This book developed out of lecture courses in Uncertain Reasoning given at the University of Manchester in 1989 and 1991 as part of the Master of Science degree in Mathematical Logic and is aimed at readers with some mathematical background who wish to understand the mathematical foundations of the subject. Thus the emphasis is on providing mathematical formulations, analyses and justifications of what I see as some of the major questions and assumptions underlying present day theories of Uncertain Reasoning whilst avoiding, as far as possible, lengthy philosophical discussions on the one hand and precise computer algorithms on the other. Much of the material presented appears already, in some form or other, in published papers, so that my main contribution is the assembling and presenting of it within a unified framework.

My hope is that by doing so I might encourage more ‘mathematicians’ to take an active interest in the subject whilst at the same time offering to those currently working on practical applications in the field easy access to some of the mathematics underlying their assumptions. In short it is the sort of book which I wish had been available to me when I first entered the area.

The subject of Uncertain Reasoning (also referred to as Approximate Reasoning and Reasoning under Uncertainty) dates back to Plato, if not beyond, but has seen an exponential expansion in the last decade with the drive towards intelligent computers, especially so called expert systems.