Note: c(e) refers to the logical content of e (not including tautologies).
If p(h|e) is greater than p(h), then p(e|h) = 1.
If p(e|h) = 1, then c(e) is a subset of c(h).
If c(h1) = c(h) - c(e), then c(h1) + c(e) = c(h)
Therefore, h = h1 + e
We can now replace any instance of h with h1 + e.
If p(h1 + e|e) is greater than p(h1 + e), then p(e|h1 + e) = 1.
However, if we break apart h1 + e, we can isolate h1: that part of h which is not equal to e. We can then ask whether h1 is supported by e,
Is p(e|h1) = 1?
Since h1 is derived from c(h) minus c(e), then the answer is no.
In other words, that portion of h which goes beyond e is not made more probable by e. The apparent increase in probability of all of h arises from treating hypotheses as indivisible elements -- a fallacy of composition. What probabilities you get, and what supports what, just depends on how you say it.
Edit: I haven't actually tried writing this out in this form before, so I probably (heh) made some mistakes, knowing me.