Definition:Conditional/Semantics of Conditional

Definition
Let $p \implies q$ where $\implies$ denotes the conditional operator. $p \implies q$ can be stated thus:


 * $q$ is true if $p$ is true.


 * (The truth of) $p$ implies (the truth of) $q$.


 * (The truth of) $q$ is implied by (the truth of) $p$.


 * $q$ follows from $p$.


 * $p$ is true only if $q$ is true.

The latter one may need some explanation. $p$ can be either true or false, as can $q$. But if $q$ is false, and $p \implies q$, then $p$ can not be true. Therefore, $p$ can be true only if $q$ is also true, which leads us to our assertion.


 * $p$ may be true unless $q$ is false.


 * $q$ is true whenever $p$ is true.


 * $q$ is true provided that $p$ is true.


 * $p$ is true therefore $q$ is true.


 * $q$ is true because $p$ is true.