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?)
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.
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.
nit: B1 should be D1 here
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?)
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.
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.
With the other problem resolved, I can confirm that adding an A=∅ escape clause to the multiplicative definitions works out.