---
tag: C3RZY20ETK
kind: concept
name: Modal logic
msc: [03B45]
sources:
  - url: https://en.wikipedia.org/wiki/Modal_logic
same_as:
  wikidata: Q210841
  wikipedia: Modal logic
---

## Summary

Modal logic extends [[propositional logic|09K9MWD2QQ]] with operators $\square$ and $\lozenge$, read for example as "necessarily" and "possibly".
Different readings of $\square$, such as "it is known that", "it will always be true" or "it is provable that", give different modal logics.
Its usual semantics uses [[Kripke models|NGHWZ9Y7ZV]], which are sets of possible worlds with an accessibility relation.

## Explanations

### L1 human

Ordinary logic says whether a statement is true.
Modal logic can also say whether it must be true, or could be true.
The formula $\square p$ says "$p$ is necessary", and $\lozenge p$ says "$p$ is possible".
For example, $\square p \to p$ says "what is necessary is true", which holds for necessity but fails if $\square$ means "it is believed that".