Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-05T05:33:05.159Z Has data issue: false hasContentIssue false

Cut Elimination in the Presence of Axioms

Published online by Cambridge University Press:  15 January 2014

Sara Negri
Affiliation:
Department of Philosophy, University of Helsinki, Helsinki, FinlandE-mail: [email protected]
Jan von Plato
Affiliation:
Department of Philosophy, University of Helsinki, Helsinki, FinlandE-mail: [email protected]

Abstract

A way is found to add axioms to sequent calculi that maintains the eliminability of cut, through the representation of axioms as rules of inference of a suitable form. By this method, the structural analysis of proofs is extended from pure logic to free-variable theories, covering all classical theories, and a wide class of constructive theories. All results are proved for systems in which also the rules of weakening and contraction can be eliminated. Applications include a system of predicate logic with equality in which also cuts on the equality axioms are eliminated.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 1998

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1] Bernays, P., Review of Ketonen [9], this Journal, vol. 10 (1945), pp. 127–130.Google Scholar
[2] Dragalin, A., Mathematical intuitionism: Introduction to proof theory, American Mathematical Society, Providence, Rhode Island, 1988.Google Scholar
[3] Dyckhoff, R., Dragalin's proof of cut-admissibility for the intuitionistic sequent calculi G3i and G3i′ , Research Report CS/97/8, Computer Science Division, St. Andrews University, 1997.Google Scholar
[4] Gentzen, G., Untersuchungen über das logische Schliessen, Mathematische Zeitschrift, vol. 39 (19341935), pp. 176–210 and 405431.CrossRefGoogle Scholar
[5] Gentzen, G., Neue Fassung des Widerspruchsfreiheitsbeweises für die reine Zahlentheorie, Forschungen zur Logik und zur Grundlegung der exakten Wissenschaften, N. S., vol. 4 (1938), pp. 1944.Google Scholar
[6] Gentzen, G., The collected papers of Gerhard Gentzen, North-Holland, Amsterdam, 1969.Google Scholar
[7] Girard, J.-Y., Proof theory and logical complexity, Bibliopolis, Naples, 1987.Google Scholar
[8] Hallnäs, L. and Schroeder-Heister, P., A proof-theoretic approach to logic programming. I. Clauses as rules, Journal of Logic and Computation, vol. 1 (1990), pp. 261283.Google Scholar
[9] Ketonen, O., Untersuchungen zum Prädikatenkalkül, Annales Academiae Scientiarum Fennicae, Series A, vol. I (1944), no. 23.Google Scholar
[10] Negri, S., Sequent calculus proof theory of intuitionistic apartness and order relations, Archive for Mathematical Logic, in press.Google Scholar
[11] von Plato, J., On the proof theory of classical logic, submitted, 1998.Google Scholar
[12] Troelstra, A. and Schwichtenberg, H., Basic proof theory, Cambridge University Press, Cambridge, 1996.Google Scholar
[13] Uesu, T., An axiomatization of the apartness fragment of the theory DLO+ of dense linear order, Logic colloquium '84, Lecture Notes in Mathematics, no. 1104, 1984, pp. 453475.Google Scholar