Abstract
We investigate cut-elimination and cut-simulation in impredicative (higher-order) logics. We illustrate that adding simple axioms such as Leibniz equations to a calculus for an impredicative logic — in our case a sequent calculus for classical type theory — is like adding cut. The phenomenon equally applies to prominent axioms like Boolean- and functional extensionality, induction, choice, and description. This calls for the development of calculi where these principles are built-in instead of being treated axiomatically.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Andrews, P.B.: Resolution in type theory. Journal of Symbolic Logic 36(3), 414–432 (1971)
Andrews, P.B.: General models and extensionality. Journal of Symbolic Logic 37(2), 395–397 (1972)
Andrews, P.B.: An Introduction to Mathematical Logic and Type Theory: To Truth Through Proof, 2nd edn. Kluwer Academic Publishers, Dordrecht (2002)
Benzmüller, C.E.: Equality and Extensionality in Automated Higher-Order Theorem Proving. PhD thesis, Saarland University (1999)
Benzmüller, C.E., Brown, C.E., Kohlhase, M.: Higher-order semantics and extensionality. Journal of Symbolic Logic 69(4), 1027–1088 (2004)
Benzmüller, C.E., Brown, C.E., Kohlhase, M.: Semantic techniques for higher-order cut-elimination. Seki Report SR-2004-07, Saarland University (2004)
Benzmüller, C.E., Brown, C.E., Kohlhase, M.: Cut-simulation in impredicative logics (extended version). Seki Report SR-2006-01, Saarland University (2006)
Brown, C.E.: Set Comprehension in Church’s Type Theory. PhD thesis, Department of Mathematical Sciences, Carnegie Mellon University (2004)
Church, A.: A formulation of the simple theory of types. Journal of Symbolic Logic 5, 56–68 (1940)
Hintikka, K.J.J.: Form and content in quantification theory. Acta Philosophica Fennica 8, 7–55 (1955)
Huet, G.P.: A mechanization of type theory. In: Proceedings of the 3rd International Joint Conference on Artificial Intelligence, pp. 139–146 (1973)
Russell, B.: Mathematical logic as based on the theory of types. American Journal of Mathematics 30, 222–262 (1908)
Smullyan, R.M.: A unifying principle for quantification theory. Proc. Nat. Acad Sciences 49, 828–832 (1963)
Smullyan, R.M.: First-Order Logic. Springer, Heidelberg (1968)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Benzmüller, C.E., Brown, C.E., Kohlhase, M. (2006). Cut-Simulation in Impredicative Logics. In: Furbach, U., Shankar, N. (eds) Automated Reasoning. IJCAR 2006. Lecture Notes in Computer Science(), vol 4130. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11814771_20
Download citation
DOI: https://doi.org/10.1007/11814771_20
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-37187-8
Online ISBN: 978-3-540-37188-5
eBook Packages: Computer ScienceComputer Science (R0)