An untyped higher order logic with Y combinator
James H. Andrews
Source: J. Symbolic Logic Volume 72, Issue 4
(2007), 1385-1404.
Abstract
We define a higher order logic which has only a notion of sort rather than a notion of type, and which permits all terms of the untyped lambda calculus and allows the use of the Y combinator in writing recursive predicates. The consistency of the logic is maintained by a distinction between use and mention, as in Gilmore’s logics. We give a consistent model theory, a proof system which is sound with respect to the model theory, and a cut-elimination proof for the proof system. We also give examples showing what formulas can and cannot be used in the logic.
First Page:
Show
Hide
Full-text: Access denied (no subscription
detected)
We're sorry, but we are unable to provide
you with the full text of this article because we are not able to identify
you as a subscriber.
If you have a personal subscription to
this journal, then please login. If you are already logged in, then you
may need to update your profile to register your subscription. Read more about accessing full-text
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.jsl/1203350794
Digital Object Identifier: doi:10.2178/jsl/1203350794
Mathematical Reviews number (MathSciNet): MR2371213
Zentralblatt MATH identifier: 1134.03010
References
James H. Andrews, A weakly-typed higher order logic with general lambda terms and Y combinator, Proceedings, works in progress track, 15th international conference on theorem proving in higher order logics (TPHOLs '02), NASA Conference Publication CP -2002-211736, Hampton Roads, Virginia, August2002, pp. 1--11.
--------, Cut elimination for a weakly typed higher order logic, Technical Report 611, Department of Computer Science, University of Western Ontario, Decembe r2003.
Peter Apostoli and Akira Kanda, Parts of the continuum: towards a modern ontology of science, Poznan Studies in the Philosophy of Science and the Humanities, 1996, Accepted for publication.
Luca Cardelli, Type systems, CRC handbook of computer science and engineering, CRC Press, 1996, pp. 2208--2236.
Weidong Chen, Michael Kifer, and David S. Warren, HiLog: A first-order semantics of higher-order logic programming constructs, Proceedings of the North American conference on logic programming, Octobe r1989, pp. 1090--1114.
Alonzo Church, A formulation of the simple theory of types, Journal of Symbolic Logic, vol. 5 (1940), pp. 56--68.
Mathematical Reviews (MathSciNet): MR1931
Digital Object Identifier: doi:10.2307/2266170
Project Euclid: euclid.jsl/1183387805
Zentralblatt MATH: 0023.28901
The Coq Development Team, The Coq proof assistant reference manual version 7.2, Technical Report 255, INRIA, 2002.
Thierry Coquand, An analysis of Girard's paradox, First IEEE symposium on logic in computer science (Cambridge, Massachusetts), Jun e1986, pp. 227--236.
Paul C. Gilmore, NaDSyL and some applications, Proceedings of the Kurt Gödel colloquium (Vienna), Lecture Notes in Computer Science, vol. 1289, Springer, 1997, pp. 153--166.
Mathematical Reviews (MathSciNet): MR1602316
--------, An intensional type theory: Motivation and cut-elimination, Journal of Symbolic Logic, vol. 66 (2001), no. 1, pp. 383--400.
Mathematical Reviews (MathSciNet): MR1825191
Digital Object Identifier: doi:10.2307/2694928
Project Euclid: euclid.jsl/1183746377
Zentralblatt MATH: 0980.03007
--------, Logicism renewed: Logical foundations for mathematics and computer science, Lecture Notes in Logic, no. 23, Association for Symbolic Logic / A K Peters, Ltd., Wellesley, MA, 2005.
M. J. C. Gordon and T. F. Melham, Introduction to HOL: A theorem proving environment for higher order logic, Cambridge University Press, 1993.
Mathematical Reviews (MathSciNet): MR1232656
Leon Henkin, Completeness in the theory of types, Journal of Symbolic Logic, vol. 15 (1950), pp. 81--91.
Mathematical Reviews (MathSciNet): MR36188
Digital Object Identifier: doi:10.2307/2266967
Project Euclid: euclid.jsl/1183730860
Zentralblatt MATH: 0039.00801
J. Roger Hindley and Jonathan P. Seldin, Introduction to combinators and lambda calculus, London Mathematical Society Student Texts, no. 1, Cambridge University Press, Cambridge, 1986.
Mathematical Reviews (MathSciNet): MR879272
Zentralblatt MATH: 0614.03014
Fairouz Kamareddine, A system at the cross-roads of functional and logic programming, Science of Computer Programming, vol. 19 (1992), pp. 239--279.
Mathematical Reviews (MathSciNet): MR1198464
Digital Object Identifier: doi:10.1016/0167-6423(92)90037-C
Zentralblatt MATH: 0779.68011
Gaisi Takeuti, Proof theory, North-Holland, Amsterdam, 1987.
Mathematical Reviews (MathSciNet): MR882549
