View source for False (logic)

{{Short description|Possessing negative truth value}}
{{see also|False statement}}
{{Refimprove|date=March 2024}}
In [[logic]], '''false'''<ref>Its noun form is [[Truth|falsity]].</ref> or '''untrue''' is the state of possessing negative [[truth value]] and is a [[nullary]] [[logical connective]]. In a [[truth function|truth-functional]] system of propositional logic, it is one of two postulated truth values, along with its [[negation]], [[logical truth|truth]].<ref>Jennifer Fisher, ''On the Philosophy of Logic'', Thomson Wadsworth, 2007, {{ISBN|0-495-00888-5}}, [https://books.google.com/books?id=k8L_YW-lEEQC&pg=PT27 p. 17.]</ref>  Usual notations of the false are [[0 (number)|0]] (especially in [[Boolean Logic|Boolean logic]] and [[computer science]]), O (in [[Polish notation|prefix notation]], O''pq''), and the [[up tack]] symbol&nbsp;<math>\bot</math>.<ref>[[Willard Van Orman Quine]], ''Methods of Logic'', 4th ed, Harvard University Press, 1982, {{ISBN|0-674-57176-2}}, [https://books.google.com/books?id=liHivlUYWcUC&pg=PA34 p. 34.]</ref><ref>{{Cite web|title=Truth-value {{!}} logic|url=https://www.britannica.com/topic/truth-value|access-date=2020-08-15|website=Encyclopedia Britannica|language=en}}</ref>

Another approach is used for several [[theory (mathematical logic)|formal theories]] (e.g., [[intuitionistic propositional calculus]]), where a propositional constant (i.e. a nullary connective),&nbsp;<math>\bot</math>, is introduced, the truth value of which being always false in the sense above.<ref>[[George Edward Hughes]] and D.E. Londey, ''The Elements of Formal Logic'', Methuen, 1965, [https://books.google.com/books?id=JbwOAAAAQAAJ&pg=PA151 p. 151.]</ref><ref>Leon Horsten and Richard Pettigrew, ''Continuum Companion to Philosophical Logic'', Continuum International Publishing Group, 2011, {{ISBN|1-4411-5423-X}}, [https://books.google.com/books?id=w_abLTXIFkcC&pg=PA199 p. 199.]</ref><ref>[[Graham Priest]], ''[[An Introduction to Non-Classical Logic|An Introduction to Non-Classical Logic: From If to Is]]'', 2nd ed, Cambridge University Press, 2008, {{ISBN|0-521-85433-4}}, [https://books.google.com/books?id=rMXVbmAw3YwC&pg=PA105 p. 105.]</ref> It can be treated as an absurd proposition, and is often called absurdity.

== In classical logic and Boolean logic ==<!-- linked from #Consistency -->
In [[Boolean logic]], each variable denotes a [[truth value]] which can be either true (1), or false (0).

In a [[classical logic|classical]] [[propositional calculus]], each [[proposition]] will be assigned a truth value of either true or false. Some systems of classical logic include dedicated symbols for false (0 or <math>\bot</math>), while others instead rely upon formulas such as {{math|{{mvar|p}} ∧ ¬{{mvar|p}}}} and {{math|¬({{mvar|p}} → {{mvar|p}})}}.

In both Boolean logic and Classical logic systems, true and false are opposite with respect to [[negation]]; the negation of false gives true, and the negation of true gives false.

{| class="wikitable" style="text-align: center"
|-
!<math>x</math>
!<math>\neg x</math>
|-
!true
|false
|-
!false
|true
|}

The negation of false is equivalent to the truth not only in classical logic and Boolean logic, but also in most other logical systems, as explained below.

{{expand section|date=February 2012}}

== False, negation and contradiction ==
In most logical systems, [[negation]], [[material conditional]] and false are related as:
: {{math|¬{{mvar|p}} ⇔ ({{mvar|p}} → ⊥)}}
In fact, this is the definition of negation in some systems,<ref>Dov M. Gabbay and Franz Guenthner (eds), ''Handbook of Philosophical Logic, Volume 6'', 2nd ed, Springer, 2002, {{ISBN|1-4020-0583-0}}, [https://books.google.com/books?id=JyewdfGhNAsC&pg=PA12 p. 12.]</ref> such as [[intuitionistic logic]], and can be proven in propositional calculi where negation is a fundamental connective. Because {{math|{{mvar|p}} → {{mvar|p}}}} is usually a theorem or axiom, a consequence is that the negation of false ({{math|¬ ⊥}}) is true.

A [[contradiction]] is the situation that arises when a [[statement (logic)|statement]] that is assumed to be true is shown to [[entailment|entail]] false (i.e., {{math|φ ⊢ ⊥}}). Using the equivalence above, the fact that φ is a contradiction may be derived, for example, from {{math|⊢ ¬φ}}. A statement that entails false itself is sometimes called a contradiction, and contradictions and false are sometimes not distinguished, especially due to the [[Latin]] term ''[[wikt:falsum#English|falsum]]'' being used in English to denote either, but false is one specific [[proposition]].

Logical systems may or may not contain the [[principle of explosion]] (''ex falso quodlibet'' in [[Latin]]), {{math|⊥ ⊢ φ}} for all {{math|φ}}. By that principle, contradictions and false are equivalent, since each entails the other.

== Consistency ==
{{main|Consistency}}
A [[theory (mathematical logic)|formal theory]] using the "<math>\bot</math>" connective is defined to be consistent, if and only if the false is not among its [[theorem]]s. In the absence of [[wikt:propositional constant|propositional constants]], some substitutes (such as the ones [[#In classical logic and Boolean logic|described above]]) may be used instead to define consistency.

==See also==  
{{Wikiquote|Falsehood}}
* [[Contradiction]]  
* [[Logical truth]]  
* [[Tautology (logic)]] (for symbolism of logical truth)
* [[Truth table]]

== References ==
{{reflist}}

{{Logical connectives}}
{{Logical truth}}
{{Common logical symbols}}

[[Category:Logical connectives]]