CHRIS MORTENSEN. Inconsistent Geometry. Studies in Logic; 27. London: College Publications, 2010. ISBN 978-1-84890-022-6. Pp. ii+162.

Abstract

Many years ago, in 1999 to be precise, I wrote a review for this very same journal [Van Bendegem, 1999] of Chris Mortensen's first book on inconsistent mathematics. In this book the focus was mainly on inconsistent arithmetic — part of that work was jointly written with Robert Meyer, who unfortunately passed away recently — and its development into an inconsistent theory of the reals, leading up to a form of calculus where the abhorred Dirac delta-function — and not its reformulation in terms of distributions — does indeed have a place and a role to play. If it is allowed to use the ancient dichotomy of mathematics into counting and measuring, then it was inevitable that a companion volume had to appear on geometry, and that is precisely the book under review. In order to keep the length of this review reasonable, I will not, in contrast to the 1999 review, present an elaborate sketch of what paraconsistent logic is all about. It seems equally reasonable to suppose that logicians, philosophers, and mathematicians alike today (should) know something about such logics and perhaps a thing or two about their applications. If not, plenty of sources exist at present; see, e.g., [Priest and Tanaka, 2009] as a perfect starting point, or else the ‘classic’ [Priest et al., 1989]. Therefore the following characterization should do: a paraconsistent logic L is any logic that does not accept the ECQ (Ex Contradictione Quodlibet), i.e., A,∼A⊢B. As noted by Mortensen, page 3,1 if one accepts ECQ, this implies that ‘the inconsistent has no structure’. To a certain extent, the whole book can be read as a defense of the thesis that the inconsistent does indeed have plenty of structure and that paraconsistent logic is the best-suited instrument to study these structures. However, what usually follows from a negative definition (as it excludes a rule) is that there is no reason to expect paraconsistent logic to be unique. Quite the contrary, the wealth of different systems that have been developed is quite staggering (as, incidentally, seems to be the case for any interesting logical idea; just think of modal logic). In addition, there is an ongoing discussion about the philosophical nature of contradictions and inconsistencies that can be summarized as whether they are ‘in our head’ or ‘out there’. Graham Priest's dialetheism is the perfect illustration of the latter view, whereas Chris Mortensen sides quite explicitly with the former: ‘It must be stressed that this sort of motivation for the theory of inconsistency is epistemic or cognitive in character’ (p. 4).