 We described how generalized quantifiers can be added to FO, resulting in more expressive logics. A logic in this sense roughly consist of a set of sentences, a class of models, and a truth relation (or a satisfaction relation) between sentences and models. Such logics are often called model-theoretic logics, since they are defined semantically in terms of models and truth, rather than proof-theoretically in terms of a deductive system for deriving  theorems.[11]  Here we restrict attention to logics of the form \(\FO(Q_1,Q_2,\ldots)\), formed by adding generalized quantifiers to FO, where each quantifier comes with a formation rule and a semantic clause for the truth definition as described in  section 5  above.