> For the complete documentation index, see [llms.txt](https://mandober.gitbook.io/math-debrief/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://mandober.gitbook.io/math-debrief/300-logic/logic-principles/bivalence.md).

# The principle of bivalence

<https://www.wikiwand.com/en/Principle_of_bivalence>

> The principle of bivalence states that every declarative sentence expressing a proposition has exactly one truth value, true or false, not both, not neither.

A logic that acknowledges this principle is called a **two-valued** or **bivalent** logic.

In formal logic, the principle of bivalence becomes a property of semantics - given semantics may or may not possess it.

Bivalence is not the same as **the law of excluded middle (EM)**; there are logics that recognize EM without being bivalent.

The principle of bivalence is related to the EM, although EM is a syntactic expression of the language of a logic with the form $$P \lor \lnot P$$.

For example, the three-valued *Logic of Paradox (LP)* upholds the law of excluded middle, but not **the law of non-contradiction**, ¬(P ∧ ¬P), and its intended semantics is not bivalent.

In *classical two-valued logic* both the law of excluded middle and the law of non-contradiction hold.

Many modern logic programming systems replace the law of the excluded middle with the concept of **negation as failure (NAF)**.