Predicate logic

Summary

Predicate logic extends propositional logic with variables for objects, predicates and quantifiers. Its formulas can say "for all $x$" with the universal quantifier and "there exists $x$" with the existential quantifier. In first-order logic, the quantifiers range over objects, not over sets of objects or predicates.

Remarks

Other names are first-order logic and predicate calculus. An example of a formula is $\forall x \, \exists y \, (y > x)$. Most of mathematics can be written in first-order logic with the axioms of Zermelo–Fraenkel set theory.

Explanations

L1 human

Propositional logic cannot say "every number has a larger number". Predicate logic can: it has variables such as $x$ and $y$, and words such as "for all" and "there exists". This is the logic of most mathematical statements.