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constructing an Aronszajn tree of perfect sets. Both constructions produce y-sets that are not strongly meager, the second one produces a set all of whose subsets are also y-sets.

A classical theorem of Sierpifiski states that every analytic (El) set A is measurable, so it can be written as A = B U N where B is Borel and N is null. In Mauldin's very short paper it is shown (by using a set in the plane that is universal for all Borel sets) that this theorem cannot be improved by requiring N to be in the a-ideal generated by closed measure zero sets. MARTIN GOLDSTERN

CHRIS MORTENSEN. Inconsistent mathematics. Mathematics and its applications, vol. 312. Kluwer Academic Publishers, Dordrecht, Boston, and London, 1995, ix + 155 pp.

CHRIS MORTENSEN and PETER LAVERS. Category theory. Therein, pp. 101-108. WILLIAM JAMES. Closed set sheaves and their categories. Therein, pp. 115-124. CHRIS MORTENSEN and JOSHUA COLE. Foundations. provability, truth and sets. Therein, pp.

135-146. As it is clear from its title, this book is concerned with the development of an inconsistent mathematics.

Such an undertaking, ever since the formulation of the first paraconsistent systems, is naturally of great importance. Indeed, its chief moral, to some extent, could be encapsulated in the assertion that inconsistent structures are not only worth studying, but need to be investigated as well.

A system is said to be inconsistent if some sentence and its negation are both theorems; it is trivial if all sentences are theorems. In classical logic these two concepts coincide. But in relevant logic, for example, they do not. This fact opens the way to constructing contradiction-tolerant, or paraconsistent, systems.

The book starts with an informal chapter devoted to a brief historical examination of paraconsistency, to a non-technical outline of the main results to be later formulated, and to the presentation of certain philosophical implications from them. This chapter can be fruitfully read by philosophers and those logicians with no previous background in paraconsistent logic. Its main conclusions can be placed under five headings:

(a) Paraconsistent logic, as is known, has concentrated greatly on the logical features of contradiction- containment. Unless it sustains a fertile, distinctive mathematics, however, it might well be deemed, from the mathematical community's perspective, as nothing more than a sheer curiosity. To this extent, paraconsistent logic needs inconsistent mathematics.

(b) From the author's viewpoint, the driving force to construct inconsistent mathematics derives from logicians' inquiries into contradiction-containment in the foundations of mathematics, such as those developed by Brady, da Costa, Priest, and Routley, and related semantical studies by Kripke and Feferman. Nevertheless, the book considers foundational issues, for instance those related to the paradoxes of set theory and semantics, only in the last chapter. Inconsistency is thus taken as mathematically interesting in its own respect, independent of what might be thought about foundations. It is emphasized that neither classical mathematics, nor incomplete theories as arising in intuitionist mathematics, is being discarded. The project attempts to extend conceptions of what is possible in mathematics, not to jettison the extant mathematics.

(c) There are three sorts of arguments for studying inconsistent mathematical structures. First, there is the argument from pure mathematics, related to the claim that these structures should be examined because they are every bit as legitimate a part of mathematics as is any other pure mathematical structure. Secondly, there is the ontological argument, which is the assertion made by Priest and others that some contradictions, such as the Liar sentence, are true. The present book does not depend on such a claim, except to the extent that one regards pure mathematics as containing true claims about its subject matter. Thirdly, there is the epistemological argument, presented by Belnap and others, to the effect that information systems are always open to the possibility of encountering inconsistencies between their sources.

(d) Nonetheless, according to Mortensen, the position of consistency or inconsistency issues in mathematics is secondary to that of functionality (substitutivity of identity into all atomic contexts). He maintains that, while questions related to consistency are the domain of logic, mathematics is concerned with functionality in a quite general way. This is not to dispute that mathematicians are, as it were, on the consistency side; in the author's view, though, such a trait does not reflect an essential feature of mathematical reasoning. As he points out, "philosophers have hitherto attempted to understand the nature of contradiction, the point however is to change it" (p. 7).

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(e) The existence of inconsistent structures insinuates, Mortensen notes, that necessary truth might have no place as an explanation of mathematical reasoning, and further that Platonism about mathe- matical objects is likewise flawed.

Chapter 2 starts with a brief account of one of the historical points of departure for inconsistent mathematical structures, namely R. K. Meyer's 1976 non-triviality proof for the relevant arithmetic R#. Meyer has argued that this result sets the stage for a revived Hilbert program by showing how a reinterpretation of the second G6del incompleteness theorem is required which escapes some of the consequences of the original in its classical logic setting. Though one should note that this book is not particularly about model theory, this chapter then sets up model theory adequate for demonstrating contradiction-containment. Finally several of the author's results on inconsistent number-theoretic structures arising from the natural numbers, integers, rationals, rings, and fields are summarized.

Chapter 3 elaborates on the device of modulo arithmetic, extending it to non-standard models, where the modulus is an infinite number. Of particular interest is the case where the modulus is an infinite prime, which as in the finite case, produces a field. The author presents his already published results on both the consistent and corresponding inconsistent cases.

Chapter 4 examines order. First it shows that the well-known result that the first-order theory of dense order without endpoints is No-categorical breaks down in the inconsistent case, and thus breaks down for theories of entailment logics such as R and E. In other words, the usual proof of this result depends on importing classical logic assumptions. (Some other results in the book are invariant with regard to change of background logic, and this suggests a program to determine which properties are logic-invariant and which not.) The chapter then looks at the combination of order with arithmetical operations, as with ordered rings and fields.

Chapter 5 presents an account of inconsistent and incomplete theories of the calculus, based on infinitesimals. The theories are obtained without the employment of set theory. As is known, the historical debates over infinitesimals, centuries before Robinson set them on a rigorous basis, were characterized by some confusion and a degree of inconsistency. Even lately, it is also known that if a pair of hyperreal numbers are identified and the resultant theory is stipulated to be functional, then the theory is trivial. That is, contradiction-containment techniques of paraconsistent logicians, such as relinquishing ex contradictions quodlibet, are not strong enough to prevent the uncontrolled spread of contradiction if the theory is functionally strong enough. The present chapter shows, nonetheless, that with limited restriction of the operations, it is possible to obtain inconsistent models in which all the usual theorems of elementary calculus hold. Results on integration are also achieved.

To this point in the book, known results, mostly by the author, are being surveyed. The remainder of it, apart from some material in the last chapter, consists of new results. Chapter 6 defines a class of functions which in the consistent case are discontinuous (step functions), but which from an inconsistent point of view can be regarded as continuous. Chapter 7 extends this discussion to the derivatives of such functions, the so-called Dirac delta "function." This is known not to be a function, but is useful in quantum mechanics. The author shows that from an inconsistent perspective there is a function here, which has the desired properties.

Chapter 8 investigates another inconsistent case which has been known for a long time, inconsistent systems of linear equations. The author shows that there are ways of producing solutions for such systems in an inconsistent space, though the author admits that the situation remains not wholly satisfactory. Some applications, such as control theory, are then delineated.

Chapter 9 considers projective spaces, after a preliminary look at vector spaces. Projective spaces turn out to be more easily made inconsistent than the vector spaces from which they can be defined. A particularly interesting case is the inconsistent field modulo an infinite prime, defined in Chapter 3, which yields a theory in which standard projective duality results can be strengthened.

Chapter 10 studies quotient constructions in topology. The author shows that the functional proper- ties of inconsistent theories here interact with the properties (for instance, Ti or T2) of the underlying classical topological space.

Chapter 11 briefly surveys foundational problems of inconsistency in category theory, and then proceeds to topos theory. The usual logical story is that topos theory stands to intuitionist (open set) logic as set theory stands to Boolean logic. By defining a certain dual of a topos, called a complement topos, it is seen that the natural logic is closed set logic, which is paraconsistent. Closed set logic does not have a natural implicational dual to intuitionist implication (pseudo difference), but there is a natural S5-style implication with a categorial definition here.

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Chapter 12, written by W James, extends this discussion to the theory of sheaves, and demonstrates the existence of sheaves over spaces of closed sets. Chapter 13 obtains duality results concerning the Routleys' *-operation on theories, as well as related operations. It was discovered by the Routleys that inconsistent and incomplete theories are *-duals of one another. The results of Chapter 11 strengthen this idea by showing that certain inconsistent and incomplete theories are topological duals of one another. Then in this chapter even further interesting duality results are obtained. This leads to the conclusion that inconsistent and incomplete mathematical theories are of equal interest and mathematical respectability.

The final chapter discusses the foundations as traditionally conceived. First results are obtained about the G6del sentence in inconsistent arithmetical theories; and one shows that this sentence becomes, in a certain sense, the Liar sentence. Fixed-point methods are then examined. Kripke's results to the effect that a truth predicate can be defined for an inconsistent theory are spelled out in a clear and economical way. Finally, earlier fixed-point results in naive set theory, due to Gilmore and Brady, are presented. It is known that the Russell set can be constructed without triviality by adopting these methods. In the case of both truth theory and set theory, the duality results of the preceding chapters make it clear that it is an easy transformation to move from incomplete theories to their inconsistent duals.

Before closing the present review, we wish to make some brief points. Of course other approaches to inconsistent mathematics are possible. (Perhaps, it would be better to say "paraconsistent mathematics" rather than "inconsistent mathematics.") Most constructions studied in the book presuppose classical mathematics, especially classical set theory; but one can first develop a paraconsistent set theory and then apply it to build an inconsistent mathematics (and appropriate semantics for certain paraconsistent logics as well), including the topics covered by the book.

The author has written a valuable book showing that paraconsistent mathematics is both possible and interesting. In order that the subject survives as any kind of mathematics, however, it is necessary that it be investigated and developed by professional mathematicians. We are sure that the present work will contribute to this goal.

A last remark: Errata may be viewed on the world wide web at http://www.etu.adelaide.edu.au/ philosophy/inconMaths.html. NEWTON C. A. DA COSTA and OTAvIo BUENO

NORMAN M. MARTIN and STEPHEN POLLARD. Closure spaces and logic. Mathematics and its applications, vol. 369. Kluwer Academic Publishers, Dordrecht, Boston, and London, 1996, xvii + 230 pp.

A closure space consists of a set S and a closure operation Cl on S, that is, a function mapping subsets of S to subsets of S that preserves the inclusion order and satisfies X C Cl(X) = Cl(Cl(X)), for all X C S. The authors are delighted by the fact that these simple properties capture both logical closure operations as well as topological ones. In their Preface, the authors assert that "At the least, the present book shows that topological methods are of use in logical theory." As the current generation says, "Hello?" I believe this close connection is something that has been noticed for at least seventy years. But the authors here seem to be unaware of the rather extensive literature on this topic. My initial annoyance at not finding my name in the bibliography was assuaged when I noticed that my absence was shared by Log, Suszko, W6jcicki, and others of the Polish school who have been studying closure spaces and logic for years. Very few of the standard references in this subject appeared, nor was there a reference to a 1970 Ph.D. dissertation by Donald J. Brown (Abstract logics) at Stevens Institute on exactly this topic.

The 224 pages of text are divided into nine chapters: Logic and topology, Basic topologicalproperties, Some theorems of Tarski, Continuous functions, Homeomorphisms, Closed bases and closure semantics I, Theory of complete lattices, Closed bases and closure semantics II, Truth functions.

It is not clear for whom the book is intended. The authors review extremely elementary topology and logic, but assume that the reader will appreciate a common generalization. Several chapters end with a plea to the reader to supply a raison d'etre: "We have offered only a glimpse at continuous functions and their applications in logical theory. Our main hope is that some readers will find the topic worth pursuing further" (page 83). "So consider this little more than an invitation to supply a context in which the question does make sense" (page 106). "There seems to be an upsurge of interest in logical applications of closure spaces. We look ahead with eager anticipation!" (page 224).

The authors use a primitive typesetting system which has produced such barbaric line breaks as V- '[T] = Cl(0), instead of V- 1 [T] = Cl(0). The symbol fi is always aligned so that its lowest point never falls below the line, resulting in a symbol that seems to be floating away, as in B E P.