# 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:

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