I am reading through Marker's book on Model Theory and have arrived at a definition I can't unpack.
Definition 1.1.5:
The set of $\mathcal{L}$-formulas is the smallest set $\mathcal{W}$ containing the atomic $\mathcal{L}$-formulas such that
- if $\phi$ is in $\mathcal{W}$ then $\neg \phi$ is in $\mathcal{W}$
- if $\phi$ and $\psi$ are in $\mathcal{W}$, then $(\phi \land \psi)$ and $(\phi \lor \psi)$ are in $\mathcal{W}$
- if $\phi \in \mathcal{W}$, then $\exists v_i~ \phi$ and $\forall v_i~ \phi$ are in $\mathcal{W}$
What is going on in bullet 3?