Ernest Nagel once remarked that it is fortunately not necessary to be clear about scientific philosophy and methodology in order to practice good science. He went on to say, "Even eminent scientists can make unholy spectacles of themselves when they don the mantle of philosophy and attempt to discuss the broad implications of their specialized labors." Feynman's recent venture into the philosophy of science is, unfortunately, a lucid illustration of the validity of Nagel's observations. The book is a rather literal (...) transcription of seven semi-popular lectures delivered by Feynman at Cornell in November of 1964. Feynman is a wonderful iconoclast, a legendary virtuoso of the bongo drums and—by the way—a Nobel laureate and one of the great theoretical physicists. But this is all the more reason to expect something a little more intellectually solid than is offered in this volume. It was obviously Feynman's intent to discuss some compelling problems which lie on the borderline of physics and philosophy with an intelligent lay audience. But few will be enlightened by Feynman's clever, if uninformative, discourse about such issues as the relation of mathematics to physics, symmetry and conservation laws, and probability and uncertainty. Feynman's finest hour undoubtedly comes in the lecture on quantum mechanics, in which some aspects of the wave-particle duality are presented. But the value of much of the rest of the book is severely qualified by the author's penchant for imprecise and confusing statements and, in some cases, outright mystery-mongering. This is a great pity because theoretical physics, especially in the hands of Richard Feynman, is a source of ideas which are most worth discussing.—H. P. K. (shrink)

This is a very fine elementary-to-intermediate level text of mathematical logic. The initial chapter of the book consists of a good discussion of standard topics in modern formal logic including arguments and argument forms, logical functors, validity, proof, the axiomatic method, interpretations, and logical systems. The book then proceeds in subsequent chapters to a development of increasingly rich systems of sentential logic, systems of natural deduction, and a chapter on consistency and completeness of formal systems. This takes the student through (...) approximately one half of the book. The second half is devoted to more advanced topics such as non-standard systems of propositional logic, and predicate logic of the first order. Finally, there are chapters on higher functional calculi, the logical paradoxes, and non-standard functional systems. A good deal of attention is paid throughout to completeness and consistency properties, the decision problem for various systems, and algebraic structure. There are no frills or "gimmicks" in this book, but good intelligent discussions of important mathematical issues, and many theorems which are left as exercises to the student. Some items which one wishes had been included are normal forms, and elementary discussions of the theories of recursion and logical models.—H. P. K. (shrink)

This is undoubtedly one of the most well-conceived and nicely executed introductory books on special relativity ever written. The authors take the view that relativity theory is no longer an advanced and esoteric branch of physics, but ought to be part of the basic intellectual equipment of any bright college student. To this end, the theory of special relativity is presented as a complete and unified set of concepts and not merely as a kind of gloss on classical Newtonian mechanics. (...) This is accomplished in a manner which captures the flavor and excitement of theoretical physics and makes it available to persons of modest mathematical means, and at the same time avoids the overpopularizing and lack of rigor which so often afflict works of this kind. The text is accompanied by numerous non-trivial problems, many with solutions, and a large number of helpful drawings. There is also a reasonable amount of discussion of matters of philosophic interest, and of the most recent experimental inquiries, including a section on the significance of the Mössbauer effect. Altogether, a first-rate job.—H. P. K. (shrink)

The material contained in this book is based on lectures given by Cohen at Harvard in 1965. It consists of a presentation of logic, set theory and other material, culminating in Cohen's ingenious proof of the independence of the continuum hypothesis and the axiom of choice. Since this proof is certainly one of the major developments in modern mathematics, Cohen's book is something of a necessity for every serious student of the foundations of set theory and mathematics. In addition to (...) Cohen's proof, the book also contains a chapter on Gödel's proof of the consistency of the continuum hypothesis and the axiom of choice, and a good deal of other material of general interest.—H. P. K. (shrink)

This revision of an important and lucid account of the various systems of axiomatic set theory preserves the basic format and essential ingredients of its highly regarded original. Quine's innovative exploitation of the virtual theory of classes in order to develop a considerable portion of set theory without ontological commitment to the existence of classes remains unchanged. So, too, does the list of topics treated--the theory of sets up to transfinite ordinal and cardinal numbers, the axiom of choice and its (...) equivalents, and a beautiful comparison of the various major formulations of set theory. There have, however, been a number of important changes, generally in the interest of greater elegance and clarity. One of the most notable of these is in the treatment of ordinal numbers. In the fourth section of that chapter an axiom schema is added to the effect that if α is a function and x an ordinal number, then there exists a class identical with the image of x under α. Although Quine, in general prefers to limit his existence assumptions as severely as possible, this added item possesses two attractive and closely related advantages: it yields the existence of all classes up to the size of any ordinal, and it greatly simplifies the subsequent treatment of ordinals and of transfinite recursion. The treatment of this latter subject, incidentally, has also been extensively modified to square with some difficulties first pointed out by Charles Parsons in The Journal of Symbolic Logic in 1964. Another important change consists in a considerable reworking of the section on infinite cardinals. The result is a more simplified and natural treatment. Finally, some new material on the theory of types and consequences of the axiom schema of comprehension has been added. Particularly with regard to the theory of types, this new material--due largely to the work of David Kaplan--has enabled Quine to drop, at one point, the assumption that there exists an object of lowest type. Altogether, this is a generally improved version of an originally masterful and brilliant work.--H. P. K. (shrink)

Russell writes with wit, candor, and uncommon honesty about his Victorian childhood, his painful adolescence, and his extracurricular amorous conquests. On the credit side, it must be said that it offers a remarkable insight into the development of a remarkable man. But for the benefit of those with more than a casual interest in the philosopher named Bertrand Russell, it should be mentioned that this book suffers from chronic intellectual malnutrition. Not that there isn't a lot of name-dropping and folksy (...) correspondence between the intellectual giants of the early twentieth century. There is a perfectly mad letter from Georg Cantor, who refers to Kant as "this abominable mummy"; and a perfectly charming one from Norbert Weiner's father, who, by way of introduction, states that his son has studied philosophy under "professors Royce, Perry, Palmer, Munsterberg, Schmidt, Holt, etc." In a delicious jest, Russell adds as a footnote, "Nevertheless he turned out well." There is, however, nothing substantial about philosophy in this book. At its close, Russell had completed Principia. He had returned to Cambridge already a most important figure in philosophy and mathematics. But there is no real discussion of the ideas which led up to and leavened this period in Russell's life. There is instead the dreary story of Whitehead's financial troubles. One hopes that a proper sense of the importance of things will be restored in future volumes of Russell's autobiography.—H. P. K. (shrink)

In this series of four lectures a scientist examines some philosophical implications of what one might loosely call "the modern scientific view of life". Bronowski has produced numerous books and articles on everything from the poetry of Blake to the clock paradox of relativity theory. It is therefore to be regretted that his latest work suffers from an annoying amount of oversimplification, and a tendency to substitute rhapsodizing for reasoning. Those concerned with the "two cultures" debate and related matters will (...) nevertheless find this book valuable and enjoyable, especially for the author's argument—which is convincing—that science and the arts need not be antithetical either in subject-matter or in spirit.—H. P. K. (shrink)

This book about philosophical and methodological problems in psychoanalytic theory is surely one of the best pieces of literature on this subject of recent vintage. The author, a psychiatrist on the faculty of Harvard Medical School, displays considerable logical skill and philosophical sophistication, in addition to the expected familiarity with the psychoanalytic literature. The major purport of the book is a logical and philosophical defense of the claim that psychoanalytic explanations of human behavior--if constructed with proper and adequate regard for (...) relevant clinical data--are methodologically sound and genuinely scientific. One useful and novel feature of Sherwood's argument is the particular type of illustrative case material which he employs: instead of mustering various bits and instances of behavior from large numbers and varieties of cases, Sherwood considers the psychiatric explanation actually given by Freud in one of his recorded clinical case histories--the famous "Karl Lorenz" or "rat man" case. Sherwood devotes two chapters to a detailed description of this case, and an analysis of the logic of Freud's explanations of the behavior presented therein. From the point of view of logical and methodological considerations, there is an obvious advantage in such an approach, for it enables the critic to avoid leveling charges against the theory which might originate simply from extracting portions of explanatory material from their proper context. Sherwood also includes a chapter of criticism of what he calls "the thesis of the separate domain"--the view that there are fundamental differences in the logic of explanations of human behavior, cognition, and other phenomena involving reasons and intentions on the one hand, and other forms of explanation characteristic of the physical sciences, on the other. Philosophically, this is the most mature chapter of the book, and will be of interest to philosophers of mind, as well as philosophers of science, particularly for its criticism of the views of R. S. Peters. Unfortunately, however, Sherwood's argument is not unmarred by serious flaws and deficiencies. These include a highly psychologistic analysis of the logic and structure of explanation whose subsequent use repeatedly involves the author in context-of-discovery vs. context-of-justification confusions. Moreover, an apparent lapse into a non-Humean account of causality in the second section of the fifth chapter leads Sherwood to an unfortunate petitio principii on the issue of causes vs. reasons. Sherwood's critical discussion of the classical "covering-law" model of scientific explanation in the penultimate chapter exhibits numerous failures to come to grips with the essential aspects of the Braithwaite-Hempel-Oppenheim theory. Finally, an important sin of omission: Sherwood completely ignores the highly germane and decisive issues raised by authors such as Nagel and Arthur Pap, who charge the theoretical concepts of psychoanalysis with lack of cognitive and empirical significance. In sum, this is a flawed but eminently worthwhile book in the philosophy of behavioral science.--H. P. K. (shrink)

The "new age" to which Sir Harrie refers in his title is none other than the contemporary one of particle accelerators, electronic devices, high-powered rockets, space probes, nuclear reactors and other fancy "things" which have their intellectual foundations in the theories of relativity, quanta and particle physics. This volume is a sort of intelligent layman's guide to the huge enterprise which is modern physics. The author refuses to recognize any artificial distinction between experimental and theoretical physics, but considers them to (...) be complementary aspects of one science. His technique is to consider various experimental items and recent technological developments, and to explain them in terms of the laws and theories which they bring into play. In general, Sir Harrie's exposition is excellent on the description of experiments and their results, and on applications of theories to technology, but it is a bit off the pace when it comes to exegesis of theories and answers to "why?" questions. This is understandable to the extent that the book was written for the mathematical novice, but there is a residual and glib thinness to the text. But despite its shortcomings the book will provide the scientifically curious layman with stimulating reading, an overview of a rich and complex subject, and a nonrigorous "working knowledge" of contemporary physical science.—H. P. K. (shrink)

One would like to be able to say that a large part of the material in this book is so obvious as to be unnecessary. Unfortunately, this book is worthwhile precisely because one cannot. Bandman's problem may be stated as follows: By what canons and criteria does one evaluate arguments in the philosophy of education? How, in particular, do we correctly evaluate arguments about what ought to be taught? The book is thus an exercise in the "metaphilosophy" of education. In (...) the first, polemical half of the volume the author convincingly demonstrates that numerous and typical patterns of argument employed in the philosophy of education by many authors violate some of the most basic and pervasive logical rules. The really striking thing here is that Bandman persuasively charges numerous well respected educational thinkers with informal fallacies that a freshman logic student would not commit. The constructive side of Bandman's work is a bit more dubious. Initially, he distinguishes between two senses of "argument": Sense I and Sense II. These are, respectively, a "special evaluative type of argument, which does not intend to demonstrate the conclusion, and is used in giving intellectual backing to [it]," and the usual sense of argument common to formal logic. Bandman's thesis is that arguments in Sense I are "capricious," inconclusive, and typically unsound, while educational arguments in Sense II are either unsound or irrelevant. He proposes a middle way between the horns of this dilemma by urging a third sense of "argument" as the provision of "good enough" and "cogent" reasons. At least two lines of attack on Bandman's position are available. In the first place, in dismissing the utility of Sense II arguments, he makes rather facile use of the noncognitivist view that moral statements are neither true nor false. Secondly, Bandman does not adequately spell out his Sense III notion of "argument," and the few necessary conditions that he does provide fail to clarify in what sense the premisses support the conclusion and how Sense III really differs from Sense I. Despite these criticisms, the merits of the book are many, and it deserves a wide reading by philosophers and educators alike.—H. P. K. (shrink)

This is a well chosen anthology of articles, both modern and classical, on logical, epistemic, and metaphysical issues of time. The editor, whose own work on the philosophy of time is well known, has provided interesting and informative introductory essays to each of the five sections of the book. Topics dealt with include general philosophical inquiry into the nature of time ; static versus dynamic theories of time, including Donald Williams' celebrated article, "The Myth of Passage," as well as two (...) articles by Findlay and Smart which are practical prerequisites for any subsequent inquiries into modal tense logic; a section on "open future" theories, leading off with Aristotle's famous passage from De Interpretatione. This section also includes a brilliant article on temporal logic by Nicolas Rescher. Section concerns problems of "human" time. Philosophically, this is the least substantial group of selections in the book. Worthy of mention, however, is Grünbaum's controversial theory of temporal becoming, represented by an extract from the first chapter of his recent book on Zeno's paradoxes. These paradoxes, finally, comprise the topic of the final section of this book, and here again Grünbaum scores high with his article "Modern Science and Zeno's Paradoxes of Motion." Also contained in this final section are articles by Bergson and J. F. Thomson. The book has a bibliography which unfortunately is arranged textually rather than indexically. This makes it difficult to use.—H. P. K. (shrink)

The philosophical problems discussed by the author of this scientifically erudite work concern the usual and much debated questions of the role of causality in microphysics and the "completeness" and "indeterminism" of statistical theories in natural science. Blokhintsev, the author of a highly-regarded Russian text on quantum theory, advocates the direct opposite of the Einsteinian thesis; and furthermore, he seems to interpret the alleged irreducibly statistical nature of physical theory in a quite literal ontological sense. "We must now accept," he (...) says, "that we cannot ignore the element of games of chance in the behavior of the universe." The materials adduced in support of this thesis consist almost entirely of various essays in theoretical physics, which, while undoubtedly relevant to the question at issue, will be accessible in their details and content only to those philosophers with better than average mathematical acumen, particularly in classical analysis. Unfortunately, as so often happens in discussions of this type, the author's scientific expertise is not matched by his degree of philosophical rigor and awareness of important conceptual distinctions. Thus, although the concept of measurement in quantum mechanics and the differences between "direct" and "derived" measurement play a central role in his argument, the author seems unaware of the important logical differences which are brought out by rigorous postulational treatments of measurement theory. Moreover, when the author argues, convincingly enough, that even classical Newtonian mechanics is not a fit candidate for the label of "deterministic theory" if such factors as random fluctuations in the initial conditions and lack of isolation of actual physical systems are taken into account, he seems to forget that this is not the basis on which deterministic status is properly assigned to classical physical theory in the first place. It is physical theory as applied to suitably idealized physical systems which is held to be deterministic, and the same sort of idealizations are commonly assumed when quantum mechanics is said to be indeterministic.--H. P. K. (shrink)

The critical reader who possesses the patience requisite for wading through this linguistic morass can expect to be rewarded with a number of interesting insights and observations, primarily concerning phenomenological aspects of science and scientific method. Sikora's argument is informed by a Thomist philosophy of science-especially as exemplified in the writings of Jacques Maritain. His task is an analysis of the distinction and relation between physical science and its philosophy—an interesting and important problem. But unfortunately as in steamship cruises to (...) exotic and far away lands, so also in philosophy: getting there is half the fun.—H. P. K. (shrink)

Andrzej Mostowski, whose previous book on Gödel's incompleteness theorem is widely acclaimed as a modern classic on the subject, now gives us a series of sixteen beautifully clear lectures on the development of logic and foundations of mathematics during the past thirty years. A very wide range of subjects is treated in this book, from the intuitionistic logic and the Gödel work on the incompleteness of arithmetic to the set-theoretical results of Cohen and the algebraic theory of direct and reduced (...) products. The notable thing about all this is that the book is everywhere readable by the non-expert. Anyone who has had an elementary course in mathematical logic and set theory can read this book with a great deal of profit. Moreover, the exposition sparkles with Mostowski's remarks, conjectures and speculations on the philosophy of mathematics and the future of foundational studies. There are very few rigorously proved theorems in the book—as this was not its purpose—but a bibliography of 244 pertinent items is furnished at the end of the book for those who would like to pursue the material to greater depth.—H. P. K. (shrink)

This book is a contribution to both the study of the logical and philosophical foundations of physics, and the investigation of applied formal axiomatic systems. Basri uses the techniques of logic and set theory in order to construct a rigorous physical theory whose theorems turn out to be those of the general theory of relativity or else arbitrarily close approximations thereof. Whether Basri's approach turns out to be fruitful for the analysis of foundational problems in physics remains to be seen, (...) but he has clearly rendered an important service in providing at least one rigorously unified way of talking about such things as particles, events, signals, clocks, rigid measuring instruments, etc. A novel feature of the book is that the basis of Basri's system is rather phenomenalistic: he uses primitives like "the class of unbiased human observers" and "the class of simultaneous sensations of observer H." The obvious philosophical problems about perception and the external world which are consequent upon this sort of approach are not treated by Basri, but he is certainly not to be faulted on this account. There are, however, a number of errors and imprecisions in the underlying logistic system, some of which are more serious than others. But in general, this book deserves to be read and its thesis considered by all students interested in serious and responsible efforts in the philosophy of physics.—H. P. K. (shrink)

This recent addition to the well-known "Studies in Logic" series is sure to be of first importance to serious students of inductive logic, confirmation theory, and related issues. The book is an anthology of fourteen papers, which are classified under five different headings: "Extensions of Inductive Logic," "Induction and Information," "Prospects of Confirmation Theory," "The Paradoxes of Confirmation," and "Probability and Foundational Problems." Needless to say, all of the papers are of uniformly high quality. Especially worthy of mention are two (...) papers by Hintikka and Hilpinen which deal with an interesting generalization of Carnap's continuum of inductive methods, and which lead to a somewhat more plausible treatment of non-logically-true closed sentences. Suppes contributes two papers dealing with the recently revived Bayesian methods and their significance for concept formation in psychology and confirmation theory in the philosophy of science. Also under the heading of confirmation theory, one might mention Max Black's paper, "Notes on the Paradoxes of Confirmation," a lucid and rather novel approach to this problem. There is also an important paper by Dana Scott and Peter Krauss entitled "Assigning Probabilities to Logical Formulas." This piece of work presents a set-theoretical method for bridging the gap between the standard treatment of probability by means of axioms defining a real measure over a Boolean algebra and the usual logical inquiries into "degree of confirmation" and the like. The reference value of the book is enhanced by a subject index of terms and concepts dealt with in the various papers.—H. P. K. (shrink)

A small but comprehensive textbook on Boolean algebra, sentential logic, and lattice theory; this book will be of interest to students of logic and foundational studies in mathematics, particularly with respect to algebraic representations of propositional logic and elementary metamathematics of algebra. The book contains a self-dual set of postulates for Boolean algebras, with proofs of its completeness and independence. The book is written on an elementary to intermediate level, contains numerous exercises, a short index, and an even shorter bibliography.—H. (...) P. K. (shrink)

The strange and fascinating philosophical problems associated with the quantum theory are brought within the grasp of the nonspecialist by this brilliant and lucid little book. The author is one of the outstanding Soviet theoretical physicists. By a judicious use of drawings and diagrams he has been able to present some high-powered physics without reliance upon equally potent mathematical methods. In fact, anyone with a fertile imagination, intellectual curiosity, and a background in high-school algebra can read this book with profit. (...) At the same time, the chapters on the uncertainty principle, quantized fields, and Dirac's theory of the electron will surely give philosophically sensitive readers a good deal to mull over; e.g., "The uncertainty principle in no way denies the existence of momentum and position as precise quantities; it merely asserts that they cannot exist simultaneously as precise quantities." The book may even be of value to sophisticated students of physics as an aid in "visualizing" some of the rather abstract concepts of quantum theory.—H. P. K. (shrink)

Yet another introductory text of symbolic logic and a very good one, at that. The four principal chapters of this book treat propositional logic and predicate calculus, devoting two chapters to each. Each of these subsections is roughly organized as follows: An initial chapter presents the basic notational devices, translation methods, and intuitive discussion of arguments and validity. The subsequent chapter gives exact formation and transformation rules, proofs and metalinguistic considerations of questions of consistency and completeness. Lemmon's notation is similar (...) to that of Suppes, and his system bears some similarity to the rules of inference in Fitch's 1951 text, although Lemmon's system is a "standard" one. Two appendices, dealing with normal forms and elementary set theory, and an annotated bibliography of relevant works complete this high-quality text. An obviously promising career was cut tragically short by the premature death of the author in 1966.—H. P. K. (shrink)

The issues treated in this book derive a certain degree of unification from their relation to the general theme of the completeness of scientific theories. Unfortunately, when a philosopher addresses himself to the question of the completeness of an empirical theory, it is far from clear at the outset just what the problem is. Schlegel, to be sure, explicates three different notions of completeness which may be relevant here: the logical, physical, and pragmatic aspects. By the first, Schlegel means the (...) usual set of issues having to do with the deductive and semantic structure of formal theories, the second refers to "the problems that are to be answered in terms of what is discovered by science about the natural world," and the third refers to "the factors in the completeness of science that seem to be chiefly dependent on the interests of scientists." As these formulations suggest, the level of logical and philosophical rigor is hardly commensurate with the kind of investigation one would expect. An early chapter deals with an epistemology of physics in which a class of entities called "constructs" plays the central role. Unfortunately, it is never clear whether constructs denote objects, properties, facts, sentences, or something else. On the other hand, the materials in this chapter are largely irrelevant to the subsequent discussion. Schlegel does an admirable job of scientific exegesis on the present state of quantum physics and cosmology. With regard to the former, he is an advocate of the "Copenhagen view," and he argues that a sort of natural limit of description and explanation has been reached. As to cosmology, Schlegel looks forward to a narrowing of the range of alternate theories as more data become available. In the chapter on purely formal questions a number of oversights seem to have been committed. The exposition of Gödel's incompleteness theorem is inaccurate in that it is not shown that provability can be defined intra-systematically. Schlegel's discussion of how the theorem might apply to scientific theories is suggestive, but also confusing. The chapter on infinity treats the continuum hypothesis as a proven fact, thus ignoring the very important questions raised by the work of Paul Cohen and others. Finally, and most distressingly, the Hebrew letter aleph is printed upside down at all of its occurrences.--H. P. K. (shrink)

This recent addition to the Studies in Logic series is a systematic treatise on the set-theoretic, or semantic, approach to mathematical logic and axiomatic method. The basic notions for the discussion are those of different kinds of languages, their realizations, and the models of a formula. The book begins with a preliminary "chapter 0," giving some general theorems about classes of functions defined by finite schemas. These results are directly applicable to the language of truth-functional propositional logic, and such application (...) is accomplished in detail in chapter 1, in which interpolation, finiteness and compactness theorems are proved for propositional calculus. Chapters 2 and 3 deal with the predicate calculus without and with identity, respectively. More advanced topics dealt with in subsequent chapters include elimination of quantifiers. Certain algebraic structures such as real closed fields and some Boolean rings are given in the form of axiomatic systems in which every formula is equivalent to a quantifier-free formula. This leads to completeness. Also covered are type theory and alternate methods for developing predicate logic, the theory of definability, and the theory of principal models and infinite formulas. Appendices deal with some additional matters of philosophic interest, including a detailed discussion of set-theoretic vs. combinatorial foundations of mathematics. The book lacks an index and a bibliography, but these are not serious defects. This is a highly rewarding work.—H. P. K. (shrink)

This is a rather extensive revision of Hick's well-known work of 1957, and is certainly a welcome addition to the literature on this subject—especially in view of the recent resurgence of interest in epistemological problems in the philosophy of religion. Hick has added a good deal of new material, including a chapter dealing with the traditional Thomist view of religious faith as a propositional attitude, and an extensively revised section dealing with the author's theory of faith as "the interpretative element (...) within the religious person's claimed awareness of God." While Hick argues for a definite philosophical thesis he has illuminating things to say about matters of general epistemological concern. Especially worthy of mention are his exposition and critique of the views of William James and his discussion of issues raised by positivistic criteria of meaning. It might be mentioned that the logical problem Hick raises concerning Aquinas' description of the faith of demons can be dispelled by the simple device of holding that the demons see things that humans don't. Altogether, this is a highly recommended book, quite well suited, incidentally, for use in a one-semester course in philosophy of religion.—H. P. K. (shrink)

It is difficult to describe this book without praising it. Collected here in one volume are some thirty-six high quality translations into English of the most important foreign-language works in mathematical logic, as well as articles and letters by Whitehead, Russell, Norbert Weiner and Post. The contents of the volume are arranged in chronological order, beginning with Frege's Begriffsschrift—translated in its entirety—and concluding with Gödel's famous "On Formally Undecidable Propositions" and Herbrand's "On the Consistency of Arithmetic". The translation of the (...) Gödel article is a small masterpiece, and is especially welcome in view of some of the rather poor efforts which have been made to render this important work into English. But this is more than a paste-pot job of anthologizing. Van Heijenoort and others have written prefaces to each of the included selections, in which are provided background historical and bibliographical information, comments on the intellectual currents which led to the production of the article, and, in many cases, a digest of the important theorems and concepts in the article itself. Each selection is a classic in its own right. In reading through the selections in this volume, one is struck by the extent to which purely mathematical problems and issues have controlled recent logical inquiry and have determined the development of various philosophies of mathematics. In the early stages, Frege and Peano were concerned with the logical reconstruction of arithmetic, and this led to the work of Russell and the logicist philosophy. Already in the development of the new logic there were certain problems which were resolved with Russell's theory of types—a device which many mathematicians felt to be rather artificial. The use of set theory, beginning with Cantor and reaching its maturity in Zermelo's 1904 proof of the well-ordering theorem, gave rise to a new set of difficulties concerning the axioms of choice, the continuum problem, and infinite sets. The two traditions are brought together in a common nexus of foundational problems by Weiner in 1914 and by Skolem's form of the Löwenheim theorem, given in 1920. Hereafter, the way is open for the rise of mathematical inquiry into the foundations of number systems, set theory and logic by Post, Skolem, von Neumann, Hilbert, Bernays, Gödel and many others. The work was far from completed by 1931, but a new and important tradition in mathematical inquiry had been established, and had come into its own. This book is, in effect, the record of an important chapter in the history of thought. No serious student of logic or foundations of mathematics will want to be without it.—H. P. K. (shrink)

This somewhat unusual introductory logic text has been clearly designed to bring the student into contact with the mathematical aspects and problems of logical systems as quickly and naturally as possible, at the expense of "fundamental" discussions of logical theory, language and philosophy. In the introductory chapter, the student is introduced to elementary logical technique via Gentzen-type rules of inference, given the requisite set-theoretical background, given a preliminary orientation with respect to the concept of an axiomatic theory, and then shown (...) the full first-order language of the basic predicate calculus. The following chapter develops an axiomatic version of first-order logic with identity, and exhibits proofs of the deduction theorem, completeness theorem, and various applications of equivalence and replacement theorems and other materials to formal number theory. Finally, in what is undoubtably the most valuable part of the book, a good treatment of first-order mathematical theories is given in the third chapter. This includes an extensive discussion of Tarski's semantical theory of truth and a detailed version of Henkin's completeness theorem. Sections on decidability and Gödel's theorem complete this fine introduction to basic mathematical logic. There are numerous exercises throughout the book.—H. P. K. (shrink)

This is a book about statistical theory without sample theory. A very substantial portion of the modern theory of statistics can be treated without involving oneself in problems of analysis consequent upon the treatment of sampling. Accordingly, Rozeboom has written a book which, while sophisticated, does not demand any high-powered mathematical knowledge or competence. A good deal of the theories of distribution, statistical regression, factor analysis, variance structure, reliability, and miscellaneous applications of probability theory is covered. The author concentrates upon (...) questions of algebraic structure and foundations of the theory, although this is a "practical" text in the best sense of the word. To that end, Rozeboom has designed a special notation which facilitates comprehension of the text, although it does not mix well with the standard modes of logical and mathematical expression. In general, this book is quite well suited for use by students of the social and behavioral sciences and the philosophy of science who have a special interest in the theory and technique of statistical inference. There are a large number of typographical errors in the notation, which are corrected in a six-page addenda supplied with the book.—H. P. K. (shrink)

Here is a fine semipopular book about the ideas which have motivated the much-talked-about revolution in the theories of information, control and communication. Jagjit Singh is one of those rare science writers who knows how to present intricate technical concepts to the less-than-expert reader without compromising the original sense or significance. The book begins, appropriately enough, with a discussion of the concept of information, culminating in the technical definition which enables us to assign numerical values to its quantity. The following (...) chapters present the conceptual heart of the Shannon-Weaver theory, the theories of automatic computing machines and the intriguing neurological theories of Warren McCulloch and John Von Neumann. The chapter on Turing Machines does not pay enough attention to the purely mathematical applications of these devices for the present reviewer's taste. The closing chapters deal with mathematical theories of intelligence and the brain. The text is accompanied by many good drawings and appropriate mathematical demonstrations. A book of this quality could have been produced only by an author with absolute command of his topic.—H. P. K. (shrink)

The contents of this book were originally published in the September 1966 issue of Scientific American. A more appropriate-but perhaps less "catchy"—title would have been "Computers." The book is all about them and what they can—and cannot—do. Conspicuously missing is a chapter on the underlying mathematical theories of information, control, and computation. Nevertheless, there is a good deal of interesting material between the covers of this book. Of greatest concern to philosophers are the chapters on "Information" by John McCarthy, "Computer (...) Logic and Memory" by David C. Evans, "The Uses of Computers in Science" by Anthony G. Oettinger, "The Uses of Computers in Education" by Patrick Suppes, and "Artificial Intelligence" by Marvin L. Minsky. The first of these consists of a brief history of computing machines, an outline of the general principles of digital computers, and an interesting discussion of some of the moral and social issues which such machines raise. Evans' article is a popular description of binary logics and logic devices. The present reviewer has used some of the material in this chapter as embellishments in an undergraduate course in logic. Oettinger's article describes the novel way in which computers may aid in the formation of scientific theories. The paper by Suppes ought to be required reading for members of the "Teaching machines undermine intellectual development" school of thought. Finally, a philosopher of mind might have a field day with some of the remarkable and outrageous notions in Minsky's article; what, for example, is the difference between "thinking" and "exhibiting intelligence"? In sum, this is a stimulating book about a technological development which, for good or ill, is going to affect all of us.—H. P. K. (shrink)

This is a very high quality book with a slightly misleading title. It is difficult to see how it could serve as an introduction for anyone except the mathematically mature or, at least, a student who has already been introduced to formal logic through the lower predicate calculus. Not that these topics are not covered in the book—they comprise the first 92 pages; but the discussion quickly moves into intellectual high gear with sophisticated treatments of the independence of systems of (...) propositional logic, first-order theories, isomorphism of interpretations, and completeness and decidability. With this caveat, Mendelson's book is one of the best of the "second generation" of books on mathematical logic. It is lucid and rigorous, and contains no "frills." It does, however, contain numerous exercises in which the student will have the pleasure of developing for himself some of the more important theorems in the various subjects treated. Among these are chapters on formal number theory, including Gödel's incompleteness theorem, recursive undecidability, Tarski's theorem; a chapter on axiomatic set theory with Hartog's theorem, the axiom of choice, and the axiom of restriction ; and a truly excellent chapter on effective computability which takes off from the notion of an algorithm, and then uses this as a basis upon which to develop the theories of Markov algorithms, Turing algorithms, the theories of Herbrand and Gödel, recursively enumerable sets and undecidable problems. An appendix contains a version of K. Schütte's consistency proof for formal number theory.—H. P. K. (shrink)

The serious formal investigation of n-valued systems of logic for n>2 dates back to Post's 1921 doctoral dissertation. The primary use for such structures, however, has been as model-theoretic devices in the investigation of systems of lower order. Ackermann's short book now comes as a welcome addition to the literature dealing with the formal properties and applications of n-valued systems in their own right. Ackermann begins with a general discussion of implicational calculi in which fundamental ideas of validity, well-formedness, and (...) other basic notions are set forth elegantly and clearly. While assuming the classical implicational calculus as an underlying substructure, Ackermann next motivates a consideration of possible modifications of the formal system in the direction of greater generality by means of the standard Lewis objections to the reading of "⊃" as "implies." This leads to the well known story of strict entailment, introduction of modal operators, and then a four-valued matrix as an underlying semantics for the expanded system. There then follows a good general discussion of the properties of many-valued logics, various nonstandard propositional calculi, and an admirable presentation of the Lukasiewicz-Tarski propositional calculus, accompanied by a number of important metatheorems. Ackermann also includes chapters on alternative multivalued [[sic]] systems and a glance at their applications. One would like to see, however, some discussion of such controversial matters as n-valued quantificational theory and the well-known objections to the entire program of nonstandard logistic theory. There is a good bibliography of important books and articles in the field.—H. P. K. (shrink)

The "textbook explosion" in recent years in the field of logic has frequently been deplored on the grounds that it has resulted in a proliferation of repetitious and sometimes antiquated material. A sober evaluation of this volume, unfortunately, supports this thesis. Despite the subtitle, the bulk of the material in this book wasn't "modern" two centuries ago. Why should 196 pages of a 355 page text be devoted to such topics as "Logic and Psychology," "Types of Statements in Logical English," (...) "The Syllogism," "Translation into Logical English," and "Informal Fallacies"? Philosophic issues are often obscured by the unmanageable clutter of syllogistic form, squares of opposition, sorites, and other baggage. Moreover, why should an entire appendix be devoted to a "block diagram" of a machine which computes the logical relationship between any two sentences? Especially when it is known that no such machine is possible. There is material of genuine philosophical interest in this book and there is a rather desultory treatment of elementary symbolic logic by means of a confused set of natural deduction rules.—H. P. K. (shrink)

The contents of this book, pedagogically sound and intellectually rigorous, live up to the high standards one would expect of its author. A two or three semester course based upon this book will carry the student through all of the requisite foundational material to many of the important contemporary results in recursion theory, nonstandard arithmetic, and other more esoteric areas. The book combines features of a rigorous logic text and a book on the foundations of mathematics and elementary recursion theory. (...) In the first chapter of this latter part, Kleene gives a discussion of such topics as countable sets, Cantor's diagonal method, the paradoxes, axiomatic vs. intuitive thinking in mathematics, metamathematics, formal number theory and various formal systems. The material here parallels that in Part I of the author's previous work, Introduction to Metamathematics, but is more reliant upon logic and less upon informal set theory. The chapter on recursion theory begins with the concept of the Turing machine, and then uses this to develop the incompleteness and undecidability results of Church and Gödel. Additional topics include Gödel's completeness theorem for various types of formal systems, Skolem's paradox and nonstandard arithmetic, Herbrand's theorem and Craig's interpolation theorem. Many results and theorems of the present work bear the same indexical number as the corresponding item in IM, and the book as a whole can serve as a prolegomenon to that more advanced work.—H. P. K. (shrink)

This book is a clear, concise, and conceptually unified treatment of various problems, both formal and philosophical, of inductive logic and probability. Ackermann's main concern throughout the book is the problem of adducing inductive support for various hypotheses, and of deciding between two competing hypotheses which is more reasonable given the available evidence. The author begins with a general consideration of the criteria to be met by satisfactory rules of inductive inference: accordance with intuitive notions of reasonableness in simple cases, (...) sensitivity to the given evidence, capacity for unambiguous intersubjective application. Ackermann quickly disposes of the argument that valid inductive rules must be made to rest upon a "postulate of the uniformity of nature": "If past projections of regularities suddenly begin to go wrong, nondeductive rules satisfying the criteria suggested above will begin to select regularities other than the ones which have failed." This clears the way for a consideration of the general problem of evidential support or confirmation, with its well-known paradoxes and difficulties. Ackermann, like Goodman, locates the source of these in problems bearing on the semantics of scientific language, and cautiously points to Goodman's theory of projection and entrenchment as the way out of most of the muddle. Succeeding chapters deal with the notion of probability of hypotheses, statistical inference, and the application of Bayesian techniques and decision theory to science. A well-chosen bibliography completes this modest but rewarding little book.—H. P. K. (shrink)

Among the dubious theses "proved" by the author of this classic example of pseudo-scientific literature are: relativistic physics is incorrect, the quantum theory is incorrect, "the success of quantum theory is purely mathematical," time, like space, has three dimensions, inductance and mass are equivalent, there is such a thing as absolute motion, and "All properties which are possessed by either space or time individually are.... properties of both space and time." What makes this book unusual is the fact that the (...) immense number of citations and footnotes refer to almost all of the important books and articles in the field of the philosophical foundations of physics. Larson therefore has accomplished one outstanding task: he has succeeded in immersing himself in some of the most philosophically and scientifically compelling literature of recent times and has emerged unscathed by a single intelligible idea.—H. P. K. (shrink)

The various papers and short "discussions" contained in this latest addition to the "Studies in Logic" series were presented at the 1965 International Colloquium in the Philosophy of Science, in London. Of the nine "problems" considered in this symposium, seven have directly to do with philosophy, one is an historical study of the origins of Euclid's axiomatics, and the last is an interesting—if one-sided—discussion of the "new math" controversy in the pre-college curriculum. Happily, this book demonstrates that the important issues (...) in the philosophy of mathematics somewhat outrun the usual ism-mongering debate between representatives of various schools. Unhappily, it also demonstrates that a good deal of undeserving material seems to find its way between the covers of scholarly books. After reading certain of the articles in this volume one is left with the feeling that the authors have not come to grips with the issues which they raise, that a lot of effort has been expended on peripheral matters, and that, in some instances, a sledge hammer has been brought down upon an egg. Abraham Robinson's article, "The Metaphysics of the Calculus," deals with a series of important linguistic and ontological problems concerning the foundations of analysis. Moreover, these are problems which can no longer be ignored, especially in view of the evolution of non-standard analysis. But, unfortunately, the interested reader would do far better to turn to Robinson's recent book on the subject, and pay special attention to the second, third, and last chapters therein. "On a Fregean Dogma" by Fred Sommers gives a rather curious non-quantificational algorithm for the monadic fragment of syllogistic. This Sommers does in the interest of "a conservative effort to restore categorical statements to predicative status." Despite the heat and the occasional interesting remarks which the discussion generated, one is inclined to ask, "So what?" The sequence of three papers by Mostowski, Bernays, and Körner is best considered as a group, since each of these is concerned with the philosophical significance of recent results in set theory and logic. Students of the philosophy of mathematics would do well to read this part of the book very carefully, although one wishes that the authors would elaborate a bit on their specific philosophical views. Finally, there is Georg Kreisel's difficult and amazing paper, "Informal Rigor and Completeness Proofs," in which is presented a mathematical treatment of the problem of selecting the primitive rules and concepts of mathematical systems. Kreisel's ideas are refreshing, unorthodox, dubious, and a genuine contribution to the philosophy of mathematics. In sum, this book is a mixed bag, uneven in the quality of its contents, but worth reading.—H. P. K. (shrink)

The publication of this book constitutes a real service to students of logic and of the foundations and philosophy of mathematics. Here, "under one roof," are translations of seventeen of the most important papers on logic and metalogic by Ajdukiewicz, Chwistek, Jaskowski, Jordan, Lesniewski, Lukasiewicz, S upecki, Sobocinski, and Wajsberg. All but two of them appear in English for the first time. Notably absent are papers by Alfred Tarski, but this omission is fully justified in view of the publisher's well-known (...) companion volume devoted exclusively to this most important logician. Within the contents of this volume, the major topics covered are various systems of sentential logic ; philosophically-motivated foundational studies ; miscellaneous papers on logical syntax, intuitionism and metalogic; and historical studies of the development of Polish logic. One cautionary note: the protothetic notation of Lesniewski is extremely difficult to read. Before undertaking that author's articles, the reader will do well to turn to the explanatory footnotes in the subsequent articles by Sobocinski. The book has a bibliography, but no index. In general, this is a worthwhile and long overdue book.—H. P. K. (shrink)

This imposing textbook bears the subtitle, "Readings in Metaphysics from Classical Philosophy to Existentialism," and appears to be uniquely designed for courses in metaphysics as taught in predominantly Catholic colleges and universities, although the selections reflect a distinct catholicity of concerns. In fact, when Bertrand Russell, A. J. Ayer and Rudolf Carnap get wind that some of their most polemical and positivistic pieces have been reprinted in a book of metaphysics, they are likely to reflect that Ecumenism has gone too (...) far. Their inclusion is justified, however, by the introductory assertion that "There is such a thing as antimetaphysical metaphysics, and empiricism specializes in this theory." But whatever the merits of this statement, the open-minded inclusion in this volume of spokesmen for "the opposition" is certainly to be applauded—the more so since this section contains a long and perceptive introduction by Jerzy Wojciechowski. Other sections of the book display classical and Christian thought on knowledge, being, God, potency and act; the dialectical tradition ; American pragmatism and naturalism ; and existentialism and phenomenology as represented by Heidegger, Sartre, Tillich, Marcel, Berdyaev and Buber. In addition to the generally good introductions to each of the five sections, the editors have provided glossaries of philosophical terms, lists of topics for discussions and term papers, and usually well-chosen "recommended readings."—H. P. K. (shrink)

This uncommonly fine textbook of the modern theory of functions of a real variable is particularly well-suited for mathematically mature students in the fields of philosophy and foundations of mathematics, philosophy of physics, probability theory, and statistics. Those who wish to achieve first-hand acquaintance with the quantum theory will also need to have a grasp of the material presented in this book. The first chapter presents a capsule survey of topics in abstract set theory and algebra, including a discussion of (...) relations, functions, the axiom of choice, and the well-ordering theorem, and leading up to the construction of the fields of real and complex numbers. The second chapter is an introduction to topology and spaces of topological functions. The approach is set-theoretical. Chapter three deals with the theory of the Lebesgue integral and measure theory. This chapter is quite comprehensive, rigorous and embellished with many exercises for the student. Subsequent sections include the theories of function spaces, Banach spaces, abstract Hilbert spaces, and differentiation. This is clearly one of the more worthwhile texts in its field.—H. P. K. (shrink)

A hypothetical scholar who came to read Kierkegaard after a thorough study of Hegel, might be struck by the parallelism between many pivotal concepts of Kierkegaard and corresponding concepts in Hegel. Just taking Hegel’s Phenomenology as an example, we note the following similarities: Kierkegaard’s idea of the "eternal moment" and Hegel’s concept of the universal "now," "Absolute Dread" in Kierkegaard’s Concept of Dread, and the "Absolute Fear" of the Slave Self-Consciousness, the three Kierkegaardian Stages and the three Hegelian stages, the (...) opposition to "Historical Proofs" in both Kierkegaard and Hegel, the importance of "the task" in both thinkers, etc., etc. But since Kierkegaard scarcely ever mentions Hegel except to criticize him, such a hypothetical scholar, if his suspicions were aroused, might turn to a book such as Thulstrup’s, for some enlightenment as to just how much influence, overt or covert, Hegel actually had on Kierkegaard’s thinking. But Thulstrup himself has apparently been strongly influenced by some of Kierkegaard’s biases. For example, he states in chapter 5 his sympathy with those scholars who "cannot imagine that such a shrewd genius as Kierkegaard could possibly have let himself be taken in by Hegel". And Thulstrup’s book is concerned precisely with showing how Kierkegaard, in each work that he wrote, managed to avoid being "taken in by Hegel." In some cases he is successful in doing this. For example, in chapter 4 he shows that Kierkegaard’s statement in From the Papers of One Still Living about Hegel’s "great attempt to begin with nothing" shows simply an ignorance of Hegel’s Logic, which begins with Being rather than Nothing. But in other cases, Thulstrup is too cavalier in dismissing any connection. (shrink)

This book begins by differentiating metaphysical entities from first-order entities for the knowledge of which some initial empirical encounter is required. Metaphysical "entities" belong to the class of entities which come into being because people decide to think in terms of them and talk about them. If talk about them desisted, metaphysical entities would themselves cease to exist. The three pivotal metaphysical entities that Sprague considers in this book are the person, world, and God.

Context: This conceptual paper tries to tackle the advantages and the limitations that might arise from including second-order science into global climate change sciences, a research area that traditionally focuses on first-order approaches and that is currently attracting a lot of media and public attention. Problem: The high profile of climate change research seems to provoke a certain dilemma for scientists: despite the slowly increasing realization within the sciences that our knowledge is temporary, tentative, uncertain, and far from stable, the (...) public expectations towards science and scientific knowledge are still the opposite: that scientific results should prove to be objective, reliable, and authoritative. As a way to handle the uncertainty, scientists tend to produce “varieties of scenarios” instead of clear statements, as well as reports that articulate different scientific opinions about the causes and dynamics of change for this specific field of research. (shrink)

"There are no paradoxes in mathematics," says Kurt Gödel. Moreover, Gödel seems to be right on this count. That is, there are no paradoxes, in the strict sense of the word, internal to the known and available body of mathematical knowledge. But while there are no paradoxes in mathematics, there certainly is an embarrassing bag of difficulties when we come to the application of mathematical concepts to the physical world. Of these, perhaps the most unruly offenders of all are the (...) problems proposed about 2500 years ago by Zeno and which, in modern idiom, have to do with the proper mathematical treatment of phenomena involving change and motion, and the physical meaning—if any—which may be assigned to convergent infinite series. At each stage in the growth of mathematical knowledge since their formulation Zeno's paradoxes have been the subject of some alleged resolution. Now Grünbaum contributes to this tradition, and it must be said that he makes a strong case. The weapons in the intellectual arsenal which Grünbaum brings to bear upon Zeno include such sophisticated devices as measure theory, but the weak link in the argument is in the first chapter where the author presents his somewhat psychologistic theory of temporal becoming. Grünbaum has a great many interesting things to say in later chapters about infinite processes, quantum mechanics, and Hilary Putnam. This is a first-rate contribution to the philosophy of physics.—H. P. K. (shrink)