How did Gottlob Frege’s Begriffsschrift influence predicate logic?

Gottlob Frege’s 1879 work, Begriffsschrift, is widely recognized as the foundational text for modern predicate logic. Its influence on the field is primarily due to several revolutionary innovations:

• Quantification and Variables: Frege invented the notation of quantifiers and variables. By using variable letters (such as x) to fill argument places and prefixing them with quantifiers like "for every" or "for some," he enabled the analysis of sentences involving multiple generality—a problem that had historically hindered the advancement of logic.123
• Functional Analysis: He introduced the concept of predicates as functions, which provided a rigorous framework for logical analysis that departed from the Boolean tradition.13
• Formal Rigor: Although his unique, two-dimensional notation was difficult for contemporaries to read, the work established a clear separation between the formal characterization of logical laws and their semantic justification.12

While his specific notation was not widely adopted, his logical insights became essential to the development of modern mathematical logic and continue to serve as a primary starting point for contemporary philosophy of language.23