Definition:Conditional/Semantics of Conditional

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


 * If $p$ is true then $q$ is true.


 * $q$ 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.


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


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


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


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


 * Given that $p$ is true, $q$ is true.


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


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


 * $q$ is true in case $p$ is true.


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


 * $q$ is true on the condition that $p$ is true.

Further colloquial interpretations can often be found in natural language whose meaning can be reduced down $p$ only if $q$, for example:


 * $p$ is true as long as $q$ is true


 * Example:
 * "Mummy, can I go to the pictures?"
 * "As long as you've done your homework. Have you done your homework? No? Then you cannot go to the pictures."
 * In other words:
 * "You can go to the pictures only if you have done your homework."
 * Using the full language of logic:
 * "If it is true that you are going to the pictures, it is true that you must have done your homework."


 * $p$ is true as soon as $q$ is true


 * "Are we going to this party, then?"
 * "As soon as I've finished putting on my makeup."
 * The analysis is the same as for the above example of as long as.