We consider many propositions about the world as either true or not true.
However, in my view, human knowledge is not a collection of true and false propositions. What we consider as evidently true are only propositions about the world that we believe to have a very small probability to be false.
The above is still compatible with an objective Reality, however most of our knowledge about Reality is highly mediated, so that we can rarely be fully certain about something.
Mount Everest
I "know" that the highest mountain in the world is Everest (from the sea level). However, I never seen that mountain. Even if I could visit and measure mount Everest, it is not realistic for me to check all the other mountains in the world to verify that there is no higher mountain somewhere. Think Popper.
I only believe that Everest is the highest mountain in the world because a lot of sources tells me this (externalism). It is unlikely that any of my sources verified personally that there is no higher mountain than Everest. However, they also believe other people that checked the other mountains and reported their height.
Few people are equipped to actually measure the altitude of a mountain. Even if someone has the knowledge and the tools to measure a mountain, he/she will likely need to base the measurement on other believes that they cannot personally check - like the pressure at the sea and the law of pressure variation with the altitude.
Therefore, I cannot have 100% certainty that the highest mountain on earth is Everest. However, as I find this information from many sources, it becomes less and less probable that the opposite would be true. It is unlikely that no one discovered a higher mountain. It is very likely that whoever would discover a higher mountain that Everest would proudly report about such discovery.
I base the above assumption on my beliefs on the human nature, based on my limited sample of people. It is still possible for a guild of people that actively hides certain information from the other people, but my experience tells me that this possibility is very unlikely.
Turkey's inference
I found Bayes inference to be very appealing to model the way we can create our beliefs based on previous beliefs. As we get more and more data about certain relations about facts, the belief on a certain proposition increases.
However, the Bayes inference is not without caveats. A turkey can see that each day it receives good food and shelter and no harm is done to it. From this data, the turkey can never infer that it will be sacrificed on Thanksgiving (credits Nassim Taleb). How can we really create reliable inferences about the world?
Reverse inference verification
I think that it is more useful to reverse the Bayes formula. A rational turkey would need to actively check for instances that would refute his inferred rule that it will live happily until old ages - to falsify his belief. The turkey should move his inference about him to a higher space, of all the turkey in the yard.
The turkey's hypothesis is: all turkeys (that are well cared in this yard) will live until their old age.
We can write this as a P -> Q proposition:
P: is turkey in this yard
Q: turkey will live a long life without problems
In order to verify P->Q, one should check for instances of P that are not Q. Of course, we need to be able to investigate the life of sufficient P. This is theoretically achievable only if turkeys would be able to communicate with each-other.
Taking the case of P (turkeys in the yard), we should actively check the probability for them to have the Q property. We can exclude young turkeys that cannot tell us if the Q will be true for them in fact. We need to focus on old enough turkey that can still testimony about their life.
Failing to find old turkeys in the yard makes it impossible to verify the hypothesis. Even if there are couple of old turkeys in the yard (think president's pardon), a careful examination would still reveal that there are not enough old turkeys to sustain the hypothesis that all/most turkey live a long and happy life.
The sudden disappearing of some turkeys are also increasing the doubt about the hypothesis. Even if we don't find evidence that a turkey will likely end in an oven, the inference based on the life before cannot be sustained by observation on other individuals.
The KK problem revised
In my view, there is no dichotomy from having knowledge on P and believing that P. Knowledge is for me a degree of belief in a proposition P, based on some information you received and some other beliefs you hold. If that proposition P has the potential for promoting adaptive actions, we can say you have a certain knowledge about the world. A few of the beliefs might be innated.
It is possible for you to believe by accident on something "true" by the wrong reasons. While you cannot get credits for your conclusion, you are still in possession of knowledge. You actions based on your beliefs can benefit from that "knowledge".
We should differentiate between a reliable method of obtaining a knowledge and having that knowledge. If you happen to be born with a fear of snakes, you possess a native knowledge telling you that many snakes are dangerous - even if nobody told you this.
When you also have arguments for your belief, you possess a skill that can provide you with other reliable beliefs. This skill is arguably more valuable than a belief for the wrong reasons, but this does not take the value of the belief that increase the adaptation of the individual. Such "beliefs for the wrong reasons" might come from various religions for example. We still talk about the likelihood outcome of some actions and not about absolute truths.
For example, If you believe that daemons from a ill person can enter your body and stay away from such ill people, you are in possession of a piece of knowledge about the mechanics of transmissible diseases, even if you have a wrong explanation for it. As this belief can have surviving advantages, it still counts as knowledge for me - even if the explanation is not accurate in this case.
I my view, a belief can only be evaluated from outside based on observable actions. If you act like you posses a certain belief, we can tell that you hold that belief at that moment. Beliefs that do not translate into actions cannot be proven to objectively exist.
Knowledge from other people
Most of our "knowledge" comes from other people.
Let's take a person X that tells you a proposition P.
For example X is a colleague and tells you proposition P="your brother called you". It is implicit that he wants to transmit that "your brother called you" is true.
If X tells you that P is true, you don't actually have knowledge the P is true. You just know that X told you that P is true.
Most likely (but not necessary), you can assume that X wants you to believe that P is true. If you don't have suspicions on X wanting to misinform you, you can safely assume that X believes that P is true.
You don't yet believe that P is true.
Depending on your believe in the competence on X on P domain, you can have different degree of belief that P is true - normally less than the belief in that X believes that P is true.
Therefore, knowledge is mostly based on high degree of belief in a certain propositions that helps you adapt to environment, often based on beliefs of others.
The most rational way to form strong beliefs is to mentally evaluate the likelihood for the opposite to be true.
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