There's a labeling problem here. E is an event. The extra information you're updating on, the evidence, the thing that you are certain of, is not "E is true". It's "E has probability p". You can't actually update until you know the probability of E.
What the joint probability give you is by how much you have to update your credence in H, given E. Without P(E), you can't actually update.
P(H|E) tells you "OK, if E is certain, my new probability for H is P(H|E)". P(H|~E) tells you "OK, if E is impossible, my new probability for H is P(H|~E)". In the case of P(E) = 0.5, I will update by taking the mean of both.
Updating, proper updating, will only happen when you are certain of the probability of E (this is different form "being certain of E"), and the formulas will tell you by how much. Your joint probabilities are information themselves: they tell you how E relates to H. But you can't update on H until you know evidence about E.
It seems like in order to go from P(H) to P(H|E) you have to become certain that E. Am I wrong about that?
Say you have the following joint distribution:
P(H&E) = a
P(~H&E) = b
P(H&~E) = c
P(~H&~E) = d
Where a,b,c, and d, are each larger than 0.
So P(H|E) = a/(a+b). It seems like what we're doing is going from assigning ~E some positive probability to assigning it a 0 probability. Is there another way to think about it? Is there something special about evidential statements that justifies changing their probabilities without having updated on something else?