This book is a translation of some of the more important parts of the Grundgesetze of Frege: the introduction, the first part of the first volume which gives an exposition of the construction, rules, axioms of Frege's formal system, and two appendices, one of which is from the second volume and gives Frege's analysis of the paradox found by Russell in his system. The editor has provided a long introduction "for those not familiar with Frege," although it will benefit those (...) who have something more than acquaintance with his name as well. Frege's original two-dimensional symbolism has been preserved, but the editor has provided enough discussion in his introduction so as to make the going easier. Next to a complete translation of the Grundgesetze, this work is the most useful introduction to the magnum opus of the nineteenth century's greatest logician.—P. J. M. (shrink)

The second volume in the Pittsburgh Series in the Philosophy of Science, this collection of papers covers a wide range of topics: the development of Newton's First Law comes under scrutiny in papers by Hanson and Ellis; Putnam attempts to clarify certain conceptual issues at the foundations of quantum theory; David Hawkins discusses the relation of teleology and thermodynamics from a neo-Aristotelian viewpoint; Morrison examines certain topics in astronomy; empiricism is studied by Feyerabend from a number of aspects, and is (...) found wanting; finally, Rescher looks at the connections of ethical principles and scientific research, with special reference to the goals, means, and standards of proof, and the dissemination of results of the latter. On the whole, a worthy successor to the first volume, just as certain to raise interest within diverse areas of philosophy.—P. J. M. (shrink)

This concise work is a study of the semantical aspects of various paradoxes arising in formal logic. The author constructs a second-order system T with an interpretation in order to provide apparatus for stating and dodging the antinomies. After presenting a number of paradoxes, the author discusses a semantic vicious-circle principle, and provides a clarification of the problems by its application. He then discusses semantic aspects of some classical meta-mathematical results of Gödel, Tarski, Kleene, and Turing on unsolvable problems. Also (...) treated is the topic of impredicative definition in mathematics, especially as it relates to the vicious-circle principle.—P. J. M. (shrink)

This is the first book-length study published of the structure of reasoning and argument dependent on hypotheses. It encompasses far more than the, by now, familiar discussion of contrafactual conditional—this is but one chapter—since it ranges over such topics as the nature of hypothetical inference, belief-contravening hypotheses, contrafactual conditionals and modality, and entailment of conclusion from premisses under restriction. There are three appendices which concern, respectively, the historical roots of hypothetical reasoning and its attendant perplexities, the difficulty concerned in the (...) mutual compatibility of hypotheses in a proof, and certain properties of laws of nature expressed in conditional form. There is a fair-sized bibliography listing all the more important works on hypothetical reasoning. Rescher's logical apparatus designed to facilitate hypothetical reasoning and its study is controversial; but anyone wishing to understand the logic of the problems set forth in current philosophical discussion must begin with Rescher's book.—P. J. M. (shrink)

The first of the authors is an engineer, the second a logician, and they have collaborated to produce a systematic and comprehensive treatise and textbook on the theory of automata—computing machines viewed abstractly—which presupposes only a slight familiarity with logic; there is a long first chapter which develops propositional and predicate logic; the stipulation of logical operators, the "nets" constructed therefrom, and their physical realization comprise the next two chapters. The representation of automata in input-output tables and flow diagrams, and (...) the study of operators in structures come next; the last two chapters concern various practical ways of synthesizing automata and measuring the complexity of logical nets. This is one of the series on logic and foundations of mathematics; the translation from Russian was edited by J. C. Shepherdson.—P. J. M. (shrink)

The emphasis in this collection is clearly on logic, and this is one reason why it lacks the overall diversity and richness of the 1960 Stanford volume. However, the eight sections do contain much interesting material; in the mathematical logic section Kochen and Specker continue their study of logics appropriate for quantum theory, Vaught presents several new results about the Löwenheim-Skolem theorem, and Büchi studies second-order ordinal theory from the viewpoint of automata theory; the section on foundations of mathematical theories (...) contains papers on higher-order logic by Kaplan and Montague, model theory and ultra-products by Keisler, definability in set theory by Lévy. The topic of the justification of formal theories was the essential topic of the philosophy of logic section: an especially interesting paper is that of A. Robinson in which the present state of Hilbertean Formalism is analyzed, and in which the notion of potential truth is formalized. The section on the philosophy of science contains a single paper, by Hesse, on metaphor and explanation; Braithwaite, Hintikka, Jeffrey, and Kyburg contribute to a lively section on probability and induction; the section on methodology in physical science takes as topics the theory of relativity, causality, and irreversibility. The last two sections, on the philosophy of the life sciences, and on history of logic and philosophy of science, contain, respectively, papers by Davidson and Suppes on meaning and concept formation, and essays by Church and Geach on existential import historically considered and intentionality among the medievals. The collection would have been better if more of the contributed papers had been included.—P. J. M. (shrink)

As inductive logic and the philosophy of probability theory have become of wider interest, it has become clear that a book of readings in these and related topics would be useful for courses since most of the important articles are scattered and inaccessible. The editors have fashioned an extensive collection of papers in four main areas: the meaning of probability, confirmation theory, simplicity of theories and structures, the justification of induction. Each chapter is preceded by an introduction which sets out (...) the basic problems of the topic under consideration. There are thirty-six papers in all, two-thirds of them in the first and last chapters. The first chapter includes articles by Ramsey, Carnap, Nagel, and Reichenbach. The second chapter is dominated by the work of Hempel, Oppenheim, and Kemeny; the third chapter features a long article by D. J. Hillman which takes as its basis the work of Goodman, and there are other papers by Bunge, Quine, and Barker. The discussion of induction and its justification contains articles by Hume and Mill, but the bulk of the papers are contemporary. There is a bibliography for each chapter at the end of the book.—P. J. M. (shrink)

Although this excellent introductory and intermediate level text is intended for students of mathematics, it could serve well in any course for philosophers on that level. The first two chapters present the propositional and predicate calculi, along with an informal discussion of some of the set-theoretic concepts needed to study logic. The third chapter discusses what exactly an axiomatic system is, and examples of various mathematical systems cast in axiomatic form are provided; the discussion here, as elsewhere in the book, (...) is generally self-contained and presupposes only an ability to deal with abstract ideas. The study of the propositional calculus is resumed in chapter four, but its formulation is now more rigorous and thoroughly axiomatic; in a similar manner the predicate calculus is treated in the next chapter, and the usual metatheorems are proved about each, but the specific techniques differ from the usual. The last chapter concentrates on the completeness proof of quantification theory, relating it to some of the axiomatic algebraic systems considered earlier. An appendix sketches some sophisticated results of Vaught and Robinson on model theory and completeness. In sum, this is one of the very best of a plethora of logic texts on the market, and it should serve well for years to come.—P. J. M. (shrink)

What is attempted in this book is a presentation of various areas of science in such ways that their attendant philosophical problems are displayed, and their philosophical relevance is made evident. Essentially, there are three parts to the book: the first, comprising chapters on the nature of number, geometry, and the mathematical treatments of motion and measurement, presents the usual problems of conventionalism in geometry, physical vs. formal geometry, but also discusses Turing machines and information theory. The next five chapters (...) analyze the laws of motion, probability, chance, and credibility, order in nature, and the conceptual structure of modern physics. The last four chapters constitute a study of evolutionary theory, psychology, ethics, and economic theory, the last topic not often appearing in studies of scientific philosophy. The selection of topics is judicious and balanced, but the author often overuses certain ideas and theories, for example, information theory, as both heuristic devices and as factually true of various situations; in this sense he is misleading and conceptually unrigorous. A number of illustrations, visually interesting and enlightening, help to add interest.—P. J. M. (shrink)

Subtitled "Studies in Ethical Analysis," this collection of eleven essays, most of which have previously appeared in journals, deals with a number of problems central to modern ethical theory: the emotive interpretation of ethical language, persuasive definitions and their role in ethical reasoning, the cognitive versus emotive conceptions of ethics: many of these problems were first raised and examined by Stevenson in his earlier book Ethics and Language. Other essays are of a less retrospective nature: studies on Moore and Dewey, (...) naturalism and relativism in ethics, and a general discussion of the relations of linguistic analysis to philosophy as a whole. Stevenson is mainly concerned with analytical, as opposed to descriptive, ethics; and he completely avoids the topics of normative ethics except for a brief survey. The approach to ethics is therefore restricted, but there is enough here to interest the philosopher whose main area lies outside of ethics; although it presupposes no acquaintance with the author's previous work, some of the questions on emotivism and persuasive language are more motivated when seen in the context of that work.—P. J. M. (shrink)

This is the collection of essays presented to Bochenski on his 60th birthday, and it contains, as a mirror of Bochenski's own work, a broad spectrum of studies ranging from formal logic and history of logic, to the philosophy of logic and language, and to the methodology of explanation in Greek philosophy. Of the seventeen articles, these are some of the more important to the reviewer: "Betrachtungen zum Sequenzen Kalkül" by Paul Bernays, which is an extensive study of Gentzen-type formulations (...) of logic; "Remarks on Formal Deduction," H. B. Curry, a further discussion of sequenzen-logics; "Marginalia on Gentzen's Sequenzen Kalkül" by Hughes Leblanc; "Method and Logic in Presocratic Explanation," Jerry Stannard; "On the Logic of Preference and Choice," H. S. Houthakker, a suggestive presentation of decision and utility theory in logical form; "Leibniz's Law in Belief Contexts," Chisholm; "On Ontology and the Province of Logic," R. M. Martin; and "N. A. Vasilev and the Development of Many-valued Logics," G. L. Kline, an important addition to the history of logic. Other contributors are: Storrs McCall, Albert Menne, E. W. Beth, Benson Mates, Ivo Thomas, J. F. Staal, F. R. Barbò, A.-T. Tymieniecka, and N. M. Luyten. There is a bibliography of Bochenski's writings through 1962.—P. J. M. (shrink)

The infinitary languages studied in this book are those in which quantification of infinitely many variables simultaneously, and conjunctions or alternations of infinitely many are permitted. Infinitary concatenation and infinitary propositional logics are first discussed, and a completeness theorem is proved about the latter. The later chapters deal with infinitary predicate languages and Scott's proof of incompleteness is introduced. Throughout the discussion, unsolved problems are mentioned and areas undergoing current development are emphasized. A short bibliography lists most recent articles on (...) this new branch of logic.—P. J. M. (shrink)

Dopp is one of the very few logicians writing in French today, and so there are few textbooks of logic in that language. This is the newest one, and it is concerned essentially with the propositional and first-order predicate calculi, from both their historical as well as contemporary aspects. After examining the concepts of classical logic, Dopp presents the propositional calculus as a calculus of truth-functions and then gives it an axiomatic underpinning. In the treatment of quantification, first traditional logic (...) is presented, and then modern first-order logic with identity is examined in some detail, with attention to the technique of semantic tableaux. An appendix provides a number of alternative systems of the propositional calculus in axiomatic form.—P. J. M. (shrink)

This is a reprinted version of Born's 1948 Waynflete Lectures at Oxford; there are several appendices: one elaborates in much greater detail the elements of physical theory developed in the lectures; the second is bibliographical; the last concerns the role of symbols in the construction of physical theory.—P. J. M.

This book is one of the series entitled "Rare Masterpieces of Philosophy and Science" and it is entitled to both distinctions. The papers collected here are virtually unobtainable except in the most complete libraries; and de Morgan's work is clearly that of a master-between Boole and Frege, he is the leading figure in formal logic. The papers found herein include the series of six on the syllogism published between 1846 and 1868, together with three shorter notes concerning logical phraseology, a (...) syllabus for a system of logic, and an encyclopaedia article whose topic is logic. The editor has added a useful introduction which, among other important services, allows us to see de Morgan, not, as we today see him, as a founder of symbolic logic, but as a rival of William Hamilton, engaged in controversy much of his life.—P. J. M. (shrink)

This volume is published concurrently with the one reviewed below and together they unite a number of Quine's previously scattered papers into two compact volumes; this volume deals with his more philosophical work while the other is concerned with more purely technical logical studies. The twenty-one essays cover the period 1934-1964 and none have appeared between hard covers before. Several of the articles—"The ways of paradox," "Foundations of mathematics," "On the application of modern logic," and "Necessary truth"—are essentially popular expositions. (...) The others are generally more restricted in both scope and appeal, and deal with the ontology of the sentential calculus, truth by convention, implicit definition, modal logic, and ontological reduction in the sciences. Several articles concern the philosophy of science directly: "On simple theories of a complex world," "Posits and reality," and "The scope and language of science." This fine collection will be of significant help in presenting the work of a distinguished philosopher to a wider audience, as well as providing the professional with a source of discussion.—P. J. M. (shrink)

Although conceived as a textbook, this extraordinary work contains a great deal of material which is either completely new or which has not appeared before in book form. It is intended as an upperlevel text for those with some familiarity with the subject already. After the introduction, there is a long chapter on formal systems which contains new material on algorithms and the theory of definition; epitheory of formal systems is then discussed, followed by an elegant algebraic treatment of logic. (...) Curry then formulates systems for negation and implication in the next two chapters, follows them by quantification theory, and ends with a sketch of modal logic. What distinguishes this from other logic texts which try to cover about the same ground is this: Curry exercises virtually exquisite care in his analysis of some of the more difficult points, variables and substitution, for example, that others often tend to gloss over. Each chapter has a section dealing with supplementary but related topics so as to give the reader some idea where the subject goes. There is an enormous bibliography and hundreds of references, including historical ones; these also increase its scholarly value. The author takes an informal semantical viewpoint about logic—trying to treat meaning as well as form as essential to logic. This view and a pellucid style make things move freely in the most difficult spots; only Curry's occasionally peculiar terminology might be confusing.—P. J. M. (shrink)

All the essays contained herein, with the exception of the last two—"On Suicide" and "On the Immortality of the Soul"—have appeared in the author's Essays, Moral, Political, and Literary ; the others were published posthumously. In this wide-ranging collection Hume addresses himself to aspects of aesthetics and literary criticism, the philosophy of history, philosophical "types", human nature and belief. The volume conveys a side of Hume too often forgotten in our present admiration of his foreshadowing of analytical philosophy: the man (...) of letters. The editor has provided a useful introduction setting Hume's work in its eighteenth century context.—P. J. M. (shrink)

This well-written introduction to the theory of recursive functions and effective computability is an English translation of the 1960 German edition. The seven chapters deal with all the usual material, beginning with a treatment of Turing machines and their relation to the intuitive idea of computability, through general recursive functions, to a chapter on such diverse topics as the hierarchy of arithmetical predicates and Fitch's basic logic system. Rather than try to cover the whole subject sketchily, the author confines himself (...) to a narrower range of subjects which he elucidates with admirable clarity. The treatment of Turing machines is similar to that of Davis' book in using quadruples as instruction units in machine "programs," but the treatment differs in some details which might interest the connoisseur. In between technical discussions there are numerous remarks, often of more than a page in length, dealing with the more purely philosophical aspects of the theory under development. Each chapter terminates with a short bibliography. The publishers are to be congratulated for making this valuable work available to a wider range of readers.—P. J. M. (shrink)

The first edition of this now classical work appeared in 1953, the second heavily revised edition in 1961; this most recent edition is a revision in detail only of the previous one. The book is divided into three parts, the first two dealing with finite and infinite sets, infinite cardinals and their arithmetic, and related remarks on non-standard mathematics and the equivalence of various definitions of finitude. The third part considers ordered sets and isomorphism types, the special case of linearly (...) ordered sets, well-ordered sets in general, the relations of ordinals and cardinals. There is a very large bibliography and also a further list of literature pertaining to the companion volume. Fraenkel intended the book for philosophers as well as for mathematicians, and has kept to studying the most general and interesting problems of set theory; metamathematical results are kept in the background and the author works in an informal system of Zermelo-Fraenkel set theory. There is relatively little symbolism and Fraenkel's arguments are therefore more intuitive than rigorous, but clearly everything can be formalized in a proper language. There are many exercises in the book; these help to elaborate points only lightly touched upon in the main text and provide practice necessary for the growth of one's set-theoretical intuition. There are many alternative proofs of important theorems and alternative definitions for essential concepts; this lack of conceptual rigidity should especially please those who reject the now fairly widely held notion that Z-F set theory is the "true" one, the others proposed being ad hoc. Anyone who works his way through this delightfully written text will come away thoroughly prepared to attack more advanced work in the subject.—P. J. M. (shrink)

This is not a textbook in mathematical physics—excepting for one chapter one need not possess much more than geometry and elementary algebra—rather it is a philosophically reflective examination of the cardinal features of special relativity theory. Throughout the book Bohm is not merely doing physics, but thinking about doing physics as well. This metatheoretical reflexion appears in chapters concerning pre-Einsteinian notions of relativity, attempts to save the aether theories, the "ambiguity" of space-time measurements in the new cosmology, "common sense" notions (...) of space and time, and the falsification of scientific theories. There is a long appendix dealing with physics and perception—the relation between scientific objects and perceptual processes. One will, however, by working through the text learn plenty of physics in a rigorous and concise fashion. The author wisely does not attempt to cover the mathematically far more difficult and philosophically more profound General Theory of Einstein. In the area chosen, Bohm has written clearly and felicitously; this should serve a model for others who like to take their physics with a dollop of philosophy.—P. J. M. (shrink)

This interesting collection is the Festschrift presented to W. Britzelmayr on his seventieth birthday, and it contains several excellent papers which ought to interest the logician and philosophical analyst alike. The most exciting paper is one by Stegmüller in which a system of set theory combining ideas from Bernays and Quine is formulated; one by Kurt Schütte discusses the limitations imposed by constructive logic on the theory of trans finite arithmetic; there are papers by each of the editors: the first (...) analyzes the deduction theorem for the first order predicate calculus, while the second treats logics with predicate quantifiers. Other papers include studies by Behmann on the philosophico-logical status of identity; by Beth on the application of his semantic tableaux to classical logic; by Wilhelmy on the semantics of quantified, many-valued logics; and by E. M. Fels concerning the work of Markov and his school in the theory of canonical systems. There are still further papers which take as their topics the logical analysis of legal concepts and reasoning, the logic of questions, the semantics of implication in ordinary language, and the relation of the theory of logical types to Frege's work. In general, this is an excellent collection which affords a wide view of the logico-linguistic concerns of recent German philosophy.—P. J. M. (shrink)

Abraham Robinson has twice been the initiator of trends in the foundations of mathematics which have later become recognized as profound and important, although they were generally ignored at first: metamathematical problems of algebra and non-standard analysis. This book considers the second topic; before we begin analysis—especially a treatment of the classical theorems of calculus—we need basic results from logic, model theory in particular. Robinson then sketches non-standard arithmetic and proceeds to develop the usual properties and relations of differentiability and (...) integrability of functions, convergence; he then examines general topology in this new context and fortified with these results develops "modern" functional analysis in some detail: measure theory and functions of a real variable, functions of a complex variable, normed and linear spaces, topological groups and some harmonic analysis. The last two chapters cover selected topics in analysis drawn principally from applied mathematics and the history of the calculus with special reference to infinitesimals. Robinson views the development of non-standard mathematics as the renaissance of ideas about infinitesimals as serious topics of mathematical study. Robinson primarily has confined himself to analysis and not attempted to cover non-standard algebraic systems; several important results of Alling concerning non-standard arithmetic and rings of continuous functions appeared too late to be included. Of course, the real test of non-standard analysis will come in allowing us to prove new theorems about standard analysis, but although Robinson does not attempt to do this here—although he has obtained results in this direction—other mathematicians have already begun to do so: the future of this new branch of mathematics already looks secure.—P. J. M. (shrink)

There is little doubt that in the actual practice of science, models, metaphors, analogies, reasoning by similar cases, and other "parallel" forms of argument are often essential for the discovery of new phenomena and their theoretical interpretation. The author has assembled in five essays, culled and developed from previous ones, her ideas on some basic questions concerning models and analogies. The first chapter considers in dialogue form the role of models in science; the next section is an exploration of the (...) questions of what is an analogy and under what conditions analogical arguments are valid. The formal logic of analogy is then developed in the third part. The theory concerning analogy put forth by Aristotle is next discussed; the last chapter treats of the explanatory function of metaphor—we are brought back to our starting point, hopefully enlightened.—P. J. M. (shrink)

This monograph is the first really systematic study of the model theory of many-valued logic. The authors develop model theory for systems of logic whose truth-values lie in a compact topological space; the results are analogous to those for two-valued logic—they yield the two valued logics as special cases—but often the methods of proof are more complicated and tend to reveal some of the deep structure of these logics. There is presupposed a fair knowledge of naive set theory and point-set (...) topology, but no knowledge of classical logic is required although it will be of help in seeing the motivation behind various results. The first three chapters are concerned with preliminaries on topology, model theory, and continuous logic. The next chapter examines the relation of elementary equivalence among models, including the downward Skolem-Löwenheim theorem; the fifth chapter contains the generalizations of such classical results as the compactness and upward S-L theorems. The authors specialize their work in the sixth chapter to consider certain particular kinds of models: saturated models, universal models. The last chapter considers classes of models closed under various algebraic operations. There is a bibliography, historical notes, and indices of exercises, symbols, and definitions. This book is the prolegomenon to any future study of many-valued logic.—P. J. M. (shrink)

This collection of twenty-three papers from the period 1934-1960 is concerned with formal number theory and syntax, axiomatic set theory, truth functions, and quantification theory. In the first group appear "Concatenation as a basis for arithmetic" and "Definition of substitution," among others; the second includes "Set-theoretic foundations for logic," "On ω-inconsistency," and "Element and number." Quine's important articles "Completeness of the propositional calculus" and "Cores and prime implicants of truth functions" are in the third section; the last one includes "A (...) proof procedure for quantification theory" and "Church's theorem on the decision problem". These are but a few of many important articles. All the articles have been reset in a new, perspicuous type-face which makes study easier. Several important papers which were omitted should, in the reviewer's opinion, have been included, but this is a minor quibble. Quine's stature in modern logic can be gauged by examining these papers.—P. J. M. (shrink)

This is a text for a one or two semester course on axiomatic set theory; the goal is to introduce and develop one system of set theory in a complete and thorough way, presupposing only the elusive "mathematical maturity" of the reader. There are nine chapters which begin with a development of propositional and predicate logic oriented toward set theory and develop the Zermelo-Fraenkel system in exceptional detail. The book starts slowly, the first 120 pages being devoted to logical preliminaries (...) and the introduction of axioms; it gathers speed, however, and the remaining chapters include treatment of the algebra of classes, relations and functions, order of various sorts and Zorn's lemma, real numbers and equivalence classes, the equipollence of sets, similarity as a relation of sets, the ordinal numbers and their arithmetic, the cardinal numbers and their arithmetic. These later chapters are extremely detailed and comprehensive, presenting a wealth of material. There are numerous exercises, many quite difficult, at the end of each chapter, along with a summary of that chapter's content. Because of the single-mindedness of the book, there is little reference to other systems of set theory, but this is not a drawback at all. The presentation is orthodox, but not dull; careful, but not generally pedantic or repetitive. This makes it a good text for self-study.—P. J. M. (shrink)

This book originally appeared in 1920 under the title Relativitätstheorie und Erkenntnis Apriori and was the first of Reichenbach's numerous writings on the philosophical problems of relativity theory, space, and time. In this book the author attempted to show how Kant's theory of the a priori, especially concerning the concept of the a priori as "constituting the concept of [the] object" in question, comes into irrevocable conflict with certain facts of both the General and Special theories of relativity; and that, (...) therefore, we must amend Kant's treatment of the role of the a priori in knowledge accordingly. The translator, Maria Reichenbach, has provided a long and useful introduction, placing the work in the context of Reichenbach's philosophical career and activities, and providing background material on the author himself. This is the second of a continuing series of works by Reichenbach to appear under the California Press imprint.—P. J. M. (shrink)

For this volume Professor Davis has assembled a number of the most important papers on undecidable propositions, unsolvable problems and computable functions. Several papers appear here in print for the first time: Gödel's remarks at the Princeton Bicentennial Conference on Problems in Mathematics, and Post's paper on Absolutely Unsolvable Problems. Other authors whose work is included are Church, Turing, Rosser, and Kleene. Gödel's classic "On Formally Undecidable Propositions..." appears in a new translation, and all papers have been corrected, many such (...) corrections provided by the original authors. The editor has contributed brief notes preceding each paper, these notes are intended to clarify and otherwise expatiate upon the contents of those articles. A valuable collection both for original source material as well as historical formulations of current problems.—P. J. M. (shrink)

The aim of the authors is to present a comprehensive study of the basis of intuitionistic mathematics by means of modern meta-mathematical devices. The first author, for whom this book is a capstone of twenty years' work on the subject, contributes three chapters on a formal system of intuitionistic analysis, notions of realizability, and order in the continuum; the second provides an analysis of the intuitionistic continuum. An extensive bibliography which includes references to almost every article on the subject makes (...) this book especially valuable; for those interested in intuitionism and its relations to classical mathematics this work will be essential.—P. J. M. (shrink)

Hitherto unavailable except in the original French, Bergson's Durée et Simultanéité is an engaging contribution to the philosophy of relativity theory, space, and time. The book appeared during a period of great debate on the philosophical status of Einstein's Special Theory, and it treats, therefore, of it to the exclusion of the more conceptually difficult General Theory. Bergson is mainly concerned with trying to explicate the problems of the twin and clock 'paradoxes' which are presently again under some critical discussion. (...) Bergson tries to show that there is a symmetry in the aging of the two twins, not the usual asymmetry. Although Bergson is not usually considered a philosopher of science, those interested in the conceptual foundations of a modern physical theory might do well to consider his refreshing conclusions. The publishers are to be congratulated for making available this stimulating book.—P. J. M. (shrink)

This short monograph is a formulation and study of a system of first-order predicate logic of terms with a Hibert-Bernays selection-symbol E. The author proceeds through the primitive notation to recursive definitions of terms, relates the usual predicate calculi to this logic of terms, discusses an appropriate theory of models for the system, and finally proves it complete. This paper is one in a series of informal lecture notes in mathematics; hopefully more such studies in logic will appear in the (...) series.—P. J. M. (shrink)

This collection of seven papers by six authors centers about the relation of free action to determinism as both a metaphysical as well as a methodological hypothesis. Chisholm is concerned with the problem of whether a human whose behavior is completely controlled by outside forces can still be considered free; Danto studies the old problem of whether it can be said that actions of men are caused in the same way as the activities of other objects. Richard Taylor examines the (...) problem of free will in light of the apparent asymmetry of the prevention and the postvention—the doing of an act not ruled out by antecedent conditions—of events; Ginet develops a notion of "contingent necessitation" and proceeds to show that humans do not have a choice as to whether their actions satisfy certain conditions, these conditions following by contingent necessity from events not themselves in an individual's control. Wilfrid Sellars contributes two papers; the first is a study of the locution of "intending that," desiring and willing, intending, believing, the condition of being disposed to, and the practical syllogism; the second paper continues the tenor of the first—Sellars formalizes the notions of various kinds of "necessary" states of affairs, and uses this to make a distinction between being able to will and being able to do an act. The editor has the last paper which concerns the matter of whether there can be an empirical disproof of determinism.—P. J. M. (shrink)

To celebrate the four-hundredth anniversary of Galileo's birth, the University of Rochester held a series of lectures on the thought and influence of Galileo; there were six contributors and their work groups itself into three areas. The first of these is the importance and relevance of Galileo in modern thought and society: these were discussed by Giorgio di Santillana and Gilgerto Bernardino respectively. Norwood Hanson and E. W. Strong study the work of Galileo in dynamics and his theory of measurement. (...) The last two papers are concerned with the role of science in society—they spring from the confrontation of Galileo and the Church, the first indication of the special problems the scientific disciplines would face in their intercourse with the rest of society—and are by Philip Abelson and Erich Kahler. Although this brief collection cannot provide a full view of the scope of Galileo's activities and their subsequent impact, we are still able to feel that some of the vast gap between ourselves and the initiator of modern physics has been spanned, and that he can be seen more clearly for it.—P. J. M. (shrink)

This book is an attempt to relate the operative and constructive formulation of symbolic logic carried out by Lorenzen—and to a lesser degree Kolmogorov and Markov—to both Wittgenstein's philosophy of logic as set forth in the Tractatus and later modified in the Investigations, and to Brouwer's critique of classical logic, especially the principles of excluded middle. The first chapter contains an exposition of Wittgenstein's critical analysis of the "mythical" views of Russell and Frege; and it develops his own "operative" theory (...) of elementary logic. The second chapter concerns the relations of the theories of Brouwer and Wittgenstein; the last section is a detailed survey of chiefly Lorenzen's operative logic in which the presentation of a dialogue-like interpretation of logic is exhaustively analyzed: the notion of a "winning strategy" in such a two-person dialogue is set forth, the technique is extended to classical logic from its original, weaker operational form, and its relations to Gentzen's Sequenzen-kalkül is discussed. The work was the author's Habilitationsschrift at Innsbruck University.—P. J. M. (shrink)

In this compact volume the author gives a sprightly introduction to modern symbolic logic, at no time side-stepping philosophical problems concerning the nature of formal logic. The first chapter is a brief comparison of traditional syllogistic logic and modern "logistic"; the next three chapters deal with the nature of logic as illustrated through various elementary logical systems: logic as ontology, logic as theory of language, logic as methodology of deductive sciences. Hasenjaeger then examines richer systems—many-sorted, those with definite descriptors, etc.—and (...) then the paradoxes. A chapter follows concerning logic developed in a combinatory style and its connections with the idea of a generative theory of formal structures—structures as calculi of concepts. The last chapter discusses formal logic as a branch of probability theory and inductive logic. This well-presented book would serve well in a study of the philosophy of mathematics and logic.—P. J. M. (shrink)

This is a collection of papers, all but one of which were previously published, by one of England's leading logicians. Goodstein has described his position in the philosophy of mathematics as that of a "constructive formalist": leaning toward the Hilbert school, but emphasizing the constructive nature of mathematical entities. The papers are more or less technical and symbolic; those most difficult are "The Nature of Mathematics," "The Decision Problem," and "The Definition of Number." Other titles are "Proof by Reductio ad (...) Absurdum," "Logic Paradoxes", "Language and Experience," "The Meaning of Counting," "Mathematical Systems," "The Axiomatic Method," "Pure and Applied Mathematics," and "The Significance of Incompleteness Theorems"; the last-mentioned paper is also quite difficult, but it constitutes not only one of the best semi-formal presentations of Gödel's theorems, but also helps make quite clear what their relevance is to considerations of the expressive power of formal languages in general. This collection constitutes a fine introduction to the philosophy of mathematics.—P. J. M. (shrink)

In this collection the authors have attempted to bring together a number of the essential papers in the subjective interpretation of probability theory; several of them—Borel's "Apropos of a theory on probability" and de Finetti's "Foresight: its logical laws, its subjective sources"—have never appeared before in English. Other articles include Venn's pioneering study as well as the more recent work of Ramsey, Koopman, and Savage. The editors provide an introduction which presents the three basic elements of any subjectivistic theory: probability (...) as degree of belief, the coherence of beliefs of an individual, and the notion of exchangeable events. A bibliography includes references to virtually all the more important works in subjective probability, with special emphasis on the development of the mathematical side of the theory. This anthology belongs on the shelf of any philosopher concerned with inductive logic, statistical inference, or the foundations of probability theory.—P. J. M. (shrink)

The author here constructs a system of simple type theory in which the type hierarchy does not extend merely to any finite height, but to an infinite height; this added part allows him to prove the existence of infinite sets within the theory, instead of taking it as an axiom in the usual simple type theory. The system has been presented in such sufficient generality so as to make it able to accommodate current scientific theories; the author has turned in (...) the direction of using type theory rather than set theory as the underlying logic of scientific theories. The system is formalized in Church's lambda-calculus notation; the semantics of the system are treated in great detail and appear to be the most complete study so far in print. The system is shown to contain at each type a "basic logic" somewhat similar to first-order logic; the author also shows that Peano's axioms for arithmetic appear as theorems when cast in suitable form. This book is one of a number of recent studies of type theory; together they represent a revival of interest in the subject, both as a foundation for mathematics and as an object of interest in itself. The book stems from Andrews' Princeton doctoral thesis.—P. J. M. (shrink)

Ever since the first edition appeared in 1952, Wilder's book has been a mainstay of courses in the philosophy and foundations of mathematics, and deservedly so, for it covers most of the topics which provide an insight into the nature of this formal science. There are two parts: the first is a rapid but thorough survey of the axiomatic method, set theory, especially infinite sets, cardinal and ordinal numbers, the linear continuum, and the theory of groups with reference to foundational (...) problems. The second part develops various views on the nature of the foundations of mathematics: it begins with a look at the naïve 19th-century approach and its transformation into Zermelo's axiomatic set theory; the Logicistic thesis is covered in great detail; intuitionism and Hilbert's formalistic view come next; the last chapter discusses the cultural setting of mathematics. The changes since the first edition have been in the way of making certain presentations more relevant to current problems. There is an excellent bibliography. This book could be used in a one semester course on philosophy or on foundations of mathematics, or in a longer course in which symbolic logic was developed extensively.—P. J. M. (shrink)

The author is interested in discussing various aspects of the propositional calculus; in particular, the relationships among the various propositional connectives in various systems of logic such as Intuitionistic and modal are scrutinized. The first three chapters survey the notation to be used and describe the general notion of logistic system; the author then describes the concept of a deductive system in exceptional generality, then treats the connexions of equivalence and independence among such deductive systems in what are essentially algebraic (...) terms. Building on the latter work, he then treats the idea of one system's being an extension of the other, going into the details concerning the fine structure of exactly what sort of extension it is—conservative, consistent, and thus arrives at Lindenbaum's lemma. The equivalence relations of congruence and equipollence among propositional systems form the central core around which Porte's discussion of axiomatizability of systems of connectives constituting various propositional calculi is based. Porte is one of the best of the several young logicians now working in France and this book is useful both as a technical treatise containing a sophisticated treatment of sentential logic, and as a guide to at least one direction and trend of contemporary French logical thought.—P. J. M. (shrink)

This is a textbook in symbolic logic comprising sentential and quantificational theory only. The logic of the propositional calculus is developed in a natural-deduction form reminiscent of Fitch's technique; therefore, most of the theorems take the form of metamathematical assertions and possess corresponding meta-proofs. The classical propositional calculus SCc is then formulated in the Hilbert-style axiomatic way which naturally leads to consistency, completeness, and decidability theorems for the system. The theory of quantifiers is also first set up in natural deduction (...) form, and then reformulated classically. Leblanc spends considerable time in a careful treatment of satisfiability and validity in first-order logic, this leading naturally to the Löweheim-Skolem [[sic]] theorem and related completeness results. There is an appendix containing some relevant results from set theory. In general this is a most thorough and neat job which succeeds admirably in giving in semantical form a rigorous introduction to logic.—P. J. M. (shrink)

Descartes considered the methods of reasoning put forth in the Discourse to be correct because, among other justifications, he had examples of scientific theories in which the techniques were successful: the Optics, Meteorology, and Geometry. The chief value of this edition is to have the Discourse back in its proper setting, as well as the more obvious one of having available three works of importance in the history of the exact sciences in one compact and readable edition. The Optics is (...) especially useful for the study of Descartes' theory of sight and perception. There are a large number of diagrams and figures which are quite helpful in making some rather schematic presentations more complete.—P. J. M. (shrink)

The author's central thesis is that a knowledge of set theory can be put to good use by the linguist interested in the syntax of natural languages. The author first points out the role of set theory in formal science, and then gives a short summary of some of the more important ideas. He then develops certain relations in set theory which are of special importance in the study of languages. A fair number of examples—admittedly in rather trivial form—which occur (...) in the study of language are exhibited to show some of the applications in mind. The last few pages are devoted to a review of how his thesis has fared. Although it may seem obvious to some that set theory is the means par excellence for the formal study of any discipline, many linguists apparently do not yet see this; this book will go some way toward demonstrating this.—P. J. M. (shrink)

Even though integrity is widely considered to be an essential aspect of research, there is an ongoing debate on what actually constitutes research integrity. The understanding of integrity ranges from the minimal, only considering falsification, fabrication and plagiarism, to the maximum, blending into science ethics. Underneath these obvious contrasts, there are more subtle differences that are not as immediately evident. The debate about integrity is usually presented as a single, universal discussion, with shared concerns for researchers, policymakers and ‘the public’. (...) In this article, we show that it is not. There are substantial differences between the language of research integrity in the scientific arena and in the public domain. Notably, scientists and policymakers adopt different approaches to research integrity. Scientists tend to present integrity as a virtue that must be kindled, while policy documents and newspapers stress norm enforcement. Rather than performing a conceptual analysis through philosophical reasoning and discussion, we aimed to clarify the discourse of ‘scientific integrity’ by studying its usage in written documents. To this end, large numbers of scientific publications, policy documents and newspaper articles were analysed by means of scientometric and content analysis techniques. The texts were analysed on their usage of the term ‘integrity’ and of frequently co-occurring terms and concepts. A comparison was made between the usage in the various media, as well as between different periods in which they were published through co-word analysis, mapping co-occurrence networks of significant terms and themes. (shrink)