A confusion regarding search and matching model in Pissarides's "Equilibrium Unemployment Theory" book I was studying chapter 1 of the book "Equilibrium Unemployment Theory" and I got confused about the way Pissarides has defined the probability of a firm not finding a worker in a short time interval $δt$ as $1-q(θ)δt$  (page 7 of the book). 

The first reason I got confused is that, to me that cannot represent the probability because if we take $δt$ large enough, the result can be negative.

Second, if we take $q(θ)δt$ as the rate (NOT the probability) of a firm finding a match in a time interval $δt$, then the rate of the firm not finding a match in the same time interval shouldn't be $[1-q(θ)]δt$ instead of $1-q(θ)δt$?

Any help would be much appreciated.
$q(\theta)$ is defined as the job-filling rate. Note that market tightness $\theta$ is not necessarily constant over time (Pissarides makes a dynamic analysis at some point). It may help to denote it $\theta_t$. 
As an approximation, $q(\theta_t)\delta t$ is a probability for a firm to meet a worker between $t$ and  $t+\delta t$ for $\delta t$ small enough. 

1) If you choose a large $\delta t$, this approximation is not valid anymore. In other words, assuming a large $\delta t$ prevent you from interpreting $q(\theta_t)\delta t$ as a probability. To define an equilibrium, Pissarides will anyway take the limit $\delta t\to 0$.

2) $q(\theta_t)$ is a rate (that is not constrained to be lower than 1), whereas $q(\theta_t)\delta t$ is a probability. Thus, the probability of a firm not meeting a worker between time $t$ and $t+\delta t$  is $1-q(\theta_t)\delta_t$. 

$1-q(\theta_t)$ has no clear interpretation (it can even be negative). I guess (but I would like confirmation) that the rate of a firm not finding a worker cannot be defined in this case (or it would be $+\infty$).