Dov M. Gabbay and Karl Schlechta.
A new perspective on nonmonotonic logics.
Springer International Publishing, New York, NY, 2016. 365 pp.
ACM CR categories:
(P)I.2.3 Nonmonotonic reasoning and belief revision
F.4.0 Mathematical logic - general
I.2.0 Cognitive simulation, philosophical foundations
I.2.4 KR formalisms and methods - relation systems, semantic networks

Nonmonotonic reasoning derives plausible conclusions from a theory. The theory is sometimes called background knowledge. If new information becomes available showing that some conclusions are wrong, one has to retract them. A common application is to avoid the frame problem pointed out by Minsky, in which one has to specify many things remaining unchanged in a large scenario where only a few things change.

Over the last decade, the authors have published their research on nonmonotonic logics in the form of books aimed at experts in the area: this is the third one after two previous works [1,2]. Similar themes recur in the books. The present book has chapters giving new results on interpolation, on abstract independence and on deontic logic. Another chapter revises some of the authors' earlier work on rules appearing in the Talmud. There are condensed introductions to earlier work such as Reiter's defaults, Moore's autoepistemic logic, the Alchourrón-Gärdenfors-Makinson (AGM) postulates on theory revision, and Chisholm's contrary-to-duty obligations, as well as Edelman's ideas on how the brain functions [3], which the authors offer as something computer scientists should know about.

The authors also introduce defeasible inheritance networks, which specify default rules (in terms of Makinson [4]), and preference structures, which allow default valuations (again see [4]). They point out that the former are expressive, natural, and robust, but mathematically impoverished, while the latter have excellent algebraic properties, but are rigid and prone to collapse under small changes. The book's centrepiece, as emphasized by the authors, is a chapter giving an order construction which combines these two ideas.

At a grossly oversimplified level, this goes as follows. Birds normally have the default property of flying. Penguins are exceptional birds, they normally have the default property of not flying. They come next on the scale of normality. Assume a language containing these two classes. Other birds which fail the default property (such as ostriches) are surprises; they come after penguins. Penguins which do fly are double surprises; they come after ostriches. Using ideas from theory revision, the authors' construction deals with several different defaults.

In their earlier book [2], the authors said that a bold logic must offer, in addition to its language, proof theory and semantics, a justification. In this book they repeatedly point out how their construction satisfies several desiderata. It takes some thought to realize that Edelman's ideas about neural connections [3] are used to make choices regarding the construction. There are only informal remarks regarding the new ideas on interpolation and independence and the complexity of finding minimal models. The connections to complexity are yet to be explicitly made.

As a broader theme, the authors suggest that logic must use language (not just mathematics) to connect to reality. This idea occurs in other areas in artificial intelligence, for instance when graphical models were used by Pearl [5] to describe joint probability distributions. One can think of such graphs in terms of small linguistic descriptions. More recently, Parikh used splitting languages to model independence in belief revision [6].

After putting in some effort, the reader can follow the authors' construction. Having read their earlier book [1], I must flag the authors' propensity, under the guise of making a book self-sufficient, of packing in a lot of material from their earlier books. In a research paper, this would only merit a reference. I do acknowledge that this helped me think of the context and the connections to earlier work, and I view this book as a thought-provoking discourse. I look forward to the second author's forthcoming textbook [7] for a more introductory treatment.

[1] Gabbay, D.M.; Schlechta, K. Logical tools for handling change in agent-based systems, Springer, Berlin, 2009.
[2] Gabbay, D.M.; Schlechta, K. Conditions and modularity in general logics, Springer, Berlin, 2011.
[3] Edelman, G.M. The remembered present, Basic Books, NY, 1989.
[4] Makinson, D. Bridges from classical to nonmonotonic logic, College Publications, London, UK, 2005.
[5] Pearl, J. Probabilistic reasoning in intelligent systems, Morgan Kaufmann, Sab Francisco, CA, 1988.
[6] Parikh, R. Belief, belief revision, and splitting languages, in Proc. Logic, language and computation (vol. 2) (Moss, L.S., Ginzburg, J. and de Rijke, M., Eds.) CSLI Publications, Stanford, CA, 1999, 266-278.
[7] Schlechta, K. Formal methods for nonmonotonic and related logics, Springer, 2017, graduate textbook in preparation.