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Dubislav and Classical Monadic Quantificational Logic

  • Christian ThielEmail author
Chapter
Part of the Boston Studies in the Philosophy and History of Science book series (BSPS, volume 273)

Abstract

Among the members of the Berlin group, the scholar most devoted to the philosophy of logic and mathematics was Walter Dubislav (1891–1937). In a paper written in 1929, titled “Elementarer Nachweis der Widerspruchslosigkeit des Logik-Kalküls” (“Elementary proof of the consistency of the calculus of logic”), Dubislav made use of a procedure of “quasi truth valuation” (suggested by Emil L. Post in 1921) that was meant to prove the consistency of quantificational logic. Dubislav then employed this procedure as a “purely calculatory criterion for definitions” in the third edition (1931) of his monograph Die Definition. In a book written in 1932, entitled Die Philosophie der Mathematik in der Gegenwart (Contemporary Philosophy of Mathematics), Dubislav claimed that his approach provides a decision procedure for quantificational logic, explaining its application via examples from classical monadic quantificational logic. Whereas the decidability of the latter had been known since 1915, the undecidability of the full range of quantificational logic could not have been known to Dubislav in 1932. The fact that Dubislav’s procedure does not qualify as a decision procedure for even the monadic case was, after some cautious criticism raised in the 1930s, clearly stated for the first time in Reichenbach’s Elements of Symbolic Logic in 1947. This paper offers an elementary and detailed elucidation of the history surrounding these developments, and of their theoretical background.

Keywords

Decision Procedure Functional Calculus Quantificational Logic Propositional Function Consistency Proof 
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.

References

  1. Behmann, Heinrich. 1922. Beiträge zur Algebra der Logik, insbesondere zum Entscheidungsproblem. Mathematische Annalen 86: 163–229.CrossRefGoogle Scholar
  2. Dubislav, Walter. 1926. Über die definition. Berlin-Schöneberg: Hermann Weiß.Google Scholar
  3. Dubislav, Walter. 1927. Über die definition. 2nd rev ed. Berlin-Schöneberg: Hermann Weiß.Google Scholar
  4. Dubislav, Walter. 1928. Zur kalkülmäßigen Charakterisierung der Definitionen. Annalen der Philosophie und philosophischen Kritik 7: 136–145.Google Scholar
  5. Dubislav, Walter. 1929. Elementarer Nachweis der Widerspruchslosigkeit des Logik-Kalküls. Journal für die reine und angewandte Mathematik 161: 107–112.Google Scholar
  6. Dubislav, Walter. 1931–32. Les recherches sur la philosophie des mathématiques en Allemagne (Aperçu général). Recherches Philosophiques 1: 299–311.Google Scholar
  7. Dubislav, Walter. 1931. Die definition. Third, completely revised and enlarged edition. Leipzig: Felix Meiner (Beihefte der „Erkenntnis“, 1).Google Scholar
  8. Dubislav, Walter. 1932. Die Philosophie der Mathematik in der Gegenwart. Berlin: Junker und Dünnhaupt (Philosophische Forschungsberichte, Heft 13).Google Scholar
  9. Dubislav, Walter. 1981. Die definition. 4th ed. With an introduction by Wilhelm K. Essler. Hamburg: Felix Meiner.Google Scholar
  10. Fraenkel, Adolf A. 1929. [Notice of Dubislav 1929]. Jahrbuch über die Fortschritte der Mathematik 55 I (Heft 1, publ. 1931), 33.Google Scholar
  11. Gödel, Kurt. 1930. Die Vollständigkeit der Axiome des logischen Funktionenkalküls. Monatshefte für Mathematik und Physik 37: 349–360.CrossRefGoogle Scholar
  12. Gödel, Kurt. 1931. Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I. Monatshefte für Mathematik und Physik 38: 173–198.CrossRefGoogle Scholar
  13. Hilbert, David, and Wilhelm Ackermann. 1928. Grundzüge der theoretischen Logik. 2nd rev ed. Berlin: Julius Springer (Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, Band XXVII).Google Scholar
  14. Hilbert, David, and Paul Bernays. 1934. Grundlagen der Mathematik. Erster Band. Berlin: Springer (Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete, Band XL).Google Scholar
  15. Löwenheim, Leopold. 1915. Über Möglichkeiten im Relativkalkül. Mathematische Annalen 76: 447–470.CrossRefGoogle Scholar
  16. Post, Emil Leon. 1921. Introduction to a general theory of elementary propositions. American Journal of Mathematics 43: 163–185.CrossRefGoogle Scholar
  17. Quine, Willard Van Orman. 1938. Review of Hilbert/Ackermann 1938. Journal of Symbolic Logic 3: 83–84.CrossRefGoogle Scholar
  18. Quine, Willard Van Orman. 1945. On the logic of quantification. Journal of Symbolic Logic 10: 1–12.CrossRefGoogle Scholar
  19. Quine, Willard Van Orman. 1948. [Review of Reichenbach 1947]. Journal of Philosophy 45(6): 161–166.CrossRefGoogle Scholar
  20. Reichenbach, Hans. 1932. Wahrscheinlichkeitslogik. Sitzungsberichte der Preußischen Akademie der Wissenschaften. Physikalisch-mathematische Klasse 1932:476–488.Google Scholar
  21. Reichenbach, Hans. 1947. Elements of symbolic logic. New York: Macmillan.Google Scholar
  22. Russell, Bertrand. 1919. Introduction to mathematical philosophy. London/New York: Allen and Unwin/Macmillan.Google Scholar
  23. Schmidt, Arnold. 1932. [Review of Dubislav 1931]. Zentralblatt für Mathematik und ihre Grenzgebiete 2(Heft 1): 1–2.Google Scholar
  24. Schmidt, Arnold. 1933. [Review of Dubislav 1932]. Zentralblatt für Mathematik und ihre Grenzgebiete 5(Heft 4): 145.Google Scholar
  25. Scholz, Heinrich. 1933. Logistik. Vorlesung [Münster i.W.] Winter-Semester 1932/33, Sommer-Semester 1933. Mimeographed lecture courses.Google Scholar
  26. Scholz, Heinrich. 1934. [Review of Dubislav 1932]. Jahresbericht der Deutschen Mathematiker-Vereinigung 43:88–90 of the section “Literarisches” [italic pagination].Google Scholar
  27. von Neumann, Johann J. 1927. Zur Hilbertschen Beweistheorie. Mathematische Zeitschrift 26: 1–46.CrossRefGoogle Scholar
  28. Wittgenstein, Ludwig. 1921. Logisch-Philosophische Abhandlung. Annalen der Naturphilosophie 14 (Heft 3/4): 185–262. Bilingual monographic edition: Tractatus Logico-Philosophicus. London: Kegan Paul, Trench, Trubner & Co. 1922.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht. 2013

Authors and Affiliations

  1. 1.Institute of PhilosophyUniversity of Erlangen–NürnbergErlangenGermany

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