So, I was explaining some logic ideas to a guy on IRC, and he was struggling with the contrapositive rule, which says "if A implies B, then not-B implies not-A." He asked me what this looked like as a set diagram.

I began to teach him how "A implies B" means that whenever you have A, you know you have B. The picture is that B can be bigger, but the situations with A are contained within it, so the picture looks like this: "A is a subset of B." (In symbols, A → B becomes A ⊆ B.)

Now you need to understand that "not A" is the entire region outside the circle of A, and "not B" is the entire region outside the circle of B. And then you have to understand a crazy perspective: that the entire space outside B is now a subset of the entire space outside A.

He was very confused about this, so I explained it this way: There is a story about a physicist, an engineer, a mathematician, and a farmer. The farmer asks all three for a fence containing the largest space for his sheep.

The engineer is first up, builds a square fence using one of the walls of the barn to get a little extra space fenced in. The farmer seems pretty satisfied with this, so they both go to meet the physicist, who has tethered a cord to a peg and is drawing a large circle. "Circles," he points out, "minimize their surface-to-area amount. Actually, I could probably do the same with your barn there, get you a little more space by chopping a chord out of the circle." The farmer says "no, this looks like it will take too long."

They both come over to see the mathematician, who has apparently gotten tangled up in the fence! They start to work to get him out of there -- the farmer asks, "what were you thinking, why did you bend the fence this way, what is wrong with you?!" The mathematician says, "you don't understand -- this is the outside of this fence!"

Suddenly, the idea of flipping "inside" and "outside" seemed to make sense, and I was able to show him that yes, if you take this perspective, the corresponding rule is Bᶜ ⊆ Aᶜ, thus not-B → not-A.

(Another strategy which often works is to leverage moral intuitions, but 'permissions' and 'implications' are opposite arrows. So A → B means "if I know A, then I also know B." Apply this to "Santa only gives presents to good children". Your intuitions all will work much better, save one: you probably would write the above statement as "good child => get presents", following what is permissible for good children, but in logic it actually states "got present → good child", if Santa gave a present then you know that the child was good, but Santa might not give a good child a present, especially if that child is, say, Jewish.)