First-order logic (FOL) provides the formal language in which most of mathematics is written. Unlike propositional logic, which deals only with true/false statements, FOL introduces quantifiers and predicates that allow us to express much more complex ideas.

The two fundamental quantifiers in FOL are:

  • Universal quantifier (∀): “for all”
  • Existential quantifier (∃): “there exists”

These powerful tools allow us to express statements like:

  • ∀x ∃y (y > x) - “For every number, there exists a larger number”
  • ∀x (x = x) - “Everything is equal to itself”

The precise syntax and semantics of FOL provide the rigor needed for mathematical proofs while remaining intuitive enough to express natural mathematical ideas.