Definition 8: Given statements \phi and \psi , we may use the abbreviation (\phi \Leftrightarrow \psi) \hspace{3mm} \textup{for} \hspace{3mm} \big( (\phi \Rightarrow \psi) \wedge (\psi \Rightarrow \phi) \big), using implication and conjunction, and we refer to this an equivalence or biconditional statement, with \phi being the left side and \psi the right side. This is often expressed as \textup{"}\phi \textup{ if and only if } \psi \textup{"} \hspace{3mm} \textup{or as } \hspace{3mm} \textup{"}\phi \textup{ is equivalent with } \psi \textup{"}. Moreover, we may use the following recursively defined notation:
(a) Base cases: We read (\alpha \Leftrightarrow \beta \Leftrightarrow \gamma) as \big( (\alpha \Leftrightarrow \beta) \wedge (\beta \Leftrightarrow \gamma) \big) .
(b) Recursion: If (\alpha \Leftrightarrow \beta \Leftrightarrow \cdots \Leftrightarrow \psi) is understood, we may further use the abbreviation (\alpha \Leftrightarrow \beta \Leftrightarrow \cdots \Leftrightarrow \psi \Leftrightarrow \omega) \hspace{3mm} \textup{for} \hspace{3mm} \big( (\alpha \Leftrightarrow\beta \Leftrightarrow \cdots \Leftrightarrow \psi) \wedge (\psi \Leftrightarrow \omega) \big). A statement of this form may be referred to as a generalized equivalence, or refer to it as stating several “equivalences“.