Graham Priest. Towards non-being: the logic and metaphysics of intentionality. Oxford University Press, 2005, xi + 190 pp.

Abstract

type-amalgamation property in the book. The model theoretic analysis also shows that the class of structures which have a smoothly approximable expansion can be characterized as the class of structures which satisfies nine model theoretic properties, a few of which have just been mentioned. Many of the results presented depend on weaker assumptions than smooth approximability, such as א0-categoricity and finite rank. As shown in [Kantor, Liebeck, Macpherson, op. cit.], every smoothly approximable structure M has the following property, called pseudofiniteness: If φ is a sentence which is true in M then φ is true in a finite substructure of M . It follows that the complete theory of a smoothly approximable structure is not finitely axiomatizable. By developing further the method of envelopes and techniques of G. Ahlbrandt and M. Ziegler (used for strongly minimal totally categorical theories) it is shown that the complete theory of any smoothly approximable structure is quasifinitely axiomatizable, meaning that the theory is axiomatized by a finite number of axiom schemes. This extends earlier results about pseudofiniteness and quasifinite axiomatizability for א0-categorical א0-stable theories. In the last chapter, which builds on methods used for quasifinite axiomatizability, decidability problems are considered. For instance, given a language with a finite signature containing only relation symbols and a sentence in this language, it is decidable whether the sentence has a stable homogeneous model. Also, in a language with finite signature, one can decide whether a sentence has a finite model with a given number of 4-types. For some results the classification of finite simple groups is used and a discussion of this is given at the end of the book. The book is technically difficult, proofs are often quite compressed and it expects that the reader is familiar with various notions from model theory and algebra; so the book demands plenty of the reader. An error (the statement of Lemma 2.4.8) is corrected in [G. Cherlin, M. Djordjević, E. Hrushovski, The Journal of Symbolic Logic, vol. 70 (2005), pp. 1359–1364]. Vera Koponen Department of Mathematics, Uppsala University, Box 480, 75106 Uppsala, Sweden. vera@math.uu.se.