Is this statement "Every element in the empty set is greater than itself." true or its negation is true?
Solution 1:
The negation of $\forall x \in S, x > x$ is $\exists x \in S, x \leq x$. If $S = \varnothing$, then there can't exist such an element in $S$, because there exists no element in $S$. Hence, $p$ is true. In general, every "$\forall$" property is true unless there exists a counter exemple.
Solution 2:
Another approach based on the principle that anything follows from a falsehood or contradiction:
$\forall a (\neg a\in\emptyset)$ (by definition)
$x\in \emptyset$ (assume falsehood)
$\neg P(x)$ (assume)
$\neg x\in \emptyset$ (universal specification, 1)
$x\in\emptyset \wedge\neg x\in\emptyset$ (2, 4)
$\neg\neg P(x)$ (conclusion 3, 5)
$P(x)$ (6)
$\forall a (a\in \emptyset\rightarrow P(a))$ (conclusion 2, 7)
where $P$ is any unary predicate.