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 $f:[0,1]\to [0,1]$

If $f$ is continuous then it must have a fixed point. Even if it has finitely many discontinuities, it must have an "almost fixed" point. An $x$ such that $\forall ϵ>0:in{f}_{y\in (x-ϵ,x)}f\left(y\right)\le x\le {sup}_{y\in (x,x+ϵ)}f\left(y\right)$

However some $f$ have multiple such points.

Has "almost fixed" points at $0$, $\frac{1}{2}$ and $1$.

A similar continuous $f$ is

With

Having every point fixed.

Consider ${\varphi }_n="f\left(E_n\right({\varphi }_n\left)\right)"$

These functions make ${\varphi }_n$ 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 $E_n\left({\varphi }_n\right)$ uniquely defined or does it depend on the implementation details of the logical inductor? Does it tend to a limit as $n\to \infty$ ? Is there a sense in which

causes $E_n\left({\varphi }_n\right)$ has a stronger attractor to $0.1$ than it does to $0.83$?

Can $E_n\left({\varphi }_n\right)$ be 0.6 where

because the smallest variation would force it to be $0.1$?