 Now let us equip \(L\) with a model-theoretic semantics. That is, we associate with \(L\) a class of interpretations, and we make available the notions “valid in \(L\) in the interpretation \(M\)” (a.k.a.: “true in \(L\) in \(M\)”) and “semantically equivalent in \(L\) relative to \(M\).” Let the notions “valid in \(L^{+}\) in \(M\)” and “semantically equivalent in \(L^{+}\) relative to \(M\)” result when the semantics of \(L\) is supplemented with that of definition \(\mathcal{D}\). The criteria of Conservativeness and Eliminability can now be made precise thus: