Notre Dame Journal of Formal Logic

The Fact Semantics for Ramified Type Theory and the Axiom of Reducibility

Edwin D. Mares


This paper uses an atomistic ontology of universals, individuals, and facts to provide a semantics for ramified type theory. It is shown that with some natural constraints on the sort of universals and facts admitted into a model, the axiom of reducibility is made valid.

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Notre Dame J. Formal Logic, Volume 48, Number 2 (2007), 237-251.

First available in Project Euclid: 16 May 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03B15: Higher-order logic and type theory 03C85: Second- and higher-order model theory

Bertrand Russell higher-order logic logicism


Mares, Edwin D. The Fact Semantics for Ramified Type Theory and the Axiom of Reducibility. Notre Dame J. Formal Logic 48 (2007), no. 2, 237--251. doi:10.1305/ndjfl/1179323266.

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  • [1] Anderson, C. A., "Russellian intensional logic", pp. 69--103 in Themes from Kaplan, edited by J. Almog, J. Perry, and H. Wettstein, Oxford University Press, Oxford, 1989.
  • [2] Church, A., "Comparison of Russell's resolution of the semantical antinomies with that of Tarski", The Journal of Symbolic Logic, vol. 41 (1976), pp. 747--60. Reprinted in R. L. Martin, editor, Recent Essays on Truth and the Liar Paradox, Oxford University Press, Oxford,1984, pp. 289--306.
  • [3] Hazen, A. P., and J. M. Davoren, "Russell's 1925 logic", Australasian Journal of Philosophy, vol. 78 (2000), pp. 534--56.
  • [4] Hazen, A. P., "Predicative logics", pp. 331--408 in Handbook of Philosophical Logic. Vol. 1. Elements of Classical Logic, 2d edition, edited by D. M. Gabbay and F. Guenthner, D. Reidel, Dordrecht, 1983.
  • [5] Jeffrey, R., Formal Logic: Its Scope and Limits, McGraw Hill, New York, 1967.
  • [6] Kaplan, D., "The ramified theory of types", unpublished manuscript.
  • [7] Landini, G., Russell's Hidden Substitutional Theory, Oxford University Press, New York, 1998.
  • [8] Leblanc, H., "That Principia Mathematica, first edition, has a predicative interpretation after all", Journal of Philosophical Logic, vol. 4 (1975), pp. 67--70. Reprinted in H. Leblanc, Existence, Truth, and Provability, SUNY Press, Albany, 1982, pp. 236--39.
  • [9] Leblanc, H., and G. Weaver, "Truth-functionality and the ramified theory of types", pp. 148--67 in Truth, Syntax and Modality (Proceedings of the Conference on Alternative Semantics, Temple University, Philadelphia, 1970), vol. 68 of Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam, 1973.
  • [10] Lewis, D., On the Plurality of Worlds, Blackwell, Oxford, 1986.
  • [11] Linsky, B., Russell's Metaphysical Logic, vol. 101 of CSLI Lecture Notes, CSLI Publications, Stanford, 1999.
  • [12] Russell, B., The Problems of Philosophy, Oxford University Press, Oxford, 1959.
  • [13] Russell, B., An Inquiry into Meaning and Truth, George Allen and Unwin, London, 1980.
  • [14] Russell, B., ``Theory of knowledge: The 1913 manuscript,'' pp. 1--178 in The Collected Papers of Bertrand Russell. Vol. 7, edited by E. R. Eames, George Allen and Unwin, London, 1984.
  • [15] Russell, B., The Philosophy of Logical Atomism, Open Court, La Salle, 1985.
  • [16] Urquhart, A., "The theory of types", pp. 286--309 in The Cambridge Companion to Bertrand Russell, edited by N. Griffin, Cambridge Companions to Philosophy, Cambridge University Press, Cambridge, 2003.
  • [17] Whitehead, A. N., and B. Russell, Principia Mathematica, 2d edition, Cambridge University Press, Cambridge, 1925.