# [Question] Logical inductors in multistable situations.

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 . 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.

• 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 goes from be­ing pos­i­tive to nega­tive. If you take a se­quence of con­tin­u­ous func­tions that con­verge poin­t­wise to then there will ex­ist a se­quence such that and .

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