Temporal logic. From ancient ideas to artificial intelligence
This book belongs to a certain tradition of temporal logic that is traced back to J. M. E. McTaggart, H. Reichenbach and, especially, to A. N. Prior (1914-1969). A large part of the book can be seen as an extensive survey and commentary of Prior's work, containing numerous references to and quotations from his writings. In this tradition emphasis is laid on the conceptual and philosophical aspects of the notions of time and logic, rather than on the technical and mathematical ones which are usually stressed in the tradition built by the founders of mathematical logic. So the book differs essentially, both in scope and style, from other established monographs and surveys on the subject, like \textit{R. Goldblatt}'s ``Logics of time and computation'' (1987; Zbl 0635.03024), \textit{J. P. Burgess}' ``Basic tense logic'' [in: D. Gabbay et al. (eds.), Handbook of philosophical logic. Vol. II: Extensions of classical logic, 89-133 (1984)] and \textit{D. Harel}'s ``Dynamic logic'' [ibid. 497-604 (1984; Zbl 0572.03003)]. The only other book known to me that, beside its mathematical competence, seems to share some of the philosophical concerns and the spirit of the present one, is \textit{J. van Benthem}'s ``Logic of time'' (1983; Zbl 0508.03008); revised edition (1991; Zbl 0758.03012). In the aforementioned work the emphasis is laid primarily on those mathematical techniques that make it possible to capture the in-time located truth; it is also on the subtleties of proof-theoretic and semantic issues raised by these logics; and it is finally on current applications of temporal logic to areas of great theoretical and practical importance, such as the logic of programs, concurrency, etc. In contrast, the present book, in its greatest part, offers to the reader on the one hand a fascinating tour over the history of the ideas and arguments in which the notions of time, tense, necessity and truth are involved, from Antiquity up to the present day; and on the other hand, the application of temporal thinking in the treatment of such philosophical questions as determinism and human freedom, the truth of counterfactual statements, tense logic and special relativity, and the particularly interesting distinction between logical and causal implications revealed by Peirce's paradox (discussed in chapter 3.6). Some topics from computer science are also considered. The reader has the freedom and divine foreknowledge and other topics that are hardly touched in mathematically oriented monographs. The book is divided into three parts, a division stemming from the authors' view about what constitutes the great periods in the development of logical thinking. Part I is entitled ``Time and logic -- from synthesis to dissociation'' and surveys the development of temporal notions from Aristotle and Diodorus to the Middle Ages, and from there to the 19th century. To give the reader an idea of the contents of this part, I cite the titles of its chapters: ``The sea-fight tomorrow'', ``The master argument of Diodorus Cronus'', ``The study of tenses in Middle Ages'', ``Temporal ampliation'', ``The duration of the present'', ``The logic of beginning and ending'', ``Time and consequentia'', ``Temporalis -- the logic of `while' '', ``Human freedom and divine foreknowledge'', ``The downfall of medieval tense logic'' and ``Logic of timeless science''. Part II is entitled ``Time and logic reunited'' and surveys the second period of logic development. The starting point is, as expected, the appearance of Boolean logic. The authors go through such issues as ``The 19th century and Boolean logic'', ``C. S. Peirce on time and modality'', ``Lukasiewicz's contribution to temporal logic'', ``A three-point structure of tenses'', ``A. N. Prior's tense logic'', ``The idea of branching time'', ``Tense logic and special relativity'', ``Some basic systems of temporal logic'', ``Four grades of tense-logical development'' and `` Metric tense logic''. Part III, entitled ``Modern issues in temporal logic'', contains the following chapters: ``Two paradigms of temporal logic'', ``Indeterministic tense logic'', ``Leibnizian tense logic'', ``Tense logic and counterfactual reasoning'', ``Logic of durations'', ``Graphs for time and modality'', and ``Temporal logic and computer science''. The book is obviously ambitious in the sense that the authors consciously try to cover almost every aspect of the subject. As they say in the preface ``\dots We wish to make it clear what we are ourselves aware of some limitations of this book. But we also believe that it does contain an unusually comprehensive exposition of the study of time.'' It does not come as a surprise that all books of such a wide scope face an inherent difficulty: They have to try hard to compromise between the two rather conflicting demands, namely conceptual-philosophical (and historical) completeness on the one hand, and technical thoroughness on the other. Usually one of these demands dominates over the other, depending on the author's particular training and inclination. Concerning the present book, the philosophical aspect, clearly, overshadows the mathematical one. This is natural and rather expected. We have, however, some objections concerning the kind of mathematics that the authors choose in order to expose the ideas of (modern) temporal logic in chapter 2.9, which seems to form the core of the technical part of the book. To be specific, it is true that the distinction between A-logic and B-logic, going back to McTaggart, is worth mentioning, for historical reasons, as it has been the predecossor of what we today call syntax and semantics of temporal logic. But instead of simply citing that as a historical reference and then pass to an up to date treatment of syntax and semantics, and the relevant soundness and completeness results, the authors embark in a full exposition of Prior's system concerning the mathematical treatment of these ideas. Prior had introduced what he called ``four grades of tense-logical involvement'', the A- and B-logic forming the first two of the four. Prior's ideas of course are important but his system is unnecessarily complicated and very hard to follow. There seems to be a confusion between language and metalanguage, and between provability and truth. For example the authors give the following quotation of Prior's definition (p. 219): ``What I shall call the third grade of tense logical involvement consists in treating the instant-variables \(a\), \(b\), \(c\) etc. as representing propositions.'' The uninitiated reader is very unlikely to understand that ``the third grade'' is nothing but the germ of what we call today ``the canonical model''. The authors write: ``We are not going to demonstrate completeness in a traditional mathematical way, but we intend to show that a result very similar to completeness can be obtained in the context of Prior's third grade.'' (p. 221). This result, given in an old-fashioned, not quite legitimate (by the present standards of rigor) mathematical formalism is hard to follow and even harder to estimate its meaning and significance in the form in which it is stated. Nevertheless, I think that, in present day terms, it essentially corresponds to the completeness of a temporal logic \(\Lambda\) with respect to its canonical model \(M^\Lambda\). For example, R. Goldblatt [loc. cit.] proves this result in one and a half page (p. 25) with incomparable elegance, clarity and mathematical economy. (Strangely, Goldblatt's book is not cited in the bibliography of the present one.) The overall technical exposition adopted by the authors belong to what J. van Benthem calls ``the dark ages of temporal thinking''. It is commonplace that the form in which important ideas are conceived by pioneers is far from clear. To insist in exposing temporal logic in the way Prior or McTaggart and other pioneers have originally formalized it, looks like wanting to expose modern Analysis in the style of Cauchy or Fourier. To give a more striking recent example, it looks like exposing the method of forcing in set theory in the way P. Cohen had conceived and exposed it in his original 1966 book. No beginner is likely to understand even a bit of the core of forcing ideas from his contact with Cohen's work. However, within less than 10 years from their first appearance, Cohen's ideas had been completely reformulated, simplified and clarified in terms of Boolean algebras to the effect that they soon became accessible even to a sophomore. If the authors considered the presentation of Prior's ideas in their original form as inescapable, they could just have outlined them in chapter 2.9, and then proceed in Part III, which is supposed to contain the modern developments of the subject, to an updated presentation in the light of recent elaborations of the notions involved. The above constitutes our criticism about the presentation of the main technical results of the book. Another obviously weak point is the last chapter 3.7, which is supposed to survey the relations of temporal logic with computer science. Surprisingly this chapter contains essentially only an example of how temporal thinking can be applied to produce a program for the solution of the Hanoi towers problem. We believe that this chapter should be either enriched and extended considerably in order to include a discussion of dynamic logic and other relevant issues, or it should be omitted altogether. In its present form it offers a very poor, and hence inexact and misleading, picture of the supposed-to-be relationships and applications of temporal logic to computer science. The preceding comments by no means intend to doubt the merit of the book. They are just hints and suggestions for improvements concerning future editions. The book is very well written, with no typographical or other kinds of errors. As an overall estimate and independently of the above criticism, the book is a valuable source of information for everyone interested in the philosophy and mathematics of time, as well as in the semantics of natural language. I myself found a lot of stimulating discussions concerning the truth of natural language propositions, in contrast to that of mathematical ones. Mathematicians often forget that temporal expressions can never concern mathematical facts, but only facts sensitive in time change. And it is doubtful whether the ``logic'' of facts is the logic as we know it from the mathematical practice. The logic of facts is the logic of \textit{this} world, whereas the logic of mathematics is the logic of every ``possible world''. I found of particular interest chapter 3.4 on counterfactuals and chapter 3.6 on Peirce's paradox concerning causal versus logical implication. I cannot refrain from casting a final minor comment. At the end of the book (p. 413) the authors cite a list of 33 names labeled ``Some important logicians and philosophers''. The list opens with Aristotle and closes with A. N. Prior. I confess that going through the list I felt somewhat philosophically and/or logically illiterate, as, besides the names of Bacon, Descartes, Boole, Frege, Lukasiewicz etc., I came across such names as Ibn Sina, Peter Lombardus, Peter of Spain (later Pope John XXI), Richard Kilvington, Joachim Jungius, Juan Luis Vives etc., that I never heard before. Of course this shows nothing but my own limited philosophical training. On the other hand, I hopelessly tried to find some other legendary names, known to everybody as the founders of present day logic and philosophy of mathematics, like Russell, Tarski, Gödel, Turing, Skolem etc. I just wonder whether this omission is due to an accident, or to a conscious attitude as to what constitutes ``real logic'' and ``real philosophy''.
scientific article; zbMATH DE number 907666