Principle of bivalence

In logic, the semantic principle of bivalence states that every proposition is either true or false. The dual semantic principle, the principle of contravalence, states that no proposition is both true and false. The principle of bivalence is related to the excluded middle though the latter is a syntactic expression of the language of a logic of the form "P or ¬P". The difference between the principle and the law is important because there are logics which validate the law but which do not validate the principle, and vice versa. For example, the "Logic of Paradox" (LP) of Graham Priest validates the law of excluded middle though its intended semantics is not bivalent.

The intended semantics of classical logic is bivalent, but this is not true of every semantics for classical logic. Classical logic may be characterized by the class of boolean algebras and many-valued matrices in which propositions (sentences) may take one of more than two truth values. The two-valued semantics has a special status, however, besides being the intended one. E.g. every finite boolean algebra is isomorphic to a power of the two-valued boolean algebra.

Bivalence and non-contradiction

The principle of bivalence is a property of the semantics of logic, one that finds itself dual to the principle of contradiction:

• Principle of bivalence:

For any proposition P, either P is true or P is false

• Principle of contradiction:

For no proposition P is both P true and ¬P true.

Parallel to these semantic principles, there are laws that may be axioms or theorems of a logic.

• Law of the excluded middle, parallel to bivalence:

For every proposition P, the formula "P or ¬P" is true.

• The law parallel to non-contradiction:

For every proposition P, the formula "¬(P and ¬P)" is true.

Typically the logical law is validated by a semantics where the principle holds.

In second-order propositional logic, second-order quantifers are available to bind the propositional variables, allowing the formula scheme to be replaced by

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

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

These analogues of the law of the excluded middle are not valid in intuitionistic logic, although the weaker ∀P¬¬(P ∨ ¬P) is an intuitionistic theorem; the rejection of excluded middle is founded in intuitionists' constructivist as opposed to Platonist conception of truth and falsity.

On the other hand, in linear logic, formal analogues of both excluded middle and non-contradiction are valid, though using linear logic's semantically different "multiplicative" negation and conjunction/disjunction.

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 disputed^[1].

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 here asserts:

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

One might say that the truth of this means that if we wait until the end of tomorrow, then we can find out whether there has been such a sea battle. But commitment to bivalence does not validate this dilemma in such a conditional manner: it asserts that either there is now a determinate fact of the matter that there will be a sea battle, or another determinate fact of the matter holding now that says there won't be.

Aristotle hesitated to embrace bivalence for such future contingents; Chrysippus, the Stoic logician, did embrace bivalence for this and all other propositions. The controversy continues to be of central importance in both metaphysics and the philosophy of logic.

Vagueness

Such puzzles as the Sorites paradox have raised doubt as to the applicability of classical logic and the principle of bivalence to concepts that may be vague in their application. Fuzzy logic and some other multi-valued logics have been proposed as alternatives that handle vague concepts better. 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.

Algebraic semantics

The algebraic semantics of classical logic directly supports bivalence in the finite case: because every finite Boolean algebra is isomorphic to a powerset algebra, two-valued truth tables tell the whole story about (finitary) propositional classical logic.

The infinitary case is trickier, however. The Stone representation theorem tells us that in the general case, Boolean algebras are subalgebras of powerset algebras. This means that the intuition that classical logic is about a set of independent true-false alternatives is not right, but is projecting an independence between Boolean-valued alternatives that does not have a basis in semantics.

See also

Thinking portal

• Exclusive disjunction
• Degrees of truth
• Anekantavada
• False dilemma
• Fuzzy logic
• Logical disjunction
• Logical equality
• Logical value
• Multivalued logic
• Propositional logic
• Relativism
• Quantum logic
• Perspectivism
• Rhizome (philosophy)

Notes

1. ^ Crispin Wright, 1986

References

• Wright, Crispin, 1986. Realism, Meaning & Truth. Oxford: Blackwell.