---
tag: 0ZNX6DQ3EJ
kind: definition
name: Implication
msc: [03B05]
defines: 0NMG7W7Q11
uses: [0ZGC7ZESYN]
sources:
  - url: https://en.wikipedia.org/wiki/Material_conditional
    locator: lead section
---

## Definition

Let $p$ and $q$ be [[propositions|0ZGC7ZESYN]].
The implication from $p$ to $q$ is the proposition $p \to q$, read "if $p$, then $q$".
The proposition $p$ is the hypothesis, and the proposition $q$ is the conclusion.
The proposition $p \to q$ is false when $p$ is true and $q$ is false. In all other cases, $p \to q$ is true.

| $p$ | $q$ | $p \to q$ |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |

## Remarks

Another notation for $p \to q$ is $p \Rightarrow q$.
The proposition $p \to q$ has the same truth value as $\neg p \lor q$.
When $p$ is false, $p \to q$ is true. This is [[vacuous truth|0V4QXPGJQK]].
The [[converse|0RVYE75E2M]] of $p \to q$ is $q \to p$. It can have a different truth value.

## Explanations

### L1 human

Think of $p \to q$ as a promise: "if it rains, I take an umbrella".
You break the promise only when it rains and you take no umbrella.
When it does not rain, you keep the promise, whatever you do.