Wednesday, November 25, 2015

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.
Creative Commons License
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 United States License.