Why did medieval Europeans believe having lice was a sign you were healthy?

Medieval Europeans noticed something real: there were rarely any lice on sick people. From that honest observation, they drew a confident but completely backwards conclusion: that lice were good for you, and that a person got sick precisely because their lice had abandoned them. The truth runs the other way entirely. Lice are extremely sensitive to body temperature, and a small rise in temperature, like the one that comes with a developing fever, sends them looking for another host. The medical thermometer didn't exist yet, so nobody could detect that early temperature rise directly; by the time obvious symptoms of illness appeared, the lice had already left, making it look like their departure had caused the sickness rather than merely preceded it.

This is the logical trap known as mistaking correlation for causation: two things happening together doesn't tell you which one, if either, is causing the other. Sometimes the arrow of causation simply points the opposite way from what seems obvious. Faster-spinning windmills are correlated with stronger wind, but wind makes windmills spin, not the reverse; wind exists perfectly well without any windmills around and existed long before anyone invented one.

More often, a hidden third factor causes both things you're looking at, which is called the third-cause fallacy. Sleeping with your shoes on is genuinely correlated with waking up with a headache, but the real cause of both is usually a third thing: going to bed drunk. A famous 1999 study linked children sleeping with a nightlight to developing nearsightedness later on, and it made headlines, but a follow-up study found the actual cause was nearsighted parents, whose children were more likely to become nearsighted and who were also more likely to leave a light on in their kids' room. Ice cream sales and drowning deaths rise together every year for the same reason: hot summer weather drives people to buy more ice cream and to go swimming more, not because ice cream causes drowning.

Sometimes two things line up with no causal link at all, just coincidence: an NFL team's last home game before a US presidential election correctly predicted the outcome of every single election from 1936 to 2000, a streak that had absolutely nothing to do with politics and finally broke in 2004.