Law of Excluded Middle - Examples

Examples

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 it is not the case that Socrates is mortal.

is true by virtue of its form alone. That is, the "middle" position, that Socrates is neither mortal nor not-mortal, is excluded by logic, and therefore either the first possibility (Socrates is mortal) or its negation (it is not the case that Socrates is mortal) must be true.

An example of an argument that depends on the law of excluded middle follows. We seek to prove that there exist two irrational numbers and such that

is rational.

It is known that is irrational (see proof). Consider the number

Clearly (excluded middle) this number is either rational or irrational. If it is rational, the proof is complete, and

and

But if is irrational, then let

and

Then

and 2 is certainly rational. This concludes the proof.

In the above argument, the assertion "this number is either rational or irrational" invokes the law of excluded middle. An intuitionist, for example, would not accept this argument without further support for that statement. This might come in the form of a proof that the number in question is in fact irrational (or rational, as the case may be); or a finite algorithm that could determine whether the number is rational or not.