Notre Dame Journal of Formal Logic

A Simple Proof that Super-Consistency Implies Cut Elimination

Gilles Dowek and Olivier Hermant

Abstract

We give a simple and direct proof that super-consistency implies the cut-elimination property in deduction modulo. This proof can be seen as a simplification of the proof that super-consistency implies proof normalization. It also takes ideas from the semantic proofs of cut elimination that proceed by proving the completeness of the cut-free calculus. As an application, we compare our work with the cut-elimination theorems in higher-order logic that involve V-complexes.

Article information

Source
Notre Dame J. Formal Logic, Volume 53, Number 4 (2012), 439-456.

Dates
First available in Project Euclid: 8 November 2012

Permanent link to this document
https://projecteuclid.org/euclid.ndjfl/1352383225

Digital Object Identifier
doi:10.1215/00294527-1722692

Mathematical Reviews number (MathSciNet)
MR2995413

Zentralblatt MATH identifier
1203.03086

Subjects
Primary: 03F05: Cut-elimination and normal-form theorems
Secondary: 03B99: None of the above, but in this section 03B15: Higher-order logic and type theory 03C90: Nonclassical models (Boolean-valued, sheaf, etc.)

Keywords
deduction modulo super-consistency cut elimination simple type theory

Citation

Dowek, Gilles; Hermant, Olivier. A Simple Proof that Super-Consistency Implies Cut Elimination. Notre Dame J. Formal Logic 53 (2012), no. 4, 439--456. doi:10.1215/00294527-1722692. https://projecteuclid.org/euclid.ndjfl/1352383225


Export citation

References

  • [1] Andrews, P. B., “Resolution in type theory,” Journal of Symbolic Logic, vol. 36 (1971), pp. 414–32.
  • [2] Church, A., “A formulation of the simple theory of types,” Journal of Symbolic Logic, vol. 5 (1940), pp. 56–68.
  • [3] De Marco, M., and J. Lipton, “Completeness and cut-elimination in the intuitionistic theory of types,” Journal of Logic and Computation, vol. 15 (2005), pp. 821–54.
  • [4] Dowek, G., “Truth values algebras and proof normalization,” pp. 110–24 in Types for Proofs and Programs, vol. 4502 of Lecture Notes in Computer Science, Springer, Berlin, 2007.
  • [5] Dowek, G., T. Hardin, and C. Kirchner, “HOL-lambda-sigma: an intentional first-order expression of higher-order logic,” pp. 21–45 in Theory and Applications of Explicit Substitutions, edited by D. Kesner, vol. 11 of Mathematical Structures in Computer Science, Cambridge University Press, Cambridge, 2001.
  • [6] Dowek, G., T. Hardin, and C. Kirchner, “Theorem proving modulo,” Journal of Automated Reasoning, vol. 31 (2003), pp. 33–72.
  • [7] Dowek, G., and O. Hermant, “A simple proof that super-consistency implies cut elimination,” pp. 93–106 in Term Rewriting and Applications, vol. 4533 of Lecture Notes in Computer Science, Springer, Berlin, 2007.
  • [8] Dowek, G., and B. Werner, “Proof normalization modulo,” Journal of Symbolic Logic, vol. 68 (2003), pp. 1289–1316.
  • [9] Girard, J.-Y., “Une extension de l’interprétation de Gödel à l’analyse et son application à l’élimination des coupures dans l’analyse et la théorie des types,” pp. 63–92 in Proceedings of the Second Scandinavian Logic Symposium (Oslo, 1970), vol. 63 of Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam, 1971.
  • [10] Hermant, O., and J. Lipton, “A constructive semantic approach to cut elimination in type theories with axioms,” pp. 169–83 in Computer Science Logic, vol. 5213 of Lecture Notes in Computer Science, Springer, Berlin, 2008.
  • [11] Hermant, O., and J. Lipton, “Completeness and cut-elimination in the intuitionistic theory of types, II,” Journal of Logic and Computation, vol. 20 (2010), pp. 597–602.
  • [12] Okada, M., “A uniform semantic proof for cut-elimination and completeness of various first and higher order logics,” Theoretical Computer Science, vol. 281 (2002), pp. 471–98.
  • [13] Prawitz, D., “Hauptsatz for higher order logic,” Journal of Symbolic Logic, vol. 33 (1968), pp. 452–57.
  • [14] Schütte, K., “Syntactical and semantical properties of simple type theory,” Journal of Symbolic Logic, vol. 25 (1960), pp. 305–26.
  • [15] Tait, W. W., “A nonconstructive proof for Gentzen’s Hauptsatz for second order predicate logic,” Bulletin of the American Mathematical Society, vol. 72 (1966), pp. 980–83.
  • [16] Takahashi, M., “A proof of cut-elimination theorem in simple type-theory,” Journal of the Mathematical Society of Japan, vol. 19 (1967), pp. 399–410.
  • [17] Troelstra, A. S., and D. van Dalen, Constructivism in Mathematics: An Introduction, Vol. I, vol. 121 of Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam, 1988. Vol. II, vol. 123 of Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam, 1988.