Semantics and Proof Theory of the Epsilon Calculus

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10119)


The epsilon operator is a term-forming operator which replaces quantifiers in ordinary predicate logic. The application of this undervalued formalism has been hampered by the absence of well-behaved proof systems on the one hand, and accessible presentations of its theory on the other. One significant early result for the original axiomatic proof system for the \(\varepsilon \)-calculus is the first epsilon theorem, for which a proof is sketched. The system itself is discussed, also relative to possible semantic interpretations. The problems facing the development of proof-theoretically well-behaved systems are outlined.


Choice Function Intuitionistic Logic Proof Theory Sequent Calculus Proof Tree 
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  1. 1.
    Abadi, M., Gonthier, G., Werner, B.: Choice in dynamic linking. In: Walukiewicz, I. (ed.) FoSSaCS 2004. LNCS, vol. 2987, pp. 12–26. Springer, Heidelberg (2004). doi: 10.1007/978-3-540-24727-2_3 CrossRefGoogle Scholar
  2. 2.
    Abiteboul, S., Vianu, V.: Non-determinism in logic-based languages. Ann. Math. Artif. Intell. 3(2–4), 151–186 (1991)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Asser, G.: Theorie der logischen Auswahlfunktionen. Z. Math. Logik Grundlag. Math. 3, 30–68 (1957)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Avigad, J., Zach, R.: The epsilon calculus. In: Zalta, E.N. (ed.) The Stanford Encyclopedia of Philosophy (Summer 2016th edn. (2016).
  5. 5.
    Bell, J.L.: Hilbert’s epsilon-operator and classical logic. J. Philos. Logic 22, 1–18 (1993)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Blass, A., Gurevich, Y.: The logic of choice. J. Symbolic Logic 65, 1264–1310 (2000)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    DeVidi, D.: Intuitionistic epsilon- and tau-calculi. Math. Logic Q. 41, 523–546 (1995)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    von Heusinger, K.: The reference of indefinites. In: von Heusinger, K., Egli, U. (eds.) Reference and Anaphoric Relations, pp. 247–265. Kluwer, Dordrecht (2000)CrossRefGoogle Scholar
  9. 9.
    von Heusinger, K.: Choice functions and the anaphoric semantics of definite NPs. Res. Lang. Comput. 2, 309–329 (2004)CrossRefMATHGoogle Scholar
  10. 10.
    Hilbert, D.: Neubegründung der Mathematik: Erste Mitteilung. Abhandlungen aus dem Seminar der Hamburgischen Universität 1, 157–77 , series of talks given at the University of Hamburg, July 25–27, 1921. English in [14], pp. 198–214 (1922)Google Scholar
  11. 11.
    Hilbert, D., Bernays, P.: Grundlagen der Mathematik. Springer, Berlin (1939)MATHGoogle Scholar
  12. 12.
    Leisenring, A.: Mathematical Logic and Hilbert’s \(\epsilon \)-symbol. MacDonald Technical and Scientific, London (1969)Google Scholar
  13. 13.
    Maehara, S.: The predicate calculus with \(\epsilon \)-symbol. J. Math. Soc. Japan 7, 323–344 (1955)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Mancosu, P. (ed.): From Brouwer to Hilbert. The Debate on the Foundations of Mathematics in the 1920s. Oxford University Press, New York (1998)Google Scholar
  15. 15.
    Meyer Viol, W.P.M.: Instantial Logic. An Investigation into Reasoning with Instances. ILLC Dissertation Series 1995–11. ILLC, Amsterdam (1995)Google Scholar
  16. 16.
    Mints, G., Sarenac, D.: Completeness of indexed epsilon-calculus. Arch. Math. Logic 42, 617–625 (2003)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Mints, G.: Heyting predicate calculus with epsilon symbol. J. Soviet Math. 8, 317–323 (1977)CrossRefMATHGoogle Scholar
  18. 18.
    Moser, G., Zach, R.: The epsilon calculus and herbrand complexity. Stud. Logica. 82(1), 133–155 (2006)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Otto, M.: Epsilon-logic is more expressive than first-order logic over finite structures. J. Symbolic Logic 65(4), 1749–1757 (2000)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Wessels, L.: Cut elimination in a Gentzen-style \(\varepsilon \)-calculus without identity. Z. Math. Logik Grundlag. Math. 23, 527–538 (1977)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Yashahura, M.: Cut elimination in \(\epsilon \)-calculi. Z. Math. Logik Grundlag. Math. 28, 311–316 (1982)MathSciNetCrossRefGoogle Scholar

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© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Department of PhilosophyUniversity of CalgaryCalgaryCanada

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