Excluded middle frames and when to doubt them

by Collin Lysford

In mathematics, you have to write down things even when they are extremely obvious. So, we have the Law of the Excluded Middle, which states that for every proposition, either it’s true or it’s negation is true. If you say I’m going to wear a silly hat today!”, either you will wear a silly hat today or you will not wear a silly hat today; exactly one of those statements will be true. It’s really that straightforward, there’s no trick here. But like many things that are extremely obvious, the Law of the Excluded Middle is often a bad way to think about things.

Scott Alexander has collected a number of these examples in his article Heuristics That Almost Always Work. A security guard hears a noise: it’s [a robber and needs intervention] or [not a robber and does not need intervention]. The queen of a volcanic island notices the lava looking a little weird: it’s [an eruption that requires immediate evacuation] or it’s [not an eruption that requires immediate evacuation]. These are excluded middle frames - you’re asked to compress all of possibility space into it happened” or it didn’t”, and evaluate accordingly. For the cases listed, the first thing almost never happens, so people always say no intervention is needed” are almost always right”, even if it also means there’s no real point to having a person in the loop instead of a rock that just says nothing abnormal ever happens”.

Scott calls these heuristics that almost always work”, but I think that’s the wrong way to look at these examples. The idea that these heuristics are working” is a product of the excluded middle frame. Each decision is presented as an individual yes/no choice, correctness is making the correct yes/no choice, and then you tally them up. But the obvious implication of Scott’s post is that this is a silly way to think about risk. It’s not a question about heuristics; it’s about the framing of the problem.

For whatever reason, absence of evidence is not evidence of absence” is a statement people will be comfortable reciting but tend not to recognize in the wild. If you hold to an excluded middle frame when the outcomes are lopsided, then not only is absence of evidence being treated as evidence of absence (it wasn’t an eruption! That’s evidence against eruptions!), but it’s basically the only sort of evidence you get until it’s too late. The correct analysis isn’t a matter of making predictions for individual phenomena on individual days; it’s to look at the mechanics of how the thing can become non-normal. It might be useful when developing that theory to look at cases of protracted normality to contrast them with non-normal cases and see if you can find a necessary condition to reach non-normality. But what you’re not doing is making a bet on every single situation for whether it will be normal or not and tallying up your score. So the heuristic situations that are usually normal will always be normal” might get you the high score in a betting pool, but all that shows is that a betting pool is not what serious players in search of understanding will be looking at.

This is why I’m generally not a fan of prediction markets. Excluded middle frames work okay when events frequently fall on both sides of the frame - if you have lots of days with robberies and lots of days of non-robberies, then someone who correctly predicts which days are which is probably on to something. Alternatively, rare events can still be priced in” by the house for things like sports or horse races that are symmetric contests - absence of evidence that horse A is faster than horse B really is evidence of absence. But the more lopsided the probabilities are, and the more different the mechanics of the non-normal state are from the normal state, the less useful the excluded middle frame becomes. If you want to understand rare events, you’re not going to get there by counting up how often they don’t happen.