A semantic thesis according to which every proposition is either true or false, i.e., the statement that there are no truth value gaps. In the case of formal logic, the principle of bivalence is frequently interpreted as the thesis that there should exist only two truth values, precluding, e.g., many-valued logics, although on some interpretations, a semantics for a logic may satisfy bivalence while enjoying more than two truth values.

For example, Suszko’s thesis—articulated by logician Roman Suszko (1919–1979)—provides a means of reimposing bivalence on many logics, such as many-valued logics, whose semantics do not immediately reflect the principle (by, e.g., allowing truth value gaps). In the case of many-valued logics, although there may be a plethora of truth values—apparently violating bivalence—a formula $\phi$ either takes a designated value or it does not, i.e., for any model $\mathfrak{M}$ and formula $\phi$, either $\mathfrak{M}$ is a model of $\phi$ or it is not, a fact that bears a strong resemblance to the principle of bivalence.

Although related, the principle of bivalence is distinct from the principle of excluded middle (i.e., the axiom scheme $\phi \vee \neg \phi$). For example, although excluded middle is not a theorem of intuitionistic logic, some interpretations of intuitionistic logic—e.g., Kripke semantics—insist that it remains bivalent. On such interpretations, that $\phi \vee \neg \phi$ fails is not to say that $\phi$ is neither true nor false, but only to say that neither disjunct is assertable (due to, e.g., insufficient evidence, unavailability of a proof, etc). Conversely, supervaluationist semantics provide a clear case of a consequence relation in which excluded middle holds but bivalence fails.

The principle of bivalence, as a semantic thesis, is intimately related with a dual notion, the principle of non-contradiction.