahresbericht der Deutsche Mathematiker-Vereinigung, vol. 20, pp. 222–255.Google Scholar
  75. Schönfinkel, M. 1924 Über die Bausteine der mathematischen Logik, Mathematische Annalen, vol. 92, pp. 305–316; translated in van Heijenoort 1967, pp. 355–366.CrossRefGoogle Scholar
  76. Schröder, E. 1890 Vorlesungen über die Algebra der Logik, vol. 1, Teubner, Leipzig.Google Scholar
  77. Shapiro, S. 1983 Remarks on the development of computability, History and Philosophy of Logic, vol. 4, pp. 203–220.CrossRefGoogle Scholar
  78. Shapiro, S. 1985 Second-order languages and mathematical practice, The Journal of Symbolic Logic, vol. 50, pp. 714–742.CrossRefGoogle Scholar
  79. Shapiro, S. 1990 Second-order logic, foundations, and rules, The Journal of Philosophy, vol. 87, pp. 234–261.CrossRefGoogle Scholar
  80. Shapiro, S. 1991 Foundations without foundationalism: A case for second-order logic, Oxford University Press, Oxford.Google Scholar
  81. Skolem, T. 1920 Logisch-kombinatorische Untersuchungen über die Erfüllbarkeit oder Beweisbarkeit mathematischer Sätze nebst einem Theoreme über dichte Mengen, Videnskapsselskapets shifter I. Matematisk-naturvidenskabelig klasse, no. 4; section 1 translated in van Heijenoort 1967, pp. 252–263.Google Scholar
  82. Skolem, T. 1922 Einige Bemerkungen zur axiomatischen Begründung der Mengenlehre, Matematikerkongressen i Helsingfors den 4-7 Juli 1922, Akademiska Bokhandeln, Helsinki, pp. 217–232; translated in van Heijenoort 1967, pp. 291–301.Google Scholar
  83. Skolem, T. 1928 Über die mathematische Logik, Norsk matematisk tidsskrift, vol. 10, pp. 125–142; translated in van Heijenoort 1967, pp. 508–524.Google Scholar
  84. Skolem, T. 1930 Einige Bemerkungen zu der Abhandlung von E. Zermelo: ‘Über die Definitheit in der Axiomatik’, Fundamenta Mathematicae, vol. 15, pp. 337–341.Google Scholar
  85. Skolem, T. 1933 Über die Unmöglichkeit einer vollständigen Charakterisuerung der Zahlenreihe mittels eines endlichen Axiomsystems, Norsk matematisk forenings skrifter, series 2, no. 10, pp. 73–82.Google Scholar
  86. Skolem, T. 1934 Über die Nicht-charakterisierbarkeit der Zahlenreihe mittels endlich oder abzählbar unendlich vieler Aussagen mit ausschliesslich Zahlenvariablen, Fundamenta Mathematicae, vol. 23, pp. 150–161.Google Scholar
  87. Skolem, T. 1941 Sur la porté du théormè de Löwenheim-Skolem, in Gonseth 1941, pp. 25–52.Google Scholar
  88. Skolem, T. 1950 Some remarks on the foundation of set theory, Proceedings of the international congress of mathematicians (Cambridge, Massachusetts), American Mathematical Society, Providence, 1952, pp. 695–704.Google Scholar
  89. Skolem, T. 1958 Une relativisation des notions mathématiques fondamentales, Colloques internationaux du Centre National de la Recherche Scientifique, Paris, pp. 13–18.Google Scholar
  90. Skolem, T. 1961 Interpretation of mathematical theories in the first-order predicate calculus, Essays on the foundations of mathematics, Dedicated to A. A. Fraenkel (Y. Bar-Hillel et al, editors), Magnes Press, Jerusalem, pp. 218–225.Google Scholar
  91. Tharp, L. 1975 Which logic is the right logic?, Synthese, vol. 31, pp. 1–31.CrossRefGoogle Scholar
  92. Thiel, C. 1977 Leopold Löwenheim: Life, work, and early influence, Logic colloquium 76 (R. Gandy and M. Hyland, editors), North-Holland, Amsterdam, pp. 235–252.CrossRefGoogle Scholar
  93. Van Heijenoort, J. 1967 From Frege to Gödel, Harvard University Press, Cambridge, Massachusetts.Google Scholar
  94. Van Heijenoort, J. 1967a Logic as calculus and logic as language, Synthese, vol. 17, pp. 324–330.CrossRefGoogle Scholar
  95. Veblen, O. 1904 A system of axioms for geometry, Transactions of the American Mathematical Society, vol. 5, pp. 343–384.CrossRefGoogle Scholar
  96. Von Neumann, J. 1925 Eine Axiomatisierung der Mengenlehre, Journal für die reine und angewandte Mathematik, vol. 154, pp. 219-244; translated in van Heijenoort 1967, pp. 393–413.Google Scholar
  97. Von Neumann, J. 1927 Zur Hilbertschen Beweistheorie, Mathematische Zeitschrift, vol. 26, pp. 1–46.CrossRefGoogle Scholar
  98. Von Neumann, J. 1928 Über die Definition durch transfinite Induktion und verwandte Fragen der allgemeinen Mengenlehre, Mathematische Annalen, vol. 99, pp. 373–391.CrossRefGoogle Scholar
  99. Wagner, S. 1987 The rationalist conception of logic, Notre Dame Journal of Formal Logic, vol. 28, pp. 3–35.CrossRefGoogle Scholar
  100. Wang, H. 1974 From mathematics to philosophy, Routledge and Kegan Paul, London.Google Scholar
  101. Wang, H. 1987 Reflections on Kurt Gödel, MIT Press.Google Scholar
  102. Weston, T. 1976 Kreisel, the continuum hypothesis and second-order set theory, Journal of Philosophical Logic, vol. 5, pp. 281–298.CrossRefGoogle Scholar
  103. Weyl, H. 1910 Über die Definitionen der mathematischen Grundbegriffe, Mathematischnaturwissenschaftliche Blätter, vol. 7, pp. 93–95, 109–113.Google Scholar
  104. Weyl, H. 1917 Das Kontinuum, Veit, Leipzig.Google Scholar
  105. Whitehead, A. N., and B. Russell 1910 Principia Mathematica, vol. 1, Cambridge University Press, Cambridge, England.Google Scholar
  106. Zermelo, E. 1904 Beweis, dass jede Menge wohlgeordnet werden kann, Mathematische Annalen, vol. 59, pp. 514–516; translated in van Heijenoort 1967, pp. 139–141.CrossRefGoogle Scholar
  107. Zermelo, E. 1908 Neuer Beweis für die Möglichkeit einer Wohlordnung, Mathematische Annalen, vol. 65, pp. 107–128; translated in van Heijenoort 1967, pp. 183–198.CrossRefGoogle Scholar
  108. Zermelo, E. 1908a Untersuchungen über die Grundlagen der Mengenlehre. I, Mathematische Annalen, vol. 65, pp. 261–281; translated in van Heijenoort 1967, pp. 199–215.CrossRefGoogle Scholar
  109. Zermelo, E. 1929 Über den Begriff der Defintheit in der Axiomatik, Fundamenta Mathematicae, vol. 14, pp. 339–344.Google Scholar
  110. Zermelo, E. 1930 Über Grenzzahlen und Mengenbereiche: Neue Untersuchungen über die Grundlagen der Mengenlehre, Fundamenta Mathematicae, vol. 16, pp. 29–47.Google Scholar
  111. Zermelo, E. 1931 Über stufen der Quantifikation und die Logik des Unendlichen, Jahresbericht der Deutsche Mathematische Verein, vol. 31, pp. 85–88.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2001

Authors and Affiliations

  • Stewart Shapiro

There are no affiliations available

Personalised recommendations