only bad witches are ugly

The title of this entry is a famous quote from The Wizard Of Oz. Yes, I realize it's awful to spread the idea that goodness is correlated with beauty. But that's not today's topic.

Today's topic is the logic of sets. When a witch says "only bad witches are ugly", which meaning is she communicating in regard to which witches are in which sets? She means that for any witch in the "ugly" set, that witch is also in the set of "bad". This implies that the ugly witches are a subset of the bad witches. Membership in the ugly set is always accompanied by membership in the bad set. Also, any witch that is not bad (good) is therefore not ugly (beautiful); for they could only be ugly if they were also bad.

But the more notable distinction is what the witch isn't claiming about these sets. The witch said nothing certain about whether all beautiful witches are good. All she told us is that each ugly witch is bad. That doesn't contradict the possibility of a beautiful witch who is also bad. Not all the bad witches are necessarily ugly. It's like a deck of playing cards. Each heart is a red card. But that doesn't mean all the red cards are hearts.

In related news, Oz the Great and Powerful has some slow-paced sections that might cause theater-goers' thoughts to wander.