Paired forcing name ($\nu (\tau ,{\tau }^{\prime }{)}_G=\{{\tau }_G,{\tau }_G^{\prime }\}$)

For the condition-first forcing name convention, pair every condition with each of two names $\tau ,{\tau }^{\prime }$. Under any nonempty generic filter, the evaluation of a forcing name is then $\{{\tau }_G,{\tau }_G^{\prime }\}$, the unordered pair of the two values. Equal values give a singleton. With the name-first convention, reverse the two components of each syntactic pair.