The Law or Principle of Excluded Middle

Originally proposed by Aristotle: "There is nothing between asserting and denying." Generally given as "A is B or A is not B;" i.e., that an object (A) either has or lacks a given property (B). An alternate formulation of this (with propositions instead of objects) is "p or not p" - but not both. It is not the same as the Law of Bivalence, that everything is either true or false. The excluded middle is a logical argument, while bivalence is a semantic argument involving the idea of "truth." The excluded middle is one of the three Laws or Principles of Thought in Aristotelian logic: Identity (A is A), Contradiction (A is not both B and not B), and the Excluded Middle.

One problem with the excluded middle is with vague properties such as "bald." If we are to determine whether a person is bald or not bald, where is the line drawn? At 10 hairs, or 10000 hairs? In this situation, the excluded middle is actually more useful than bivalence; one of "A is bald" or "A is not bald" does not hold for all definitions of bald. However, it is true that for all definitions of bald, one of the two must hold. Intuitionism and many-valued logics also deny the excluded middle, instead preferring a version of bivalence: "No proposition is neither true nor false."

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