Anything above 90% eggplant is rounded to 100%, anything below 10% eggplant is rounded to 0%, and anything between 10% and 90% is unexpected, out of spec, and triggers a “Wait, what?” and the sort of rethinking I’ve outlined above, which should dissolve the question of “Is it really eggplant?” in favor of “Is it food my roommate is likely to eat?” or whatever new question my underlying purpose suggests, which generally will register as >90% or <10%.
Note that in the example we never asked the question “Is it really an eggplant?” in the first place, so this isn’t a question for us to dissolve. The question was rather how to update our original belief, or whether to update it at all (leave it unchanged). You are essentially arguing that Bayesian updating only works for beliefs whose vagueness (fuzzy truth value) is >90% or <10%. That Bayesian updating isn’t applicable for cases between 90% and 10%. So if we have a case with 80% or 20% vagueness, we can’t use the conditionalization rule at all.
This “restricted rounding” solution seems reasonable enough to me, but less than satisfying. First, why not place the boundaries differently? Like at 80%/20%? 70%/30%? 95%/5%? Heck, why not 50%/50%? It’s not clear where, and based on which principles, to draw the line between using rounding and not using conditionalization. Second, we are arguably throwing information away when we have a case of vagueness between the boundaries and refrain from doing Bayesian updating. There should be an updating solution which works for all degrees of vagueness so long as we can’t justify specific rounding boundaries of 50%/50%.
Do note that the difficulty around vagueness isn’t whether objects in general vary on a particular dimension in a continuous way; rather, it’s whether the objects I’m encountering in practice, and needing to judge on that dimension, yield a bunch of values that are close enough to my cutoff point that it’s difficult for me to decide. Are my clothes dry enough to put away? I don’t need to concern myself with whether they’re “dry” in an abstract general sense.
This solution assumes we can only use probability estimates when they are a) relevant to practical decisions, and b) that cases between 90% and 10% vagueness are never decision relevant. Even if we assume b) is true, a) poses a significant restriction. It makes Bayesian probability theory a slave of decision theory. Whenever beliefs aren’t decision relevant and have a vagueness between the boundaries, we wouldn’t be allowed to use any updating. E.g. when we are just passively observing evidence, as it happens in science, without having any instrumental intention with our observations other than updating our beliefs. But arguably it’s decision theory that relies on probability theory, not the other way round. E.g. in Savage’s or Jeffrey’s decision theories, which both use subjective probabilities as input in order to calculate expected utility.
Note that in the example we never asked the question “Is it really an eggplant?” in the first place, so this isn’t a question for us to dissolve. The question was rather how to update our original belief, or whether to update it at all (leave it unchanged). You are essentially arguing that Bayesian updating only works for beliefs whose vagueness (fuzzy truth value) is >90% or <10%. That Bayesian updating isn’t applicable for cases between 90% and 10%. So if we have a case with 80% or 20% vagueness, we can’t use the conditionalization rule at all.
This “restricted rounding” solution seems reasonable enough to me, but less than satisfying. First, why not place the boundaries differently? Like at 80%/20%? 70%/30%? 95%/5%? Heck, why not 50%/50%? It’s not clear where, and based on which principles, to draw the line between using rounding and not using conditionalization. Second, we are arguably throwing information away when we have a case of vagueness between the boundaries and refrain from doing Bayesian updating. There should be an updating solution which works for all degrees of vagueness so long as we can’t justify specific rounding boundaries of 50%/50%.
This solution assumes we can only use probability estimates when they are a) relevant to practical decisions, and b) that cases between 90% and 10% vagueness are never decision relevant. Even if we assume b) is true, a) poses a significant restriction. It makes Bayesian probability theory a slave of decision theory. Whenever beliefs aren’t decision relevant and have a vagueness between the boundaries, we wouldn’t be allowed to use any updating. E.g. when we are just passively observing evidence, as it happens in science, without having any instrumental intention with our observations other than updating our beliefs. But arguably it’s decision theory that relies on probability theory, not the other way round. E.g. in Savage’s or Jeffrey’s decision theories, which both use subjective probabilities as input in order to calculate expected utility.