I am trying to find the negation of the statement "Some operating systems always crash"
I know that the negation of "some" is "all" so:
All operation systems always crash ? Or: All operation systems never crash ?
I don't understand what to do with the "always" in this statement. Does anyone know the answer to this?
One way to look at the statement "some operating systems always crash":
$\exists O \in \textrm{OS}: \forall t: c(O,t)$, where I use OS for the set of operating systems and $t$ quantifies over times, $c(O,t)$ means the OS $O$ crashes at time $t$.
If you agree that this is the intended translation (this is a matter of linguistic discussion) then its logical negation is
$\forall O \in \textrm{OS}: \exists t: \lnot c(O,t)$, by the usual rules for first order logic, which in English I would translate as "every operating system sometimes doesn't crash".