 Disjunctive syllogism is also invalid (or better quasi-valid) in Priest’s Logic of Paradox (LP). In PL both \(1\) and \(\#\) are designated values (i.e., preserved in valid inferences), where # stands for both true and false. More formally, validity is defined as follows: \(\Sigma \models \phi\) iff there is no valuation \(v\) such that for all \(\psi \in \Sigma: v(\psi) = 1\) or # and \(v(\phi) =0\). Adopting the strong Kleene tables for disjunction and negation presented above, we obtain that \(\phi, (\neg \phi \vee \psi) \not\models \psi\), because \(\phi\) and thus also \((\neg \phi \vee \psi)\) might be both true and false, while \(\psi\) is false. The schema \(\phi , (\neg \phi \vee \psi) \models \psi\) however only fails in case \(\phi\) is a dialetheia (both true and false), in this sense is the scheme quasi-valid. Priest (2006: ch. 8) then argues that applications of disjunctive syllogism can be justified/are reasonable in case \((\phi \wedge \neg \phi)\) is rationally rejectable. 