1

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?

CUPA
  • 79
  • 2
    The negation of "some" is not "all". "Some" means "at least one". Thus, the negation of "some" is "not some", i.e. "all not". – Mauro ALLEGRANZA Feb 01 '19 at 13:33
  • "always" is tricky here; if you have not a specific need to express the temporal fact, we can symply say : "Some operating systems always-crash", i.e. $\exists x (\text {OpSys}(x) \land \text {Crash}(x))$. – Mauro ALLEGRANZA Feb 01 '19 at 13:38

1 Answers1

0

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".

Henno Brandsma
  • 242,131