 Forcing name for the complement of a generic filter

The forcing name $\{(\check{p},q):p,q\in \mathbb{P},q\perp p\}$ evaluates to $\mathbb{P}\setminus G$. If $p\notin G$, genericity applied to $\{q:q\le p\text{ or }q\perp p\}$ supplies an incompatible member of $G$. If $p\in G$, directedness prevents such a member.