 The second major historical contribution to the theory of generalized quantifiers came from the “inventor” of modern logic, Gottlob Frege, in the 1870s. In fact, Frege’s contribution is twofold. As every philosophy student knows, he introduced the language of predicate logic, with sentential connectives, identity, and the variable-binding operator \(\forall\) (though his 2-dimensional logical notation is no longer used). These are the quantifiers that logicians during the 1950’s began to “generalize”. But Frege also explicitly formulated the abstract notion of a quantifier as a second-order relation, or, as he called it, a second level concept (“Begriff zweiter Stufe”). He was well aware that the four Aristotelian quantifiers were prime examples, but he wanted to avoid the focus on subject-predicate form, which he (with much justification) saw as having been a major obstacle to the development of logic after Aristotle. It was therefore an important discovery that these quantifiers could all be defined in terms of \(\forall\) and sentential operators (replacing all\((A,B)\) by \(\forall x(A(x) \rightarrow B(x)\)), some\((A,B)\) by \(\neg\forall x(A(x) \rightarrow \neg B(x)\)), etc.).