In logic, the law of excluded middle (also known as the principle of excluded middle or excluded middle or excluded third) is the principle that for any proposition, either that proposition is true, or its negation is. The principle can be expressed in either a logical or a semantical form. The semantical form uses the nonlogical word "true", as above. The logical form uses only logical expressions "either", "or"^{[1]} and can be expressed by the formula "P ∨ ¬P": "either P or not P", where "P" is schematic ^{[2]} for any proposition such as 'snow is white', 'Socrates is running' and so on.
The earliest known formulation of the principle is in the book On Interpretation by Aristotle,^{[3]} 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.^{[4]} He also states it as a principle in the Metaphysics book 3, saying that it is necessary in every case to affirm or deny,^{[5]} and that it is impossible that there should be anything between the two parts of a contradiction.^{[6]}
The principle should not be confused with the principle of bivalence, which states that every proposition is either true or false, and only has a semantical formulation.
Some systems of logic have different but analogous laws. For some finite nvalued logics, there is an analogous law called the law of excluded n+1th. If negation is cyclic and "∨" is a "max operator", then the law can be expressed in the object language by (P ∨ ~P ∨ ~~P ∨ ... ∨ ~...~P), where "~...~" represents n−1 negation signs and "∨ ... ∨" n−1 disjunction signs. It is easy to check that the sentence must receive at least one of the n truth values (and not a value that is not one of the n). Other systems reject the law entirely.
The law is also known as the law (or principle) of the excluded third, or, in Latin, principium tertii exclusi. Yet another Latin designation for this law is tertium non datur: "no third (possibility) is given".
The Principle of Excluded Middle, along with its complement, the Law of Contradiction, are correlates of the Law of Identity; the first principle of thought (reason). Because the principle of identity intellectually partitions the Universe into exactly two parts: ‘self’ and ‘other, it creates a dichotomy wherein the two parts are ‘mutually exclusive’ and ‘jointly exhaustive’. The principle of contradiction is merely an expression of the mutually exclusive aspect of that dichotomy, and the principle of excluded middle is an expression of its jointly exhaustive aspect.
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