According to Bertrand Russell (at least at one point in his career), not only is a belief not knowledge if it can't be substantiated with arguments, reasons or other beliefs, but it's not knowledge if it is derived from false beliefs which may have acted as reasons for the new belief.
So, in theory, I could derive the true belief that the earth isn't flat from the true belief that most swans are white. Clearly, in this case, there's no clear material connection between the two beliefs. If they weren't connected in some way by the subject, then such connections would have been either bogus or irrelevant – that is, not genuine connections.
This may be why logicians say that "from a true premise anything can be derived". That is, unless we require a material or relevant derivation from the true premise.
A true and materially connected belief for "The earth is not flat" may be "The horizon shows us that the earth is not flat’" Here, of course, there is a material and relevant connection with initial true belief.
Similarly, it's quite possible to deduce a true belief from a false belief. In this case, even if the false belief is somehow materially connected to the true belief, one still can't acceptably deduce a true belief from a false belief.
So, for example, we may derive from ‘Fish can fly’ the true belief that ‘Fish generally have scales’. Here there seems to be some kind of material connection in that both beliefs are about fish. But that isn't enough in this case.
For a start, they are about different aspects of fish. So, in that sense, there is still no genuine material connection. Also, we must repeat that one can't derive a true belief from a false belief.
In all these examples, it's clear that in order to gain knowledge, certain logical principles have to be adhered to. These principles guarantee knowledge.
For example, if the premises are true, then all correct inferences from these true premises must result in true conclusions. These true conclusions, then, would be examples of knowledge within a logical context. Of course the premises and conclusions must have material content and not just be examples of formal logic. That is, the premises must themselves be the conclusions of previous empirical investigation.
To have knowledge, according to the above, seems to mean that some kind of cognitive work has been done in order to reach a belief that is true. That is, Russell’s man is correct to believe that the then Prime Minister’s last name begins with B. However, this is because he thinks that the then Prime Minister was Balfour. In that sense, he was wrong. But he was, after all, only articulating a belief about the first letter of the Prime Minister’s last name. Though if he thought that Balfour is the Prime Minister, and the actual Prime Minister was Bannerman, then he clearly hadn't done any cognitive work to establish the first letter of the last name of the then Prime Minister. His true belief, therefore, was based on false knowledge. That is, he came to that belief for false reasons. Therefore he had no true knowledge.
To put this another way: Russell’s man had no justification for believing what he believed, even though what he believed is true. In a sense, his belief, though true, was based on pure guesswork. Knowledge, on the other hand, requires reasons for believing what one believes.
Though what if he did have reasons? He might have had the wrong reasons for believing his true belief. He might have thought, for example, that it was about time that a Prime Minister’s last name began with ‘B’. After all, up until then, no Prime Minister’s last name had begun with ‘B’. So, on his probabilistic reasoning, it was very likely that the Prime Minister’s last name would begin with ‘B’.
This kind of reasons is of course ridiculous. However, ridiculous or not, he might still have offered us reasons for believing what he believed. The problem is that they were simply bad reasons for believing what he believed. The problem now is: what would constitute, in this case, good reasons for believing what he believed about the then Prime Minister’s last name? This is, of course, another issue.
Does the above amount to saying that a belief can't be true, or can't be true knowledge, if it is believed for the wrong reasons? Similarly, a true belief may not be knowledge if it is produced by the wrong method:
"… a true belief cannot be called knowledge when it is deduced by a fallacious process of reasoning, even if the premises from which it is deduced are true. If I know that all Greeks are men and that Socrates was a man, and I infer that Socrates was a Greek, I cannot be said to know that Socrates was a Greek, because, although my premises and my conclusion are true, the conclusion does not follow from the premises." (131-2, Problems of Philosophy)
Are we to say that nothing is knowledge except what is validly deduced from true premises?
So it's not just false premises or prior beliefs that we need to watch out for; but also fallacious processes of reasoning. That is, there is more to reasoning that true premises and true conclusions. We could go from ‘Most birds fly’ to ‘Snow is white’; though clearly this would not instantiate a correct process of reasoning. However, is relying on the star charts or clairvoyance an example of correct reasoning simply because we take the premises to be true and the conclusion to be true? Russell's own example of faulty reasoning starts off with true premises and ends up with a true conclusion. However, no correct method of inference is used in this example: therefore the conclusion isn't an example of knowledge.
According to David Lewis, we can have examples of knowledge that don't depend on true beliefs, true premises or correct methods of inference. Sometimes we just know something and we don’t know why. For example, when we may have arrived at that piece of knowledge by justificatory reasoning in the past. Now, however, we have completely forgotten these cognitive processes; though we still, nevertheless, have knowledge.
Also, according to Lewis, we may have knowledge even though we haven't eliminated all possibilities of not-P, where P is our piece of knowledge. Lewis simply claims that it's impossible to eliminate all examples of not-P. Primarily because we are simply not aware of all examples of not-P. We couldn't be aware of all examples of not-P. In addition, some examples of not-P must be simply disregarded if we are to get the epistemic ball rolling.
For example, I believe that ‘snow is white’. But is it possible that my sensory receptors have inverted the colours I receive from the world due to a brain damage of some kind? Similarly, and more fashionably, I could be a ‘brain in a vat’ who is being fed sensory distortions care-of a mad scientist. Though I think that Lewis may see here that the brain in a vat hypothesis must be ignored ‘(proper ignoring’) simply because there is no known way of disproving that possibility. (Putnam does claim, however, that it is possible to prove that we aren't brains in a vat.)
There are similar sceptical stratagems that, according to certain epistemologists, we can't disprove. And in many cases, the reality of the sceptical challenge is essentially indistinguishable from the non-occurrence of the sceptical possibility.
Bertrand Russell himself gave the example of the earth being created five minutes ago with all its examples of fossils and remains exactly as they would be if the earth had been had it been around for billions of years. Though because the differences between the sceptical possibility and the present actuality are zero, it's simply counter-productive to take these sceptical challenges seriously. Again, if not-P were taken seriously, we couldn't get going in any epistemic process. Not only is there a possible infinite amount of not-Ps, but some not-Ps are deemed, by some, to be basically nonsensical.