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<blockquote data-quote="Hawkman" data-source="post: 445164" data-attributes="member: 23715"><p><strong>Barbershop Paradox </strong></p><p></p><p>The <strong>barbershop paradox</strong> was proposed by <a href="https://en.wikipedia.org/wiki/Lewis_Carroll" target="_blank">Lewis Carroll</a> in a three-page essay titled "A Logical Paradox", which appeared in the July 1894 issue of <em><a href="https://en.wikipedia.org/wiki/Mind_(journal)" target="_blank">Mind</a></em>. The name comes from the "ornamental" short story that Carroll uses in the article to illustrate the paradox. It existed previously in several alternative forms in his writing and correspondence, not always involving a barbershop. Carroll described it as illustrating "a very real difficulty in the Theory of Hypotheticals". From the viewpoint of modern logic, it is seen not so much as a <a href="https://en.wikipedia.org/wiki/Paradox" target="_blank">paradox</a> than as a simple <a href="https://en.wikipedia.org/wiki/Formal_fallacy" target="_blank">logical error</a>. It is of interest now mainly as an episode in the development of <a href="https://en.wikipedia.org/wiki/History_of_logic#Algebraic_period" target="_blank">algebraic logical methods</a> when these were not so widely understood (even among logicians), although the problem continues to be discussed in relation to theories of <a href="https://en.wikipedia.org/wiki/Logical_consequence" target="_blank">implication</a> and <a href="https://en.wikipedia.org/wiki/Modal_logic" target="_blank">modal logic</a>.</p><p></p><h2>The paradox</h2><p></p><p>In the story, Uncle Joe and Uncle Jim are walking to the barber shop. They explain that there are three barbers who live and work in the shop—Allen, Brown, and Carr—and some or all of them may be in. We are given two pieces of information from which to draw conclusions. Firstly, the shop is definitely open, so at least one of the barbers must be in. Secondly, Allen is said to be very nervous, so that he always brings Brown with him wherever he goes.</p><p>Now, according to Uncle Jim, Carr is a very good barber, and he wants to know whether Carr will be there to shave him. Uncle Joe insists that Carr is <em>certain</em> to be in, and claims that he can prove it logically. Uncle Jim demands this proof.</p><p>Uncle Joe gives his argument as follows:</p><p>Suppose that Carr is out. We will show that this assumption <a href="https://en.wikipedia.org/wiki/Proof_by_contradiction" target="_blank">produces a contradiction</a>. If Carr is out, then we know this: "If Allen is out, then Brown is in", because there has to be someone in "to mind the shop." But, we also know that whenever Allen goes out he takes Brown with him, so as a general rule, "If Allen is out, then Brown is out". The two statements we have arrived at are incompatible, because if Allen is out then Brown cannot be both In (according to one) and Out (according to the other). There is a contradiction. So we must abandon our hypothesis that Carr is Out, and conclude that Carr must be in.</p><p>Uncle Jim's response is that this conclusion is not warranted. The correct conclusion to draw from the incompatibility of the two "hypotheticals" is that what is hypothesized in them (that Allen is out) must be false under our assumption that Carr is out. Then our logic simply allows us to arrive at the conclusion "If Carr is out, then Allen must necessarily be in."</p><p></p><p>In modern logic theory this scenario is not a paradox. The <a href="https://en.wikipedia.org/wiki/Logical_conditional" target="_blank">law of implication</a> reconciles what Uncle Joe claims are incompatible hypotheticals. This law states that "if X then Y" is logically identical to "X is false or Y is true" (not X or Y). For example, given the statement "if you press the button then the light comes on", it must be true at any given moment that either you have <em>not</em> pressed the button, or the light is on.</p><p></p><p>In short, what obtains is not that Carr is out yields a contradiction, only that it necessitates Allen is in, because Allen is out is what actually yields the contradiction.</p><p></p><p>In this scenario, that means Carr doesn't have to be in, but that if he isn't in, Allen has to be in.</p><p></p><p><strong>Barbershop Paradox 2</strong></p><p></p><p>People often confuse Carroll's puzzle with a different, much more famous puzzle about a village barber.</p><ul> <li data-xf-list-type="ul"><strong>The Setup:</strong> In a small town, there is a barber who shaves <strong>all</strong> men, and <strong>only</strong> those men, who do not shave themselves.</li> <li data-xf-list-type="ul"><strong>The Question:</strong> Who shaves the barber?</li> <li data-xf-list-type="ul"><strong>The Paradox:</strong><ul> <li data-xf-list-type="ul">If the <strong>barber shaves himself</strong>, then he violates his own rule (since he only shaves men who <em>do not</em> shave themselves).</li> <li data-xf-list-type="ul">If the <strong>barber does not shave himself</strong>, then he fits the criteria for someone who should be shaved by the barber, meaning he <em>must</em> shave himself.</li> </ul></li> <li data-xf-list-type="ul"><strong>The Resolution:</strong> Such a barber cannot logically exist. The premise itself is contradictory.</li> </ul></blockquote><p></p>
[QUOTE="Hawkman, post: 445164, member: 23715"] [B]Barbershop Paradox [/B] The [B]barbershop paradox[/B] was proposed by [URL='https://en.wikipedia.org/wiki/Lewis_Carroll']Lewis Carroll[/URL] in a three-page essay titled "A Logical Paradox", which appeared in the July 1894 issue of [I][URL='https://en.wikipedia.org/wiki/Mind_(journal)']Mind[/URL][/I]. The name comes from the "ornamental" short story that Carroll uses in the article to illustrate the paradox. It existed previously in several alternative forms in his writing and correspondence, not always involving a barbershop. Carroll described it as illustrating "a very real difficulty in the Theory of Hypotheticals". From the viewpoint of modern logic, it is seen not so much as a [URL='https://en.wikipedia.org/wiki/Paradox']paradox[/URL] than as a simple [URL='https://en.wikipedia.org/wiki/Formal_fallacy']logical error[/URL]. It is of interest now mainly as an episode in the development of [URL='https://en.wikipedia.org/wiki/History_of_logic#Algebraic_period']algebraic logical methods[/URL] when these were not so widely understood (even among logicians), although the problem continues to be discussed in relation to theories of [URL='https://en.wikipedia.org/wiki/Logical_consequence']implication[/URL] and [URL='https://en.wikipedia.org/wiki/Modal_logic']modal logic[/URL]. [HEADING=1]The paradox[/HEADING] In the story, Uncle Joe and Uncle Jim are walking to the barber shop. They explain that there are three barbers who live and work in the shop—Allen, Brown, and Carr—and some or all of them may be in. We are given two pieces of information from which to draw conclusions. Firstly, the shop is definitely open, so at least one of the barbers must be in. Secondly, Allen is said to be very nervous, so that he always brings Brown with him wherever he goes. Now, according to Uncle Jim, Carr is a very good barber, and he wants to know whether Carr will be there to shave him. Uncle Joe insists that Carr is [I]certain[/I] to be in, and claims that he can prove it logically. Uncle Jim demands this proof. Uncle Joe gives his argument as follows: Suppose that Carr is out. We will show that this assumption [URL='https://en.wikipedia.org/wiki/Proof_by_contradiction']produces a contradiction[/URL]. If Carr is out, then we know this: "If Allen is out, then Brown is in", because there has to be someone in "to mind the shop." But, we also know that whenever Allen goes out he takes Brown with him, so as a general rule, "If Allen is out, then Brown is out". The two statements we have arrived at are incompatible, because if Allen is out then Brown cannot be both In (according to one) and Out (according to the other). There is a contradiction. So we must abandon our hypothesis that Carr is Out, and conclude that Carr must be in. Uncle Jim's response is that this conclusion is not warranted. The correct conclusion to draw from the incompatibility of the two "hypotheticals" is that what is hypothesized in them (that Allen is out) must be false under our assumption that Carr is out. Then our logic simply allows us to arrive at the conclusion "If Carr is out, then Allen must necessarily be in." In modern logic theory this scenario is not a paradox. The [URL='https://en.wikipedia.org/wiki/Logical_conditional']law of implication[/URL] reconciles what Uncle Joe claims are incompatible hypotheticals. This law states that "if X then Y" is logically identical to "X is false or Y is true" (not X or Y). For example, given the statement "if you press the button then the light comes on", it must be true at any given moment that either you have [I]not[/I] pressed the button, or the light is on. In short, what obtains is not that Carr is out yields a contradiction, only that it necessitates Allen is in, because Allen is out is what actually yields the contradiction. In this scenario, that means Carr doesn't have to be in, but that if he isn't in, Allen has to be in. [B]Barbershop Paradox 2[/B] People often confuse Carroll's puzzle with a different, much more famous puzzle about a village barber. [LIST] [*][B]The Setup:[/B] In a small town, there is a barber who shaves [B]all[/B] men, and [B]only[/B] those men, who do not shave themselves. [*][B]The Question:[/B] Who shaves the barber? [*][B]The Paradox:[/B] [LIST] [*]If the [B]barber shaves himself[/B], then he violates his own rule (since he only shaves men who [I]do not[/I] shave themselves). [*]If the [B]barber does not shave himself[/B], then he fits the criteria for someone who should be shaved by the barber, meaning he [I]must[/I] shave himself. [/LIST] [*][B]The Resolution:[/B] Such a barber cannot logically exist. The premise itself is contradictory. [/LIST] [/QUOTE]
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