Knowledge, false belief, and reductio

Recently, a number of cases have been proposed which seem to show that – contrary to widely held opinion – a subject can inferentially come to know some proposition p from an inference which relies on a false belief q which is essential. The standard response to these cases is to insist that there is really an additional true belief in the vicinity, making the false belief inessential. I present a new kind of case suggesting that a subject can inferentially come to know a proposition from an essential false belief where no truth in the vicinity seems to be present.