As I see it, they can be solved by assuming that knowledge requires something else: a set of propositions Q which are held by the subject and which represent his background of implicit beliefs.
I will make this definition clear:
Q contains all propositions q such that:
- q is not a tautology
- q is not p
- The justification of S's belief that p depends on "S believes that q"
Equipped with this set we can formulate a slightly different JTB* account for knowledge:
"S knows that p" if and only if:
- p and each one of q is true
- S believes p and each one of q in Q
- S is justified in believing that p
Note: We do not require each q to be justified. We merely assume that those beliefs are held by S. In this sense, Q represents some belief which are held and which grant a justification for p.
Let's have a look at a Gettier problem:
"Let it be assumed that Plato is next to you and you know him to be running, but you mistakenly believe that he is Socrates, so that you firmly believe that Socrates is running. However, let it be so that Socrates is in fact running in Rome; however, you do not know this."
The key here is that "S believes that q" where q is the proposition "Socrates is next to me". Intuitively this belief justifies p ("Socrates is Running"); for if q was not believed by S, he wouldn't have a justification for p. But q is false, so this doesn't qualify as knowledge, even though p is true.
Is this a possible solution for Gettier problems? Why not?