In some cases this is acceptable.
If you expected something on the table but the table has nothing on it.
For instance you complain to your friend about your wife. Я пришел домой, на столе ничего. You were expecting hot dinner but your wife didn't cook anything.
Hope this will help you.
So if I understand you correctly:
If you are simply describing a situation you use what in English would be a double negative.
But if referring to an unfulfilled expectation you use a single negative.
Something should be there but isn't = single negative. Where's my money, there is nothing in this envelope?
Life's a bitch and then you die kind of comment = double negative. I see nothing but doom and gloom.
Here is the information duo gives on this, I think this should help:
"Russian uses.... let's call it "consistent" negation. It means that in negative sentences you are required to use "nothing" instead of "anything", "nowhere" instead of "somewhere" and so on. Let's meet the first of these pronouns:
У меня ничего нет. = I don't have anything. Она ничего не ест. = She doesn't eat anything.
You'll also notice that, unlike standard English, Russian has no rule against using double negatives."
<столе> is in the prepositional case. This case is used after certain prepositions, especially <на> and <в>. This case usually marks words by making them end with a <е>, which turns <стол> into <столе>. You'll learn more about this case later, but for now just be on the watch for words ending in <е> if there's a <на> nearby!
@Ataque77 - It is actually prepositional case.
[На столе][ничего нет] // [prepositional][genitive]
На uses prepositional when you're describing the physical location of something ("I am already at the station - Я уже на станции"). На uses accusative when you're describing something's motion to something else (like, "I am going to the station - Я иду на станцию").
Am I right in thinking that even though this looks like "Nothing is not on the table," which in English (because of double negatives) would be "Everything is on the table," in Russian it's just "(super) Nothing is on the table"? I think I remember another sentence with double negatives that worked like this.