  Continuing the previous paper, we study the Strong Downward
L\"owenheim-Skolem Theorems (SDLSs) of the stationary logic and their
variations. It has been shown that the SDLS for the ordinary stationary logic
with weak second-order parameters down to $<\aleph_2$ is equivalent to the
conjunction of CH and Cox's Diagonal Reflection Principle for internally
clubness. We show that the SDLS for the stationary logic without weak
second-order parameters down to $<2^{\aleph_0}$ implies that the size of the
continuum is $\aleph_2$. In contrast, an internal interpretation of the
stationary logic can satisfy the SDLS down to $<2^{\aleph_0}$ under the
continuum being of size $>\aleph_2$. This SDLS is shown to be equivalent to an
internal version of the Diagonal Reflection Principle down to an internally
stationary set of size $<2^{\aleph_0}$. We also consider a ${\cal
P}_\kappa\lambda$ version of the stationary logic and show that the SDLS for
this logic in internal interpretation for reflection down to $<2^{\aleph_0}$ is
consistent under the assumption of the consistency of ZFC $+$ "the existence of
a supercompact cardinal" and this SDLS implies that the continuum is (at least)
weakly Mahlo. These three "axioms" in terms of SDLS are consequences of three
instances of a strengthening of generic supercompactness which we call
Laver-generic supercompactness. Existence of a Laver-generic supercompact
cardinal in each of these three instances also fixes the cardinality of the
continuum to be $\aleph_1$ or $\aleph_2$ or very large respectively. We also
show that the existence of one of these generic large cardinals implies the
"$++$" version of the corresponding forcing axiom.
