Introduction

For a long time, I was planning to write a comprehensive post patiently exploring all the problems with conventional “anthropic reasoning”. How, for historical reasons, the whole discipline went sideways at some point and just can’t recover, continuing to apply confused frameworks, choosing between several ridiculous options and accumulating paradoxes. And how one should reason correctly about all the “anthropic problems”.

I’m sorry but this post isn’t going to be that. This time I’m mostly getting the frustration out of my system because of Bentham's Bulldog’s You Need Self-Locating Evidence! which confidently reiterates all the standard confusions, even though he really should know better at this point.

So, in this post I’ll resolve only some of the confusions of “anthropic reasoning”, leaving others as well as the deeper historical analysis for the future. Frankly, maybe it’s even for the best.

Probability theory 101

My apologies, but this section is necessary, as it’s exactly the sloppy probabilistic reasoning that led us to the current miserable state. I promise to be brief with this section1.

Let’s start from explicitly defining what probabilities are. Probability theory gives us a mathematical model to approximate some causal processes from reality to some degree of uncertainty.

It’s very helpful to think in terms of maps and territories here. We look at some territory in the world and create an imperfect map of it. The less we know about the territory the more generic is the map. And when we learn new details, we add them to our map, making it more specific.

Consider a roll of a fair 6-sided die.

Imagine an infinite number of iterations where the die is rolled again and again - a probability experiment representing any roll of a fair 6-sided die. Every trial has an outcome: either ⚀ or ⚁ or ⚂ or ⚃ or ⚄ or ⚅. This set of mutually exclusive and collectively exhaustive outcomes of the probability experiment is called the sample space.

Sets of these outcomes are called events. The simplest are individual events, consisting only of a single outcome, but likewise there can be events consisting of any number of outcomes up to the whole sample space.

Events can be interpreted as statements that have truth values in every iteration of the experiment. For example, event {⚁; ⚃; ⚅} is interpreted as a statement:

“In this trial the die is even”.

Naturally, this statement is True in every iteration of the probability experiment that the die is either ⚁ or ⚃ or ⚅ and False in every other iteration. Probability of an event is a ratio of trials where this event is True to the total number of trials throughout the whole probability experiment.

With this in mind, let’s answer a simple question. What’s the probability that our die rolled a ⚅?

At first, we are completely indifferent between all of the iterations of the probability experiment. Our roll can be any of these infinitely many trials. But we know that 1/6 of them are ⚅. Therefore:

P(⚅) = 1/6

Now, suppose we’ve learned that the outcome of the roll is even. This gives us new information, makes our map more specific by eliminating half of the possible outcomes. Now we are indifferent only between the trials where the outcome of the die roll is even and 1/3 of them are ⚅, therefore:

P(⚅|⚁; ⚃; ⚅) = 1/3

If you can understand that, accept my congratulations, you understand probabilities better than most of the philosophers of probability. I wish I was joking.

Possible Worlds, Impossible Confusions

Philosophers do tend to overcomplicate things sometimes. For reasons, I’m not going to dwell on right now, instead of outcome of a probability experiment, they decided to talk about “possible words” and then “centred possible worlds”, completely confusing themselves and everyone else.

As a part of this confusion, they came up with the notions of “Self-Locating” and “Non-Self-Locating Evidence”. Here is what BB tells us about this framework:

People often think probabilities and beliefs merely concern how the world is.

I think this is wrong.

Self-locating probabilities are probabilities that concern one’s place in the world, rather than what the world is like.

We may immediately come up with couple of corrections. First of all, probabilities are not just about the way the world is. They are about some aspects of the world to the best of our knowledge. That’s why probabilities change when we learn new facts even though the territory we are describing may stay the same.

And, whether a particular person is positioned in the world is also a fact about the world, so the whole distinction makes no sense even in its own terms. A world where I’m in one city is different from the world where I’m in some other city. Obviously. So, case closed?

Oh, not so fast! You see, as an additional complication, that would confuse everyone even more, philosophers have long ago added here the notion of personal identity:

For example, imagine that there is one clone of me in a dark and murky bunker in California and another in Paris (what rotten luck). I have no special evidence concerning which one I am (for example, there are no nearby croissants or people surrendering). I should think there’s a 50% chance that I am the one in California.

This evidence is self-locating because it’s not about what the world is like. I already know what the world is like. I know that there is one copy of me in California and another in Paris. What I’m uncertain about is which one I am. That’s what self-locating information concerns: which of the people you are, not what the world is like.

That is, in what sense are two worlds different if we switch the places of two completely identical people?

And, fair enough, it’s an interesting question in its own right. We can say that “switching places” isn’t a free action. We need to exert some work, which increases entropy in the universe. Therefore, the world where such switching has happened is different from a world where it didn’t. In the very least, they have different causal stories.

But more importantly, none of this matters in the slightest when talking about probabilities.

Once again, probability theory describes some real-world situation to the best of our knowledge. In the example above the situation is “either being in one place or the other” and the best of our knowledge is “no evidence whatsoever”.

So, we have a probability experiment with two mutually exclusive outcomes. In half of the iterations, I’m in Paris and in the other half - in California and I’m uncertain between all of them. Therefore:

P(California) = P(Paris) = 1/2

That’s all. It doesn’t matter whether there is or isn’t a clone in the other location. It doesn’t affect anything. Neither we need to think about some alternative worlds and whether they are real and in which sense. It is completely irrelevant to our probabilistic model. There is absolutely no difference in methodology between this example and the 6-sided-die example from the beginning of the post. We don’t need a special category “Self-Locating Evidence” to talk about such probability experiments; it’s a completely useless concept.

The Crux

Wait, doesn’t it mean that I essentially agree with Bentham’s Bulldog? Sure, I’m annoyed with his terminology and the framework he is applying, but it’s just formalism what about the substance? He argues that “Self-Locating Evidence” is not fake and we should treat it as any other evidence:

But a number of people have suggested that self-locating evidence is sort of fake. They claim that it doesn’t make any sense to wonder who I am once I know what the world is like. After all, there’s a copy of me in each situation. What can I possibly be wondering about if not what the world is like?

In this article, I’ll explain why self-locating evidence is real.

I claim that we shouldn’t even have a separate category for this sort of stuff in the first place, because all probabilistic reasoning works the exact same way in terms of probability experiment. What am I even arguing about?

Let’s make it clear with a handy Venn Diagram:

The problem with the “Self-Locating Evidence” category is that, while some part of it is just completely normal probabilistic reasoning, the other is total nonsense that goes against the core principles of probability theory and is a source of constant stream of paradoxes.

People who say that “Self-Locating Evidence” is “sort of fake” are not wrong - a huge part of it is. But due to conversation being framed in either pro- or anti-self-locating-evidence way, expressing this nuanced point is hard.

As a result, someone like BB can come up with an example of “Self-Locating Evidence” producing valid reasoning and then falsely generalize it to a domain where it doesn’t work. And when you try to point this out, such person just says:

“What do you mean probability theory doesn’t work like that? Haven’t you heard about Self-Locating Evidence? Are you denying that I can have some credence whether I’m in Paris or in California? That’s crazy!”

That’s why the term should be abolished and we should just be talking about all the probability theoretic problems in a unified way in terms of probability experiments and their trials.

You may continue reading the post on my substack.