Re:Logical Induction
by adrianratnapala on Tuesday September 13, 2016 @06:33AM
(#12486839)
That's interesting I had never heard of Solomonoff Induction until I read your comment. But how important, philosophically, is the existence of such a system, given that the brain already does exist as the "low fidelity analogue".
The limits on what computers can do are related to logical no-go theorems such as Gödel's which are about proofs -- examples of certain knowledge. But once you have accepted the fallibility or "low fidelity" of human reasoning, then all those no-go theorems are no longer relevant in the first place.
Solomonoff induction is useful because without it there's no model of induction with infinite computational resources. "Logical induction" is not useful in the same way because without it we already have such a model: simply prove/disprove the propositions.
And what if the propositions we want to reason about are self-referencing or say something about the system itself in a Godelian way? You need probabilistic reasoning for that to actually work.
Also, "simply prove/disprove the propositions" requires infinite computational resources (we don't know how long the proofs will be or if there are any). Logical induction does not.
Yeah, the actual performance of Solomonoff Induction is uncomputable, but to me the useful point is that "induction can be done mathematically", and then what we do heuristically in our brains can be thought of as a low-fidelity analog of that. If I'm understanding the page correctly, this is the same idea but for statements based on proofs and logical theorems. Which seems to expand the scope somewhat.
(I'm really excited about this, actually, just as a person who enjoys learning about this stuff from Wikipedia. I feel like I've vaguely thought about how Solomonoff induction would work on statements that are derived from each other (or when combined with type-checking, since type-checking is closely related to theorem-proving), but had no idea how to even ask a precise question much less make anything of it.)