Note that this is meant in the sense: a has a belief to the effect that q, and as a matter of fact q is true, regardless of a's reasons for believing in q, and also regardless of whether or not it is in fact widely known that q is true.

1. As defined, knows is a propositional attitude defined in terms of believes, and things are intentionally left simple.

One common widely accepted definition of "knowledge" is "justified true belief", but there are several problems with the "justified" part, that is left out above.

One reason is that one may be justified in quite a good sense yet be mistaken, as happens in science, and as may happen to oneself if one believes one's car keys are in the trash-can because one's little daughter has put them there, whereas they are there but for another reason, and a third is that the whole notion of justifying a belief is none too clear.

2. A complication of the above, that is in fact quite welcome for many ends, is that (q) itself may be a proposition that attributes some sort of probability, as in "a knows that this fair coin has a probability of falling heads of 1/2".

This is a complication because it makes the analysis of what is involved in knowing more complicated, but it is welcome because the analysis is more subtle and anyway all thinking human beings believe in many probabilistic propositions.

3. It is quite consistent with the given definition that a sincerely believes that a knows q, and that a is completely mistaken - that is, a is right in maintaining (a believes q) but as a matter of fact (~q). In brief, one may believe one knows and be mistaken.

Indeed, since it seems quite true of everyone that one knows about some of one's earlier beliefs that they turned out to be mistaken, it follows that everyone knows something, if only about one's own beliefs, one's own lack of knowledge, or one's knowing the natural language with which one can state claims to the effect that one knows almost nothing and knows one knows this.

Now the proposed minimalistic definition of knows is interesting also in that it is a conjunction of a propositional attitude of a person (a belief) and a statement of fact - while it would seem as if the latter, even if true, cannot be known by the person except in the form of a propositional attitude.

For the reasons just given it seems quite sensible to assume that every living person knows something, if only that one is often mistaken about what one believes one knows. That is formally: (a)(Eq)(aKq), where "K" is short for "knows".

And this again is quite compatible in principle with knowing one knows some things without knowing which, except for this, or - far more realistically - with knowing that one probably knows some things and certainly is mistaken about others, without presently knowing which.

Incidentally, in these formulations a non-constructive use is made of the existential quantifier: If one knows one is mistaken about some things and doesn't know which, one cannot instantiate them other than by artiticial formulas like "Let t* be something I believe I know but am mistaken about while I don't know more about it". (The tip of the tongue phenomenon - "There is a name for this thing, I know, but I just now can't recall it" - is another example of such an existential quantifier.)