Principle of bivalence

In logic, the semantic principle of bivalence states that every proposition takes exactly one of two truth values (e.g. truth or falsehood). The laws of bivalence, excluded middle, and non-contradiction are related, but they refer to the calculus of logic, not its semantics, and are hence not the same. The law of bivalence is compatible with classical logic, but not intuitionistic logic, linear logic, or multi-valued logic.

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

For any proposition P, P is either true or false.
 * Law of bivalence:

For any proposition P, P is true or 'not-P' is true.
 * Law of the excluded middle:

For any proposition P, it is not the case that both P is true and 'not-P' is true.
 * Law of non-contradiction:

Bivalence is deepest
Through the use of propositional variables, it is possible to formulate analogues of the laws of non-contradiction and the excluded middle in the formal manner of the traditional propositional logic:


 * Excluded middle: P ∨ ¬P
 * Non-contradiction: ¬(P ∧ ¬P)

In second-order logic, second-order quantifers are available to bind the propositional variables, allowing one to formulate closer analogues:


 * Excluded middle: ∀P(P ∨ ¬P)
 * Non-contradiction: ∀P¬(P ∧ ¬P)

Note that both the aforementioned logics assume the law of bivalence. The law of bivalence itself has no analogue in either of these logics: on pain of contradiction, it can be stated only in the metalanguage used to study the aforementioned formal logics.

Analogues of excluded middle are not valid in intuitionistic logic; this rejection is founded in intuitionists' constructivist as opposed to Platonist conception of truth and falsity. On the other hand, in linear logic, analogues of both excluded middle and non-contradiction are valid, and yet it is not a two-valued (i.e., bivalent) logic.

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.

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 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 or false) are the same cases considered by two-valued logic, and the same rules apply.

Of course, it may be stated that bivalence must always be true, and that multi-valued logic is simply by definition a vague state of perception. That is, multi-valued logic is a convenient way of saying, "This instance has not been observed in enough detail to determine the truth value of P." In other words, if a pale apple is 50% red (where red is noted as P), then P can be said to be 100% true, noting that bivalence makes little delineation as to the nature of not-P aside from the given, meaning that the apple might very well be 50% white as well (when white is noted as not-P), meaning that P and not-P can both be true, but in separate instances, as they both exist as separate colours, which combine in a larger instance set in perhaps an unobservable, exceedingly subtle way to create the shade of pale red. In this case, the apple might be set S, which consisted of P and not-P to greater or lesser or equal respective degrees, as long as it is acknowledged that P and not-P are separate instances within a set instance. In this way, bivalence simply states that white cannot be red, and makes no claim about the colour of the set instance, to which is applied multi-value logic, in which case multi-value logic is simply derivative of bivalence as well.