Editing Barber paradox (section)

== In first-order logic ==

: <math>(\exists x ) (\text{person}(x) \wedge (\forall y) (\text{person}(y) \implies (\text{shaves}(x, y)  \iff \neg \text{shaves}(y, y))))</math>

This sentence says that a barber {{mvar|x}} exists. Its [[truth value]] is false, as the existential clause is unsatisfiable (a contradiction) because of the [[Universal quantification|universal quantifier]] <math>(\forall)</math>. The universally quantified {{mvar|y}} will include every single element in the domain, including our infamous barber {{mvar|x}}. So when the value {{mvar|x}} is assigned to {{mvar|y}}, the sentence in the universal quantifier can be rewritten to <math> \text{shaves}(x,x)\iff \neg \text{shaves}(x,x)</math>, which is an instance of the contradiction <math>a \iff \neg a</math>. Since the sentence is false for the biconditional, the entire universal clause is false. Since the existential clause is a conjunction with one operand that is false, the entire sentence is false. Another way to show this is to negate the entire sentence and arrive at a [[Tautology (logic)|tautology]]. Nobody is such a barber, so there is no solution to the paradox.<ref name=siegelj/><ref name="oxfordref"/>

: <math>(\exists x ) (\text{person}(x) \wedge \bot)</math>
: <math>(\exists x ) (\bot)</math>
: <math>\bot</math>