---
tag: CT4FZ84V45
kind: concept
name: Deductive closure
aliases: [closure-logic, deductive-closure-principle, deductively-closed, deductively-closed-theory, deductively-defined-theory]
sources:
  - wikipedia: en:7750539@r1302527093
    locator: Deductive closure
    note: adopted
    mention: wikipedia://en/7750539
    sections: [Summary]
same_as:
  wikidata: Q4633743
  wikipedia: Deductive closure
inventory: [ZWA2YXE8KA]
---

## Summary

In mathematical logic, a set $\mathcal{T}$ of logical formulae is **deductively closed** if it contains every formula $\varphi$ that can be logically deduced from $\mathcal{T}$; formally, if $\mathcal{T} \vdash \varphi$ always implies $\varphi \in \mathcal{T}$. If $T$ is a set of formulae, the **deductive closure** of $T$ is its smallest superset that is deductively closed.