How do I translate the following into English? $$ 1. \quad \forall x (Ax \land Bx)$$ $$ 2. \quad \forall x (Ax \to Bx)$$ $$ 3. \quad \exists x (Ax \land Bx)$$ $$ 4. \quad \exists x (Ax \to Bx)$$
Everyone are both A and B.
Every A are B.
Someone is A and B.
... ?
Thanks for your time.
The literal translation of the fourth one is 'There exists someting such that if it is an A, then it is a B'
However, that is a very awkward sentence (and indeed the reason why you'll rarely see a conditional after an existential).
To make a little more sense of the fourth one, you could rewrite it as its logical equivalent:
$\exists x (\neg A(x) \lor B(x))$
which translates as a somewhat more meaningful 'There is something that is either not an A, or it is a B$'
Still awkward ...