It is shown that all members of the proper class of canonical structures
of
a modal logic $\ll$ have the same quasi-modal first-order theory $\pl$.
The
models of this theory determine a modal logic $\ll^e$ which is the
largest
sublogic of $\ll$ to be determined by an elementary class. The canonical
structures of $\ll^e$ also have $\pl$ as their quasi-modal theory.
In addition there is a largest sublogic $\ll^c$ of $\ll$ that
is determined by
its canonical structures, and again the canonical structures of $\ll^c$
have
$\pl$ are their quasi-modal theory. Thus $\p^{\ll}=\p^{\ll^c}=\p^{\ll^e}$.
Finally, we show that all finite structures validating $\ll$ are
models of
$\pl$, and that if $\ll$ is determined by its finite structures, then
$\p^\ll$
is equal to the quasi-modal theory of these structures.