| Ch. Mortensen: Inconsistent Mathematics. Mathematics
and its Applications, vol. 302. Kluwer Academic Publishers; Dordrecht 1995. 155 pp. Inconsistency is a phenomenon that may occur in mathematics, information systems, certain physical theories and possibly somewhere else. Some cases of inconsistency occur quite naturally. It is a well-known theorem of classical logic that a formula A is a theorem of a theory T if and only if the theory T with the negation of the closure of A as an additional axiom is inconsistent. In fact, it is a routine argument that the closure of A is a theorem of T whenever A itself is a theorem of T. The above inconsistency then follows from the ex contradictione quodlibet rule (ECQ) that deduces an arbitrary formula C, from an arbitrary formula B and its negation. It suffices to take the closure of A for B. This type of inconsistencies is well understood and it is not too serious for foundationalist's view of symbolic logic and mathematics. It does not cast doubt on the validity of the ECQ rule in classical logic. In the above case we know, where the inconsistency comes from.In the general case, however, we are not in such a condition. Inconsistency of a general theory is a more complex problem. The idea that the world is or might be inconsistent has recently been gaining support. In its modern manifestation it has been the province of symbolic logic with motivations from logic, semantics and the foundations of mathematics. From the philosophical point of view, the idea has its roots in an older view that change and especially motion is contradictory. This view can be traced back through Hegel to Zeno and Heraclitus. The paradoxes of logic, semantics and set theory on one hand and the semantics of relevant logic on the other hand are two recent motivations. For the purposes of the book under review, a theory is a set of sentences closed under a deductive relation. A logic is then a theory being closed under the rule of |
substitution relative to uniform specified set of constructs
such as 'not', 'and', 'or', 'implies', 'is equivalent to',
'for all', 'there exists' and 'equals'. The uniform substitution
rule expresses the idea that the logic ought to be neutral with respect to subject
matter. Moreover, the deductive theories are presented with a logic in the background,
which is tacitly presupposed as neutral. The naive set theory with its principle of unrestricted set abstraction known as the unrestricted comprehension scheme can serve as an example of inconsistent theory. The foundationalist approach calls for a weakening of the comprehension scheme in order to insure a consistent set theory, which is seen as necessary to provide a consistent foundation for mathematics. Another possible way was given by Da Costa and Routley who suggested that the Russell set that has its origin in the unrestricted comprehension scheme, might be dealt more naturally within the frames of an inconsistent but nontrivial set theory. Here triviality means that every sentence is provable, and obviously, triviality of a theory makes it uninteresting. To keep a possibly inconsistent theory nontrivial means that the rule ex contradictione quodlibet (ECQ) should not be valid in the background logic. Such logics are called inconsistency-tolerant or paraconsistent. Another motivation came Anderson and Belnap's investigations of relevance or conceptual connection. They came with the idea that correct natural entailments should be based on conceptual connection. Then the ECQ rule could not be a universally valid principle, because its premises can be irrelevant to its conclusion. The paraconsistent doctrine is expressed in two ways, namely, as strong paraconsistentism which accepts true contradictions, and, as weak para-consistentism which is the thesis that contradictory possibilities have to be considered in the semantics of natural logic. Needless to say that the book under review is not foundationalist. It puts the problem of inconsistent mathematical theories from the paraconsistent point of view. |
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| The book discusses problems related to several inconsistent
mathematical theories, its fourteen chapters are organized as follows. Chapter 1 is
introductory, Chapters 2 and 3 are on number theories. The consistent
relevant arithmetic of Meyer is discussed in Chapter 2 and it is argued that it
gives a closer account of 'natural' arithmetics than classical Peano arithmetic does. The
classical denumerable nonstandard model of arithmetic gives rise to infinite inconsistent
models of the theory of the field Q of rational numbers with some consequences for
the Fermat's last theorem. Chapter 4 is on order, it examines the well-known
theorem stating that the theory of dense orders without endpoints is {0} -
categorical from the point of view of a three valued logic. Chapters 5 - 7
are on analysis, they deal with an inconsistent differential calculus, inconsistent
continuous functions and the delta function of Dirac. Inconsistent systems of linear
equations are treated in Chapter 8. Chapters 9 and 10 treat some
problems from geometry of projective spaces and topology. Chapter 11 (with P.
Lavers) and Chapter 12 (by W. James) are on category theory and sheaves. Chapter 13
is on duality in certain logics and Chapter 14 (with J. Cole) is on foundations. It
deals with provability, truth and sets. A short bibliography and index conclude the book. The book touches upon many mathematical problems. Although most of them have a consistent foundationalist solution, the book stresses the (usually inconsistent) paraconsistent approach. It is likely that most readers are acquainted with the consistent solutions of the mathematical problems and that the paraconsistent background logics are not known to them. For such a reader, the presentation of nonstandard background logics which is sometimes restricted to the set of axioms and rules and/or Hasse diagrams of the lattice of the truth values is rather short and insufficient. It is there wiever's opinion that a more complete presentation of paraconsistent logics which play the central role in the book would be desirable. The dedicated reader may find his or her way through numerous citations of relevant literature, however. The book is recommendable to researchers and students of Logics, Mathematics and Physics. To some a |
extent also to students and professionals in the domain of
Computer Science. Petr Štěpánek Department of Theoretical Computer Science, Charles University, Prague, Czech Republic
Peter Öhrström, Per F. V. Hasle: Temporal Logic, from Ancient Ideas to Artificial Intelligence. Studies in Linguistics and Philosophy, vol. 57. Kluwer Academic Publishers; Dordrecht, Boston, London 1995. viii + 413 pages. Time is ubiquitous and so is temporal logic. (Or, at least, it should be.) That is why the book under review has been written. It puts the problem of temporal logic in a broad perspective of time ranging from Antique through the Middle Ages to the twentieth century as well as perspective of variety of disciplines from general philosophy, ethical and theological considerations, conceptual analysis, linguistic considerations and literary fiction on one side and mathematics, physics and computer science on the other one. The pivotal discipline for linking together those various observations is logic in a broad sense, that is, informal logic and logic in a fully symbolic form. The book under review is divided into three parts. In the first part of the book the concept of time is discussed from the perspective of the history of logic. It is shown that there is a rich tradition of temporal logic from the ancient and medieval periods. This part of the book deals with the rich contribution to temporal logic in ancient and medieval philosophy as well as the crucial systematic questions within the field up to its downfall in the Renaissance. The authors take the liberty to present some of these old ideas within the framework of symbolic logic. They go into this type of explanation in spite of the continuing dispute on the application of symbolic logic to ancient and medieval logic which is considered as |
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| anachronistic and misleading by some researchers. The example
of this approach was set by the Polish School of Logic and the authors stress that some of
the brilliant insights of some ancient and medieval logicians can be fully understood only
when presented in a symbolic formalism. The second part describes the modern rediscovery of temporal logic,which is especially due to the works by A. N. Prior. In the 1950's and 1960's Prior laid out the foundations of temporal logic and showed its intimate connection with modal logic. He also revived the medieval attempt at formulating a temporal logic corresponding to natural language. On his way to this goal, he also used his symbolic formalism for investigating the ideas put forward by medieval logicians. Prior argued that temporal and modal logic are particularly relevant to a number of theological as well as philosophical problems. He also analyzed the fundamental question of determinism versus freedom of choice. Part two described all this focusing on Prior's contribution and on the work of his most important forerunners in the field of temporal logic in the 19th century and the first half of the 20th century. During the last decades, it has become clear that temporal logic also has a number of practical applications, namely in Computer Science, in Natural Language Understanding and Artificial Intelligence. To some extent, this was predicted by Prior in the late 1960's. Part three deals with modern issues in temporal logic. It starts with the tension between two philosophical conceptions of time. It deals with the problem of human freedom and determinism versus indeterminism. It is show that it is possible to establish an indeterministic tense logic which is satisfactory from a philosophical point of view, and that exhibits a close relation to natural language. Branching time models with a particular view to natural language understanding, the calculus of duration and C.S. Peirce's formalism of existential graphs that can be related to deductive databases and represent a kind of a simulation of human cognition are next topics |
discussed along with the problem of modal and temporal
reasoning. The last chapter is presenting the use of temporal logic in various parts of
Computer Science, namely, in Artificial Intelligence and Program Specification and
Verification. The book is concluded by an Appendix which gives a summary of some formal
systems discussed in previous chapters, by bibliography and index. The book under review presents history and philosophical background of temporal logic as a fascinating subject. It is particularly strong on the philosophical issues related ancient and medieval period of temporal logic and to rediscovery of temporal logic in the twentieth century. All the material is presented in a uniform fashion that allows even a freshman to the field to grasp very soon the main concepts and ideas of various systems of temporal logic. On the other hand, the presentation of applications of temporal logic to Computer Science and Artificial Intelligence reduces to two examples of temporal versions of Computer Science classics (Towers of Hanoi and a program computing the greatest common denominator) and to a few comments on such concepts as total and partial correctness, termination, deadlock, mutual exclusion, fairness and liveness. To cover these problems in depth would require more space to discuss some highly technical points of Computer Science before presenting the contribution of temporal logic to the subject. The authors went up to the point to let the reader feel that positive gain from the use of temporal logic is possible. The book is recommendable to students and researchers in Logic, Philosophy, Linguistics and History of science. It may serve as well to more theoretically minded students and research workers in Computer Science and Artificially Intelligence. The book is likely to become a standard reference for some time to come. Petr Štěpánek Department of Theoretical Computer Science, Charles University, Prague, Czech Republic |
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| Ulrich Höhle and Erich Peter Klement (editors): Non
classical Logics and their Applications to Fuzzy Subsets, A Handbook of the
Mathematical Foundations of Fuzzy Set Theory. Theory and Decision Library, series
B: Mathematical and statistical methods. Kluwer Academic Publishers; Dordrecht, Boston,
London 1995. 300 pages. The book under review is dedicated to the semantics of non-classical logics and to its application to Fuzzy Set Theory. It collects the proceedings of the 14th Linz Seminar on Fuzzy Set Theory, a research symposium dedicated exclusively to non-classical logics and their applications to fuzzy set theory. This symposium took place in September 1992 at the Bildungszentrum St. Magdalena, Linz Austria. The book presents twelve papers with special emphasis on the fields of intuitionistic logic, Lukasiewicz logic and fuzzy set theory selected from twenty six talks and three invited lectures presented at the symposium. The papers included in the book cover topics reaching from monoidal, lattice-theoretical structures and categorical aspects of non-classical logics to epistemological problems of fuzzy set theory. The advent of non-classical logics can be traced back to the classical papers published by J. Lukasiewicz, E.L. Post, A. Heyting, G. Birkhoff and J. von Neumann in the 1920's and the 1930's. Non-classical logics are fragments of classical logic. Most non-classical logics can be characterized by the abandonment of the law of excluded middle and the maintenance of the integrality, the exportation, importation and Duns Scotus law. Intuitionistic logic and Lukasiewicz logic represent two typical directions among non-classical logics, each of which has its own significant role: intuitionistic logic is of great importance in the foundations of constructive mathematics and Lukasiewicz logic, which admits antinomies,throws a light on paradoxes of set theory. The book is divided |
into three parts. The first one, Part A, deals with
lattice-theoretical and monoidal foundations of non-classical logics. It investigates
properties of so called MV-algebras, Heyting algebras, integral, commutative Girard
monoids and their relationships. However, as the book is preoccupied with the algebraic
and categorical characterization of non-classical logics, the reader interested in the
formal systems of these logics is recommended to start his reading elsewhere, in classical
papers concerning nonclassical logics. Part B is on categorical foundations of non-classical logics and their applications to fuzzy set theory. Papers of this part have a topos-theoretical flavour and deal with relations between lattice-valued maps which are viewed as generalized characteristic functions and subobjects in various categories. Part C is on applications. It deals with some general aspects of non-classical logics and comprises some fundamental investi-gations in epistemological questions of many-valued logics. It studies interesting model-theoretical properties of fuzzy logic as well as a concept of the logic programming language Prolog in many-valued language. A bibliography that covers the substantial part of the current literature on the relations between non-classical logics and fuzzy set theory and an index concludes the book. The book can be recommended to graduate students and scientists in the domains of Algebra, Category Theory, Logic and Computer Science who are interested in many-valued non-classical logics and fuzzy set theory. It will be a standard reference source for some time to come. Petr Štěpánek Department of Theoretical Computer Science, Charles University, Prague, Czech Republic |