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 $S: \mathbb U \to \set {T, F}$ be the propositional function:
 * $\forall x \in \mathbb U: \map S x \iff x \text { is shaved by $x$}$

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

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

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

Hence:
 * $\map S b \iff \map B b \iff \paren {\neg \map S b}$

and so from Biconditional is Transitive:
 * $\map S b \iff \paren {\neg \map S b}$

So from either case there derives a contradiction.

Thus the initial premises are contradictory and cannot both hold.