Bivalence

Bivalence is the principle that every declarative sentence about a given domain has exactly one truth value: true or false. In classical logic and standard semantics, this assumption underpins the notion that truth conditions are determinate.

Formally, for any proposition P, P is true or P is false; equivalently, the disjunction P ∨ ¬P is true. In a two-valued valuation, v(P) is either true or false, and logical connectives are defined accordingly. This framework supports rigorous proof theory and model theory, where structures assign definite truth values to sentences.

Historically, questions about bivalence trace back to Aristotle’s discussion of future contingents and evolved through late 19th and early 20th century logic, culminating in the development of truth-functional semantics by Frege, Russell, and others. In classical logic, bivalence is closely linked to the law of the excluded middle, though they are conceptually distinct: the former asserts determinate truth values, while the latter asserts the universal truth of P ∨ ¬P.

Variants and challenges arise in alternative logics. Intuitionistic logic rejects bivalence, maintaining that the truth of P may be undetermined until proven. Many-valued logics extend truth values beyond true and false. Dialetheism allows that both P and ¬P could be true in some cases, while theories dealing with vagueness may posit truth-value gaps. These approaches explore limits of the bivalence assumption in language, mathematics, and philosophy.

In summary, bivalence is a foundational principle of classical semantic theories, with enduring relevance and notable alternatives that address its limitations.