In logic and mathematics, a logical value, also called a truth value, is a value indicating the relation of a proposition to truth.
In classical logic, the truth values are true and false. Intuitionistic logic lacks a complete set of truth values because its semantics, the BrouwerHeytingKolmogorov interpretation, is specified in terms of provability conditions, and not directly in terms of the truth of formulae. Multivalued logics (such as fuzzy logic and relevance logic) allow for more than two truth values, possibly containing some internal structure.
Even nontruthvaluational logics can associate values with logical formulae, as is done in algebraic semantics. For example, the algebraic semantics of intuitionistic logic is given in terms Heyting algebras.
Topos theory uses truth values in special sense: the truth values of a topos are the global elements of the subobject classifier. Having truth values in this sense does not make a logic truth valuational.
In logic, the truth value of a logical statement says how much it is true. Usually, the truth value can only be "true" or "false". For example, "The car is red" is true when the car is red and false when it is not.
In multivalued logics, the truth value can be other values as well. For example, one could use a value between 0 and 1 to say how much it is true. Zero would mean that it is completely false and one would mean that is completely true. When the car is orange (half red, half yellow), the truth value could be 0.5 because the statement is half true and half false.
