Cross posted from Overcoming Bias. Comments there.

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Warning: this post is technical.

Suppose you know that there are a certain number of planets, N. You are unsure about the truth of a statement Q. If Q is true, you put a high probability on life forming on a given arbitrary planet. If Q is false, you put a low probability on this. You have a prior probability for Q. So far you have not taken into account your observation that the planet you are on has life. How do you update on this evidence, to get a posterior probability for Q? Since you don’t know which is ‘this’ planet, with respect to the model, you can’t update directly on ‘there is life on this planet’, by excluding worlds where this planet doesn’t have life. And you can’t necessarily treat ‘this’ as an arbitrary planet, since you wouldn’t have seen it if it didn’t have life.

I have an ongoing disagreement with an associate who suggests that you should take ‘this planet has life’ into account by conditioning on ‘there exists a planet with life’. That is,

P(Q|there is life on this planet) = P(Q|there exists a planet with life).

Here I shall explain my disagreement.

Nick Bostrom argues persuasively that much science would be impossible if we treated ‘I observe X’ as ‘someone observes X’. This is basically because in a big world of scientists making measurements, at some point somebody will make most mistaken measurements. So if all you know when you measure the temperature of a solution to be 15 degrees is that you are not in a world where nobody ever measures its temperature to be 15 degrees, this doesn’t tell you much about the temperature.

You can add other apparently irrelevant observations you make at the same time – e.g. that the table is blue chipboard – in order to make your total observations less likely to arise once in a given world (at its limit, this is the suggestion of FNC). However it seems implausible that you should make different inferences from taking a measurement when you can also see a detailed but irrelevant picture at the same time than those you make with limited sensory input. Also the same problem re-emerges if the universe is supposed to be larger. Given that the universe is thought to be very, very large, this is a problem. Not to mention, it seems implausible that the size of the universe should greatly affect probabilistic judgements made about entities which are close to independent from most of the universe.

So I think Bostrom’s case is good. However I’m not completely comfortable arguing from the acceptability of something that we do (science) back to the truth of the principles that justify it. So I’d like to make another case against taking ‘this planet has life’ as equivalent evidence to ‘there exists a planet with life’.

Evidence is what excludes possibilities. Seeing the sun shining is evidence against rain, because it excludes the possible worlds where the sky is grey, which include most of those where it is raining. Seeing a picture of the sun shining is not much evidence against rain, because it excludes worlds where you don’t see such a picture, which are about as likely to be rainy or sunny as those that remain are.

Receiving the evidence ‘there exists a planet with life’ means excluding all worlds where all planets are lifeless, and not excluding any other worlds. At first glance, this must be different from ‘this planet has life’. Take any possible world where some other planet has life, and this planet has no life. ‘There exists a planet with life’ doesn’t exclude that world, while ‘this planet has life’ does. Therefore they are different evidence.

At this point however, note that the planets in the model have no distinguishing characteristics. How do we even decide which planet is ‘this planet’ in another possible world? There needs to be some kind of mapping between planets in each world, saying which planet in world A corresponds to which planet in world B, etc. As far as I can tell, any mapping will do, as long as a given planet in one possible world maps to at most one planet in another possible world. This mapping is basically a definition choice.

So suppose we use a mapping where in every possible world where at least one planet has life, ‘this planet’ corresponds to one of the planets that has life. See the below image.

Squares are possible worlds, each with two planets. Pink planets have life, blue do not. Define ‘this planet’ as the circled one in each case. Learning that there is life on this planet is equal to learning that there is life on some planet.

Now learning that there exists a planet with life is the same as learning that this planet has life. Both exclude the far righthand possible world, and none of the other possible worlds. What’s more, since we can change the probability distribution we end up with, just by redefining which planets are ‘the same planet’ across worlds, indexical evidence such as ‘this planet has life’ must be horseshit.

Actually the last paragraph was false. If in every possible world which contains life, you pick one of the planets with life to be ‘this planet’, you can no longer know whether you are in ‘this planet’. From your observations alone, you could be on the other planet, which only has life when both planets do. The one that is not circled in each of the above worlds. Whichever planet you are on, you know that there exists a planet with life. But because there’s some probability of you being on the planet which only rarely has life, you have more information than that. Redefining which planet was which didn’t change that.

Perhaps a different definition of ‘this planet’ would get what my associate wants? The problem with the last was that it no longer necessarily included the planet we are on. So what about we define ‘this planet’ to be the one you are on, plus a life-containing planet in all of the other possible worlds that contain at least one life-containing planet. A strange, half-indexical definition, but why not? One thing remains to be specified – which is ‘this’ planet when you don’t exist? Let’s say it is chosen randomly.

Now is learning that ‘this planet’ has life any different from learning that some planet has life? Yes. Now again there are cases where some planet has life, but it’s not the one you are on. This is because the definition only picks out planets with life across other possible worlds, not this one. In this one, ‘this planet’ refers to the one you are on. If you don’t exist, this planet may not have life. Even if there are other planets that do. So again, ‘this planet has life’ gives more information than ‘there exists a planet with life’.

You either have to accept that someone else might exist when you do not, or you have to define ‘yourself’ as something that always exists, in which case you no longer know whether you are ‘yourself’. Either way, changing definitions doesn’t change the evidence. Observing that you are alive tells you more than learning that ‘someone is alive’.