The Law of the Excluded Middle
From Logic
Revision as of 23:11, 18 June 2007
For propositions: "A proposition, such as P, is either true or false."
We can denote this law symbolically:
P ∨ ¬P" ("P or not-P")
Example:
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.
The distinction between the principle of Bivalence and the Law of Excluded middle can be difficult to
understand — even the Oxford Companion to Philosophy conflates them. In classical logic the two
seem equivalent, with bivalence stated as every proposition is either true or false and the law of excluded middle stated as p or not-p. At first glance the two do seem equivalent but consider the following case: Bivalence means that there are only two truth-values i.e. true and false. The Law of Excluded middle, on the other hand, is consistent with 'supervalued' logics such as Fuzzy Logic
where 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. Of course, once you have the principle of bivalence, you can derive the law of
excluded middle but the opposite does not follow for the reason that the law of excluded middle is
consistent with three (or more) value logic as well as the principle of bivalence.
A reason for postulating a third truth-value 'indeterminate' is the problem of vagueness. Consider a
colour spectrum between red and orange. Let us also call the statement, 'It is red here', 'p'. Now, it is obvious that there are cases where 'p' is true (the red case) and clear cases where 'p' is false (the orange case). However, between the two extremes there seems to be a large class of colours where we just cannot say whether 'p' is true or false. Hence, some have suggested that in such cases 'p' is neither true nor false and that a third truth-value — indeterminate — is needed. Such a suggestion would rule out bivalence but retain the law of excluded middle. The best book on this distinction and the problem of vagueness is Timothy Williamson's book Vagueness.
References
- Copi, I. M, Cohen, C., (2001), "Introduction to Logic", 11th Edition.
