### 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.

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