Semicolon and omission of repetitive words

The Heracles quote would more correctly read "You're our son, Heracles, and we your family." The third comma as written above is confusing, and its elimination helps to clear that up.

I think you have things backwards. The semicolon should only be used to join what could otherwise stand alone as complete sentences. Thus you could write:
“To err is human; to forgive is divine,”
or
“To err is human, to forgive, divine,”
but NOT what you have
“To err is human; to forgive, divine.”

I think your final sentence is technically correct, but, personally, I would not drop the second "are." “You’re our son, Heracles, and we are your family,” is both clearer and more euphonious, to my ear, than what you have. Rob is right that your version is confusing, but things are NOT improved by dropping the third comma without reinstating the missing "are."

I believe the confusion with the last sentence could be avoided by replacing the first comma by a full stop and removing the second comma completely, like this:

"You're our son. Heracles and we, your family."

Anyway, thanks for this. I'm an English learner and the post has been helpful :-D

@nigel - “To err is human; to forgive, divine” - Sigurd was directly quoting the poet Alexander Pope, from his Essay On Criticism (1711), and punctuated it exactly as in the original.
http://www.ourcivilisation.com/smartboard/shop/popea/critic.htm

@Igor Ribeiro - Sorry, but your full stop has separated (the appositive) Heracles from its related noun phrase 'our son'. If you want a full stop, it would need to come after Heracles. This of course could jut be a typo.

@sigurd - I'm of the 'longer pause' school for semicolons, rather than following strict rules. So I wouldn't call your version incorrect, but personally I would use something stronger than a comma here. You have, after all, two equal but related ideas. And I would agree with Rob, it gives you too many commas, not all of which are working at the same intensity. Just read the various versions out aloud to yourself, and see which works best. I go for the version with the semicolon, myself.

Sorry, that wasn't of course an appositive; I was obviously trying to be too clever. Hoist with my own petard, perhaps. But the idea is still the same.