Editing Truth value (section)

== Intuitionistic and constructive logic ==
{{main|Constructivism (mathematics)}}

Whereas in classical logic truth values form a [[Boolean algebra]], in [[intuitionistic logic]], and more generally, [[Constructivism (mathematics)|constructive mathematics]], the truth values form a [[Heyting algebra]]. Such truth values may express various aspects of validity, including locality, temporality, or computational content.

For example, one may use the [[open set|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 <math>n</math>, and output a prime larger than <math>n</math>.

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 <math>\neg (p \neq \top \land p \neq \bot)</math> ("it is not the case that <math>p</math> is neither true nor false").<ref>[http://plato.stanford.edu/entries/intuitionistic-logic-development/#4.3 Proof that intuitionistic logic has no third truth value, Glivenko 1928]</ref>

In [[intuitionistic type theory]], the [[Type_theory#Curry-Howard_correspondence|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 {{sectionlink|Intuitionistic logic|Semantics}}.