Truth value

This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. Find sources: "Truth value" – news · newspapers · books · scholar · JSTOR (February 2012) (Learn how and when to remove this message)

In logic and mathematics, a truth value, sometimes called a logical value, is a value indicating the relation of a proposition to truth, which in classical logic has only two possible values (true or false).^[1][2] Truth values are used in computing as well as various types of logic.

In some programming languages, any

Propositional variables become variables in the Boolean domain. Assigning values for propositional variables is referred to as valuation.

Whereas in classical logic truth values form a

For example, one may use the open sets of a topological space as intuitionistic truth values, in which case the truth value of a formula expresses where the formula holds, not whether it holds.

In realizability truth values are sets of programs, which can be understood as computational evidence of validity of a formula. For example, the truth value of the statement "for every number there is a prime larger than it" is the set of all programs that take as input a number ${\displaystyle n}$, and output a prime larger than ${\displaystyle n}$.

In category theory, truth values appear as the elements of the subobject classifier. In particular, in a topos every formula of higher-order logic may be assigned a truth value in the subobject classifier.

Even though a Heyting algebra may have many elements, this should not be understood as there being truth values that are neither true nor false, because intuitionistic logic proves ${\displaystyle \neg (p\neq \top \land p\neq \bot )}$ ("it is not the case that ${\displaystyle p}$ is neither true nor false").^[5]

In intuitionistic type theory, the Curry-Howard correspondence exhibits an equivalence of propositions and types, according to which validity is equivalent to inhabitation of a type.

For other notions of intuitionistic truth values, see the Brouwer–Heyting–Kolmogorov interpretation and Intuitionistic logic § Semantics.

Not all

But even non-truth-valuational logics can associate values with logical formulae, as is done in algebraic semantics. The algebraic semantics of intuitionistic logic is given in terms of Heyting algebras, compared to Boolean algebra semantics of classical propositional calculus.

• Bayesian probability
• Circular reasoning
• Degree of truth
• False dilemma
• History of logic § Algebraic period
• Paradox
• Semantic theory of truth
• Slingshot argument
• Supervaluationism
• Truth-value semantics
• Verisimilitude

• Shramko, Yaroslav; Wansing, Heinrich. "Truth Values". In Zalta, Edward N. (ed.). Stanford Encyclopedia of Philosophy.