Definition 6: Given statements \phi and \psi , we may use the abbreviation (\phi \vee \psi) \hspace{3mm} \textup{for} \hspace{3mm} \neg (\neg \phi \wedge \neg \psi), using negation and conjunction. We call this statement the disjunction of \phi and \psi . This is often expressed by saying \textup{"}\phi \textup{ or } \psi\textup{"}, \hspace{3mm} \hspace{3mm} \textup{"at least one of } \phi \textup{ and } \psi\textup{"}, or similar. Moreover, we may use the following recursively defined notation:
(a) Base cases: We understand (\alpha \vee \beta \vee \gamma) to mean \big( (\alpha \vee \beta) \vee \gamma \big) .
(b) Recursion: If (\alpha \vee \beta \vee \cdots \vee \psi) is understood, then we may further use the abbreviation (\alpha \vee \beta \vee \cdots \vee \psi \vee \omega) \hspace{3mm} \textup{for} \hspace{3mm} \big( (\alpha \vee \beta \vee \cdots \vee \psi) \vee \omega \big).
A statement of this form may be referred to as an iterated disjunction, or just a disjunction if the context is clear. We express (\alpha \vee \beta \vee \cdots \vee \omega) in natural language as “ \alpha or \beta or \cdots or \omega “, or as “at least one of \alpha , \beta , …, \omega “, or similar.