The appearance of this small, attractive book with a picture of Valentin de Boulogne’s Die Falschspieler on the dust jacket may suggest some form of intellectual recreation. And indeed, the book is very entertaining, although it seriously attempts to tackle in a novel way the ancient problems connected with the interpretation of sentences like the Liar: “This proposition is not true.”
The introductory part, comprising three chapters, defines the problems and introduces the tools to be used. In the first chapter, the Liar and many related semantic paradoxes are introduced, and the plan of the book is outlined. It is decided that the appropriate bearers of truth are the propositions. The authors also argue quite convincingly that the customary ways to avoid these paradoxes, by a stratification of formalized languages into object languages and metalanguages, solve the problems only in a limited sense while at the same time bannning perfectly harmless cases of circularity. In the second chapter, the views of Russell and John Austin on sentences, statements, and propositions are explained, and a simple toy language to be used throughout the book is introduced. The third chapter is probably of much interest even to many who do not find the paradoxes themselves important. Here the authors give an intuitively understandable exposition of Peter Aczel’s theory of hypersets. This theory results when one replaces the Axiom of Foundation, which prevents a set from being its own member, in the ZFC system. There is also an opposite postulate called the Anti-Foundation Axiom.
In Part 2, Russellian propositions are modeled as set-theoretic objects within the frame of Aczel’s theory, the notion of truth for such propositions is defined, and then the paradoxes are analyzed using the resulting formalization. The results are not entirely satisfactory, as much of the paradoxical remains. For example, in any model of the world, the Liar itself turns out to be not true, but its falsity cannot be a part of the same model. This element of the paradox is dispelled by the Austinian account, which is similarly developed in Part 3 of the book.
I cannot expect that philosophers will unanimously hail these analyses as a final solution to the Liar paradox, but the authors must at least be credited for having introduced new, powerful tools for dealing with problems of this type. At one point they remark that the significance of a paradox is never the paradox itself, but what it is a symptom of. This observation is also relevant to the question of whether the book is of any professional interest to a computer scientist; an imaginative reader may very well beneficially apply the authors’ ideas on circular reference, fixed points of recursion equations, and negation to quite different, but analogous, situations.
The exposition is excellent. The authors have managed to present quite a substantial conceptual machinery in a way accessible to a wide readership, and the reasoning is always transparent. However, a full understanding of the material will require much more effort from the reader because proofs are often just sketched and much of the technical development has been relegated to the many exercises. In particular, I hope that a more complete account of Aczel’s theory will soon be generally available.
