Barber Paradox/Analysis 2

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