Morgensen notes (fn.4) the possibility a representor could be a fuzzy set (whereby membership is degree-valued), which could be a useful resource here. One potential worry is this invites an orthodox-style approach: one could weigh each element by its degree of membership, and aggregate across them.
We could use degrees without cardinal interpretation, e.g. they’re only ordinal/for ranking.
Even with real-valued degrees, their sum or integral might not be finite, so weighting by them wouldn’t be possible, since we can’t normalize.
But yes, if your degrees represent (possibly unnormalized) probabilities or probability densities, I think you would want to aggregate (if it’s practical to do so). I don’t see this as a “worry”: if you feel justifiably confident enough to reduce to the orthodox-style approach, I think you should. But you should try to recognize when you aren’t justified in doing so.
With coin flipping, I might not commit to a single prior for the probability, but I might be willing to say that it is symmetric around 50% (and monotonic around that). I suspect what we’d want from fuzzy sets can be achieved by having multiple priors over the things you want to assign fuzzy set membership.
We could use degrees without cardinal interpretation, e.g. they’re only ordinal/for ranking.
Even with real-valued degrees, their sum or integral might not be finite, so weighting by them wouldn’t be possible, since we can’t normalize.
But yes, if your degrees represent (possibly unnormalized) probabilities or probability densities, I think you would want to aggregate (if it’s practical to do so). I don’t see this as a “worry”: if you feel justifiably confident enough to reduce to the orthodox-style approach, I think you should. But you should try to recognize when you aren’t justified in doing so.
With coin flipping, I might not commit to a single prior for the probability, but I might be willing to say that it is symmetric around 50% (and monotonic around that). I suspect what we’d want from fuzzy sets can be achieved by having multiple priors over the things you want to assign fuzzy set membership.