Truth value
This article has multiple issues. Please help improve it or discuss these issues on the talk page. (Learn how and when to remove these messages)

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]}
Computing
In some programming languages, any
In
false
^{[3]} are sometimes called falsy (of which the complement is truthy) to distinguish between strictly typechecked and coerced Booleans (see also: JavaScript syntax#Type conversion).^{[4]} As opposed to Python, empty containers (Arrays, Maps, Sets) are considered truthy. Languages such as PHPClassical logic
⊤  ·∧·  
true  conjunction  
¬  ↕  ↕  
⊥  ·∨·  
false  disjunction  
Negation interchanges true with false and conjunction with disjunction. 
In
 ¬(p ∧ q) ⇔ ¬p ∨ ¬q
 ¬(p ∨ q) ⇔ ¬p ∧ ¬q
Propositional variables become variables in the Boolean domain. Assigning values for propositional variables is referred to as valuation.
Intuitionistic and constructive logic
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 , and output a prime larger than .
In category theory, truth values appear as the elements of the subobject classifier. In particular, in a topos every formula of higherorder 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 ("it is not the case that is neither true nor false").^{[5]}
In intuitionistic type theory, the CurryHoward 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.
Multivalued logic
Algebraic semantics
Not all
But even nontruthvaluational 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.
See also
 Agnosticism
 Bayesian probability
 Circular reasoning
 Degree of truth
 False dilemma
 History of logic § Algebraic period
 Paradox
 Semantic theory of truth
 Slingshot argument
 Supervaluationism
 Truthvalue semantics
 Verisimilitude
References
 ^ Shramko, Yaroslav; Wansing, Heinrich. "Truth Values". In Zalta, Edward N. (ed.). Stanford Encyclopedia of Philosophy.
 ^ "Truth value". Lexico UK English Dictionary. Oxford University Press. n.d.
 ^ "ECMAScript Language Specification" (PDF). p. 43. Archived from the original (PDF) on 20150412. Retrieved 20110312.
 ^ "The Elements of JavaScript Style". Douglas Crockford. Archived from the original on 17 March 2011. Retrieved 5 March 2011.
 ^ Proof that intuitionistic logic has no third truth value, Glivenko 1928
External links
 Shramko, Yaroslav; Wansing, Heinrich. "Truth Values". In Zalta, Edward N. (ed.). Stanford Encyclopedia of Philosophy.