I know. In fact, at this point, I understand that I’m basically wrong. But there’s something about this penguin that very slightly bugs me.
Say I impose an entirely reasonable burden of proof on Riemann. I’m happy with a quick look around. We go to his house, looking for penguins in the living room. He opens the door to the room, sees no penguin. He checks under the table. He checks behind the sofa. He checks behind the curtain. No penguins. He looks at me with a very tired expression, and says: “Nada. No Antarctic avian wildlife. Can we go?”
Meanwhile, I’ve been looking at my phone, and therefore not paying close attention to events in the room. I say: “Fine.” But then something catches my eye. Under the table. That’s unmistakably a penguin! And I go: "Hey! Dude, that’s a penguin right there!"
I grab the bird, and stuff it my backpack for further study. We return to the pub from whence we came. Our friends ask: “Did you find anything?” I say: “Well, **Riemann **says no. Personally, I’m not sure, philosophically speaking. But there’s this.” Blammo. From my backpack, I produce the penguin.
This doesn’t seem to work in reverse. Say it plays like this instead: We go to Riemann’s’s house. We open the door to the living room. And there’s the penguin, sitting on the sofa, eating Cheetos and watching the Discovery Channel. Well, game over. **Riemann **can’t go: “Oh, yeah? Well, look over there! In the corner! I see a non-penguin!”
But nobody is denying this part. To go back to my first reply to the original post:
When an exhaustive search becomes genuinely difficult, then the truth of the negative assertion will certainly be in doubt.
But that has nothing to do with the imposition of an unreasonable burden of proof in the negative case but not the positive, or with the notion that for some reason “penguin” must be defined with exquisite precision in the negative case but not the positive.
Well, you’re the one who brought up both of those things. All I have is a penguin. Look, say we leave the house without me noticing the bird. We have proved that there are no penguins in your living room, by our agreed standards. Right? I give you $1000.
But later, you go home. Now the penguin is sitting on the sofa, watching TV. Should you give me my money back? We proved your lack of a live-in penguin, right?
And the proof is fine by me. Except for the minor detail of the gosh-darned penguin.
You seem to be claiming that there is never such a thing as an exhaustive search, however limited in scope the negative claim is. That does not seem reasonable to me.
Yes, there is a legitimate asymmetry here, but it’s based on the nature of the evidence combined with the prior.
Suppose instead of pointing to the non-penguin in the corner, Riemann walks over and passes his hand through the penguin on the coach and his hand passes straight through. “Dude, it’s just my sister’s penguin hologram. I thought I told you about that?”
The reason that the penguin on the sofa watching Discovery channel pretty much seals the deal is that human observation is generally pretty good. We don’t have any prior expectation that a hologram would be good enough to fool us. If we did have that prior expectation then it would be a different story. Just seeing the penguin wouldn’t be enough. Is the penguin doing what we would normally expect penguins to do (like watching Discovery channel) or is it doing something we would not typically expect from penguins (like doing the macarena)? This relates to the idea of hearing hoofbeats in the US, as opposed to Africa. Think horse, not zebra.
Or another case: imagine a million bowls spread out in front of you, with a mad scientist telling you that he put the key to your shackles under one of those million. (There are computer science problems like this.) It takes effort to check under each bowl. You have to walk up to it, lift it up, look on the ground, then make sure the key’s not taped to the inside of the bowl. But if the mad scientist’s assistant tells you, “Psst, check under #347,582”, then you can do that very quickly. This is to say that finding the right answer is much more difficult problem than confirming that any given guess is the right answer. That’s the same sort of asymmetry.
Why?
Well, because supposing that the mad scientist is telling the truth that the key exists, then your prior probability of finding the key from a random first choice is one in a million. Looking under the second? One in 999,999. (The first place you looked has been ruled out.) Then the chance it’ll be under the next will be one in 999,998. And so on. What you’re doing here is the same thing as: here’s a non-penguin, and here’s a non-penguin, and here’s a non-penguin, etc. These are all legitimate pieces of evidence. But there are SO MANY PLACES that the key could be that each definitive piece of evidence (“Look! There’s a non-penguin!”) isn’t particularly informative. It doesn’t tell you much, unless you start with only two bowls. You have extremely little prior probability mass located in each of the million points, so each time you find a non-penguin under each bowl, that tiny bit of probably mass gets distributed to the other remaining choices. Yes, you are learning with every non-penguin… but you’re not learning much. Contrast that with the existence of the penguin. The assistant tells you to look under a certain bowl, you do, you find the key, and you use it to unshackle yourself. The end.
As soon as you find the key, you can infer the “existence” of a non-key under every other bowl. “Look, a non-penguin” doesn’t make much sense in that context because it’s not informative at all, not even as much as the one-in-a-million informativeness you got from looking under the first bowl. You expect a non-penguin in the corner when you’ve finally found the penguin on the sofa, just as you expect a non-key under every other bowl after you’ve lucked into finding the key under the right bowl early on. Yelling about a non-key after you’ve found the key isn’t helpful, isn’t informative.
This is a legitimate asymmetry. Not seeing the key under each of a million bowls is not very informative, whereas finding the key is perfectly informative. That’s inherent in the nature of the problem, and in most problems of the sort. The world is a big fucking place, and it’s quite easy to hide a single penguin somewhere because the search space is so big that finding non-penguins is not particularly helpful, when you consider all of the many, many, many other places that it could be. But if we want to be purely scrupulous and fair, we still have to admit that each non-penguin we find is at least some teeny, tiny evidence that there is no penguin, especially if the existence of a penguin implies other pieces of related evidence that we should be able to find.
Well, I’m not sure. You can do an infinite search, which is obviously not feasible. Or you can do a quick look around, in which case you might miss the penguin. An exhaustive search, then, must be some sweet spot between those. But how can you know when you’ve hit that sweet spot?
Don’t get me wrong: I certainly think that your hypothesis that there are no penguins in your living room is a very good hypothesis. It’s eminently falsifiable, which is good. A single observed penguin will disprove it. Assuming that you have lived in your house with no such observations happening for a reasonable amount of time, I would say that it has stood up to scrutiny and repeated testing very well indeed. I would think that the chances of a penguin suddenly appearing are very low, bordering on zero. I would happily take this hypothesis, incorporate it into a wider theoretical framework, and use that for understanding the universe.
My hypothesis that there *is *a penguin there, on the other hand, is clearly baloney. It’s not even falsifiable, since I’ll never agree that you have proved that there is no penguin. Basically, it’s nonsense. Even if a penguin does turn up, it’s still nonsense, and I just happened to get extremely lucky. And a lucky loon is still a loon.
So it’s fine if you can’t prove the lack of penguins. According to Popper, you’re not supposed to. You’re supposed to attempt to disprove it, and then trust your hypothesis more as you fail to do that.
All I’m asking is: If you do happen upon a penguin in your living room when you return from the pub tonight, could I get my money back? I was extremely drunk when I made that bet, and I could really use it to pay the rent.