How did Frege use quantifiers and predicates in his logical system?

Gottlob Frege revolutionized logic by introducing a formal system that treated predicates as functions and utilized quantifiers to analyze generality, effectively founding modern mathematical logic.1

Predicates as Functions

In Frege’s system, predicates are viewed as "unsaturated" or incomplete expressions, analogous to mathematical functions. For example, the predicate "...is a lecturer" is treated like a function that, when applied to a "saturated" term like "John," yields a complete sentence with a specific truth-value. This approach allowed Frege to isolate predicate-functions to determine the truth of complex statements based on their functional outputs.2

Quantifiers and Generality

Frege’s most significant contribution was his treatment of generality using quantifiers and variables. By using variables (such as x) as placeholders in the argument-place of a predicate and prefixing them with universal or existential quantifiers, he could "bind" the variables to express concepts like "for every x" or "there is an x." This innovation enabled the logical analysis of sentences involving multiple generality, a problem that had previously hindered the advancement of logic. Furthermore, Frege utilized "second-order" functions—functions that take other functions as arguments—to provide a rigorous account of generality.132