This discussion is somewhat confusing because it's mixing up two different things which unfortunately go by the same initials. PVA=polyvinyl Alcohol is soluble in water when brushed out thin and dried, and for that reason can be used as the colloid in dichromated colloid processes like gum printing. PVA=polyvinyl Acetate, while it can be mixed with water, is not soluble in water when brushed out thin and dried, and that's why it works for a sizing without requiring hardening. I understand it's more complicated than this if you want to get really technical chemically, but this soluble/insoluble distinction of the dried film is a good enough distinction for our purposes.
I made the mistake once when I bought something labeled PVA Glue, intending to use it to test some things about printing with PVAlcohol as the colloid rather than gum, but it turned out I'd unwittingly bought polyvinyl acetate rather than polyvinyl alcohol, and it didn't work at all for the purpose I'd intended. I haven't seen that bottle lately; I hope I didn't toss it, because now I'd like to try it for sizing, per Steve. So I'd suggest that you double-check the label if you want to try this with a different brand than the Gamblin specified here.
Katharine