The use of addition in:
G -> H
G -> (H ^ J)
is called "improper" because the letter that is added is not added to the whole line.  It turns out, however, that even though the addition rule is not correctly applied, the inference is still valid.  Hence, this inference is not called "invalid," as the others are.  As for the last example, a DeMorgans Rule will be presented that will allow us to remove parentheses preceeded by negation signs.  But even after the parentheses have been removed, the inference remains valid.

And this is nothing compared to what goes on in Malta.