Necessary Extension

Given a semantics $\sigma$, a set $E \subseteq A$ is a necessary $\sigma$-extension of an iAF $I = (A, A^?, R, R^?)$ iff $E \in \sigma(F^*)$ for every completion $F^*$.

Remarks

This is the necessary verification of (Baumeister et al., 2018) for definite sets; (Baumeister et al., 2018) also admits sets $S \subseteq A \cup A^?$ and then asks whether $S \cap A^*$ is a $\sigma$-extension of every completion.