knoweldge is not closed under entailment.

Suppose I know that P. Suppose also that P (logically) implies Q.¹ Does it follow that I know Q? Obviously not. (Or at least not by my dim and fallible lights, at any rate). Therefore knowledge is not closed under entailment. 

However, suppose that I know P. And suppose that P implies Q. It does seem to follow that I do not know ~Q. For I know P only if P is true, and if P implies Q, then ~Q is false, and since no one knows (nor could know) what is false, it follows that I know P only if I don’t know ~Q.

The same isn’t true for belief, however. Suppose that I believe P. Suppose that P implies is Q. It seems that I could still believe that ~Q even though P and ~Q are not (nor could be) both true.

¹ Barring the possibility of extreme logical pluralism—viz. the thesis that there is no valid inference rule which all possible logics have in common.