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For any proposition P, at a given time, in a given respect, there are three related laws:
It is possible to state the laws of non-contradiction and the excluded middle in the formal manner of the traditional propositional calculus:
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.
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.
A famous example is the contingent sea battle case found in Aristotle's work, De Interpretatione, chapter 9:
The law of the excluded middle clearly holds:
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.
Intuitionistic logic rejects the excluded middle.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:
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.
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