Truth value

One of the two values, "true" or "false" , that can be taken by a given logical formula in an interpretation (model) considered. Sometimes the truth value is denoted in the literature by or , and by or . If the truth values of elementary formulas are defined in a model , then the truth value of any formula can be inductively determined in the following way (for classical logic):

One sometimes considers interpretations in which logical formulas may take, besides and , other "intermediate" truth values. In such interpretations, the truth values of formulas may be, e.g., elements of Boolean algebras (so-called Boolean-valued models for classical logic, cf. Boolean-valued model), elements of pseudo-Boolean algebras (also known as Heyting algebras, cf. Pseudo-Boolean algebra) or open sets in topological spaces (for intuitionistic logic), or elements of topological Boolean algebras (for modal logic ) (cf. [2]). In a Boolean-valued model, if the truth values of elementary formulas are defined, then the truth values of compound formulas can be determined as follows

where is the complement to the element . For example, in topological models for intuitionistic logic, the truth values of compound formulas can be determined as follows:

where denotes the interior of the set .

References

[1]	P.S. Novikov, "Elements of mathematical logic" , Oliver & Boyd and Acad. Press (1964) (Translated from Russian)
[2]	E. Rasiowa, R. Sikorski, "The mathematics of metamathematics" , Polska Akad. Nauk (1963)