The aim of this thesis is to specify a fuzzy rule-based language and parallel computational mechanism for fuzzy expert systems on the basis of the unit bilattice $[0,1] \times [0,1]$. The use of the bilattice provides us with a more expressive way to describe the truth value of a statement. Where an element of the billatice can be used to capture supporting and refuting knowledge respectively about the truth of a statement. This more expressive way of describing belief also introduces the possibility of capturing conflicting or deficient knowledge about the truth of a statement. In this thesis we analyze the impact of this problem on the expert systems of this class, investigate existing approaches to dealing with the problem and finally propose and incorporate an solution to our computational mechanism developed in this thesis.