---
tag: 0T3DPBG981
kind: definition
name: Predicate
msc: [03B10]
defines: 0QHEWQD30B
uses: [0ZGC7ZESYN]
sources:
  - url: https://en.wikipedia.org/wiki/Predicate_(logic)
    locator: lead section
---

## Definition

Let $X$ be a set.
A predicate on $X$ is a statement $P(x)$ with a variable $x$ that ranges over $X$, such that $P(a)$ is a [[proposition|0ZGC7ZESYN]] for each $a \in X$.
More generally, a predicate $P(x_1, \dots, x_n)$ has $n$ variables, and $n$ is its arity.

## Remarks

The predicate "$x > 3$" on the integers gives the true proposition "$5 > 3$" and the false proposition "$1 > 3$".
A predicate on $X$ determines the set $\{x \in X \mid P(x)\}$ of the values that make it true. See [[set-builder notation|0TMZ89Q5KA]].
A predicate with two variables is a [[binary relation|08EJC6Q25H]].

## Explanations

### L1 human

A predicate is a sentence with a blank, such as "___ is prime".
It is not true or false yet.
When you fill the blank with a number, you get a proposition.