Knowledge without Belief? Justified beliefs or disbeliefs worthy of Knowledge?

According to Jonathan Jenkins Ichikawa at The Stanford Encyclopedia of Philosophy, a correct analysis of knowledge would do more than pick out the actual extension of knowledge; even if, in actual fact, all cases of S knowing that p are cases of j, and all cases of the latter are cases of the former, j might fail as an analysis of knowledge. For example, it might be that there are possible cases of knowledge without j, or vice versa. A proper analysis of knowledge should at least be a necessary truth. Consequently, hypothetical thought experiments provide appropriate test cases for various analyses, but even a necessary biconditional linking knowledge to some state j would probably not be sufficient for an analysis of knowledge, although just what more is required is a matter of some controversy. Ref Knowledge without Belief? Justified beliefs or disbeliefs worthy of Knowledge? I wrote this and a challenger responded, ” Atheism and the traditional laws of logic? Three traditional laws of logic: Law of Identity, Law of non-contradiction, Law of excluded middle. Law of Identity: ‘Whatever is, is.’ Thus, Atheism is Atheism, it is not theism nor agnosticism. Law of non-contradiction: ‘Nothing can both be and not be.’ Thus, Atheism is either only Atheism and cannot be also theism nor agnosticism. Law of excluded middle: ‘Everything must either be or not be.” Thus, Atheism must either be or not Atheism and only then can there be theism or agnosticism. And, thus because of the three traditional laws of logic the term agnostic atheism can be logically rejected for just Atheism by itself. Ref *Challenger, “Except theism and atheism are the subject...