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Law of excluded middle

In logic, the law of excluded middle or the principle of excluded middle states that for every proposition, either this proposition or its negation is true.[1][2] It is one of the three laws of thought, along with the law of noncontradiction, and the law of identity; however, no system of logic is built on just these laws, and none of these laws provides inference rules, such as modus ponens or De Morgan's laws. The law is also known as the law / principle of the excluded third, in Latin principium tertii exclusi. Another Latin designation for this law is tertium non datur or "no third [possibility] is given". In classical logic, the law is a tautology.

Not to be confused with fallacy of the excluded middle.

The principle should not be confused with the semantical principle of bivalence, which states that every proposition is either true or false. The principle of bivalence always implies the law of excluded middle, while the converse is not always true. A commonly cited counterexample uses statements unprovable now, but provable in the future to show that the law of excluded middle may apply when the principle of bivalence fails.[3]

History[edit]

Aristotle[edit]

The earliest known formulation is in Aristotle's discussion of the principle of non-contradiction, first proposed in On Interpretation,[4] 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.[5] He also states it as a principle in the Metaphysics book 3, saying that it is necessary in every case to affirm or deny,[6] and that it is impossible that there should be anything between the two parts of a contradiction.[7]


Aristotle wrote that ambiguity can arise from the use of ambiguous names, but cannot exist in the facts themselves:

Satisfying all of including

De Morgan's laws

 – foundational controversy in twentieth-century mathematics: an account on the formalist-intuitionist divide around the Law of the excluded middle

Brouwer–Hilbert controversy

 – Pattern of reasoning in propositional logic

Consequentia mirabilis

Constructive set theory

Diaconescu's theorem

 – Splitting of a whole into exactly two non-overlapping parts; dyadic relations and processes

Dichotomy

 – System including an indeterminate value

Law of excluded fourth

Law of excluded middle is untrue in  – Propositional calculus in which there are more than two truth values such as ternary logic – System including an indeterminate value and fuzzy logic – System for reasoning about vagueness

many-valued logic

 – Axioms of rational discourse

Laws of thought

 – Mathematical concept

Limited principle of omniscience

 – Type of diagrammatic or visual notation for logical expressionss: a graphical syntax for propositional logic

Logical graph

 – view that a proposition about the future is either necessarily true, or its negation is necessarily true: the application excluded middle to modal – Type of formal logic propositions

Logical determinism

Mathematical constructivism

Non-affirming negation in the  – Doctrinal distinction within Tibetan Buddhism school of Buddhism, another system in which the law of excluded middle is untrue

Prasangika

 – Axiom used in logic and philosophy: another way of turning intuition classical

Peirce's law

"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, Illinois, 1952. Cited as GB 19–20.

Aquinas, Thomas

"Metaphysics", W.D. Ross (trans.), vol. 8 in Robert Maynard Hutchins (ed.), Great Books of the Western World, Encyclopædia Britannica, Inc., Chicago, Illinois, 1952. Cited as GB 8. 1st published, W.D. Ross (trans.), The Works of Aristotle, Oxford University Press, Oxford, UK.

Aristotle

2000, Engines of Logic: Mathematicians and the Origin of the Computer, W. W. Norton & Company, NewYork, New York, ISBN 0-393-32229-7 pbk.

Martin Davis

Logical Dilemmas, The Life and Work of Kurt Gödel, A.K. Peters, Wellesley, Massachusetts, 1997.

Dawson, J.

From Frege to Gödel, A Source Book in Mathematical Logic, 1879–1931, Harvard University Press, Cambridge, Massachusetts, 1967. Reprinted with corrections, 1977.

van Heijenoort, J.

Luitzen Egbertus Jan , 1923, On the significance of the principle of excluded middle in mathematics, especially in function theory [reprinted with commentary, p. 334, van Heijenoort]

Brouwer

Andrei Nikolaevich , 1925, On the principle of excluded middle, [reprinted with commentary, p. 414, van Heijenoort]

Kolmogorov

Luitzen Egbertus Jan , 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.

Brouwer

Luitzen Egbertus Jan , 1927(2), Intuitionistic reflections on formalism,[reprinted with commentary, p. 490, van Heijenoort]

Brouwer

1952 original printing, 1971 6th printing with corrections, 10th printing 1991, Introduction to Metamathematics, North-Holland Publishing Company, Amsterdam, New York, ISBN 0-7204-2103-9.

Stephen C. Kleene

and Kneale, M., The Development of Logic, Oxford University Press, Oxford, UK, 1962. Reprinted with corrections, 1975.

Kneale, W.

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.

Alfred North Whitehead

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). Easy to read.

Bertrand Russell

The Art of Philosophizing and Other Essays, Littlefield, Adams & Co., Totowa, New Jersey, 1974 edition (first published 1968). Includes a wonderful essay on "The Art of drawing Inferences".

Bertrand Russell

Elements of Symbolic Logic, Dover, New York, 1947, 1975.

Hans Reichenbach

Machine Learning, WCB McGraw–Hill, 1997.

Tom Mitchell

Hilbert, Copernicus: Springer–Verlag New York, Inc. 1996, first published 1969. Contains a wealth of biographical information, much derived from interviews.

Constance Reid

Fuzzy Thinking: The New Science of Fuzzy Logic, Hyperion, New York, 1993. Fuzzy thinking at its finest but a good introduction to the concepts.

Bart Kosko

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

David Hume

in the Stanford Encyclopedia of Philosophy

"Contradiction" entry