Barber Paradox/Analysis 1

Analysis
This is an application of Russell's Paradox.

Let $\mathbb U$ be the set of all the men of the community.

Thus $\mathbb U$ is considered to be the universe.

Let $S: \mathbb U \to \left\{{T, F}\right\}$ be the propositional function:
 * $\forall x \in \mathbb U: S \left({x}\right) \iff x \text { is shaved by $x$}$

Let $b \in \mathbb U$ be the barber.

Let $B: \mathbb U \to \left\{{T, F}\right\}$ be the propositional function:
 * $\forall x \in \mathbb U: B \left({x}\right) \iff x \text { is shaved by $b$}$

The initial premises can be coded:
 * $(1): \quad \forall x \in \mathbb U: \left({\neg S \left({x}\right)}\right) \iff B \left({x}\right)$
 * $(2): \quad B \left({b}\right) \iff S \left({b}\right)$

Hence:
 * $S \left({b}\right) \iff B \left({b}\right) \iff \left({\neg S \left({b}\right)}\right)$

and so from Biconditional is Transitive:
 * $S \left({b}\right) \iff \left({\neg S \left({b}\right)}\right)$

So from either case there derives a contradiction.

Thus the initial premises are contradictory and cannot both hold.