## Logic: Implication

In logic, a **materical conditional** is a proposition of the form *If the proposition \( P \) is true, then the proposition \( Q \) is also true*.
A material conditional is also known as an **implication** or simply as a **conditional** proposition.

An implication is denoted as \( P \Rightarrow Q \) and vocalized as "\( P \) implies \( Q \)" or "If \( P \), then \( Q \)".

By definition, the implication \( P \Rightarrow Q \) is false only when \( P \) is true and \( Q \) is false.
Because \( P \) being true is enough to make \( Q \) true, it is also said that "\(P\) is a sufficient condition for \( Q \)".

Here's the truth table of the implication.

\( P \) |
\( Q \) |
\( P \Rightarrow Q \) |

T |
T |
T |

T |
F |
F |

F |
T |
T |

F |
F |
T |

In an implication \( P \Rightarrow Q \), the proposition \( P \) is known as the **hypothesis**, **antecedent** or **premise**.
The proposition \( Q \) is known as the **conclusion**, the **consequent** or the **consequence**.

#### Examples

Consider the propositions

$$ P: \text{The integer } x \text{ is a multiple of 8} $$
$$ Q: \text{The integer } x \text{ is divisible by 2} $$

It can be shown that for the integer \( x \) when \( P \) is true, then the proposition \( Q \) is also true. In this case the implication \( P \Rightarrow Q \) denotes the proposition

$$ R: \text{If the integer } x \text{ is a multiple of 8, then } x \text{ is divisible by 2} $$