Definition 7: Given statements \phi and \psi , we may use the abbreviation (\phi \Rightarrow \psi) \hspace{3mm} \textup{for} \hspace{3mm} \neg (\phi \wedge \neg \psi), using negation and conjunction. We call this an implication or conditional statement, with \phi being the antecedent and \psi the consequent. This is often expressed by saying that \textup{"} \phi \textup{ implies } \psi \textup{"}, \hspace{3mm} \textup{or that} \hspace{3mm} \textup{"if } \phi \textup{, then } \psi \textup{"}, \hspace{3mm} \textup{or that} \hspace{3mm} \textup{"} \psi \textup{ whenever } \phi \textup{"}. Moreover, we may use the following recursively defined notation:
(a) Base cases: We read (\alpha \Rightarrow \beta \Rightarrow \gamma) as \big( (\alpha \Rightarrow \beta) \wedge (\beta \Rightarrow \gamma) \big) .
(b) Recursion: If (\alpha \Rightarrow \beta \Rightarrow \cdots \Rightarrow \psi) is understood, we may further use the abbreviation (\alpha \Rightarrow \beta \Rightarrow \cdots \Rightarrow \psi \Rightarrow \omega) \hspace{3mm} \textup{for} \hspace{3mm} \big( (\alpha \Rightarrow \beta \Rightarrow \cdots \Rightarrow \psi) \wedge (\psi \Rightarrow \omega) \big). A statement of this form may be referred to as a generalized implication, or as “implications” if the context is clear. We may denote (\phi \Rightarrow \psi) by (\psi \Leftarrow \phi) and use the alternative notation (\alpha \Leftarrow \beta \Leftarrow \cdots \Leftarrow \psi \Leftarrow \omega) \hspace{3mm} \textup{for} \hspace{3mm} (\omega \Rightarrow \psi \Rightarrow \cdots \Rightarrow \beta \Rightarrow \alpha).