Full Idea
If you claim that S knows that P, and yet grant that S cannot eliminate a certain possibility of not-P, it certainly seems as if you have granted that S does not after all know that P. To speak of fallible knowledge just sounds contradictory.
Gist of Idea
To say S knows P, but cannot eliminate not-P, sounds like a contradiction
Source
David Lewis (Elusive Knowledge [1996], p.419)
Book Reference
Lewis,David: 'Papers in Metaphysics and Epistemology' [CUP 1999], p.419
A Reaction
Starting from this point, fallibilism seems to be a rather bold move. The only sensible response seems to be to relax the requirement that not-P must be eliminable. Best: in one epistemic context P, in another not-P.