The Extremely Simple Explanation of the Self-Indication Assumption

Why SIA doesn't depend on any theory of personal identity and is correct

The self-indication assumption (SIA) is the core engine of the anthropic argument for God’s existence. SIA holds that your present existence is rendered more probable by theories on which there are more candidates for being your present self. Suppose a coin is flipped that creates one person if heads and ten if tails. You are created as a result of this coin flip. The self-indication assumption holds: you ought to think tails is ten times likelier than heads. The coin landing tails will lead to ten times more people existing; it’s thus ten times likelier it will lead to you existing.

But what is this funny business about “candidates for being your present self”? Various critics have alleged that this term is polysemous or vague and that the core dynamics of the theory are underspecified. This isn’t true. But to explain why, I will need to introduce a bit of terminology.

Relative probability refers to how probable one thing is compared to another. You can know the relative probability of each of two things, even without knowing their absolute probability. You can know that one thing is half as likely as another even without knowing how likely either of them is.

A “center” refers to a person, at a time, in a world. A center will specify an agent, a time, and a world. So, for instance, take the famous Sleeping Beauty problem. Beauty is put to sleep on Sunday. A coin is flipped. If it comes up heads, she wakes up once—just on Monday. If it comes up tails, she wakes up once on Monday and once on Tuesday, each time with no memory of any previous days. The coin coming up tails begets two centers (one for each day she is awoken), while the coin coming up heads begets one.

SIA in a sentence: the relative probability that you’re some center doesn’t depend on the presence of other centers. Put more plainly, suppose you want to compare how likely it is that you’re each of two people at times relative to one another. SIA instructs you: only look at those two people at times. All the other people at times can be safely ignored.

Note that when I talk about centers here, I’m not requiring that we’re given some maximally precise specification of the world. SIA tells you about the world even when you don’t know everything true about it. The word “center” is compatible with any description of agents or the world, as I’m using it—it doesn’t require that the description is maximally precise. If you don’t like this looser use of the word center, feel free to sub in your own word (like partial center).

The story is the same for times. If you want to figure out the relative odds you’re awake Monday vs. Tuesday, SIA instructs you to ignore Wednesday. SIA can be used to figure out the odds it’s Monday, or the first half of Monday, or the block containing Monday and Tuesday. It doesn’t just apply to arbitrarily small slices of time. It can apply to any length of time.

Once we understand what SIA says, we can see why SIA favors theories on which there are more people. If each person occupies a bit of the probability space—despite their many compatriots—then theories on which there are more total centers will get more probability. The more centers there are, the more probability they’ll take up.

For example, suppose a coin is flipped. If it comes up heads, person 1 is created. If it comes up tails, person 2 is created and person 3 is created. You’re created but you don’t know which person you are. Following SIA, you reason:

1. The odds I’m person 1=the odds I’m person 2. After all, when comparing these relative odds, SIA says: ignore person 3. Well, if person 3 didn’t exist, then clearly it would be just as likely that you’re person 1 as person 2.

2. The odds I’m person 1=the odds I’m person 3. Once again, when comparing these relative odds, SIA says: ignore person 2. Well, if person 2 didn’t exist, then clearly it would be just as likely that you’re person 1 as person 3.

From these it follows that tails is twice as likely as heads. If you’re person 2 or 3, then the coin definitely came up tails. If you’re person 1, it definitely came up heads. All three of those events are equally probable. Two of the events entail that the coin came up tails. Thus, tails is twice as likely as heads.

Now, we’ve so far talked about “centers” rather than people. Here is why: SIA isn’t about people. It’s about centers—slices of people at times. To see this, consider the Sleeping Beauty problem. A coin is flipped Sunday. If it comes up heads, you wake up once on Monday. If it comes up tails, you wake up once on Monday, then your memory is wiped, and you’re woken up a second time on Tuesday. Put another way, if the coin comes up heads, you wake up once with no memories of prior days. If it comes up tails, you wake up twice with no memories of prior days. After waking up with no memories of prior days, how likely should you think it is that the coin came up heads?

Applying SIA, we reason:

1. The probability I’m presently the first wake up and the coin came up heads=the probability I’m presently the first wake up and the coin came up tails. After all, if we were to ignore the second wakeup that would happen if the coin came up tails, then obviously heads and tails would be equally likely. SIA instructs you: do that, ignore other centers, only consider the two centers you’re comparing when assessing their relative odds.

2. The probability I’m presently the first wake up and the coin came up heads=the probability I’m presently the second wake up and the coin came up tails. Same idea. If the tails coin flip only woke you up on Tuesday, then obviously heads and tails would be equally likely.

Thus, tails is twice as likely as heads. Each of the following events is equally probable: the coin came up heads and it’s day one, the coin came up tails and it’s day one, and the coin came up tails and it’s day two.

Let’s apply this same logic to infinity. A 100-sided die is rolled. If it comes up 1-99, one person is created. If it comes up 100, infinity people are created. Following SIA, we reason:

1. The probability I’m the first person and one person was created is 99 times greater than the probability I’m the first person and infinity people were created. After all, this is what the relative probability would be if we ignored all the other people.

2. The probability I’m the first person and one person was created is 99 times greater than the probability I’m the second person and infinity people were created. Same idea.

3. The probability I’m the first person and one person was created is 99 times greater than the probability I’m the third person and infinity people were created.

…through ∞.

Thus, we have infinity events each 1/99th as likely as the die having come up 1-99 and me being the first person. Because there are infinity of them, they gobble up all the probability space. The probability of the die coming up 1-99 falls to zero. This is not, of course, to suggest that an actual agent should be infinitely certain that there are infinite people. You should think there’s some chance that you’re mistaken about anthropics. But on the most plausible theory about how to do anthropic reasoning, your existence gives you infinitely strong evidence that there are infinite centers.

Thus, you can derive SIA from a very modest principle about how to assign probabilities. You don’t need to think that we’re all souls drawn from some great cosmic soul bank. That has never been what SIA presupposed. Similarly, questions about personal identity and origin essentialism are beside the point—there are people that exist at times, and SIA tells you how to assign probabilities to you being each.

Once one sees that SIA follows from this principle, this gives powerful reason to think SIA is right. When analyzing the relative odds that you’re each of two centers, why would the presence of other centers matter? It makes no difference to which one you are. Why would the relative odds I’m Bob vs James depend on whether or not Fred exists? Fred’s existence can make no difference to whether I’m Bob vs James.

It is an advantage of SIA that it falls out of some principled probabilistic principle, rather than arising solely from a gerrymandered attempt to fit our anthropic intuitions. SIA’s rivals are not this way. SSA, the most plausible rival of SIA, depends on positing a totally ad hoc reference class with no principled basis. Compartmentalized conditionalization has a principled—if very metaphysically contentious—basis, but it implies near certainty that you’re a Boltzmann brain and other absurdities. Minimal reference class SSA inherits the same defects as CC.

We can also see, using this method, why SIA gives you strong evidence that there are infinite people. Suppose there are two theories of physics. One of them says there are infinite people. The other says there are finite people. Let’s say they have equal prior probability. Because the infinite theory holds that there are infinite centers that you might presently be, it gobbles up all of the probability—leaving nothing left for the finite theory.

At this point, there are a few remaining questions you might have.

First, does it make any sense to talk about the probability that the present you is one person vs. another person? Yes (and see here for my longer piece about it). Suppose there are seven copies of me created. Six will be tortured one hour from now. The seventh will not. One can perfectly sensibly wonder about the probability that I will be tortured in an hour. One can assign probabilities to such a thing. The probability here ought to be 6/7.

Second, here I’ve talked about relative probabilities. But SIA is supposed to give you absolute probabilities as well. Absolute probabilities are obtained by normalizing the relative probabilities so that they sum to one and then deriving the probabilities of the individual events. For example, in Sleeping Beauty, you know that tails is twice as likely as heads. The only possible events are heads and tails. Tails thus gets 2/3 of the overall probability mass, heads gets 1/3. This process is called renormalization.

Third, does SIA run afoul of the constraint that you can’t assign a uniform probability distribution over an infinite set? No more than anyone else. Following SIA, you can sensibly hold that, in the above case, the probability that you’re any particular person is zero. You can also hold that the probability is infinitesimal (though you’ll probably have to give up countable additivity). But even if you have no probability directed at being any particular person, 100% of your probability mass still can—and should—be directed at you being one of the infinite people created by the tails coin flip.

Everyone will have to assign probabilities in some way when there are infinite people, or else admit that, if the physicists who suggest the cosmos is infinite are correct, we’re in a skeptical scenario. However the non-SIAer does this, the SIAer can as well.

The consideration I cite in this article strikes me as reasonably forceful. It may not be enough on its own to tell you which anthropic theory to adopt, but it at least illustrates one major reason to find SIA plausible. Contrary to what people often say, SIA depends on neither bizarre metaphysics nor tendentious theories of personal identity. It merely depends on the notion that the relative odds that you’re each of two person-moments doesn’t depend on other person moments. Surely that should be the default assumption.

Then, when one thinks more about anthropics and notices that alternative views imply myriad independent bizarre absurdities, they should simply think that SIA is the right view.