Abstract
In Russell's Ramified Theory of Types RTT, two hierarchical concepts dominate:orders and types. The use of orders has as a consequencethat the logic part of RTT is predicative.The concept of order however, is almost deadsince Ramsey eliminated it from RTT. This is whywe find Church's simple theory of types (which uses the type concept without the order one) at the bottom of the Barendregt Cube rather than RTT.Despite the disappearance of orders which have a strong correlation with predicativity, predicative logic still plays an influential role in Computer Science.An important example is the proof checker Nuprl, which is basedon Martin-Löf's Type Theory which uses type universes. Those type universes,and also degrees of expressions in AUTOMATH, are closely related toorders. In this paper, we show that orders have not disappeared frommodern logic and computer science, rather, orders play a crucial role in understanding the hierarchy of modern systems. In order to achieve our goal, we concentrate on a subsystem of Nuprl.
The novelty of our paper lies in: (1) a modest revival of Russell'sorders, (2) the placing of the historical system RTTunderlying the famous Principia Mathematica in a context with a modernsystem of computer mathematics (Nuprl) and modern type theories(Martin-Löf's type theory and PTSs), and (3) the presentation of acomplex type system (Nuprl) as a simple and compact PTS.
Similar content being viewed by others
References
Barendregt, H.P., 1992, “Lambda calculi with types,” pp. 117–309 in Handbook of Logic in Computer Science 2: Background: Computational Structures, S. Abramsky, Dov Gabbay, and T.S.E. Maibaum, eds., Oxford: Oxford University Press.
Barthe, G. and Sorensen, M.H., 1997, “Domain-free pure type systems,” pp. 9–20 in Logical Foundations of Computer Science, S. Adian and A. Nerode, eds., Lecture Notes in Computer Science, Vol. 1234, Berlin: Springer-Verlag.
Church, A., 1940, “A formulation of the simple theory of types,” The Journal of Symbolic Logic 5, 56–68.
Church, A., 1941, xCalculi of Lambda Conversion, Annals of Mathematical Studies, Vol. 6, Princeton, NJ: Princeton University Press.
Church, A., 1976, “Comparison of Russell's resolution of the semantical antinomies with that of Tarski,” Journal of Symbolic Logic 41, 747–760.
Constable, R.L., Allen, S., Bromley, H., Cleaveland, W., Cremer, F., Harper, R., Howe, D., Knoblock, T., Mendler, N., Panangaden, P., Sasaki, J., and Smith, S., 1986, Implementing Maths with the Nuprl Proof Development System, Englewood Cliffs, NJ: Prentice-Hall.
Coquand, T., 1986, “An analysis of Girard's paradox,” pp. 227–236 in IEEE Symposium on Logic in Computer Science, Boston, New York: IEEE.
Coquand, T. and Huet, G., 1988, “The calculus of constructions,” Information and Computation 76, 95–120.
Dowek, G., Fetly, A., Herbelin, H., Huet, G., Murthy, C., Parent, C., Paulin-Mohring, C., andWerner, B., 1993, “The Coq proof assistant user's guide, version 5.8,” Rapport de Recherche 154, INRIA.
Feferman, S., 1964, “Systems of predicative analysis,” Journal of Symbolic Logic 29, 1–30.
Frege, G., 1892 and 1903, Grundgesetze der Arithmetik, begriffsschriftlich abgeleitet, I + II, Jena: Pohle.
Gödel, K., 1931, “Ñber formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I,” Monatshefte für Mathematik und Physik 38, 173–198. English translation: 1967, pp. 592–618 in From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931, J. van Heijenoort, ed., Cambridge, MA: Harvard University Press.
Harper, R. and Pollack, R., 1991, “Type checking with universes,” Theoretical Computer Science 89, 107–136.
Hilbert, D. and Ackermann,W., 1928, Grundzüge der Theoretischen Logik, 1st edn., Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen, Band XXVII, Berlin: Springer-Verlag.
Howard, W.A., 1980, “The formulas-as-types notion of construction,” pp. 479–490 in To H.B. Curry: Essays on Combinatory Logic, γ-Calculus and Formalism, J.P. Seldin and J.R. Hindley, eds., New York: Academic Press.
Jackson, P.B., 1995a, “Enhancing the Nuprl proof development system and applying it to computational abstract algebra,” Ph.D. Thesis, Cornell University, Ithaca, NY.
Jackson, P.B., 1995b, The Nuprl Proof Development System, Version 4.1 Reference Manual and User's Guide, Ithaca, NY: Cornell University, Department of Computing Science.
Kamareddine, F. and Laan, T., 1996, “A reflection on Russell's ramified types and Kripke's hierarchy of truths,” Journal of the Interest Group in Pure and Applied Logic 4, 195–213.
Klop, J.W., 1992, “Term rewriting systems,” pp. 1–116 in Handbook of Logic in Computer Science 2: Background: Computational Structures, S. Abramsky, Dov M. Gabbay, and T.S.E. Maibaum, eds., Oxford: Oxford University Press.
Kripke, S., 1975, “Outline of a theory of truth,” Journal of Philosophy 72, 690–716.
Laan, T.D.L., 1997, “The evolution of type theory in logic and mathematics,” Ph.D. Thesis, Eindhoven University of Technology, The Netherlands.
Laan, T.D.L. and Nederpelt, R.P., 1996, “A modern elaboration of the Ramified Theory of Types,” Studia Logica 57, 243–278.
Leivant, D., 1991, Finitely Startified Polymorphism, Selections from the 1989 IEEE Symposium on Logic in Computer Science, Information and Computation 93.
Luo, Z., 1994, Computation and Reasoning, Oxford: Oxford University Press.
Martin-Löf, P., 1975, “An intuitionistic theory of types: Predicative part,” pp. 73–118 in Logic Colloquium '73, H.E. Rose and J. Shepherdson, eds., Amsterdam: North-Holland.
Martin-Löf, P., 1982, “Constructive mathematics and computer programming,” pp. 153–175 in Sixth International Congress for Logic, Methodology and Philosophy of Science, Amsterdam, Studies in Logic and the Foundations of Mathematics, Vol. 104, Amsterdam: North-Holland.
Martin-Löf, P., 1984, Intuitionistic Type Theory, Bibliopolis.
Murthy, C., 1990, “Extracting constructive conent from classical proofs,” Ph.D. Thesis, Cornell University, Ithaca, NY.
Nederpelt, R.P., Geuvers, J.H., and de Vrijer, R.C., eds., 1994, Selected Papers on Automath, Studies in Logic and the Foundations of Mathematics, Vol. 133, Amsterdam: North-Holland.
Palmgren, E., 1991, “On fixed point operators, inductive definitions and universe in Martin-Löf type theory,” Ph.D. Thesis, Uppsala University.
Parigot, M., 1992, “Lambda-mu-calculus: An algorithmic interpretation of classical natural deduction,” pp. 190–201 in Logic Programming and Automated Reasoning: International Conference LPAR '92 Proceedings, St. Petersburg, Russia, A. Voronkov, ed., Berlin: Springer-Verlag.
Quine, W.V., 1976, The Ways of Paradox and Other Essays, New York: Random House; second revised edition, 1976, Cambridge MA: Harvard University Press.
Ramsey, F.P., 1925, “The foundations of mathematics,” Proceedings of the London Mathematical Society 25, 338–384.
Russell, B., 1903, The Principles of Mathematics, London: Allen & Unwin.
Russell, B., 1908, “Mathematical logic as based on the theory of types,” American Journal of Mathematics 30, 222–262.
Schütte, K., 1977, Proof Theory, Grundlehren de mathematischen Wissenschaften, Vol. 225, Berlin: Springer-Verlag.
Terlouw, J., 1989, “Een nadere bewijstheoretische analyse van GSTT's,” Technical Report, Department of Computer Science, University of Nijmegen.
Weyl, H., 1918, Das Kontinuum, Leipzig: Veit. Also: 1960, in Das Kontinuum und andere Monographien, New York: Chelsea Publishing Company.
Tarski, A., 1936, “Der Wahrheitsbegriff in den formalisierten Sprachen,” Studia Philosophica 1, 261–405. German translation by L. Blauwstein from the Polish original (1933) with a postscript added.
van Heijenoort, J., ed., 1967, From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931, Cambridge, MA: Harvard University Press.
Whitehead, A.N. and Russell, B., 19101, 19272, Principia Mathematica, Cambridge, MA: Cambridge University Press.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Kamareddine, F., Laan, T. A Correspondence between Martin-Löf Type Theory, the Ramified Theory of Types and Pure Type Systems. Journal of Logic, Language and Information 10, 375–402 (2001). https://doi.org/10.1023/A:1011286100450
Issue Date:
DOI: https://doi.org/10.1023/A:1011286100450