The Law of the Excluded Middle

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For propositions: "A proposition, such as P, is either true or false."

We can denote this law symbolically:

P ∨ ¬P" ("P or not-P")


For example, if P is the proposition:

Socrates is mortal.

then the law of excluded middle holds that the logical disjunction:

Either Socrates is mortal or Socrates is not mortal.

is true by virtue of its form alone. I.e. it is tautologous.

Bivalence and The Law of The Excluded Middle

The principle of bivalence states that

every proposition is either true or false

and the law of excluded middle states:

p or not-p.

It is important to see that these two principles are stating entirely different things. Bivalence holds that that there are only two truth-values i.e. true and false. The Law of Excluded middle, on the other hand, is consistent with logics such as Fuzzy Logic which hold that there are more than two truth-values i.e. true, false and indeterminate.

To see this, consider that 'p' means 'it is true that p' but 'not-p' means 'it is not true that p' from which it does not immediately follow that 'p is false' as p could also be Indeterminate, at least within a supervalued logical framework.


  • Copi, I. M, Cohen, C., (2001), "Introduction to Logic", 11th Edition.
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