Definition of intensity matrix for continous markov chains

Question

I have recently picked up a book about markov chains and am working through the case of continuous-time markov chains with values in a discrete space $E$ One of the first definitions given is the definition of the sequence of intensities $(q_i)$ at which the process leaves the state $i$ after reaching it. If $J_1$ is the first jumping time, the intensity $q_i$ is defined as such : $$q_i=\frac{1}{E_i(J_1)}$$ Now the issue is that $P_i$ is not defined in the book. I assume it is the conditional probability with relation to the event ${X_0=i}$. But this conditioning does not seem to make sense when the probability of the event $X_0=i$ is null and hence the transition matrix would not be defined on the row $i$. Any help would be greatly appreciated, i can't make sense of any other way of interpreting the probability $P_i$.