To his very slight credit, Eugene was savvy enough not to claim that he was a medical doctor, but those PhDs in science that he made up to impress Negan may have been what convinced Negan that it was perfectly okay to make an example of Dr. Emmett Carson.
Negan may actually discover that killing Dr. Carson was a massive mistake.
He could definitely kill off the good doctor if there was a replacement scientist waiting in the wings clutching a pickle jar, right?
So, all in all, it was just another fun episode of happy adventures at The Sanctuary.