Blackwell’s theorem says that the conditions under which can be said to be more

generallyuseful than are precisely the situations where is a post-garbling of .

Are the indices the wrong way around here?

Karma: 397

Blackwell’s theorem says that the conditions under which can be said to be more

*generally*useful than are precisely the situations where is a post-garbling of .Are the indices the wrong way around here?

A formalisation of the ideas in this sequence in higher-order logic, including machine verified proofs of all the theorems, is available here.

- 5 Dec 2020 8:34 UTC; 11 points) 's comment on Introduction to Cartesian Frames by (

“subagent [] that could choose ”—do you mean or or neither of these? Since is not closed under unions, I don’t think the controllables version of “could choose” is closed under coarsening the partition. (I can prove that the ensurables version is closed; but it would have been nice if the controllables version worked.)

ETA: Actually controllables do work out if I ignore the degenerate case of a singleton partition of the world. This is because, when considering partitions of the world, ensurables and controllables are almost the same thing.

because whereas math-proof data-centers might result in our inadvertent death, something that refers to what humans want might result in deliberate torture.

I want to note that either case of screwing up this badly currently feels pretty implausible to me.

Great summary!

I have something suggestive of a negative result in this direction:

Let be the prime-detector situation from Section 2.1 of the coarse worlds post, and let be the (non-surjective) function that “heats” the outcome (changes any “C” to an “H”). The frame is clearly in some sense equivalent to the one from the example (which deletes the temperature from the outcome) -- I am using my version just to stay within the same category when comparing frames. As a reminder, primality is not observable in but is observable in .

**Claim:**No frame of the form is biextensionally equivalent to**Proof Idea**:

The kind of additional observability we get from coarsening the world seems in this case to be very different from the kind that comes from externalising part of the agent’s decision.

With the other problem resolved, I can confirm that adding an escape clause to the multiplicative definitions works out.

Using the idea we talked about offline, I was able to fix the proof—thanks Rohin!

Summary of the fix:

When and are defined, additionally assume they are biextensional (take their biextensional collapse), which is fine since we are trying to prove a biextensional equivalence. (By the way this is why we can’t take , since we might have after biextensional collapse.) Then to prove , observe that for all , which means , hence since a biextensional frame has no duplicate columns.

I presume the fix here will be to add an explicit escape clause to the multiplicative definitions. I haven’t been able to confirm this works out yet (trying to work around this), but it at least removes the counterexample.

How is this supposed to work (focusing on the claim specifically)?

and so

Thus, .

Earlier, was defined as follows:

given by and

but there is no reason to suppose above.

- 5 Feb 2021 10:59 UTC; 3 points) 's comment on Eight Definitions of Observability by (

It suffices to establish that

I think the and here are supposed to be and

Indeed I think the case may be the basis of a counterexample to the claim in 4.2. I can prove for any (finite) with that there is a finite partition of such that ’s agent observes according to the assuming definition but does

*not*observe according to the constructive multiplicative definition, if I take

Let

nit: should be here

and let be an element of .

and the second should be . I think for these and to exist you might need to deal with the case separately (as in Section 5). (Also couldn’t you just use the same twice?)

UPDATE: I was able to prove in general whenever and are disjoint and both in , with help from Rohin Shah, following the “restrict attention to world ” approach I hinted at earlier.

this is clearly isomorphic to , where , where . Thus, ’s agent can observe according to the nonconstructive additive definition of observables.

I think this is only true if partitions , or, equivalently, if is surjective. This isn’t shown in the proof. Is it supposed to be obvious?

EDIT: may be able to fix this by assigning any that is not in to the frame so it is harmless in the product of s—I will try this.

And it seems we do actually need in the proof to justify:

Thus it suffices to show that .

Without it, we have to show instead.

Because and are not a partition of the world here.

EDIT: but what we actually need in the proof is where the do result in a partition, so I think this will work out the same as the other comment. I’m still not convinced about biextensional equivalence between the frames without the rest of the product.

I haven’t yet figured out why it’s true under - I’ll keep trying, but let me know if there’s a quick argument for why this holds. (Default next step for me would be to see if I can restrict attention to the world then do something similar to my other comment.)

and observe that

This cannot be true. I can prove in general whenever by observing that the agent cardinalities on each side differ.

That’s right. A partial function can be thought of as a subset (of its domain) and a total function on that subset. And a (total) function can be thought of as a partition (of its domain): the parts are the inverse images of each point in the function’s image.