Problems With The Possible World Model and With The Modal Model of Knowledge
If we take the possible worlds approach to knowledge, it follows that our epistemic agent a knows all the logical consequences of his or her or its beliefs. If is a logical consequence of, then there is no possible world where is true but is not. So if a knows that, it follows that all of the logical consequences of are true of all of the possible worlds compatible with a 's beliefs. Therefore, a knows . It is not epistemically possible for a that not- given his knowledge that . This consideration was a part of what led Robert Stalnaker to develop two dimensionalism, which can arguably explain how we might not know all the logical consequences of our beliefs even if there are no worlds where the propositions we know come out true but their consequences false.
Even when we ignore possible world semantics and stick to axiomatic systems, this peculiar feature holds. With K and N (the Distribution Rule and the Knowledge Generalization Rule, respectively), which are axioms that are minimally true of all normal modal logics, we can prove that we know all the logical consequences of our beliefs. If is a logical consequence of, then we can derive with N and the conditional proof and then with K. When we translate this into epistemic terms, this says that if is a logical consequence of, then a knows that it is, and if a knows, a knows . That is to say, a knows all the logical consequences of every proposition. This is necessarily true of all classical modal logics. But then, for example, if a knows that prime numbers are divisible only by themselves and the number one, then a knows that 8683317618811886495518194401279999999 is prime (since this number is only divisible by itself and the number one). That is to say, under the modal interpretation of knowledge, when a knows the definition of a prime number, a knows that this number is prime. It should be clear at this point that a is not human. This shows that epistemic modal logic is an idealized account of knowledge, and explains objective, rather than subjective knowledge (if anything).
Read more about this topic: Epistemic Modal Logic
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