How would
No car is faster than an airplane.
look in predicate logic?
I believe it is:
¬∃ x, y [faster(car(x), airplane(y))]
But I'm not sure if you would use a for all, or there exists. Can somebody explain the difference to me? Because it seems like if something doesn't exist that already means for all can't exist either.
So the statement:
Nothing is both a human and an elephant.
Seems like you would use a ∀, but then again I believe you could also use a ¬∃ and it would still make sense. Can somebody explain this to me please?
Thanks!
No. Don't nest the predicates inside each other; that won't work. Predicates are functions which return boolean (truth) values, and this is not (typically) appropriate to be used as the argument of another predicate. You are literally trying to say "(is the thing called $x$ a car?) is not faster than (is the thing called $y$ an aeroplane?)".
You want to say: "There is no thing (call it $x$) and nothing (call it $y$), such that: $x$ is a car, $y$ is an aeroplane, and $x$ is faster than $y$."
$$\neg\exists x \, \exists y\; \Big(\operatorname {car}(x) \wedge \operatorname{aeroplane}(y) \wedge \operatorname{faster}(x, y)\big)\Big)$$
Alternatively:
$$\forall x \, \forall y\; \Big(\big(\operatorname {car}(x) \wedge \operatorname{aeroplane}(y)\big) \to \neg \operatorname{faster}(x, y)\Big)$$