Picking up on (or perhaps merely rehashing) fast's last example, how about this?

A6

P1'": Joe is not an elephant

P2": Joe is an elephant

C'"': Joe is an elephant

A7

P1'": Joe is not an elephant

P2": Joe is an elephant

C'"'': Joe is not an elephant

Both valid? My (amateur) guess is....yes. Even though (a) in both cases the conclusion is only one of two things that must follow and (b) in both cases the conclusion is also a premise.

In other words, I'm thinking fast's last example (just above) would also have been valid if the conclusion had beenanyof the premises.

If (if) so, I'm curious to know what the correct term would be to describe something which is valid but which is 'incomplete' (layman's term again) if the conclusion is not the only conclusion that must follow from the premises? In other words if a certain premise (or premises) is/are not taken into account, not used, or ignored.

I'm thinking that if you were sitting a hypothetical logic exam and the question was 'what conclusion follows from the premises in argument A5 (or A4)' you might not get full marks if you only gave either of the answers in A6 or A7 above. Possibly in some ways, though not all, similar to how you would not get full marks in a GCSE Maths exam by giving 7 as the answer to the question, 'what is the square root of 49?' (Though we could be pedantic there and say the latter question, and possibly also the former one, was/were worded misleadingly or ambiguously in implying that there is 'a' or only one, answer).