In logic, an implication, also known as a material conditional or material consequence, is a binary, logical operator that defines a relationship between two statements, an antecedent and a consequent. Implications are most often symbolized using a right arrow, →. Given two statements *p* and *q*, a conditional statement of the form *p*→*q* is read as “if p then q.” In this statement, *p* is the antecedent and *q* is the consequent.

A material conditional statement, an implication, does not imply causality. *p* does not necessarily cause *q*, but whenever *p* is true, so is *q*. An implication tells us nothing of the truth value of *p* when we only know that *q* is true. In this case *p* can be either true or false. The only way the state *p*→*q* is false is if *p* is true and *q* is false.

The truth table for *p*→*q* is:

p |
q |
p→q |
---|---|---|

T | T | T |

T | F | F |

F | T | T |

F | F | T |

As the truth table above shows *p*→*q* is equivalent to ¬*p* ∨ *q* (not *p* or *q*).

*p*→*q* is also equivalent to ¬*q*→¬*p* (not *p* implies not *q*). This is called a contraposition.

Importantly, *p*→*q* is not the same ¬*p*→¬*q.* This called an inversion, and it is not true. Not *p* **does not imply** not *q*.

¬*p*→¬*q* the same as saying *q*→*p* (*q* implies *p*), called a conversion, which is not true because, as was stated above, knowing the truth value of *q* tells us noting about the truth value of *p*.