  We ask whether $\mathbf{\Delta^1_2}$ or $\mathbf{\Sigma^1_2}$ equivalence
relations with $I$-small classes for $I$ a $\sigma$-ideal must have perfectly
many classes. We show that for a wide class of ccc $\sigma$-ideals, a positive
answer for $\mathbf{\Delta^1_2}$ equivalence relations is equivalent to the
$I$-measurability of $\mathbf{\Delta^1_2}$ sets. However, the analogous
statement for $\mathbf{\Sigma^1_2}$ equivalence relations is false:
$\mathbf{\Sigma^1_2}$ equivalence relations with meager classes have a perfect
set of pairwise inequivalent elements if and only if $\mathbf{\Delta^1_2}$ sets
have the Baire property.
