 Binary quantifiers of this sort played an important role in what is perhaps the first formal study of quantification developed by Aristotle in the Prior Analytics. The details of Aristotle’s syllogistic logic are given in the entry on  Aristotle’s Logic.  Aristotle investigated a restricted class of inferential patterns, which he called syllogisms, in which two categorical propositions served as premises and a third served as a conclusion. A categorical proposition is a sentence obtained from one of four binary quantifiers “\(all(A, B)\)”, “\(some(A, B)\)”, “\(no(A, B)\)” and “\(not \ all(A, B)\)”, when suitable predicates are substituted for \(A\) and \(B\). In a syllogism, each categorical proposition contains two of three shared predicates. The logical relations, which, according to Aristotelian logic, obtain between them are codified by  the square of opposition.  Although Aristotle’s syllogistic logic dominated logic for centuries, it eventually revealed itself inadequate for the representation of mathematical argumentation. Aristotelian logic finally became displaced by the advent of modern quantificational logic, which originated with George Boole’s algebraic approach to logic and Gottlob Frege’s approach to logic and quantification (1879).