Note that you can show, for every E, P(E|E) = 1 (the proof is left as an exercise). This means that yes, whatever you have to the right of the sign | is taken to be certain. Why is this so?
The main reason is that updating on evidence, in a sense, means translating your entire probability model to a possible world where that evidence is true. This is usually justified because you treat sensory data (readings from a gauge, the color of a ball extracted from an urn, etc.) as certainties. But nothing limits you to do this only for evidence in the sensorial meaning. You can also entertain the idea that a memory or a fictional fact is true and update your model accordingly. By the same theorem, you can move in and out of that possible world: everything is controlled by P(E). If you divide any probability by P(E), you move in, if you multiply everything by P(E), you move out.
Note that you can show, for every E, P(E|E) = 1 (the proof is left as an exercise). This means that yes, whatever you have to the right of the sign | is taken to be certain. Why is this so?
The main reason is that updating on evidence, in a sense, means translating your entire probability model to a possible world where that evidence is true. This is usually justified because you treat sensory data (readings from a gauge, the color of a ball extracted from an urn, etc.) as certainties. But nothing limits you to do this only for evidence in the sensorial meaning. You can also entertain the idea that a memory or a fictional fact is true and update your model accordingly.
By the same theorem, you can move in and out of that possible world: everything is controlled by P(E). If you divide any probability by P(E), you move in, if you multiply everything by P(E), you move out.