I have something suggestive of a negative result in this direction:
Let C be the prime-detector situation from Section 2.1 of the coarse worlds post, and let p:W→W be the (non-surjective) function that "heats" the outcome (changes any "C" to an "H"). The frame p∘(C) 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 C but is observable in p∘(C).Claim: No frame of the form ExternalV(C) is biextensionally equivalent to p∘(C)Proof Idea: Image(ExternalV(C))=Image(C)≠Image(p∘(C))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 A=∅ 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 D1 and D2 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 b1=b2, since we might have A⊇B1≠B2⊆A after biextensional collapse.) Then to prove h=hf1, observe that for all b∈B1, b∙1h(b′2)=b∙1h(b2) which means b⋆1h(b′2)=b⋆1f1, hence h(b′2)=f1 since a biextensional frame has no duplicate columns.
I presume the fix here will be to add an explicit A=∅ 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 null counterexample.
How is this supposed to work (focusing on the h=hf1 claim specifically)?
Earlier, hf1 was defined as follows:
given by gf1(b1)=b1⋅1f1 and hf1(b2)=f1
but there is no reason to suppose f1=r above.
It suffices to establish that Ensure(CTi)⊇Ensure(CTj)
I think the Ti and Tj here are supposed to be V and U
Indeed I think the A=∅ case may be the basis of a counterexample to the claim in 4.2. I can prove for any (finite) W with |W|>1 that there is a finite partition V of W such that C's agent observes V according to the assuming definition but does not observe V according to the constructive multiplicative definition, if I take C=null.
nit: B1 should be D1 here
and let b2 be an element of b2.
and the second b2 should be B2. I think for these b1 and b2 to exist you might need to deal with the A=∅ case separately (as in Section 5). (Also couldn't you just use the same b twice?)
UPDATE: I was able to prove AssumeS1(C)&AssumeS2(C)≃AssumeS1∪S2(C) in general whenever S1 and S2 are disjoint and both in Obs(C), with help from Rohin Shah, following the "restrict attention to world S1∪S2" approach I hinted at earlier.
this is clearly isomorphic to D1&…&Dm, where Di=(Bi,F,⋆i), where b⋆if=b⋆f. Thus, C's agent can observe V according to the nonconstructive additive definition of observables.
I think this is only true if VB partitions W, or, equivalently, if vB 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 s∈V that is not in VB to the frame ⊤ so it is harmless in the product of Dis -- I will try this.