Smallest proposition given a state $psi$

This terminology is not familiar to me, but I can think of only one possibility :

For Hermitian operators, there is a notion of positivity (and therefore an order). In this case we would say that a hermitian projector $P$ is smaller than another $Q$ if : $$forall |varphirangle, langle varphi|P|varphirangleleq langle varphi|Q|varphirangle$$ We see that this is equivalent to the range of $P$ being included in that of $Q$. As any closed subspace containing $|psirangle$ will contain the range of $P$, we see that $P$ is the smallest orthogonal projector with $P|psirangle = |psirangle$.