" To a certain extent I think that the phrase "I am conscious of nothing" to be a logical absurdity "
It is. That's what made the experience so interesting. But "logical absurdity" only means we don't understand it.
" I have experienced the phenomenon you're talking about, but I wouldn't call it being conscious of nothing. "
If you won't call it 'being conscious of nothing', I doubt we're talking about the same thing. But that is only a guess.
" Being conscious of nothing is something I equate with oblivion, the state my "consciousness" was in before I was born. "
To me that is best described as unconscious. To experience nothing is not the same as not experiencing anything.
" Lying there with an empty head is something I'd call extremely sedate consciousness or near-unconsciousness. "
I really don't think we're talking about the same thing. I didn't feel sedate or semi-conscious, I felt fully awake and my mind was very focussed. I was aware of everything that was going on in the room, all the smells, noises, vibrations, but somehow I felt completely detached from my senses. It was weird.
" ~~"something other than logic is required to decide which is true and which is false."~~
Evidence? Or intuition? Which do you prefer? (or need I ask?) "
According to Godel, if I'm not misinterpreting him, here is what happens: all logical systems can be used to express questions for which there are no answers within the system itself. Those answers may be found in a larger, more complex logical system, but that logical system can also be used to express questions for which there are no answers, and so on and on.
So, in answer to your question, it's neither evidence nor intuition, just more logic that pushes the problem "one level up". That is, you can never have all the answers, but you can have as many of them as you want.