What does this formula/notation mean in boolean algebra/bayesian probability? I am reading Jaynes' "probability theory: the logic of science"
He uses a notation that I do not understand.
He says that if $A_i$ and $A_j$ are two mutually exclusive events, then:
$p(A_i A_j |B) = p(A_i |B)δ_{ij} $
How am I to understand this notation? Jayne uses Boolean logic notation, so $A_i A_j$ means the event that both $A_i$ and $A_j$ are true. However if these two events are mutually exclusive, wouldn't that mean that:
$p(A_i A_j |B) = 0$
So why the weird notation with $\delta _{ij}$?

$$P(A_iA_j)=\left.\begin{cases}P(A_iA_i)=P(A_i), &\text{if } i=j\\ P(A_iA_j)=P(\emptyset)=0, &\text{if } i\neq j\end{cases}\right\}=P(A_iA_j)δ_{ij}$$ where $δ_{ij}$ is called Kronecker delta and has the purpose to indicate the event $i=j$. That is  $$δ_{ij}=\begin{cases}1, &i=j\\0, &i\neq j\end{cases}$$
