Barbershop paradox

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This article is about a paradox in the theory of logical conditionals introduced by Lewis Carroll in "A Logical Paradox." For an unrelated paradox of self-reference with a similar name, attributed to Bertrand Russell, see the Barber paradox.

The Barbershop Paradox was proposed by Lewis Carroll in a three-page essay entitled "A Logical Paradox," which appeared in the July 1894 issue of Mind. The name comes from the "ornamental" short story that Carroll uses to illustrate the paradox (although it had appeared several times in more abstract terms in his writing and correspondence before the story was published).

The story runs as follows. Two logicians -- "Uncle Joe" and "Uncle Jim" -- and their nephew -- who narrates the story -- are walking towards a barber's shop. There are three barbers who work in the shop - Allen, Brown, and Carr - but not all of them are always in the shop. Carr is a good barber, and Uncle Jim is keen to be shaved by him. He knows that the shop is open, so at least one of them must be in. He also knows that Allen's recently been very ill, and so nervous that he never goes out without Brown with him.

Uncle Joe insists that Carr is certain to be in, and then claims that he can prove it logically. Uncle Jim demands the proof. Uncle Joe reasons as follows. Suppose that Carr is out. If Carr is out, then if Allen is also out Brown would have to be in -- since someone must be in the shop for it to be open. However, we know that whenever Allen goes out he takes Brown with him, and thus we know as a general rule that if Allen is out, Brown is out. Thus, if Carr is out then the two statements "if Allen is out then Brown is in" and "if Allen is out then Brown is out" would both be true at the same time. But (Uncle Joe argues) this seems paradoxical; the hypotheticals seem "incompatible" with each other, and cannot both be true so, by reductio ad absurdum, Carr must (logically) be in.

[edit] Discussion

In modern logic theory this paradox does not exist. It is eliminated by the law of implication, which states that the statement "if A then B" is logically identical to "either A is false, or B is true". (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 not pressed the button, or the light is on.)

Applying the law of implication to the statements above alters them as follows: "If Carr is out, then if Allen is out, Brown is in" (not-C implies (not-A implies B)) is reduced to "Either Carr is in, or Allen is in, or Brown is in" (C or A or B) - this is correct, since it clearly represents the condition for there to be somebody in the shop. "If Allen is out, then Brown is out" (not-A implies not-B) is reduced to "Either Allen is in or Brown is out" (A or not-B).

So, if Carr is out, then the two statements which become true at once are: "Either Allen is in or Brown is in", and "Either Allen is in or Brown is out". It is then clear that both of these statements can become true at once in the case where Allen is in; thus there is no contradiction.

This argument is foreseen by Carroll, who includes a mention of it at the end of the story. He attempts to counteract it by arguing that the protasis and apodosis of the implication "If Carr is in..." are incorrectly divided. However, after the application of the Law of Implication, which removes the "If...", no protasis and apodosis exist and the counter-argument is rendered void.

[edit] Further reading

Lewis Carroll (1894), A Logical Paradox. Mind, New Series, Vol. 3, No. 11 (Jul., 1894), 436-438.
Bertrand Russell (1903), The Principles of Mathematics § 19 n. 1. Russell suggests a truth-functional notion of logical conditionals, which (among other things) entails that a false proposition will imply all propositions. In a note he mentions that his theory of implication would dissolve Carroll's paradox, since it not only allows, but in fact requires that both "p implies q" and "p implies not-q" be true, so long as p is not true.