I was reading about logical induction at
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.
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
Having every point fixed.
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 ?