One in sixteen

About this video

Kai asks to stay up until nine to watch a film. His big sister objects: if he stays up till nine tonight, tomorrow he'll want ten; then midnight; then he'll be too tired for school; then he'll fail all his exams. So if Kai stays up till nine, he'll fail his exams. Their dad laughs. But why, exactly, is it silly?

Kai's sister has built a chain of ifs, each one starting where the last ended. The shape is valid: if the first thing always leads to the second and the second always to the third, the first always leads to the third. Logicians call this the chain rule. The trouble is in the links. Each one is only maybe: one late night doesn't mean he will always want a later one, and Dad can say no. And maybes add up. If each of the four links were a fair coin toss, the whole chain would come true only about 1 time in 16. This animation shows the chance shrinking link by link.

An argument that one small step will lead, link by link, to something terrible, when the links are weak, is called a slippery slope. Not every slope is slippery: a small leak in a roof lets rain in, the wood rots, the roof falls, and each link is what usually happens. Check each link on its own. This video goes with The Slippery Slope lesson of Logic Level 2 on Knaph.

Transcript

1. 0:00If Kai stays up till nine tonight, he'll fail his exams.
2. 0:05Nine, then ten, then midnight, too tired, failed exams. A chain of ifs.
3. 0:12The shape is valid. So what's wrong?
4. 0:17Each link is only maybe. Half, a quarter, an eighth. Four maybes, one in sixteen.
5. 0:26Not every slope is slippery. Check each link.
6. 0:31A chain is never stronger than its links.