P1. If P, then Q
P2. Not P
C. Therefore, not Q
Since P was never asserted as the only sufficient condition for Q, other factors could account for Q. Therefore, the argument is deductively invalid.
P2. Queen Elizabeth is not an American citizen
C. Therefore, Queen Elizabeth is not a human being
With this example, both premises are true statements yet the conclusion is false. This of course is due to the fact that being an American citizen is not the only sufficient condition for being a human being.