Editing Truth value

{{short description|Value indicating the relation of a proposition to truth}}
{{Redirect|True and false}}

{{more citations needed|date=February 2012}}

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 (''[[logical truth|true]]'' or ''[[false (logic)|false]]'').<ref>{{cite SEP |url-id=truth-values |title=Truth Values |first=Yaroslav |last=Shramko |first2=Heinrich |last2=Wansing}}</ref><ref>{{OxfordDictionaries.com|Truth value}}</ref> Truth values are used in [[computing]] as well as various types of [[logic]].

== Computing ==
{{see also|Implicit type conversion}}
In some programming languages, any [[Expression (computer science)|expression]] can be evaluated in a context that expects a [[Boolean data type]]. Typically (though this varies by programming language) expressions like the number [[zero]], the [[empty string]], empty lists, and [[Null pointer|null]] are treated as false, and strings with content (like "abc"), other numbers, and objects evaluate to true. Sometimes these classes of expressions are called '''falsy''' and '''truthy'''. For example, in [[Lisp (programming language)|Lisp]], [[Null pointer|nil]], the empty list, is treated as false, and all other values are treated as true. In [[C (programming language)|C]], the number 0 or 0.0 is false, and all other values are treated as true. 

In [[JavaScript]], the empty string (<code>""</code>), <code>null</code>, <code>undefined</code>, [[NaN|<code>NaN</code>]], +0, [[−0]] and <code>false</code><ref>{{cite web |url=http://www.ecma-international.org/publications/files/ECMA-ST/ECMA-262.pdf |title=ECMAScript Language Specification |page=43 |url-status=dead |archive-url=https://web.archive.org/web/20150412040502/http://www.ecma-international.org/publications/files/ECMA-ST/Ecma-262.pdf |archive-date=2015-04-12 |access-date=2011-03-12 }}</ref> are sometimes called ''falsy'' (of which the [[Complement (set theory)|complement]] is ''truthy'') to distinguish between strictly [[Type safety|type-checked]] and [[Type conversion|coerced]] Booleans (see also: [[JavaScript syntax#Type conversion]]).<ref>{{cite web |url=http://javascript.crockford.com/style2.html |title=The Elements of JavaScript Style |publisher=Douglas Crockford |access-date=5 March 2011 |url-status=live |archive-url=https://web.archive.org/web/20110317074944/http://javascript.crockford.com/style2.html |archive-date=17 March 2011 }}</ref> As opposed to Python, empty containers (Arrays, Maps, Sets) are considered truthy. Languages such as [[PHP]] also use this approach.

== Classical logic ==
{| class="floatright" cellpadding=0 style="text-align:center;"
 |-
 |
 | style="font-size:300%; font-weight:700;" | ⊤
 | rowspan=5 |
 | style="font-size:300%;"                  | ·'''∧'''·
 |-
 |
 | true
 | <small>[[logical conjunction|conjunction]]</small>
 |-
 | style="font-size:250%; font-weight:300;" | ¬
 | style="font-size:200%; font-weight:500;" | ↕
 | style="font-size:200%; font-weight:500;" | ↕
 |-
 |
 | style="font-size:300%; font-weight:700;" | ⊥
 | style="font-size:300%;"                  | ·'''∨'''·
 |-
 |
 | false
 | <small>[[logical disjunction|disjunction]]</small>
 |-
 | colspan=4 style="padding-left:24px; padding-top:8px; padding-bottom:14px; text-align:left;" | <small>Negation interchanges <br/>true with false and <br/>conjunction with disjunction.</small>
 |}
In [[classical logic]], with its intended semantics, the truth values are ''[[logical truth|true]]'' (denoted by ''1'' or the [[verum]] ⊤), and ''[[false (logic)|untrue]]'' or ''[[false (logic)|false]]'' (denoted by ''0'' or the [[falsum]] ⊥); that is, classical logic is a [[two-valued logic]]. This set of two values is also called the [[Boolean domain]]. Corresponding semantics of [[logical connective]]s are [[truth function]]s, whose values are expressed in the form of [[truth table]]s. [[Logical biconditional]] becomes the [[equality (mathematics)|equality]] binary relation, and [[negation]] becomes a [[bijection]] which [[permutation|permutes]] true and false. Conjunction and disjunction are [[Dual (mathematics)#Duality in logic and set theory|dual]] with respect to negation, which is expressed by [[De Morgan's laws]]:
: ¬({{math|{{mvar|p}} ∧ {{mvar|q}}) ⇔ ¬{{mvar|p}} ∨ ¬{{mvar|q}}}}
: ¬({{math|{{mvar|p}} ∨ {{mvar|q}}) ⇔ ¬{{mvar|p}} ∧ ¬{{mvar|q}}}}

[[Propositional variable]]s become [[Variable (computer science)|variables]] in the Boolean domain. Assigning values for propositional variables is referred to as [[valuation (logic)|valuation]].
<!-- Also something should be written about first-order logics -->

== 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}}.

== Multi-valued logic ==
[[Multi-valued logic]]s (such as [[fuzzy logic]] and [[relevance logic]]) allow for more than two truth values, possibly containing some internal structure. For example, on the [[unit interval]] {{closed-closed|0,1}} such structure is a [[total order]]; this may be expressed as the existence of various [[degrees of truth]].

== Algebraic semantics ==
{{main|Algebraic logic}}
Not all [[logical system]]s are truth-valuational in the sense that logical connectives may be interpreted as truth functions. For example, [[intuitionistic logic]] lacks a complete set of truth values because its semantics, the [[Brouwer–Heyting–Kolmogorov interpretation]], is specified in terms of [[proof theory|provability]] conditions, and not directly in terms of the [[necessarily true|necessary truth]] of formulae.

But even non-truth-valuational logics can associate values with logical formulae, as is done in [[algebraic semantics (mathematical logic)|algebraic semantics]]<!-- if this becomes a dab page linking to [[Algebraic logic]], just remove the link -->. The algebraic semantics of intuitionistic logic is given in terms of [[Heyting algebra]]s, compared to [[Boolean algebra (structure)|Boolean algebra]] semantics of classical propositional calculus.
<!-- This also has an analogy to vector spaces instead of scalars. But is it encyclopedic enough? -->

==See also==
{{Portal|Philosophy|Psychology}}
* [[Agnosticism]]
* [[Bayesian probability]]
* [[Circular reasoning]]
* [[Degree of truth]]
* [[False dilemma]]
* {{slink|History of logic|Algebraic period}}
* [[Paradox]]
* [[Semantic theory of truth]]
* [[Slingshot argument]]
* [[Supervaluationism]]
* [[Truth-value semantics]]
* [[Verisimilitude]]

==References==
{{Reflist}}

==External links==
* {{cite SEP |url-id=truth-values |title=Truth Values |first=Yaroslav |last=Shramko |first2=Heinrich |last2=Wansing}}

{{Mathematical logic}}
{{Logical truth}}

[[Category:Concepts in logic]]
[[Category:Propositions]]
[[Category:Logical truth]]
[[Category:Concepts in epistemology]]