The issue with reflection is more basic. What exactly does reflection require of imprecise probabilists in this case? Now, it is obviously the case that each credal committee member’s prior credence is its expectation over the possible future evidence (this is a theorem of probability theory). But somehow, it is felt, the credal state as a whole isn’t sensitive to reflection in the way the principle requires. Each \(p\in P\) satisfies the principle, but the awkward symmetries of the problem conspire to make \(P\) as a whole violate the principle. This looks to be the case if we focus on \(P(H)\) as an adequate representation of that part of the belief state. But as noted earlier, this is not an adequate way of understanding the credal state. Note that while learning \(Y\) and learning \(Y'\) both prompt revision to a state where the posterior belief in \(H\) is represented as an interval by \([0,1]\), the credal states as sets of probabilities are not the same. Call the state after learning \(Y\), \(P'\) and the state after learning \(Y'\), \(P''\). So \(P' = \{p(\cdot \mid Y), p\in P\}\) and \(P'' = \{p(\cdot\mid Y'), p\in P\}\). While it is true that \(P'(H) = P''(H)\), \(P' \neq P''\) as sets of probabilities, since if \(p\in P'\) then \(p(Y) = 1\) whereas if \(p\in P''\) then \(p(Y) = 0\). So one lesson we should learn from dilation is that imprecise belief is represented by sets of functions rather than by a set-valued function (see also, Joyce 2011; Topey 2012; Bradley and Steele 2014b).