---
tag: 1HK0VYWTV7
kind: concept
name: Logical consequence
msc: [03B05, 03B10]
sources:
  - url: https://en.wikipedia.org/wiki/Logical_consequence
same_as:
  wikidata: Q374182
  wikipedia: Logical consequence
---

## Summary

A formula $\varphi$ is a logical consequence of a set of formulas $\Gamma$ when every interpretation making all of $\Gamma$ true makes $\varphi$ true.
This is written $\Gamma \models \varphi$.
It is the semantic notion of "follows from", in contrast with derivability $\Gamma \vdash \varphi$ in a proof system.
[[Soundness|4A84VHDTHV]] and [[Gödel's completeness theorem|WQ1ZE255SF]] say that the two notions agree in first-order logic.

## Explanations

### L1 human

A conclusion follows logically from premises when there is no way to make the premises true and the conclusion false.
For example, from "if it rains, the street is wet" and "it rains", it follows that "the street is wet".
This depends only on the form of the sentences, not on facts about weather.