---
tag: 086QKPF0A6
kind: concept
name: Predicate logic
aliases: [first-order-logic, predicate-calculus]
msc: [03B10]
uses: [09K9MWD2QQ, 0T3DPBG981]
sources:
  - url: https://en.wikipedia.org/wiki/First-order_logic
    locator: lead section
same_as:
  wikidata: Q4055684
  wikipedia: First-order logic
---

## Summary

Predicate logic extends [[propositional logic|09K9MWD2QQ]] with variables for objects, [[predicates|0T3DPBG981]] and quantifiers.
Its formulas can say "for all $x$" with the [[universal quantifier|0B88PQTJ66]] and "there exists $x$" with the [[existential quantifier|0JW2FHW7G3]].
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|0HE5WDQ9FJ]].

## 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.