any() and all() on empty list?

[Paul Rubin]

"Paul McGuire" <ptmcg at austin.rr._bogus_.com> writes:
> > Usually, "for all X in S, PRED(x) is true" means:
> > there does not exist X in S so that PRED(x) is false.
> >
> How do you get this "usually" stuff?  I would agree that this is usually
> implemented as a short-circuited loop through the list, that breaks out at
> the first False value.  But I would not be quick to equate "commonality of
> implementation" with "meaning".

See <http://en.wikipedia.org/wiki/For_all>:

    Generally, then, the negation of a propositional function's universal
    quantification is an existential quantification of that propositional
    function's negation; symbolically,

    \lnot\ \forall{x}{\in}\mathbf{X}\, P(x) \equiv\
    \exists{x}{\in}\mathbf{X}\, \lnot\ P(x)