You may believe that I don't know whatever I say I know, but then your belief is only a belief whereas what I know is actual knowledge.
That I can't justify that I know what I know doesn't make my knowledge disappear in a puff of logic.
Do you not know that reality exists? I hope you do. Yet, how do we justify that we do?
Me, I don't know, and I certainly don't think anyone does.
Asking me to justify that I know what I know is like asking me to walk the shortest route between where I am and where I am. Sorry, no, the shortest route is that I don't need to walk at all.
How do you explain that there is something rather than nothing? I don't think anyone can explain that but whether we can explain it is irrelevant to the existence of reality. That there is something doesn't depend on our ability to explain why.
There is no solution to the Münchhausen trilemma.
However, the trilemma is only effective against the claims to knowledge we make about things we don't actually know, such as for example whether there is a material world as we think of it, or whether the tree I am looking at really exists as I naively think it does. The absence of proper justification shows it isn't knowledge but rational belief.
Yet, we don't need to know whether there really is a material world as we naively think there is. We only ever believe there is one and this is good enough. What matters is that we should exchange on what is best to believe and that our beliefs should help us survive and prosper. So far so good.
And then if most people can't resist calling our rational beliefs "knowledge", then let them justify their claims to knowledge.
There is no difficulty justifying one's rational beliefs, at least on principle. And if our beliefs turn out to be false, we just adopt different beliefs. This is the way science works. We started with simple theories and we just adopted new theories to accommodate discoveries that falsified our initial theories.
We have a long experience now of scientific theories being falsified and replaced by what we see as better theories. We also all have first hand experience of having our personal beliefs being falsified again and again. But we are never short on new beliefs to replace them and in any case we don't know how to know the world. We just keep going regardless just because we can.
Nobody can justify that science is knowledge but there is no difficulty articulating a good justification that science is our best belief. And if that is not even true, then we may have to change our belief at some point in the future. Meanwhile, we will keep relying on science.
The solution to the Münchhausen trilemma is simple. We just have to admit that we don't know what we don't know. And I think we can all live with the fact that we still won't be able to justify that we know what we do know.
While we don't seem to know the material world as we think of it, we do know quite a few things. Broadly, what we know are our qualia. Whenever I look--or rather believe that I am looking--at the blue sky, what I cannot deny is the blueness I subjectively experience. The suggestion that I don't know this blueness at the moment that I experience it is merely nonsensical. What I don't know, is whether there is really a sky as I think of it and that this sky is really of the same blue as the blueness I experience.
Obviously, I cannot prove to anyone else I know the blueness I experience, but I also don't need to. In fact, I don't even need to justify it to myself. All I need is to fail to be able to deny this blueness.
And I also don't need to believe, let alone know, that other people experience the same blueness. As a matter of fact, I don't think they do. All we need is that we should be able to communicate effectively and cooperate, i.e. agree on whatever actions we need to carry out.
Different computers using different operation systems and different standard of internal data coding can communicate with each other. We can communicate to some surprising extent with dogs and cats and most animals. And we act on the basis of this communication. And, although it is often protracted and laborious, we seem to be able to communicate and cooperate with each other.
Indeed, it is arguable that human beings are the most successful species in that respect, with now some degree of communication and cooperation and understanding extending over a big chunk of the 7 billion people living today on Earth. No only that, but we can still read, and understand, and therefore communicate, if only one-way, with many long-dead people like Aristotle and many others.
The Münchhausen trilemma is a necessary argument because it successfully falsify our many claims to knowledge and claims to knowledge are often made in support of bad ideologies. However, it would be a mistake to interpret the dilemma as proving we don't know anything. I certainly know pain whenever I am in pain and I have no good reason to believe that other people don't.
Our individual ability to know what we know is also the basis of our cooperation as a community of rational beings. The Münchhausen trilemma is a salutary reminder that humanity exists as the cooperation of individual human beings, which itself relies on the human individuals' capacity to cooperate.