Abstract
The \(\varepsilon \)-elimination method of Hilbert’s \(\varepsilon \)-calculus yields the up-to-date most direct algorithm for computing the Herbrand disjunction of an extensional formula. A central advantage is that the upper bound on the Herbrand complexity obtained is independent of the propositional structure of the proof. Prior (modern) work on Hilbert’s \(\varepsilon \)-calculus focused mainly on the pure calculus, without equality. We clarify that this independence also holds for first-order logic with equality. Further, we provide upper bounds analyses of the extended first \(\varepsilon \)-theorem, even if the formalisation incorporates so-called \(\varepsilon \)-equality axioms.
Similar content being viewed by others
Notes
More precisely on Page 21 of [3] it is stated that “If equality is present, however, the maximal rank of critical formulas will also play a role.”.
Kindly see https://en.wikipedia.org/wiki/Drinker_paradox.
A term is fully indicated if every occurrence of the term is obtained by a replacement [27, Definition 1.6].
References
Hilbert, D., Bernays, P.: Grundlagen der Mathematik. Springer, Berlin (1934)
Hilbert, D., Bernays, P.: Grundlagen der Mathematik, vol. 2. Springer, Berlin (1939)
Moser, G., Zach, R.: The epsilon calculus and Herbrand complexity. Stud. Log. 82(1), 133–155 (2006)
Gentzen, G.: Untersuchungen über das logische Schließen I-II. Math. Z. 39(176–210), 405–431 (1934)
Gerhardy, P., Kohlenbach, U.: Extracting Herbrand disjunctions by functional interpretation. Arch. Math. Log. 44(5), 633–644 (2005)
Zach, R.: Semantics and proof theory of the epsilon calculus. In: Ghosh, S., Prasad, S. (eds.) Proceedings of the Logic and Its Applications—7th Indian Conference, ICLA 2017, Kanpur, India, January 5–7, 2017, Volume 10119 of Lecture Notes in Computer Science, pp. 27–47. Springer (2017)
Baaz, M., Leitsch, A., Lolic, A.: A sequent-calculus based formulation of the extended first epsilon theorem. In Artëmov, S.N., Nerode, A. (eds.) Proceedings of Logical Foundations of Computer Science, Volume 10703 of Lecture Notes in Computer Science, pp. 55–71. Springer (2018)
Asser, G.: Theorie der logischen Auswahlfunktionen. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 3, 30–68 (1957)
Statman, R.: Lower bounds on Herbrand’s theorem. Proc. Am. Math. Soc. 75(1), 104–107 (1979)
Orevkov, V.P.: Lower bounds for increasing complexity of derivations after cut elimination. J. Sov. Math. 20(4), 2337–2350 (1982)
Arai, T.: Exact bounds on epsilon processes. Arch. Math. Log. 50(3–4), 445–458 (2011)
Baaz, M., Zach, R.: Epsilon theorems in intermediate logics. J. Symb. Log. 87, 682–720 (2022)
Moser, G.: Ackermann’s substitution method (remixed). Ann. Pure Appl. Log. 142(1–3), 1–18 (2006)
Mints, G.: Intuitionistic existential instantiation and epsilon symbol. In Wansing, H. (ed.) Dag Prawitz on Proofs and Meaning, Volume 7 of Outstanding Contributions to Logic, pp. 225–238. Springer (2015)
Ackermann, W.: Zur Widerspruchsfreiheit der Zahlentheorie. Math. Ann. 117, 162–194 (1940)
Tait, W.W.: The substitution method. J. Symb. Log. 30, 175–192 (1965)
Avigad, J.: Update procedures and the \(1\)-consistency of arithmetic. Math. Log. Q. 48, 3–13 (2002)
DeVidi, D.: Intuitionistic epsilon- and tau-calculi. Math. Log. Q. 41, 523–546 (1995)
Maehara, S.: A general theory of completeness proofs. Ann. Jpn. Assoc. Philos. Sci. 3, 54–68 (1970)
Shirai, K.: Intuitionistic predicate calculus with \(\varepsilon \)-symbol. Ann. Jpn. Assoc. Philos. Sci. 4(1), 49–67 (1971)
Blass, A., Gurevich, Y.: The logic of choice. J. Symb. Log. 65, 1264–1310 (2000)
Abadi, M., Gonthier, G., Werner, B.: Choice in dynamic linking. In: Proceedings of the 7th FOSSACS, Volume 2987 of LNCS, pp. 12–26 (2004)
Aguilera, J.P., Baaz, M.: Unsound inferences make proofs shorter. J. Symb. Log. 84(1), 102–122 (2019)
Maehara, S.: Equality axiom on Hilbert’s \(\varepsilon \)-symbol. J. Fac. Sci. Univ. Tokyo Sect. 1 7, 419–435 (1957)
Maehara, S.: The predicate calculus with \(\varepsilon \)-symbol. J. Math. Soc. Jpn. 7, 323–344 (1955)
Yasuhara, M.: Cut elimination in \(\varepsilon \)-calculi. Zeitschrift für mathematische Logik und Grundlagen der Mathematik 28, 311–316 (1982)
Takeuti, G.: Proof Theory, 2nd edn. North-Holland, Amsterdam (1987)
Baaz, M., Fermüller, C.G.: A note on the proof theoretic strength of a single application of the schema of identity. Submitted to the Dagsthul Seminar “Proof Theory in Computer Science” (2001)
Krajíček, J.: Lower bounds to the size of constant-depth propositional proofs. J. Symb. Log. 59, 73–86 (1994)
Pudlak, P.: The length of proofs. In: Buss, S.R. (ed.) Handbook of Proof Theory, pp. 1–79. Elsevier, Amsterdam (1998)
Parikh, R.J.: Some results on the length of proofs. Trans. Am. Math. Soc. 177, 29–36 (1973)
Krajíćek, J., Pudlák, P.: The number of proof lines and the size of proofs in first-order logic. Arch. Math. Log. 27, 69–84 (1988)
Yukami, T.: Some results on speed-up. Ann. Jpn. Assoc. Philos. Sci. 6(4), 195–205 (1984)
Buss, S.R.: Introduction to proof theory. In: Buss, S.R. (ed.) Handbook of Proof Theory, pp. 1–79. Elsevier, Amsterdam (1998)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Miyamoto, K., Moser, G. Herbrand complexity and the epsilon calculus with equality. Arch. Math. Logic 63, 89–118 (2024). https://doi.org/10.1007/s00153-023-00877-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00153-023-00877-3