Editing Barber paradox

{{short description|Colloquial version of Russell's paradox}}
{{about|a paradox of self-reference|an unrelated paradox in the theory of [[logical conditional]]s with a similar name, introduced by [[Lewis Carroll]]|Barbershop paradox}}
{{Redirect|Who shaves the barber|the ''[[Fargo (TV series)|Fargo]]'' episode|Who Shaves the Barber?}}
{{more citations needed|date=March 2015}}

The '''barber paradox''' is a [[puzzle]] derived from [[Russell's paradox]]. It was used by [[Bertrand Russell]] as an illustration of the [[paradox]], though he attributes it to an unnamed person who suggested it to him.<ref name="atomism">Russell, Bertrand (1919). "The Philosophy of Logical Atomism", reprinted in ''The Collected Papers of Bertrand Russell, 1914-19'', Vol 8, p. 228</ref> The puzzle shows that an apparently plausible scenario is logically impossible. Specifically, it describes a barber who is defined such that he both shaves himself and does not shave himself, which implies that no such barber exists.<ref name="siegelj">{{Cite web |title=The Barber's Paradox |url=https://www.umsl.edu/~siegelj/SetTheoryandTopology/TheBarber.html |access-date=2023-10-21 |website=[[UMSL]]}}</ref><ref name="oxfordref">{{Cite web |title=Barber paradox |url=https://www.oxfordreference.com/display/10.1093/oi/authority.20110803095446216 |access-date=2023-10-21 |website=[[Oxford Reference]] |language=en }}</ref>

== Paradox ==
<!-- Please don't add supposed resolutions of the paradox here; see the last paragraph of this section. -->

The barber is the "one who shaves all those, and those only, who do not shave themselves". The question is, does the barber shave himself?<ref name=atomism/>

Any answer to this question results in a [[contradiction]]:
# The barber cannot shave himself, as he only shaves those who do ''not'' shave themselves. Thus, if he shaves himself he ceases to be the barber specified.
# Conversely, if the barber does not shave himself, then he fits into the group of people who the specified barber would shave, and thus, as that barber, he must shave himself.

In its original form, this paradox has no solution, as no such barber can exist. The question is a [[loaded question]] in that it assumes the existence of a barber who could not exist, which is a vacuous proposition, and hence false. There are other non-paradoxical variations, but those are different.<ref name=oxfordref/>

== History ==

This paradox is often incorrectly attributed to [[Bertrand Russell]] (e.g., by [[Martin Gardner]] in ''Aha!''). It was suggested to Russell as an alternative form of [[Russell's paradox]],<ref name=atomism/> which Russell had devised to show that [[set theory]] as it was used by [[Georg Cantor]] and [[Gottlob Frege]] contained contradictions. However, Russell denied that the Barber's paradox was an instance of his own:

{{Quotation|That contradiction [Russell's paradox] is extremely interesting. You can modify its form; some forms of modification are valid and some are not. I once had a form suggested to me which was not valid, namely the question whether the barber shaves himself or not. You can define the barber as "one who shaves all those, and those only, who do not shave themselves". The question is, does the barber shave himself? In this form the contradiction is not very difficult to solve. But in our previous form I think it is clear that you can only get around it by observing that the whole question whether a class is or is not a member of itself is nonsense, i.e. that no class either is or is not a member of itself, and that it is not even true to say that, because the whole form of words is just noise without meaning.|Bertrand Russell, ''The Philosophy of Logical Atomism''<ref name=atomism/>}}

This point is elaborated further under [[Russell's paradox#Applied versions|Applied versions of Russell's paradox]].

== In first-order logic ==

: <math>(\exists x ) (\text{person}(x) \wedge (\forall y) (\text{person}(y) \implies (\text{shaves}(x, y)  \iff \neg \text{shaves}(y, y))))</math>

This sentence says that a barber {{mvar|x}} exists. Its [[truth value]] is false, as the existential clause is unsatisfiable (a contradiction) because of the [[Universal quantification|universal quantifier]] <math>(\forall)</math>. The universally quantified {{mvar|y}} will include every single element in the domain, including our infamous barber {{mvar|x}}. So when the value {{mvar|x}} is assigned to {{mvar|y}}, the sentence in the universal quantifier can be rewritten to <math> \text{shaves}(x,x)\iff \neg \text{shaves}(x,x)</math>, which is an instance of the contradiction <math>a \iff \neg a</math>. Since the sentence is false for the biconditional, the entire universal clause is false. Since the existential clause is a conjunction with one operand that is false, the entire sentence is false. Another way to show this is to negate the entire sentence and arrive at a [[Tautology (logic)|tautology]]. Nobody is such a barber, so there is no solution to the paradox.<ref name=siegelj/><ref name="oxfordref"/>

: <math>(\exists x ) (\text{person}(x) \wedge \bot)</math>
: <math>(\exists x ) (\bot)</math>
: <math>\bot</math>

== See also ==
* [[Cantor's theorem]]
* [[Gödel's incompleteness theorems]]
* [[Halting problem]]
* [[List of paradoxes]]
* [[Self-reference]]
* [[List of self–referential paradoxes]]
* [[Double bind]]
* [[Principle of explosion]]

== References ==
<references/>

== External links ==
* [http://www.umsl.edu/~siegel/SetTheoryandTopology/TheBarber.html Proposition of the Barber's Paradox]
* Joyce, Helen. [http://plus.maths.org/issue20/xfile/index.html "Mathematical mysteries: The Barber's Paradox".] ''Plus'', May 2002.
* [http://www.cs.utexas.edu/users/EWD/transcriptions/EWD09xx/EWD923a.html Edsger Dijkstra's take on the problem]
* {{cite journal |jstor=27900748|title=The Philosophy of Logical Atomism |last1=Russell |first1=Bertrand |journal=The Monist |year=1919 |volume=29 |issue=3 |pages=345–380 |doi=10.5840/monist19192937 |url=https://zenodo.org/record/1946681 }}

{{Paradoxes}}

[[Category:Self-referential paradoxes]]
[[Category:Bertrand Russell]]
[[Category:Logical paradoxes]]