Suppose we try to formalize a logic of knowledge. Take the following axioms to be true (phi and psi stand for sentences which can be true or false and known or unknown): 1) If we know that "phi and psi" is true, then we know that "phi" is true and that "psi" is true. For example, if we know that "the sky is blue and the grass is green", then we know that "the sky is blue" and we know that "the grass is green." 2) If we know that "phi" is true, then "phi" is true. This is just part of the definition of "knowledge." In order for a belief to count as knowledge, that belief must be true. 3) If "phi" is provable from no assumptions, then "phi" is necessarily true. This just says that if we can prove that "phi" is true without making any extra assumptions, then it's logically necessary that "phi" is true, i.e. that it's logically impossible that "phi" be false. This seems obvious. 4) If it's necessarily true that "phi" is false, then it's not possible that "phi" is true. Again, this seems obvious—if it's logically necessary that "phi" isn't true, then it's not logically possible that "phi" is true. OK, so. It can be proven that any formalization of knowledge which obeys those 4 axioms falls prey to Fitch's paradox of knowability, which states that So, you have three choices: 1) There exist truths which can never, in principle, be known. 2) All truths are, in fact, known. 3) one of the 4 axioms listed is false What do you think?