Collection of inference rules viewed itself as a judgment
The interpretation of such paragraph is that rules define judgements. It's more or less the same as saying
closure: the judgment satisfies this rules
strongest: ..and this is all there is

So, for example, if you have the following two rules to define the judgement "to be a nat":

$$\frac{}{\mbox{zero nat}} \quad \frac{x \mbox{ nat}}{\mbox{suc(} x\mbox{) nat}}$$

then you are stating that:

closed:    


$\mbox{zero}$ is a $\mbox{nat}$ and  
if $x$ is a $\mbox{nat}$, then $\mbox{suc}(x)$ is a $\mbox{nat}$  


strongest: no other word is a $\mbox{nat}$