∀x(x∈G->x∈B); k∈b, therefore k∈G
I hope you see this is basically second order version of affirming the consequent:
P->Q; Q, therefore P.
Every car has 4 wheel, a cart has 4 wheel; therefore, is a cart a car?
Just because all goats have beards doesn't mean that if you are not a goat you cannot have a beard: it could be that lots of different things have beards: goats, but also pigs and salamanders. Therefore, if I tell you that Karl Marx has a beard, Karl Marx need not be a goat. Maybe Karl Marx is a salamander!
Tacit inference is the norm, not the exception. In a real sense, language is merely pointing with words, and we all have to infer referents from the inherently ambiguous symbols that others provide for us. Consider our interactions with young children: a young child will point and make a grasping gesture with a hand, and say (perhaps) "Gagrlagaaagah"; we are ...
It's incorrect because its logical form is incorrect. If we use notation similar to what's used in Tarski's world (a good program to learn the basics of first-order logic, see the lecture notes on it here, especially the first one on atomic sentences), the general form of the first two premises would be something like this:
For all x, Property1(x) -> ...
The premise ∃x∃y∀z(x = z ∨ y = z) says that there exists at most two distinct values in the domain.
The conclusion ∀x∀y(¬x = y → ∀z(x = z ∨ y = z)) says that for any two distinct values in the domain, there does not exist a third.
Clearly we need to utilise existential elimination and universal introduction.
∃-elim here will be tricky.
The 'trick' is ...