Law of excluded middle
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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.
See also
- Brouwerâ€“Hilbert controversy: an account on the formalist-intuitionist divide around the Law of the excluded middle
- Consequentia mirabilis
- Constructive set theory
- Diaconescu's theorem
- Dichotomy
- Law of excluded fourth
- Law of excluded middle is untrue in many-valued logic such as ternary logic and fuzzy logic
- Laws of thought
- Limited principle of omniscience
- Logical graphs: a graphical syntax for propositional logic
- Logical determinism: the application excluded middle to modal propositions
- Mathematical constructivism
- Non-affirming negation in the Prasangika school of Buddhism, another system in which the law of excluded middle is untrue
- Peirce's law: another way of turning intuition classical