Then how do you know that the evidence is very strong? You'd have to have strong evidence that the evidence is strong, ad infinitum. That was essentially Plato's definition of knowledge, "justified true belief." Then a guy named Gettier came around and showed that you can have strong evidence supporting a belief, and still be wrong. Then a guy named Nozick[1] came along and provided his own definition of knowledge, and that brings us up to the early 1980's.

There's an entire branch of philosophy dedicated to figuring out exactly what "knowledge" is, Epistemology. The "brain in the vat" problem has be considered in context[2].

[1]http://www.iep.utm.edu/nozick/#H3 [2]http://plato.stanford.edu/entries/epistemology/