 Frege’s logical system had several advantages over Peirce. His axiomatic presentation of a purely syntactic calculus was considerably more precise, and his analysis of the number concept went deeper. His system permitted both variables and functions to be quantified. This was a central component of his program for providing a logical foundation for arithmetic, since, in his logical system, identity, cardinal number, and mathematical induction were all defined via higher-order quantifications. In his Grundlagen (1884) he distinguishes between concepts of different order, so that if concept A falls under concept B, then B is of “second-order” (§53). In the more technical treatment in his Grundgesetze (1893) he considered third-order quantifications, though his actual derivation of arithmetic proceeded entirely within second-order logic.