The Law of the Excluded Middle

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		+	==Bivalence and The Law of The Excluded Middle==
		+	The principle of bivalence states that
-	The distinction between the principle of Bivalence and the Law of Excluded middle can be difficult to	+	''every proposition is either true or false''
-	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	+	and the law of excluded middle states:
-	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.	+
		+	''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.

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Revision as of 23:15, 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.

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.

References

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