I was reading about logical induction at

https://intelligence.org/files/LogicalInduction.pdf

and understand how it resolves paradoxical self reference, but I’m not sure what the inductor will do in situations where multiple stable solutions exist.

Let

If is continuous then it must have a fixed point. Even if it has finitely many discontinuities, it must have an “almost fixed” point. An such that

However some have multiple such points.

Has “almost fixed” points at , and .

A similar continuous is

With

Having every point fixed.

Consider

These functions make the logical inductor version of “this statement is true”. Multiple values can be consistently applied to this logically uncertain variable. None of the possible values allow a money pump, so the technique of showing that some behaviour would make the market exploitable that is used repeatedly in the paper don’t work here.

Is the value of uniquely defined or does it depend on the implementation details of the logical inductor? Does it tend to a limit as ? Is there a sense in which

causes has a stronger attractor to than it does to ?

Can be 0.6 where

because the smallest variation would force it to be ?

Different logical inductors will give different probabilities for each ϕn. The logical induction criterion does not require any answer in particular.

Any particular deterministic algorithm for finding a logical inductor (such as the one in the paper) will yield a logical inductor that gives particular probabilities for these statements, which are close to fixed points in the limit. The algorithm in the paper is parameterized over some measure on Turing machines, and will give different answers depending on this measure. You could analyze which measures would lead to which fixed points, but this doesn’t seem very interesting.