Welcome to The Nonlinear Library, where we use Text-to-Speech software to convert the best writing from the Rationalist and EA communities into audio. This is: Löbian emotional processing of emergent cooperation: an example, published by Andrew Critch on January 17, 2023 on The AI Alignment Forum.

Related to: A Löbian argument pattern for implicit reasoning in natural language: Löbian party invitations

Epistemic status: my opinion based on a mix of math, reflection, and speculation; not backed up by any systematic psychological studies.

Summary: Since my 2019 paper generalizing Löb's Theorem, a couple dozen people have asked me if the way humans naturally cooperate might be well-described by Löb's Theorem. In short, my answer is probably sometimes, and in this post I'll try using an example to convey what that means. Importantly, Löb's Theorem is not a theorem when applied to real-world humans and emotions — i.e., when its hypotheses are met, its conclusion is only sometimes true. Nonetheless, the reasoning pattern in its proof (I claim) sometimes genuinely occurs at the level of intuition in real people, whether or not they know any math or Löb's Theorem.

Introduction

There are at least two real-world patterns that could reasonably be called Löbian cooperation in humans, which I'll name here:

Functionally Löbian cooperation. Sometimes people become aware that they're anticipating (predicting) cooperation from each other, and then that anticipation causes them to cooperate, rendering the anticipation itself valid. In this pattern, the fact that anticipation of cooperation will cause cooperation is analogous to the hypothesis (main assumption) of Löb's Theorem, and the fact that the cooperation in fact emerges is analogous to the conclusion of Löb's Theorem. I call this pattern "functionally" Löbian because its input and output resemble the input (hypothesis) and output (conclusion) of Löb's Theorem.

Procedurally Löbian cooperation. Sometimes the mental procedure a person follows to anticipate and decide upon cooperation can resemble an entire proof of Löb's Theorem, as I'll describe below. In other words, instead of just the hypothesis and conclusion of Löb's Theorem matching reality, the structure in the intermediate steps of the proof also match reality, at least somewhat. I call this "procedurally" Löbian cooperation, and it's a special case of functionally Löbian cooperation because it demands a stronger analogy between the theorem and the real world. Illustrating how this might work constitutes is the bulk of content in this post.

What functionally Löbian cooperation feels like

For those who recognize the symbols involved, Löb's Theorem says that if ⊢□cc then ⊢c. I don't plan to use these symbols with their normal meanings in the rest of this post, so don't worry if you don't recognize them.

In words, functional Löbian cooperation happens when anticipation of future or unobserved cooperation causes present cooperation. So if you're interacting with someone, and you feel like they're probably going to be nice to you in the future, and that fact makes you decide to be nice to them now, I call that functional Löbian cooperation.

What procedurally Löbian cooperation feels like

Most human cooperation is probably not procedurally Löbian, and maybe not even functionally Löbian. However, I'm confident that human cooperation is sometimes procedurally Löbian, and I can even point to experiences of my own that fit the bill. To explain this, I'll be talking a lot more about feelings, because I think most unconscious processing is carried out by and/or experienced as feelings. I'll write

Feeling("Pigs can probably fly.")

for the feeling that pigs can probably fly. Such a feeling can be true or false, according to whether it correctly anticipates the real world.

In procedurally Löbian cooperation, part of the mental process will involve first feeling something uncertain to do with cooperation, then believing it, and then feeling lik...