That depends, of course, on what you mean by "know". On one well-known, though flawed, definition, to know that P is to have a justified true belief that P. Suppose that you are willing to say that a computer believes-true a proposition P if P is true and the computer has (a representation of) P stored in its database. And suppose that a computer would be justified in such a belief if it deduced it from axioms (also stored in its database). Because some computers can deduce propositions according to rules of logic, such a computer could, on this definition of "know", know that P. Sometimes, people feel that to know something is also to be aware that one knows it, or that one knows that one knows it. So, could a computer know something in this sense? That, of course, depends on what might be meant by "aware". But it is certainly possible for a computer to believe that it believes something (for example, in the sense that it not only has P stored in its database, but also has something like "I...

From a cognitive science perspective (as opposed to a purely philosophical perspective), you should take a look at Daniel Kahneman's book, Thinking Fast and Slow I have a bibliography on the topic (again from the cog sci perspective) here Hope this helps!