Suppose we have a universal affirmative statement (a statement of the form: ∀x(Px → Qx)), such as "all dogs go to heaven". Does an existential affirmative statement (a statement of the form: ∃x(Px ∧ Qx)) such as "there exists a dog, such that she went to heaven" follow from that?
I think that we need a more precise discussion of the above topic.
First : are we assuming classical logic ? From your question, it is not so clear.
But, see comment above, if we "equate" ∀x(Px → Qx) with ∀x(¬Px v Qx), then we are assuming so, because (P → Q) is equivalent to (¬P v Q) in classical logic.
Second, it seems to me that you are mixing" two different questions :
(i) Does a universal affirmation entail an existential affirmation?
(ii) does an universal affirmative statement, like : ∀x(Px → Qx) entails an existential affirmative statement, like : ∃x(Px ∧ Qx) ?
For (i), in classical logic the general answer is : YES. We have : ∀xDx |= ∃xDx, because the model-theoretic semantics for first-order logic assume that every interpretation has a not-empty domain.
Thus, if in the interpretation I with domain D we have that all things are Dogs, being the domain D not-empty, for sure there is at least one Dog.
The case of (ii) is different. Again, if we assume classical logic, we are licensed to "traslate" ∀x(Px → Qx) with ∀x(¬Px v Qx).
Now, the question is :
does ∀x(¬Px v Qx) |= ∃x(Px ∧ Qx) ?
Of course : NO. As per Hunan's comment above :
any interpretation with no dogs in it will make the universal vacuously true, because (¬Dx) is true. But if there are no dogs, then (Dx ∧ Hx) is always false.
But obviously [see (i)] we have : ∀x(Px → Qx) |= ∃x(Px → Qx).
Domain: set of dogs
H__: __ goes to heaven
'For all d, Hd' is true for every d that exists. It doesn't state that any d does exist. So no, the universal quantifier doesn't imply the existential quantifier.
However, for arbitrary domains X,Y and arbitrary predicate Fxy, 'there exists y such that for all x, Fxy' implies that 'for all x, there exists y such that Fxy' since the existing y is the same for every x, but 'for all x, there exists y such that Fxy' does not imply 'there exists y such that for all x, Fxy' since it is possible that the existing y is different for every x.
Mathematically, your first statement means: for all x, when x is P, then it is also Q (e.g. for all things x, when x is a unicorn, its skin has white color). From this it does not follow that there exists a unicorn.