Propositional, probabilistic and evidential reasoning. Integrating numerical and symbolic approaches (Q5948011)

The main objective of the book is to describe a new approach to dealing with probabilistic uncertainty and incomplete knowledge, an approach that includes several known approaches as particular cases and in this inclusion, explains the heuristic parts of the existing approaches -- and also explains why some of these semi-heuristic approaches sometimes lead to counterintuitive results. This new approach is based on the ideas of incidence calculus, in which uncertainty is describe by: (1) a set of possible worlds \(W\), (2) a probability \(\mu(w)\) assigned to each of possible worlds \(w\), and (3) for each formula \(\varphi\), the set \(i(\varphi)\) of all worlds \(w\) in which this formula is true. This description can be extracted from the data, e.g., as follows: if we have \(n\) observations, then we take these observations as possible worlds with equal probability \(\mu(w)=1/n\), and take, as \(i(\varphi)\), the set of all observations in which the property \(\varphi\) was true. In many real-life situations, we only have partial observations, in which we know the truth values of some properties but not of all of them. To cover such situations, the author proposes a generalized incidence calculus in which the truth value of a formula \(\varphi\) can be ``unknown'' in some worlds \(w\in W\). In this calculus, instead of a single set \(i(\varphi)\), we have two sets: \(i_*(\varphi)\) is the set of all the worlds in which \(\varphi\) is known to be true, and \(i^*(\varphi)\supseteq i_*(\varphi)\) is the set of all the worlds in which \(\varphi\) may be true. The author shows that several other approaches to uncertainty such as Dempster-Shafer formalism, rough sets, ATMS, etc., can be viewed as particular cases of this new approach. In several cases, the re-interpretation of the existing approaches in the new terms clarifies the existing approaches: e.g., it becomes clear why the original Dempster-Shafer's combination rule works well in some situations, but sometimes leads to counterintuitive results. The book starts with a brief description of different existing approaches. A reader should be warned that these descriptions do not intend to serve as a complete description of these approaches for readers who are completely unfamiliar with the subject: several important details are omitted. For example, the author mentions only briefly an important discussion of what is the more adequate description of the probability of implication ``if \(A\) then \(B\)'': \(P(A\to B)=P(B\vee \neg A)\) or the conditional probability \(P(B\,| \,A)\). If we follow Nilsson in using the value \(P(B\vee\neg A)\), then always \(P(B\vee \neg A)+P(A)\geq 1\), so the author's criticism on p. 8 -- that Nilsson's formulas are not applicable when \(P(B\vee \neg A)+P(A)<\) -- is not convincing; on the other hand, if we use the conditional probability, then, of course, the case \(P(B\vee \neg A)+P(A)<1\) is possible, but then Nilsson's formulas cannot be applied because they are based on a different interpretation of probability of implication. In short, readers who are not familiar with the subject should be cautious. For readers who are reasonably familiar with the existing approaches, this book provides a fresh thought-provoking view.