 This implies the statement given for consequence relations (putting “⊩” in place of “⊢”) since a generalized consequence relation, like a consequence relation, is always determined by the class of all valuations consistent with it. (But only the weaker statement above is available for consequence relations, as such cases as that of ∨ illustrates, since the consequence relation determined by the class of ∨-Boolean valuations—with respect to to which ∨ is by definition truth-functional—is also determined by other classes of valuations, with respect to it is not.) Indeed another way of putting the contrast between the consequence relations claim above and the generalized consequence relations claim is that while for consequence relations #‘s being fully determined according to ⊢ is equivalent to some class of valuations which determines ⊢ having # truth-functional with respect to it, for generalized consequence relations we can say that #‘s being fully determined according to ⊩ is equivalent to every class of valuations which determines ⊩ having # truth-functional with respect to it.