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23. To Shave or Not to Shave … Oneself:

The Question:

Figaro is a barber in Seville. He must shave all of Seville’s men who do not shave themselves…and only those. Does Figaro shave himself?

If he does, he mustn’t. If he doesn’t, he must.
To make it a bit more intelligible: If Figaro shaves himself, the barber must not shave him. If Figaro does not shave himself, the barber must shave him. The problem is, of course, that Figaro is the barber. So, what is Figaro/barber to do?

Background:

This famous paradox is based on a discovery in 1902 by the philosopher Bertrand Russell, even though it had been identified before. He formulated it in terms of the then new-fangled theory of sets that had been introduced by the mathematician Georg Cantor in the late nineteenth century and further developed by the logician Gottlob Frege. In Russell’s words: “Some classes are members of themselves, some are not: the class of all classes is a class, the class of not-teapots is a not-teapot. Consider the class of all the classes not members of themselves; if it is a member of itself, it is not a member of itself; if it is not, it is.” (Russell used ‘class’ instead of ‘set’)
A year later, in 1903, Frege had just finished writing the second volume of Grundgesetze der Arithmetik (Basic Laws of Arithmetic) in which he formalized his logic. In this work he attempted to derive all laws of arithmetic from several axioms that he considered self-evident. The two volumes together were to be the culmination of his life’s work. In fact, the second volume was about to go to print when an ominous letter arrived from across the Channel. In it Russell informed Frege of the paradox. It threw into doubt all of Frege’s work.
The logician must have been devastated. In a remarkable display of intellectual honesty, he added an appendix to his book. It began with the words "Hardly anything more unfortunate can befall a scientific writer than to have one of the foundations of his edifice shaken after the work is finished. This was the position I was placed in by a letter of Mr. Bertrand Russell, just when the printing of this volume was nearing its completion." He recognized and admitted to the problem and subsequently abandoned many of his mathematical and logical beliefs.

Dénouement:

The Barber’s Paradox is one of the notorious problems in set theory that occur when elements of a set refer to themselves. (See also the chapters /// and /// and ///.) Set theory defines sets as collections of items with a common property. The shirts in a cupboard, the books in a library, the odd numbers, are sets. An item either belongs to the set or it does not. The blue jeans do not belong to the set of shirts, the Elvis Presely CD does not belong to the set of books, the numbers 2, 4, 6, … do not belong to the set of odd numbers.
Now, a set of sets is also a set, the common property being that the items in this set are not shirts or books, but sets. A library, for example, can have a set of French books and a set of Italian books. Then the library itself is the set that contains the two sets of books.
Now let us construct a ‘library catalogue’. We define it as a booklet that lists all the sets that do not contain themselves. Obviously, ‘library catalogue’ contains the sets ‘French books’ and ‘Italian books’. Neither of these two sets contains itself so the definition is satisfied. How about the booklet ‘library catalogue’ itself? Since it is a set that does not list itself as one of its elements – its sole elements are ‘French books’ and ‘Italian books’ – it should be listed in ‘library catalogue’. But as soon as it is listed, ‘library catalogue’ does contain itself, thereby violating the booklet’s condition. So, it should not be listed.
It’s the vexing Figaro all over again, the barber who auto-shaves…or not.

* * *

Whenever a series of reasonings leads both to a conclusion and to its opposite, we have a paradox. And a problem. Something has to give. In logic or mathematics, it is possible to pinpoint the culprit. Frege based his basic laws of arithmetic on several axioms. Since their combined use leads to a paradox, one of them must be incompatible with the others. It soon became apparent that one of the axioms that Frege had stipulated, ‘Basic Law V’, was the culprit.
As it eventually turned out, however, the infamous Basic Law V is not even required to prove the laws of arithmetic. Instead, one may use ‘Hume’s Principle’. More on that below.

* * *

What does this mean for the barber of Seville? The culprit here is the word all, as in ‘all of Seville’s men’. One way out of the dilemma is to assume that the barber is a woman. Or that Figaro hails not from Seville but from another town. But that would be a copout. In fact, the answer to the dilemma is as simple as it is short: there simply is no such barber. Figaro does not, and cannot, exist.

Technical supplement…

…for scientists schooled in mathematical logic: Frege’s problematic Basic Law V postulates that the course-of-values of the function f is identical to the course-of-values of the function g if and only if f and g map every object to the same value. (Course-of-value refers to recursions that use any number of values for previous arguments.) Initially, this sounds quite harmless – at least to the mentioned scientists – but it lies at the heart of the problem. However, by replacing this law with Hume’s Principle which asserts that for any concepts F and G, the number of Fs is equal to the number of Gs if and only if there is a one-to-one correspondence between the Fs and the Gs, the paradox of self-reference is avoided.

* * *

By the way, the Barber Paradox can come in many different guises. All one needs is a suitable transitive verb (‘to blurb’) and its substantive form (‘the Blurber’). Then one can ask the question: Does the Bluber who blurbs all (and only those) who don't blurb themselves, blurb himself or herself? Try it with, say, to teach, to paint, to love, and, voilà: you have a paradox.