Where $\vee$ and $\neg$ are disjunction and negation connectives, respectively, the inference that from the joint assumptions $\phi \vee \psi$ and $\neg \psi$, one may legitimately infer $\phi$. It may be more formally represented as the inference:

When $\to$ is a material conditional connective, i.e., when $\phi \to \psi$ is equivalent to $\neg \phi \vee \psi$, disjunctive syllogism is equivalent to the validity of modus ponens for the material conditional. The validity of disjunctive syllogism is frequently rejected by proponents of paraconsistent logics, i.e., those who accept that there are models in which both $\phi$ and $\neg \phi$ are true although there exists a formula $\xi$ that is not true. In such models, the truth of $\phi$ in many paraconsistent logics entails the truth of $\phi \vee \xi$ and $\neg \phi$ is assumed true, although $\xi$ is assumed to fail.