Can you spot the barber Russell?

Russell Barber

It’s been a while since we posted the last entry! I was finishing my PhD thesis (which has been finally ended and defended) and we were pretty busy. This is a tribute to the Paradox of the Barber.

Here’s the challenge: let’s see if you can spot the barber Russell!

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Does the key open itself? Self-reference and paradox

Self-reference is one of the main ingredients in semantic paradoxes. Thus, for instance, the Liar paradox can use the following sentence that ascribes untruth to itself:

(1)       This sentence is not true

Although we can achieve self-reference in a simple way using demonstratives like ‘this’ in the example above, we have a wide enough range of different methods to make a sentence mention itself. For example, we can name the following sentence ‘the Liar sentence’:

(The Liar sentence)       The Liar sentence is not true

so that it turns out to be a self-referential sentence; it ascribes untruth to itself.

The American philosopher W.V.O. Quine (The Ways of Paradox) offered the following version of a sentence ascribing untruth to itself:

(2)     ‘Yields untruth when appended to its own quotation’ yields untruth when appended to its own quotation.

This last sentence asserts its own untruth, for note that it claims that a certain sentence yields untruth when it is following its own quotation. But it turns out that the sentence which yields untruth when following its own quotation is precisely the sentence ‘Yields untruth when appended to its own quotation’ which happens to follow its own quotation in (2). Hence, (2) ascribes untruth to itself.

Raymond Smullyan (Diagonalization and Self-Reference) also shows that we can achieve self-reference in any language that contains some name formation device for linguistic expressions (usually single quotation marks). Suppose we define the norm of an expression to be the expression followed by its own quotation. For example, if we consider the following expression:

(3)     I like apples

its norm would be (note that the norm of an expression does not need to be a grammatically correct expression):

(4)    I like apples ‘I like apples’

Now consider the following expression:

(5)    The set of untrue sentences contains the norm of ‘the set of untrue sentences contains the norm of’

(5) (which is a grammatically correct sentence) claims that a given sentence is untrue (that is, it is a member of the set of untrue sentences). Which one? the norm of ‘the set of untrue sentences contains the norm of’. But, what is the norm of ‘the set of untrue sentences contains the norm of’? Well, (5)!

Hence, (5) claims that (5) is untrue. It is, therefore, a self-referential sentence.

The idea behind the notion of normalization as used by Smullyan is similar to the method of diagonalization which Gödel used to prove his incompleteness theorems.

The Paradox of the Paradox (with apologies…)

A paradox is said to be a riddle
that might attract and catch all your attention,
then it will surely make your whole brain twiddle
and you’ll be suffering from hypertension.

The premises are true but the conclusion
is false, which makes you think the world is silly;
you tell yourself it must be an illusion
to be dissolved in coffee, drugs and spirits.

If ever you should find a nice solution,
immediately some others will be claiming
they should condemn you to electrocution
accused of lying and of being feigning.

In sum, a paradox is a conundrum
that finally will make you throw a tantrum.

The Paradox of the Barber

barber

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The paradox of the Barber was first introduced by the British philosopher Bertrand Russell (1872-1970). Russell asks us to consider a village with just one barber who has some restrictions on which people he shaves: he must shave all and only those villagers who do not shave themselves.

The paradox arises when we ask ourselves whether the Barber shaves himself or not. If he does not, then he is one of the villagers who do not shave themselves and, consequently, he must shave himself. But, if he shaves himself then he must be one of the villagers who do not shave themselves, for he only shaves such villagers. Thus, if he does not shave himself, he must do it; and if he shaves himself, he must not do it.

This paradox is usually taken to show that such a barber cannot exist.

The Liar Paradox

liardef

Llicència de Creative Commons

The Liar paradox arises when we ask ourselves whether the following sentence, called a “Liar sentence”, is true or not:

This sentence is not true.

The sentence just above this line says of itself, so to speak, that it is not true. If we suppose that the sentence is true, then what the sentence says (that it is not true) must be the case and, thus, the sentence is not true. But if we suppose that the sentence is not true, then what the sentence says (again, that it is not true) turns out to be the case and, hence, the sentence is true after all.

The Liar paradox was already known by the ancient Greeks and, after more than 2000 years, is yet to be solved.