---
tag: 0V4QXPGJQK
kind: concept
name: Vacuous truth
msc: [03B05]
uses: [0ZNX6DQ3EJ, 0B88PQTJ66]
sources:
  - url: https://en.wikipedia.org/wiki/Vacuous_truth
    locator: lead section
same_as:
  wikidata: Q1096079
  wikipedia: Vacuous truth
---

## Summary

A vacuous truth is an [[implication|0ZNX6DQ3EJ]] or a universal statement that is true because its hypothesis is never true.
An implication $p \to q$ with a false $p$ is true, and a statement $\forall x \in X, \, P(x)$ with an empty $X$ is true.

## Remarks

Example: "each element of the [[empty set|0ECVK59Y03]] is an even number" is vacuously true.
Vacuous truth explains why the empty set is a [[subset|0FE8Q5S42Y]] of every set.

## Explanations

### L1 human

"All unicorns in this room are blue" is true when the room has no unicorns.
There is nothing that can break the statement.
This convention keeps the rules of logic simple and regular.