Regarding the example, I earlier defined “action i leads to outcome j” to mean the conjunction of GL⊢Ai→Uj and GL⊬¬Ai; i.e., we check for spurious counterfactuals before believing that GL⊢Ai→Uj tells us something about what action i leads to, and we only consider ourselves “fully informed” in this sense if we have non-spurious information for each i. (Of course, my follow-up post is about how that’s still unsatisfactory; the reason to define this notion of “fully informative” so explicitly was really to be able to say more clearly in which sense we intuitively have a problem even when we’ve ruled out the problems of ambiguous counterfactuals and not enough counterfactuals.)
Given that we’re using PA+n to defeat evil problems, the true modal definition of “action i leads to outcome j” might be something like “there exists a closed formula ϕ such that N⊨ϕ, GL ⊢ϕ→(Ai→Uj), and GL ⊬ϕ→¬Ai”. But that’s an unnecessary complication for this post.
Yeah, that sounds good! Of course, by the Kripke levels argument, it’s sufficient to consider ϕ’s of the form ¬□n⊥. And we might want to have a separate notion of ”i leads to j at level n”, which we can actually implement in a finite modal formula. This seems to suggest a version of modal UDT that tries to prove things in PA, then if that has ambiguous counterfactuals (i.e., it can’t prove Ai→Uj for any j) we try PA+1 and so on up to some finite n; then we can hope these versions of UDT approximate optimality according to your revised version of ”i leads to j” as n→∞. Worth working out!
Typo on indices, should be:
then GL⊢¬A2 would imply that the agent would take action 2 (because ¬A2 implies A2→U1)
Also, isn’t your example also fully informative, since if GL⊢¬A2, then GL also proves true and spurious counterfactuals about A2?
Fixed—thanks, Patrick!
Regarding the example, I earlier defined “action i leads to outcome j” to mean the conjunction of GL⊢Ai→Uj and GL⊬¬Ai; i.e., we check for spurious counterfactuals before believing that GL⊢Ai→Uj tells us something about what action i leads to, and we only consider ourselves “fully informed” in this sense if we have non-spurious information for each i. (Of course, my follow-up post is about how that’s still unsatisfactory; the reason to define this notion of “fully informative” so explicitly was really to be able to say more clearly in which sense we intuitively have a problem even when we’ve ruled out the problems of ambiguous counterfactuals and not enough counterfactuals.)
Ah! I’d failed to propagate that somehow.
Given that we’re using PA+n to defeat evil problems, the true modal definition of “action i leads to outcome j” might be something like “there exists a closed formula ϕ such that N⊨ϕ, GL ⊢ϕ→(Ai→Uj), and GL ⊬ϕ→¬Ai”. But that’s an unnecessary complication for this post.
Yeah, that sounds good! Of course, by the Kripke levels argument, it’s sufficient to consider ϕ’s of the form ¬□n⊥. And we might want to have a separate notion of ”i leads to j at level n”, which we can actually implement in a finite modal formula. This seems to suggest a version of modal UDT that tries to prove things in PA, then if that has ambiguous counterfactuals (i.e., it can’t prove Ai→Uj for any j) we try PA+1 and so on up to some finite n; then we can hope these versions of UDT approximate optimality according to your revised version of ”i leads to j” as n→∞. Worth working out!