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Bivalence and related laws

In logic, the laws of bivalence, the excluded middle, and non-contradiction are related, but not the same. This page discusses the differences.

For any proposition P, at a given time, in a given respect, there are three related laws:

• Law of bivalence: The proposition is either true or false.
• Law of the excluded middle: (P or not-P) is true.
• Law of non-contradiction: (P and not-P) is false.

 Bivalence is deepest

It is possible to state the laws of non-contradiction and the excluded middle in the formal manner of the traditional propositional calculus:

• Excluded middle: P &or ¬P

In fact, with the law of bivalence taken for granted, the two other laws can be derived as theorems, using the rules of propositional calculus.

It is, however, not possible to state the principle of bivalence in such a way, as the traditional propositional calculus just assumes sentences are true or false.

 Why these distinctions might matter

These different principles are closely related, but there are certain cases where we might wish to affirm that they do not all go together. Specifically, the link between bivalence and the law of excluded middle is sometimes challenged.

#### Future contingents

A famous example is the contingent sea battle case found in Aristotle's work, De Interpretatione, chapter 9:

Imagine P refers to the statement "There will be a sea battle tomorrow."

The law of the excluded middle clearly holds:

There will be a sea battle tomorrow, or there won't be.

However, some philosophers wish to claim that P is neither true nor false today, since the matter has not been decided yet. So, they would say that the principle of bivalence does not hold in such a case: P is neither true nor false (yet). This view is controversial, however.

#### Vagueness

In fuzzy logic or other multi-valued logics dealing with vagueness, again it may be the case that (P or not-P) is true but P is not considered true or false. For example, suppose P is:

Joe is bald.

In this case, in multi-valued logics, this proposition may be given a truth value in between "true" or "false" (if Joe has a bit of hair, but not much). Nevertheless, under some such logics, the following statement may be considered true:

Joe is bald, or Joe is not bald.

So such logics reject bivalence, but maintain the law of the excluded middle.

• The distinction between the three laws is described here (http://www.gospelcom.net/ivpress/groothuis/logic.htm), here (http://www.philosophypages.com/dy/b2.htm#biva) and here (http://www.philosophypages.com/dy/e9.htm#exmid).