The Self-Indication Assumption: Answering an Objection

Scott Alexander vs. Bentham's Bulldog

Bentham’s Bulldog argues that the best argument for God is based on the Self Indication Assumption (SIA). The argument is that the fact that you exist gives you evidence that we live in a universe with many people rather than few people. Since it is good for people to exist and God is good, God would create a universe with many people. Therefore, the fact that you exist greatly strengthens the hypothesis that God exists and greatly weakens Naturalism. The argument is a more rigorous version of the folk argument: “There is something rather than nothing. There is life rather than just rocks and gas. These things are good. Therefore a good God probably exists.” If you’re confused, go read BB’s piece.

In reply, Scott Alexander posed the following challenge: “Suppose I notice I am a human on Earth in America. I consider two hypotheses. One is that everything is as it seems. The other is that there is a vast conspiracy to hide the fact that America is much bigger than I think - it actually contains one trillion trillion people. It seems like SIA should prefer the conspiracy theory (if the conspiracy is too implausible, just increase the posited number of people until it cancels out.)”

There is an answer to Scott’s challenge. The SIA does not require you to believe that conspirators are hiding the true population of America.

1. Normal Conspiracies vs. Truman Show Conspiracies

Let’s first distinguish between two kinds of conspiracy hypotheses, normal conspiracy hypotheses and Truman Show conspiracy hypotheses.

A Normal Conspiracy Hypothesis posits that the “facts” you think you know about some particular topic are actually mistaken beliefs that some conspirators tricked you into believing.

Example: the hypothesis that George Bush and his associates secretly orchestrated the 9/11 attacks. We think we know that those attacks were perpetrated by Islamic fundamentalists, but the evidence that led us to that conclusion was actually planted by the devious Mr. Bush.

A Truman Show Conspiracy Hypothesis posits that almost everything you think you know about the world - or at least a great amount of what you think you know - are actually mistaken beliefs that some conspirators tricked you into believing.

Example: The film The Truman Show. The premise of the film is that unbeknownst to Truman, his whole life is a TV show. Everyone in his town except him is an actor. In fact, his town is inside a giant TV set. Every book he has read was a prop placed in the set.

If you are the victim of a Truman Show Conspiracy, then everything you think you know about America is probably wrong. Maybe there is really no such place as America, or maybe the place called “America” is so different from what you think that it does not really match up with your concept of America. If the conspirators made fake science books for you to read, then everything you think you know about the world and the Universe is probably wrong as well.

2. Ways in which the true population of America could be hidden

2.1 Statistical Error

Before considering conspiracy hypotheses, let’s consider a more modest hypothesis. Maybe the official estimate of the American population is slightly inaccurate just because of statistical error. Suppose the official estimate of the American population is 300 million with a standard deviation of 1 million. To simplify things, let’s collapse the probability distribution into just three cases, 299 million (prob=16%), 300 million (prob=68%), and 301 million (prob=16%). Under these assumptions the expected number of people in America is

\((0.16 * 299 million) + (0.68 * 300 million) + (0.16 * 301 million) = 300 million\)

However, if we add the SIA to our calculation we should expect the prior probability of the 301 million case to be slightly higher than the prior probability of the 300 million case which should have a slightly higher prior probability than the 299 million case. With the SIA the expected number of people in America would (I think?) be given by this formula

\((0.16 * \frac{299}{300} * 299 million) + (0.68 * \frac{300}{300} * 300 million) + (0.16 * \frac{301}{300} * 301 million) = 300.001 million\)

So in cases where statistical error is a possibility (which is all the time) the SIA requires you to believe that the the true population is higher than the officially estimated population, but only by a very small amount. In this case, we got an estimate under SIA which is only 1/1000 SD higher than the non-SIA estimate, which is a trivial difference.

If it seems strange to shade official population estimates based on the evidence of your existence, consider how you would react if the official estimate of the American population was zero. You would know that that estimate must be wrong because you know that you exist.

2.2 Normal Conspiracies

Suppose that the officially reported population of America is 300 million but the actual population is 600 million. Suppose there is a conspiracy of statisticians and government officials who suppress knowledge of America’s true population. The conspiracy hypothesis posits twice as many people as the non-conspiracy hypothesis so according the SIA we would conclude that the probability of the conspiracy hypothesis was 2 thirds if we ignored base rates. But we should not ignore base rates. We have strong prior evidence that conspiracies of this kind are very unlikely, so even though the 600 million Americans hypothesis is strengthened by the SIA, its probability remains very low. The far more probable case is that the population is 300 million as officially reported.

2.3 Truman Show Conspiracies

One popular conspiracy theory is that George Bush participated in a conspiracy to orchestrate the 9/11 attacks. While this theory is very unlikely, it is not impossible. Politicians are aware that tragedies increase their approval ratings and give them political capital, and it is well known that agencies like the CIA engage in secret and morally dubious activities. The probability is low, but it is higher than the probability that the Earth is flat or that Asia does not exist, or that the Moon is made of cheese.

It would seem that the prior probability that you are in a Truman Show Conspiracy situation is actually greater than the prior probability of many less ambitious conspiracy hypotheses, because the only way for you to be deceived about a very large number of subjects is for the conspirators to control your whole environment, and if they went to the trouble of controlling your whole environment, it is unlikely that they would deceive you about just a few subjects. To me, the hypothesis that I am in a Truman Show seems more probable than the hypothesis that the Moon is made of cheese. (Or to be precise, the hypothesis that the Moon is made of cheese BUT everything else I think I know is true, is less likely than the hypothesis that I am in a Truman Show. In some small fraction of the Truman Show scenarios, I would be in the Truman Show AND the Moon would be made of cheese - which would be just one of the many facts the conspirators tricked me about.)

It is not clear what prior probability one should assign to the hypothesis that you are in a Truman Show situation. On the one hand it sounds very unlikely, but on the other hand, how would you know? Under normal circumstances we impose a no-Truman-Show condition before assessing probabilities. For example, suppose I read a paper in the Journal of Sciencey Science that says that Drug X has been found to have a statistically significant effect in treating malaria with a p value of 0.05. Strictly speaking, I would not say that the probability that Drug X is effective is 95%. Rather, I would say:

1. I assume that I am not a brain in a vat.

2. I assume that I am not in a Truman Show.

3. I assume that the Journal of Sciencey Science has not been corrupted by Big Pharma or other malicious actors.

4. Conditional on those three assumptions I am 95% confident that Drug X has an effect.

2.4 Scott’s Trillion Trillion Americans Argument

Scott asks us to consider if there is a conspiracy which hides the fact that the population of America is one trillion trillion. If the American population is one trillion trillion, then a huge portion of what you know about America, the size of the Earth, and humanity in general are wrong. The Trillion Trillion Americans hypothetical posits a Truman Show conspiracy, not a normal conspiracy. Scott is asking us to consider whether we might be mistaken about everything we know about the world, not merely what we know about the American population.

The reason the trillion trillion Americans conspiracy hypothesis sounds so absurd is partly because it is so specific. When the hypothesis is reframed as the nearly equivalent “You are living in a Truman Show and everything you think you know about your country and the world is wrong” it sounds more plausible (to philosophers at least).

The question of what probability you should assign to the hypothesis that you’re living in a Truman Show is an interesting philosophical question, but it’s not a unique problem to the SIA.

3. Rebuttals

One could argue that while the Truman Show problem is not unique to the SIA, the SIA interacts with the Truman Show problem in ways that cause problems which non-SIA believers don’t have to deal with.

3.1 Even if the probability of a Truman Show is tiny, the postulated number of people can be made arbitrarily large

Scott raised this potential rebuttal when he said that the posited number of people can be made as high as possible in order to cancel out the very low probability of a conspiracy.

However, there are some reasons to think that the SIA should make us lower the credence we put on the possibility that we are in a Truman Show situation. In the classic Truman Show scenario, there are hundreds of conspirators and only one Truman. According to the SIA, you would be much more likely to be one of the many conspirators than the one Truman, so the fact that you are not a conspirator should make you think that no Truman Show conspiracy is occurring.

Also, the SIA lowers the probability of certain conspiracy scenarios. While the SIA would increase your credence that a conspiracy is hiding the fact that America’s population is actually several billion rather than 300 million, the SIA would decrease your credence that a conspiracy is hiding the fact that America’s population is actually several thousand rather than 300 million.

So the SIA raises the probability of some Truman Show scenarios and lowers the probability of other Truman Show scenarios. It’s not clear how this will effect the overall probability that any Truman Show scenario is occurring.

You can make the posited number of people arbitrarily large, but the larger you make that number of people, the more elaborate and less probable the conspiracy would have to be. It could be that the probability of a conspiracy hiding N people is so low, that the probability converges to zero as N goes to infinity.

3.2 If the Interaction of the SIA and the Truman Show possibility is a problem, it’s only a problem for atheists.

In ordinary reasoning, we start with a no-Truman-Show assumption. That assumption would rule out Scott’s trillion trillion Americans hypothetical, because hiding a trillion trillion Americans would be a Truman Show conspiracy rather than a normal conspiracy.

But let’s say you’re not convinced that the no-Truman-Show assumption is reasonable. Let’s say that you think the probability of a Truman Show scenario is high enough to be considered, and is high enough that it does not converge to zero as the postulated number of hidden people goes to infinity. In that case the interaction between the Truman Show possibility and the SIA would indeed lead to the problem that Scott described.

But theists have an easy way out of this problem. Theists can just say that God would not allow large numbers of people to be as colossally deceived as Truman from the Truman show. This “God is not a deceiver” idea is the same line of reasoning that led DesCartes to dismiss the possibility of DesCartes’ Demon.

Bentham’s Bulldog made an argument from the SIA to the existence of God, and if Scott is correct that the SIA requires atheists to believe that we are in a Truman Show, that’s yet another argument for the existence of God.

Conclusion

Scott asked us to consider whether there is a conspiracy which hides the fact that the true population of America is one trillion trillion. On reflection, we see that in order for us to be the victims of such a deception, we would have to be deceived about nearly everything, so what Scott is proposing is a Truman Show conspiracy hypothesis, not a normal conspiracy hypothesis. (If it were a normal conspiracy hypothesis, namely that the conspiracy was deceiving us about just the American population and nothing else, then we could immediately reject this possibility since such a conspiracy would be extraordinarily unlikely, and its probability would converge to zero as the number of hidden people N went to infinity)

Literally all of our beliefs about the external world are always premised on a no-Truman-Show condition. You are invoking this condition when you say you are >99% confident that the Earth is round. Following this no-Truman-Show assumption which we already apply to all our beliefs, we would reject Scott’s trillion trillion Americans conspiracy hypothetical as impossible. If you don’t think we can take the no-Truman-Show assumption for granted, then the Truman Show scenario is indeed a problem you’d have to grapple with, but this is a problem for everyone, not just people who believe in the SIA.

There is a potentially problematic interaction between the SIA and the Truman Show Problem because the SIA raises the probability of Truman Show scenarios which posit a very large population, which is what Scott pointed out in his original comment. However, the SIA reduces the probability of other Truman Show scenarios (those that posit a low population). The SIA might also militates against Truman Show scenarios generally if Truman shows require the number of conspirators to be larger than the number of dupes. Maybe the probability of a Truman Show conspiracy converges to zero as the number of hidden people goes to infinity.

Finally, even if the interaction between the SIA and the Truman Show scenario is a problem, it’s only a problem for atheists, in which case Scott’s hypothetical actually strengthen’s Bentham’s Bulldog’s argument that the SIA points to God’s existence. Theists can get around the Truman Show problem by saying that God would not allow vast numbers of people to be deceived, which is the same kind of argument that DesCartes invoked to conclude that his sense experience was not a trick played on him by an evil demon.