We propose an epistemic logic with so-called graded modalities, in which certain types of knowledge are expressible that are less absolute than in traditional epistemic logic. Beside ‘absolute knowledge’ (which does not allow for any exception), we are also able to express ‘accepting ϕ if there at most n exceptions to ϕ’. This logic may be employed in decision support systems where there are different sources to judge the same proposition. We argue that the logic also provides a link between epistemic logic and the more quantitative (even probabilistic) methods used in AI systems. In this paper we investigate some properties of the logic as well as some applications.