## Logical Fallacies As Weak Bayesian Evidence: Argumentum Ad Populum (Appeal to Popularity)

03 Sep

This is another post in my, I guess, series where I explore typical logical fallacies in a Bayesian context. This time I’ll be looking at the appeal to popularity.

Appeal to popularity, of course, is a logical fallacy because in bare bones logic the conclusion must follow from the premises necessarily. As in, 100% of the time. But again, we don’t live in a world of deductive certainty where we can have 100% certainty. We live in a world of induction; the world of uncertainty; the world of probability (I like abusing semicolons).

So an example of the logical fallacy appeal to popularity: Most of the world believes in god, so therefore god exists. That is an obvious example of the fallacy but I’ll pick a less controversial one; at least, one where the conclusion is more than likely true, but the argument itself is fallacious:

Most of the world believes in evolution, therefore evolution is true.

So to analyze this in a Bayesian fashion we need our three variables. The prior probability, the success rate, and the false positive rate. The prior probability is the probability that the hypothesis is true before looking at any specific evidence.

Next is the success rate. This is the probability the majority of the world believes in something given that it is true. This in turn means our evidence (the Total Probability) is “the majority of the world believes it”. Our third variable is the false positive rate. This is the probability that the majority of the world believes in something given that it is false.

Now, we don’t have any hard numbers for the success rate/false positive rate. But in general, we can make a good guess at how many times the majority of people have believed in something given that it is true and contrast it to the frequency that the majority of people have believed in something given that it is false. Taking the entirety of human history into account, it seems like the false positive rate vastly outnumbers the success rate. The majority has been wrong a lot more than its been right. This means that dividing the success rate by the false positive rate returns a quotient less than 1. And that means that, generally, an appeal to popularity is usually Bayesian evidence against some hypothesis.

So using evolution as our example, the fact that the majority of people on the planet accept the theory of evolution means that this is Bayesian evidence against evolution; appeals to popularity in and of themselves are strong Bayesian evidence against some proposition. The good thing is that the entire point of Bayes Theorem is to continually update the probability of your hypothesis when you encounter new evidence. And there is plenty of evidence in support of evolution to update against. Unfortunately, the same doesn’t happen with god belief. The fact that the majority of the world believes in god is also Bayesian evidence against the existence of god. You would, again, need some other evidence to update against, but since the existence of god is unfalsifiable, that is highly unlikely.

But there are also different populations that we can appeal to. Instead of appealing to humanity as a whole, we could appeal to more specific groups. Like doctors, or lawyers, or family members, or what have you. If the majority of surgeons say that the best instrument to cut a sternum is so-and-so knife, what is the probability that they would say so given that they are correct, and what is the probability that they would say so given that they are incorrect? In this case, the success rate vastly outperforms the false positive rate. Thus, appeals to popularity are pretty good Bayesian evidence when you appeal to specialists in some field, or in general, to a population who is usually correct. In general, humanity as a whole is usually incorrect so appealing to a general consensus is probably not a good idea.

This seems counter intuitive. Why would we have a natural tendency to follow the herd if it weren’t beneficial in an evolutionary context? That’s exactly it; an evolutionary context is only good at helping you not end up dead and have lots of kids… not to uncover complex information about the true nature of reality. If the majority of people in your tribe 100,000 years ago believed in something, and you went against the grain, chances are that you would be ostracized or kicked out. Getting kicked out = death. Getting ostracized = less access to mates. Something like belief in god might not be true, but it certainly seems to help with having lots of kids.

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Posted by on September 3, 2012 in Bayes, logical fallacies as weak bayesian evidence

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