The context for this paper is Feferman's theory of explicit mathematics, $\mathbf{T_0}$. We address a problem that was posed in [6]. Let $\mathbf{MID}$ be the principle stating that any monotone operation on classifications has a least fixed point. The main objective of this paper is to show that $\mathbf{T_0} + \mathbf{MID}$, when based on classical logic, also proves the existence of non-monotone inductive definitions that arise from arbitrary extensional operations on classifications. From the latter we deduce that $\mathbf{MID}$, when adjoined to classical $\mathbf{T_0}$, leads to a much stronger theory than $\mathbf{T_0}$.