Law of the excluded middle

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. Another Latin designation for this law is tertium non datur: "no third (possibility) is given".

The earliest known formulation is in Aristotle's discussion of the 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:

${\displaystyle \mathbf {*2\cdot 11} .\ \ \vdash .\ p\ \vee \thicksim p}$.

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