Abstract
The paper first formalizes the ramified type theory as (informally) described in the Principia Mathematica [32]. This formalization is close to the ideas of the Principia, but also meets contemporary requirements on formality and accuracy, and therefore is a new supply to the known literature on the Principia (like [25], [19], [6] and [7]).
As an alternative, notions from the ramified type theory are expressed in a lambda calculus style. This situates the type system of Russell and Whitehead in a modern setting. Both formalizations are inspired by current developments in research on type theory and typed lambda calculus; see [3].
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References
S. Abramsky, Dov M. Gabbay, and T.S.E. Maibaum, editors, 1992, Handbook of Logic in Computer Science, Volume 2: Background: Computational Structures. Oxford Science Publications.
H. P. Barendregt, 1984, The Lambda Calculus: its Syntax and Semantics, North-Holland, Amsterdam, revised edition.
H. P. Barendregt, 1992, Lambda calculi with types, In [1], Oxford University Press, 117–309.
P. Benacerraf and H. Putnam, editors, 1983, Philosophy of Mathematics, Cambridge University Press, second edition.
E.W. Beth, 1959, The Foundations of Mathematics, Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam.
A. Church, 1940, A formulation of the simple theory of types, The Journal of Symbolic Logic 5, 56–68.
A. Church, 1976, Comparison of Russell's resolution of the semantic antinomies with that of Tarski, The Journal of Symbolic Logic 41, 747–760.
R.L. Constable et al, 1986, Implementing Mathematics with the Nuprl Proof Development System, Prentice-Hall, New Jersey.
H.B. Curry and R. Feys, 1958, Combinatory Logic, volume I of Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam.
H.B. Curry, J.R. Hindley, and J.P. Seldin, 1972, Combinatory Logic, volume II of Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam.
A.A. Fraenkel, Y. Bar-Hillel, and A. Levy, second edition, 1973, Foundations of Set Theory, Studies in Logic and the Foundations of Mathematics 67, North Nolland, Amsterdam.
G. Frege, 1879, Begriffschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens, Nebert, Halle. Also in [18].
G. Frege, 1892, Grundgesetze der Arithmetik, begriffsschriftlich abgeleitet, volume I, Jena. Reprinted 1962 (Olms, Hildesheim).
G. Frege, 1903, Grundgesetze der Arithmetik, begriffsschriftlich abgeleitet, volume II, Pohle, Jena. Reprinted 1962 (Olms, Hildesheim).
C.I. Gerhardt, editor, 1890, Die philosophischen Schriften von Gottfried Wilhelm Leibniz, Berlin.
K. Gödel, 1931, Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I, Monatshefte für Mathematik und Physik 38, 173–198, German; English translation in [18].
K. Gödel, 1944, Russell's mathematical logic, in P.A. Schlipp, editor, The Philosophy of Bertrand Russell, Evanston & Chicago, Northwestern University. Also in [4].
J. van Heijenoort, editor, 1967, From Frege to Gödel: A Source Book in Mathematical Logic, Harvard University Press, Cambridge, Massachusetts, 1879–1931.
D. Hilbert and W. Ackermann, first edition, 1928, Grundzüge der Theoretischen Logik, Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen, Band XXVII, Springer Verlag, Berlin.
P. B. Jackson, 1995, Enhancing the Nuprl Proof Development System and Applying it to Computational Abstract Algebra, PhD thesis, Cornell University, Ithaca, New York.
F. Kamareddine and T. Laan, 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.
S. Kripke, 1975, Outline of a theory of truth, Journal of Philosophy 72, 690–716.
T.D.L. Laan, 1994, A formalization of the Ramified Type Theory, Technical Report 33, TUE Computing Science Reports, Eindhoven University of Technology.
W. Van Orman Quine, 1963, Set Theory and its Logic, Harvard University Press, Cambridge, Massachusetts.
F.P. Ramsey, 1925, The foundations of mathematics, Proceedings of the London Mathematical Society, 338–384.
B. Russell, 1903, The Principles of Mathematics, Allen & Unwin, London.
B. Russell, 1908, Mathematical logic as based on the theory of types, American Journal of Mathematics 30. Also in [18].
B. Russell, 1919, Introduction to Mathematical Philosophy, Allen & Unwin, London.
J. Terlouw, 1989, Een nadere bewijstheoretische analyse van GSTT's, Technical report, Department of Computer Science, University of Nijmegen.
H. Weyl, 1960, Das Kontinuum, Veit, Leipzig, 1918, German; also in: Das Kontinuum und andere Monographien, Chelsea Pub. Comp., New York.
H. Weyl, 1946, Mathematics and logic: A brief survey serving as preface to a review of “The Philosophy of Bertrand Russell”, American Mathematical Monthly.
A. N. Whitehead and B. Russell, 19101, 19272, Principia Mathematica, Cambridge University Press.
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Supported by the Co-operation Centre Tilburg and Eindhoven Universities. 1[32], Introduction, Chapter II, Section I, p. 37.
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Laan, T., Nederpelt, R. A modern elaboration of the ramified theory of types. Stud Logica 57, 243–278 (1996). https://doi.org/10.1007/BF00370835
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DOI: https://doi.org/10.1007/BF00370835