  In an attempt to realize Leibniz’s ideas for a universal formal language and a rational calculus, Frege developed a formal notation for regimenting thought and reasoning. Though this notation was first outlined in his Begriffsschrift (1879), the most mature statement of Frege’s system was in his 2-volume Grundgesetze der Arithmetik (1893/1903). Frege’s two systems are best characterized as term logics, since all of the complete expressions are denoting terms. Frege analyzed ordinary predication in these systems, and so they can also be conceived as predicate calculi. A predicate calculus is a formal system (a formal language and a method of proof) in which one can represent valid inferences among predications, i.e., among statements in which properties are predicated of objects.