Originally Posted by

**polypus** what is the truth value of a universal quantification over an empty set?

$\displaystyle

\forall x \in \varnothing, P(x)

$

since it is equivalent to

$\displaystyle

\lnot \exists x \in \varnothing, \lnot P(x)

$

it seems that it would be vacuously true, however it is intuitively rather difficult to accept. for instance if i want to prove that the null set is a subset of any arbitrary set then all i need to prove is that for any element of the null set that element is also in any given set. however by the above this is always true and it seems that there is no connection whatsoever between the predicate and the quantifier, and the statement hence seems contrived.

any thoughts?