How did Giuseppe Peano influence the symbols used in modern logic and set theory?

Giuseppe Peano was a pivotal figure in the development of symbolic logic, and his influence on modern mathematical notation was profound, often exceeding that of his contemporaries like Gottlob Frege and Charles Sanders Peirce.1

Peano’s primary contributions to logical and set-theoretic symbolism include:

• Set Membership: He introduced the use of the Greek letter epsilon ($ϵ$) to denote membership in a set, a symbol that remains standard in set theory today.12
• Quantifiers: While Peano adopted the concept of quantifiers from the work of Ernst Schröder (who had been influenced by Peirce), Peano’s specific notation for these quantifiers was adopted by Bertrand Russell and Alfred North Whitehead in their influential work, Principia Mathematica.134
• Formalization: Through his Formulaire de mathématiques, Peano sought to develop mathematics entirely from fundamental postulates using his own logic notation. Although his specific system was sometimes criticized for being difficult to read, his notation became a foundational element for later symbolic logic.51

By bridging the gap between the older algebraic tradition of logic and the modern nonalgebraic approach, Peano helped establish the formal language that underpins contemporary mathematics.1