The Ugly Witch Problem

This is a flaw in reasoning which I have not seen discussed much, but I wanted to have a resource available to explain it to people. Feel free to link this page yourself if you don't want to be bothered explaning it over and over!

I did not name this problem, but I can't recall where I was first introduced to it. The name comes from The Wizard of Oz film (1939). The explanation may spoil the plot. When Dorothy (Judy Garland) lands in Oz she is asked by Glinda the Good Witch (Billie Burke) if she is a good witch or a bad witch. Dorothy protests that she is not a witch at all and she explains that, in her worldview, witches are ugly. She explains this with the implication that all witches are ugly. Glinda explains that only bad witches are ugly. She then asks Dorothy again if she is a good witch or a bad witch.

Now, if only bad witches are ugly why might she still need to ask? Dorothy is by no description ugly. However, to assume all bad witches are ugly would be a flaw in reasoning. Although all ugly witches are bad this should not imply that all bad witches are ugly.

We can think of this more abstract that a subset of a group can not be used to describe the greater accurately group in broad terms. If all 𝒷 are 𝒶 then all 𝒶 are not necessarily 𝒷. This could also be thought of in terms that if some 𝒶 are 𝒷 it should not be used to imply that all 𝒷 are 𝒶.