When to use conjunction and when to use implication? (First Order Logic)?

Question

I'm learning First Order Logic at the moment. I'm looking at this example: Some dogs bark ∃x (dog(X) Λ bark(x)) All dogs have four legs ∀x (dog(x) -> have_four_legs(x)) My question is: is it possible for the second example to be: ∀x (dog(x) Λ have_four_legs(x)) And why can't the first example be: ∃x (dog(X) -> bark(x))

Answer 1

It is not possible because universal quantifiers do not admit the existence of the things they are quantifying over, whereas existential quantifiers do admit the existence of the things they are quantifying.

When one says that "All dogs have four legs," it has to be translated formally as "For all x, if x is a dog, then x has four legs" rather than "For all x, x is a dog and x has four legs". The former translation does not admit the existence of the dogs while the latter does. This is so because "All dogs have four legs" could be true even if there are no dogs existing. And so the latter translation is too strong/restrictive since it doesn't allow the proposition to be true in a world in which there are no dogs.

On the other hand, "Some dogs bark" does commit one to posit that there are these dogs existing. So one ought to translate it as "There exists some x, such that x is a dog and x barks", rather than "There exists some x, such that if x is a dog, then x barks". The latter translation is too weak, for it could be true in a world where there are no dogs but only cats, for example. This is so because it is true that if an existing cat were a dog, then it would bark. But this is not what "Some dogs bark" communicates, since it suggests that some dogs do in fact exist.

Hope that helps!