  We give a level-by-level analysis of the Weak Vop\v{e}nka Principle for
definable classes of relational structures (WVP), in accordance with the
complexity of their definition, and we determine the large-cardinal strength of
each level. Thus, in particular we show that WVP for $\Sigma_2$-definable
classes is equivalent to the existence of a strong cardinal. The main theorem
shows, more generally, that WVP for $\Sigma_n$-definable classes is equivalent
to the existence of a $\Sigma_n$-strong cardinal. Hence, WVP is equivalent to
the existence of a $\Sigma_n$-strong cardinal, all $n <\omega$.
