Definition:Conditional/Semantics of Conditional

From ProofWiki
Jump to navigation Jump to search


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.


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.