\begin{definition}[Definition:Conditional/Semantics of Conditional]
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'''.
\end{definition}