Definition 11: Given a statement \phi and a variable x , we may write \forall x \phi \hspace{3mm} \textup{for} \hspace{3mm} \neg \exists x \neg \phi ,using negation and existential quantification, and refer to this a universally quantified statement or universal statement. We shall express this as \textup{"for all } x \textup{, we have } \phi \textup{"}. We may write “each“, “every” or “any” instead of “all”, and may add “that” after “we have”. We may use the following recursively defined notation:
(a) Base cases: We understand \forall a, b \phi to abbreviate \forall a \forall b \phi .
(b) Recursion: If \forall b, c, ..., z \phi is understood, we may further write \forall a, b, c, …, z \phi \hspace{3mm} \textup{for} \hspace{3mm} \forall a \forall b, c, …, z \phi.
A statement of this form is called a generalized universal statement, or just a universal statement if the context is clear. We may express \forall a, b, ..., z \phi as “for all a, b, ..., z , we have \phi “, or similar.