Predicate Logic: disproving equivalence

Question

How can I use a tree proof to determine whether or not the two propositions are equivalent?

And if not how would I tell whether one is true or the other is false?

∀x(P x → Gc)

∀xPx → Gc

All help is appreciated!

Answer 1

As per Hanno's comment above, a counterexample will do and truth tree method is exactly a way to produce a counterexample (if any).

We have that $∀x(P x → Gc) \vDash ∀xPx → Gc$ and you can prove it with a simple tree.

We have instead that $∀xPx → Gc \nvDash ∀x(P x → Gc)$ and we can generate the counterexample developing the tree starting with:

1. $∀xPx → Gc$

2. $\lnot ∀x(P x → Gc)$

3. $\lnot (Pa → Gc)$ --- from 2)

4. $Pa$ --- from 3)

5. $\lnot Gc$ --- from 3)

6. $Pa → Gc$ --- from 1)

7a) $\lnot \forall x Px$ --- from 6)

7b) $Gc$ --- from 6)

As we can see, the right branch (7b) closes and thus we have to go on with the right one (7a):

8. $\lnot Pb$

and this is enough to define the countermodel: $Pa, \lnot Pb, \lnot Gc$.