Barber Paradox/Analysis 2

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Paradox

There exists a community, one of whose members is a barber.

This barber operated under an unusual rule: his task was to shave every man in the community who did not shave himself, and only those men.

Who shaves the barber?

If he does not shave himself, then he must shave himself.

But if he shaves himself, he must not shave himself.


Analysis

Let $M \left({x}\right)$ be defined as:

$x$ is a man in the community.

Let $S \left({x, y}\right)$ be defined as:

$x$ shaves $y$.

Let $b$ be the barber.

Suppose $M \left({b}\right)$.

Suppose that:

$\forall x, y: \left({S \left({x, y}\right) \implies M \left({x}\right), M \left({y}\right)}\right)$


Suppose to the contrary that:

$\forall x: \left({M \left({x}\right) \implies \left({S \left({b, x}\right) \iff \neg S \left({x, x}\right)}\right)}\right)$

For $x = b$ we obtain the contradiction:

$S \left({b, b}\right) \iff \neg S \left({b, b}\right)$

Therefore, it must be false that:

$\forall x: \left({M \left({x}\right) \implies \left({S \left({b, x}\right) \iff \neg S \left({x, x}\right)}\right)}\right)$