Barbershop paradox
 | This article needs attention from an expert in Logic. The specific problem is: Lacks details as to the historical context, in particular as to whether this was considered a valid problem. (February 2014) |
The barbershop paradox was proposed by Lewis Carroll in a three-page essay titled "A Logical Paradox", which appeared in the July 1894 issue of Mind. The name comes from the "ornamental" short story that Carroll uses in the article to illustrate the paradox. It was in existence 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".[1] From the viewpoint of modern logic it is seen not so much as a paradox than as a simple logical error. It is of interest now mainly as an episode in the development of algebraic logical methods when these were not so widely understood (even among logicians), although the problem continues to be discussed in relation to theories of implication and modal logic.[2]
The paradox
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 never leaves the shop unless Brown goes with him.
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 certain 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 produces a contradiction. 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".
- ^ Carroll, Lewis (July 1894). "A Logical Paradox". Mind. 3 (11): 436–438.
- ^ Carroll, Lewis (1977). Bartley, William Warren (ed.). Symbolic Logic, Parts I and II. Harvester Press. ISBN 0855279842.