# Bivalence and related laws

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

 Contents

## The laws

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

• Law of bivalence: P 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 ∨ ¬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. (But that does not necessarily mean that it has some other truth-value, e.g. indeterminate, as it may be truth-valueless). This view is controversial, however.

Intuitionistic logic rejects the excluded middle.

### Vagueness

Multi-valued logics and fuzzy logic have been considered better alternatives to bivalent systems for handling vagueness. Truth (and falsity) in fuzzy logic, for example, comes in varying degrees. Consider the following statement.

The apple on the desk is red.

Upon observation, the apple is a pale shade of red. We might say it is "50% red". This could be rephrased: it is 50% true that the apple is red. Therefore, P is 50% true, and 50% false. Now consider:

The apple on the desk is red and it is not red.

In other words, P and not-P. This violates the law of noncontradiction and, by extension, bivalence. However, this is a only a partial rejection of these laws because P is only partially true. If P were 100% true, not-P would be 100% false, and there is no contradiction because P and not-P no longer holds.

However, the law of the excluded middle is retained, because P and not-P implies P or not-P, since "or" is inclusive. The only two cases where P and not-P is false (when P is 100% true and when P is 100% false) are the same cases considered by two-valued logic, and the same rules apply.

• The distinction between the three laws is described by Douglas Groothuis (http://www.gospelcom.net/ivpress/groothuis/logic.htm), in the Philosophical Dictionary (here (http://www.philosophypages.com/dy/b2.htm#biva) and here (http://www.philosophypages.com/dy/e9.htm#exmid)), and by Peter Suber (http://www.earlham.edu/~peters/courses/logsys/pnc-pem.htm). The latter also defines a Principle of Exclusive Disjunction for Contradictories which is the logical conjunction of the Law of excluded middle and the Law of non-contradiction.

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