# 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 $\map M x$ be defined as:

- $x$ is a man in the community.

Let $\map S {x, y}$ be defined as:

- $x$ shaves $y$.

Let $b$ be the barber.

Suppose $\map M b$.

Suppose that:

- $\forall x, y: \paren {\map S {x, y} \implies \map M x, \map M y}$

Suppose to the contrary that:

- $\forall x: \paren {\map M x \implies \paren {\map S {b, x} \iff \neg \map S {x, x} } }$

For $x = b$ we obtain the contradiction:

- $\map S {b, b} \iff \neg \map S {b, b}$

Therefore, it must be false that:

- $\forall x: \paren {\map M x \implies \paren {\map S {b, x} \iff \neg \map S {x, x} } }$