There’s a famous story about swans that Nasim Taleb used for the title of his recent book. European zoologists had seen swans, and all the swans they had seen had white feathers, so they said that all the swans in the world were in fact white. Then a European went to Australia and saw swans with black feathers. Taleb’s point is that no matter how much we know, we don’t know what we don’t know, and overconfidence in our knowledge can make us rigid and vulnerable.
Sentences like “There are swans,” or “there are black swans” are what logicians call existential statements. “All swans are white” is a universal statement. In science, the two are very different and require different kinds of evidence.
All it takes to make an existential statement is a single observation. I know that there are swans because I’ve seen at least one. But to say that all swans are white with certainty requires us to have seen every swan. The European zoologists made that universal statement without universal observations, and all it took was one observation of a black swan to prove them wrong.
Now here’s a point that seems to be missed: averages are universal statements. Actually, they’re made with some relatively simple arithmetic applied to data from all members of a category, so they entail universal statements. If the Regal Swan Foundation says “Mute Swans’ wingspans average 78 inches,” we take that to mean someone measured all the mute swans (or a representative sample; I’ll get into that in another post). If someone says that and then you find out they only measured swans in their local park, you’d feel deceived, wouldn’t you?
Here’s another one: percentages are universal statements. How can it be a universal statement to say that ten percent of people are left-handed? Well, you have to look at all of them to know for sure that the rest are not. And the same thing goes for “most,” “the majority” and other words that entail statements about fractions. I think this is why Robyn Hitchcock said, “the silent majority is the crime of the century”: if the silent majority really is silent, there’s no way for the person making the claim to know what they think.
On the other hand, statements about units and fractions of units are only existential statements. I have two cats, and I ate half a sausage roll for dinner last night. Words like “many,” “some” and “a few” are also existential statements. I’ve seen many white swans, but I claim no knowledge of the ones I haven’t seen. The only way you could falsify that statement is by process of elimination: showing that there aren’t enough swans in my experience to count as “many.”
So the bottom line is that in order to make a universal statement, including a percentage or an average, you need to have looked at all the members of the group. But in order to make an existential statement, you only need one.
Now here’s why I’m writing this post: this stuff sounds simple, and I feel like I’m writing it for kids, but there are a lot of people who don’t follow it. Either they’ve never been taught about the standards of evidence for universals, or they’ve been taught to ignore them. Most of the times I’ve read “all,” people seem to get that it’s a universal and don’t say it unless they can reasonably claim to have data from all of the things concerned. But a lot of people have trouble with “most” and averages, and especially percentages. You can’t say “most” unless you’ve seen them all.
Well, there actually is a way you can say “most” without seeing them all. It’s called induction, but it’s hard to do, harder than a lot of people seem to think. I’ll talk about that in a future post.
(Update, August 19, 2015: I liked the black swan drawing above so much I decided that it should exist on T-shirts. And you too can call into existence T-shirts, throw pillows, travel mugs – all with the Existential Black Swan on them! I’ll get a cut of the money. Click here for details.)