The paradox is a refutation of the assertion that such a hygienist exists (assuming the implied fact that he only cleans the teeth of those who don't clean their own, or there's no paradox). Since the hygienist does not exist, there is no-one who cleans the hygienist's teeth.

If P is the asserted property of the hygienist, R the relation x R y = x cleans y's teeth, and Q x the property "x cleans a hygienist's teeth", the matter can be formalised as follows.

P is defined by: P x =def (∀y. x R y ⇔ ¬(y R y))

P x implies (by substituting x for y in the quantifier) x R x ⇔ ¬(x R x), which is equivalent to false. Therefore no x satisfies P x.

Q is defined by: Q x =def ∃y. P(y) ∧ x R y

Since P(y) is always false, no x satisfies Q x.