We study the following problem: Given a transition system T and its quotient T/∼ under an equivalence ∼, which are the sets L, L′ of Hennessy-Milner formulae such that: if ϕ ∈ L and T satisfies ϕ, then T/∼ satisfies ϕ; if ϕ ∈ L′ and T/∼ satisfies ϕ, then T satisfies ϕ.