No such thing as conditional events? I recently answered a question (Confusion over marginalising out probability) on conditional probability and one particular bit of terminology I used seems to have evoked a visceral reaction in some people. However, I still can't see what's wrong with it. Let's say there are three events, $A$, $B$ and $C$. It is pretty clear what is meant when we say $P(A,B|C)$. And it is also clear what we mean when we say $P(A|C)$ and $P(B|C)$. In this sense, I think of $A|C$ as a conditional event and $P(A|C)$ as its probability. Taking this further, I can write:
$$P(A,B|C) = P(A|C,B|C)$$
But some people seem to think there is no such thing as a conditional event. My problem with this statement is that probabilities are always defined on events. If there is no such thing as a conditional event, how can there be a conditional probability? And if there indeed is "no such thing", what exactly prevents me from defining it? I just want to understand where that user was coming from and what I might be missing.

I understand your notation but I can also see where others are coming from when they object to calling $A|B$ an event.
If $A$ and $B$ are events in some probability space $\Omega$, we can define the conditional probability $P(A|B)$.  If we want to interpret $A|B$ itself as an event, then it would have to be as the event $A \cap B$ in the space $B$, not in the original probability space $\Omega$. When people say $A|B$ is not an event, they probably mean that it's not an event in $\Omega$.
Now, when you write a joint probability expression like $P(A|C, B|C)$, it makes sense because both of the individual $A|C$ and $B|C$ can be interpreted as events on the space $C$.  However, we cannot write something like $P(A|C, B|D)$, because now we have 2 events in two different probability spaces $C$ and $D$.
