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 ?

I see no almost fixed point for the function that is 1 until 0.5 and 0 after.

0.5 is the almost fixed point. Its the point where f(x)−x goes from being positive to negative. If you take a sequence of continuous functions fn(x) that converge pointwise to f(x) then there will exist a sequence yn such that fn(yn)=yn and limn→∞yn=0.5.

That definition makes more sense than the one in the question. :)