When you think about it, how can something be 'all-knowing'? Say that I am all-knowing (although I'm not). Now, that would mean that I know that I am all-knowing, because I know everything. But how can one know that one knows everything? If one doesn't know something, then one doesn't know that one doesn't know it, hence, one could think one knows everything, but actually not know everything. Hence, is it actually logically possible for a being to be all-knowing at all?
It is not logically-impossible for a being to be all-knowing. The difficulty would be for any being to know for certain that they are all-knowing, since he cannot know whether there are still things he does not know. However, for such an all-knowing being to not know that he is in fact all-knowing would be a contradiction.
Therefore, an all-knowing being would also have to be somehow certain that he knows everything (through means unimaginable to us who are dangerously close to the other end of the spectrum). In this case there would be two types of beings who consider themselves all-knowing: the ones who only think they are all-knowing but aren't, and the ones who think they are all-knowing and are.
My 5 cents - not very scientific and most likely flaw logic:
'all knowing' being must have abilities:
1. Means of observing in all the things in the universe that he is in.
2. Means of processing all the information that he is observing at once (to know).
To do that he would have to take up all the space in that universe, and be universe itself (bigger the universe, larger the being has to be to observe and process everything what's going on).
This could be reduced to: that being that is the only thing in the universe that he is in that knows everything about itself could be considered 'all knowing'. He would then be all knowing as there would not be anything he does not know (in that universe).
The possibility of an all-knowing being appears to lead to a contradiction because the being would have to have a "mental" representation that included itself (including the "mental" representation and any parts of itself that are not part of that representation) and of any parts of the universe that are not parts of the being itself. In short, the being would have to be bigger than itself, which is a contradiction.
There are two simple (philosophically, if not practically) solutions to that contradiction:
- The being is itself the entirety of all it is, and the entirety of all that is is its own representation, which is a possible view in Pantheism, for example; or
- The being is infinite, because Leibniz showed than an infinity can contain itself and other things too. In a finite universe, that would mean that the being could not be a material being, but in, for example, the Abrahamic religions, God is identified as being "spirit". It's not clear what that is, but it's pretty clear that it's not material and so is not constrained by finite limits on the material (and might get around the issues of causality entailed in knowledge of the future).
The concept of an all-knowing being is therefore not logically contradictory, although it does have some major implications for what such a being could be like.