Let $E$ be a formal ethical system with propositions $E_P$, conclusions $E_C$ such that:
$$
\forall c \in E_C, \exists P \subseteq E_P : (\prod_{i=0}^{|P|} p_i \vdash c), p_i \in P
$$
Where $\prod_{i=0}^{|P|} p_i \equiv p_1 \land p_2 \land … \land p_{|P|}$
Let $A_U$ be the universal action space, the set of all possible actions, $I(E) \Leftrightarrow E$ is an inconsistent ethical system, $G(a) \Leftrightarrow a$ is a good action under $E$ and $X(a) \Leftrightarrow a$ is an evil action under $E$.
$$
\begin{align}
&\text{Lemma } 1.1) \quad \left( \exists p \in E_P : E_P \vdash (p \land \neg p) \right) \Rightarrow I(E) \\
&\text{Lemma } 1.2) \quad I(E) \Rightarrow G(a) \in E_C, \forall a \in A_U \\
&\text{Lemma } 1.3) \quad \left( \exists a \in A_U : X(a) \right) \\
&\text{Lemma } 1.4) \quad I(E) \Rightarrow \left( G(a) \in E_C, \forall a \in A_U \right) \land \left( \exists a \in A_U : X(a) \right) \dashv (1.2 \land 1.3) \\
&\text{Lemma } 1.5) \quad \text{JustifiesEvil}(E) \Leftrightarrow (\exists a \in A_U : E(a) \land G(a) \in E_C) \\
&\text{Therefore}) \quad I(E) \Rightarrow \text{JustifiesEvil}(E) \dashv (1.4 \land 1.5)
\end{align}
$$
It has hence been shown that an inconsistent ethical system necessarily justifies evil.
