Notre Dame Journal of Formal Logic

Logic in Russell's Principles of Mathematics

Gregory Landini


Unaware of Frege's 1879 Begriffsschrift, Russell's 1903 The Principles of Mathematics set out a calculus for logic whose foundation was the doctrine that any such calculus must adopt only one style of variables–entity (individual) variables. The idea was that logic is a universal and all-encompassing science, applying alike to whatever there is–propositions, universals, classes, concrete particulars. Unfortunately, Russell's early calculus has appeared archaic if not completely obscure. This paper is an attempt to recover the formal system, showing its philosophical background and its semantic completeness with respect to the tautologies of a modern sentential calculus.

Article information

Notre Dame J. Formal Logic Volume 37, Number 4 (1996), 554-584.

First available: 16 December 2002

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Digital Object Identifier

Zentralblatt MATH identifier

Primary: 03-03: Historical (must also be assigned at least one classification number from Section 01)
Secondary: 01A60: 20th century 03A05: Philosophical and critical {For philosophy of mathematics, see also 00A30} 03B05: Classical propositional logic


Landini, Gregory. Logic in Russell's Principles of Mathematics . Notre Dame Journal of Formal Logic 37 (1996), no. 4, 554--584. doi:10.1305/ndjfl/1040046142.

Export citation


  • [1] Byrd, M., ``Russell, logicism and the choice of the logical constants,'' Notre Dame Journal of Formal Logic, vol. 30 (1989), pp. 343--361.
  • [2] Church, A., Introduction to Mathematical Logic, Princeton University Press, Princeton, 1956.
  • [3] Church, A., ``Russell's theory of the identity of propositions,'' Philosophia Naturalis, vol. 21 (1984), pp. 515--522.
  • [4] Frege, G., ``On Herr Peano's Begriffsschrift and my own,'' Australasian Journal of Philosophy, vol. 47 (1969), pp. 1--14.
  • [5] Hylton, P., Russell, Idealism and the Emergence of Analytic Philosophy, Oxford University Press, Cambridge, 1990.
  • [6] McGuiness, B., Gottlob Frege: Philosophical and Mathematical Correspondence, University of Chicago Press, Chicago, 1980.
  • [7] Peano, G., (ed.), Formulaire de Mathèmatiques, Bocca, Turin, 1889.
  • [8] Peano, G., ``Arithmetices Principia,'' Novo Methodo Exposita, Turin, 1889. Reprinted in van Heijenoort, From Frege to Gödel, Cambridge, 1967, pp. 83--97.
  • [9] Quine, W.V.O., Set Theory and Its Logic, Harvard University Press, Harvard, 1963.
  • [10] Russell, B., The Principles of Mathematics, W.W. Norton \mbox& Company, New York, 1996.
  • [11] Russell, B., ``The Theory of Implication,'' American Journal of Mathematics, vol. 28, 1906, pp. 159--202.
  • [12] Russell, B., ``On `Insolubilia' and their solution by symbolic logic,'' pp. 90--214 in Essays in Analysis, edited by D. Lackey, George Braziller, New York, 1973.
  • [13] Russell, B., ``On fundamentals,'' pp. 359--413 in The Collected Papers of Bertrand Russell, vol. 4, edited by A. Urquhart and A. Lewis, Routledge, London, 1994.
  • [14] Russell, B., ``Mathematics and the Metaphysicians,'' pp. 59--74 in Mysticism and Logic, Longmans, New York, 1963.
  • [15] van Heijenoort, J., ``Logic as language and logic as calculus,'' Synthese, vol. 17 (1967), pp. 324--330.
  • [16] van Heijenoort, J., ``Systéme et Mètasystéme ches Russell,'' pp. 111--122 in Logic Colloquium 85, edited by Paris Logic Group, North-Holland, Amsterdam, 1987.
  • [17]Vuillemin, J., Lecons sur la Premiére Philosophie de Russell, Colin, Paris, 1968.
  • [18] Whitehead, A. N., and B. Russell, Principia Mathematica vol. 1, Cambridge University Press, Cambridge, 1910.