Law of excluded middle  

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Train wreck at Montparnasse (October 22, 1895) by Studio Lévy and Sons.
Train wreck at Montparnasse (October 22, 1895) by Studio Lévy and Sons.

In logic, the law of excluded middle (or the principle of excluded middle) is the third of the three classic laws of thought. It states that for any proposition, either that proposition is true, or its negation is true.

The law is also known as the law (or principle) of the excluded third, in Latin principium tertii exclusi. Yet another Latin designation for this law is tertium non datur: "no third (possibility) is given".

The earliest known formulation is Aristotle's principle of non-contradiction, first proposed in On Interpretation, where he says that of two contradictory propositions (i.e. where one proposition is the negation of the other) one must be true, and the other false. He also states it as a principle in the Metaphysics book 3, saying that it is necessary in every case to affirm or deny, and that it is impossible that there should be anything between the two parts of a contradiction. The principle was stated as a theorem of propositional logic by Russell and Whitehead in Principia Mathematica as:

<math>\mathbf{*2\cdot11}. \ \ \vdash . \ p \ \vee \thicksim p</math>.

The principle should not be confused with the semantical principle of bivalence, which states that every proposition is either true or false.


  • Aquinas, Thomas, "Summa Theologica", Fathers of the English Dominican Province (trans.), Daniel J. Sullivan (ed.), vols. 19–20 in Robert Maynard Hutchins (ed.), Great Books of the Western World, Encyclopædia Britannica, Inc., Chicago, IL, 1952. Cited as GB 19–20.
  • Aristotle, "Metaphysics", W.D. Ross (trans.), vol. 8 in Robert Maynard Hutchins (ed.), Great Books of the Western World, Encyclopædia Britannica, Inc., Chicago, IL, 1952. Cited as GB 8. 1st published, W.D. Ross (trans.), The Works of Aristotle, Oxford University Press, Oxford, UK.
  • Martin Davis 2000, Engines of Logic: Mathematicians and the Origin of the Computer", W. W. Norton & Company, NY, ISBN 0-393-32229-7 pbk.
  • Dawson, J., Logical Dilemmas, The Life and Work of Kurt Gödel, A.K. Peters, Wellesley, MA, 1997.
  • van Heijenoort, J., From Frege to Gödel, A Source Book in Mathematical Logic, 1879–1931, Harvard University Press, Cambridge, MA, 1967. Reprinted with corrections, 1977.
  • Luitzen Egbertus Jan Brouwer, 1923, On the significance of the principle of excluded middle in mathematics, especially in function theory [reprinted with commentary, p. 334, van Heijenoort]
  • Andrei Nikolaevich Kolmogorov, 1925, On the principle of excluded middle, [reprinted with commentary, p. 414, van Heijenoort]
  • Luitzen Egbertus Jan Brouwer, 1927, On the domains of definitions of functions,[reprinted with commentary, p. 446, van Heijenoort] Although not directly germane, in his (1923) Brouwer uses certain words defined in this paper.
  • Luitzen Egbertus Jan Brouwer, 1927(2), Intuitionistic reflections on formalism,[reprinted with commentary, p. 490, van Heijenoort]
  • Stephen C. Kleene 1952 original printing, 1971 6th printing with corrections, 10th printing 1991, Introduction to Metamathematics, North-Holland Publishing Company, Amsterdam NY, ISBN 0-7204-2103-9.
  • Kneale, W. and Kneale, M., The Development of Logic, Oxford University Press, Oxford, UK, 1962. Reprinted with corrections, 1975.
  • Alfred North Whitehead and Bertrand Russell, Principia Mathematica to *56, Cambridge at the University Press 1962 (Second Edition of 1927, reprinted). Extremely difficult because of arcane symbolism, but a must-have for serious logicians.
  • Bertrand Russell, An Inquiry Into Meaning and Truth. The William James Lectures for 1940 Delivered at Harvard University.
  • Bertrand Russell, The Problems of Philosophy, With a New Introduction by John Perry, Oxford University Press, New York, 1997 edition (first published 1912). Very easy to read: Russell was a wonderful writer.
  • Bertrand Russell, The Art of Philosophizing and Other Essays, Littlefield, Adams & Co., Totowa, NJ, 1974 edition (first published 1968). Includes a wonderful essay on "The Art of drawing Inferences".
  • Hans Reichenbach, Elements of Symbolic Logic, Dover, New York, 1947, 1975.
  • Tom Mitchell, Machine Learning, WCB McGraw-Hill, 1997.
  • Constance Reid, Hilbert, Copernicus: Springer-Verlag New York, Inc. 1996, first published 1969. Contains a wealth of biographical information, much derived from interviews.
  • Bart Kosko, Fuzzy Thinking: The New Science of Fuzzy Logic, Hyperion, New York, 1993. Fuzzy thinking at its finest. But a good introduction to the concepts.
  • David Hume, An Inquiry Concerning Human Understanding, reprinted in Great Books of the Western World Encyclopædia Britannica, Volume 35, 1952, p. 449 ff. This work was published by Hume in 1758 as his rewrite of his "juvenile" Treatise of Human Nature: Being An attempt to introduce the experimental method of Reasoning into Moral Subjects Vol. I, Of The Understanding first published 1739, reprinted as: David Hume, A Treatise of Human Nature, Penguin Classics, 1985. Also see: David Applebaum, The Vision of Hume, Vega, London, 2001: a reprint of a portion of An Inquiry starts on p. 94 ff

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