I’ve been thinking a bit about the how to get a “uniform rather than pointwise” version of the logical induction stuff.

It seems like a lot of the challenge of the problem is generic to Bayesianism, and not particular to “logical” or “mathematical” outcomes.  Anyone who reads my posts on this stuff knows I have an axe to grind about how, outside of specialized small domains, you don’t have a complete sigma-algebra to put probabilities on.  (Because you aren’t logically omniscient, you don’t know all the logical relations between hypotheses, which means you don’t know all of the subset relationships between sets in your algebra.)

The case of “logical induction” forces Bayesians to think about this even if they wouldn’t otherwise, since the prototypical/motivating examples involve math, a world in which we are continually discovering facts of the form “A implies B.”  So assuming logical omniscience would be assuming we know all the theorems at the outset, in which case we wouldn’t need LI to begin with.

But the problem that “A may imply B even though you don’t know it does” is generic and comes up for Bayesian inference about real-world events, too.  A good solution to this problem would be very interesting and important (?) even if it didn’t, in itself, handle the “logical” aspects of “logical induction” (like “what counts as evidence for a logical sentence”).

I have to imagine there is work on this problem out there, but I have had a hard time finding it.

The basic mathematical setup would have to involve some “incomplete” version of a sigma-algebra (generically, a field of sets), where not all of the union/intersection information is “known.”  This is a bit weird because when we talk about a collection of sets, we usually mean we know what is in the sets, and that information contains all the relations like “A is a subset of B” (i.e. B implies A), when we want to make some of them go away.

A Boolean algebra is like a field of sets where we forget what the sets contain, and just leave them as blank symbols that happen to have union/intersection (AKA “join/meet”) relations with one another.  That seems closer to what we want, except that we need some of the join/meet operations to give undefined results.  There are Boolean algebras where not everything has a join/meet (those that aren’t complete, in the complete lattice sense), but this seems like a thing having to do with inf/sup stuff in infinite spaces and isn’t really what we want.  (Despite my username, I know very little about algebra and am just flying blind on Wikipedia here.)

An example of the sort of thing I want to do is the following.  Say we are assigning probabilities in (0,1) to P(A), P(B), P(A=>B), and P(B=>A).  Suppose P(A=>B) > P(B=>A), that is, we think it’s more likely that A implies B than the reverse (and in particular, more likely than A<=>B).

Now consider P(A) and P(B).  The probabilities above say we’re most likely to be in a world where A=>B and not vice versa, in which case we should have P(A) > P(B), or we’ll be incoherent.  So it seems like we should have P(A) > P(B) right now.  Of course, this will make us incoherent if it turns out that we are in the B=>A or A<=>B worlds, but we think those are less likely.  In betting terms, the losses we might incur from incoherence in a likely world should outweigh those we’d incur from incoherence in an unlikely world.

What we’re really doing here, I guess, is treating the implication (i.e. subset relation) as a random event, so implicitly there is a second, complete probability space whose events (or outcomes?) include the subset relations on the first, incomplete probability space (the one discussed above).  Maybe you could just do the whole thing this way?  I haven’t tried it, I’m curious what would happen

Anyway, I can’t help but think there must be the right math tools out there for doing this kind of thing, and I just don’t know about it.  Anyone have pointers?