# 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.

## Examples

The statement:

If I pass this course, then (it shows that) I have studied hard for it.

may be rephrased as:

I will pass this course only if I have studied hard for it.
To prove that I have studied hard for this course, it is sufficient to know that I passed it.
For me to pass this course, it is necessary for me to study hard for it.