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 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)

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)

This rather compendious volume contains twelve articles, eleven of which have been published in the last twenty years; the last, from which the book takes its title, appears in print for the first time. There are four chapters: "Confirmation, Induction, and Rational Belief" contains the paper "Inductive Inconsistencies" as well as "Studies in the Logic of Confirmation"; "Empiricist Criteria of Cognitive Significance" appears in the section "Conceptions of Cognitive Significance"; the very well-known "The Theoretician's Dilemma" appears in the third chapter—"Structure (...) and Function of Scientific Concepts and Theories"—along with "Typological Methods in the Natural and Social Sciences"; "Scientific Explanation," the fourth chapter, contains "Studies in the Logic of Explanation" and "The Logic of Functional Analysis," as well as the title article. "Aspects" runs to nearly book length and is concerned to survey and contribute to the range of scientific explanation—from physics to history. Hempel discusses the Deductive-Nomological approach and compares it to certain forms of Statistical Explanation; the Covering-law Model of Explanation is introduced and defended against proponents of Explanation-by-Concept, Dispositional Explanation, and Genetic Explanation: Hempel asserts that these three can be subsumed under the Covering-law model. The relation of rationality and decision to Explanation, and the related pragmatic aspects of Explanation are also considered. These are postscripts to certain articles which bring their discussions up to date. There are several rather confusing misprints, but overall the presentation is workmanlike. Hempel's contributions in the philosophy of science have been influential and important, and it is gratifying to have the most important of these collected together.—P. J. M. (shrink)

Although Machiavelli was never a military commander, he was throughout much of his life deeply concerned with the conduct of martial affairs; in short, a Renaissance Herman Kahn. This book is an essay on the technique of war: how on army is organized, who make the best soldiers, field manœuvers and battle formations, logistics, internal stability and control of military units, techniques of siege; these are considered both historically with reference to the ancients, as well as the present—the contemporary applications (...) of these techniques in the author's own time. The editor has provided a long critical and historical introduction—amounting to a small monograph in itself—which helps to place Machiavelli's ideas in the tradition of military-political theorists of Western Europe.—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)

A translation of the 1913 volume Dernières Pensées, this collection of papers contains much material of interest to the logician and the philosopher of science. In "The Logic of Infinity" Poincaré clarifies the notion of "predicative set" and discusses Zermelo's and Russell's approaches to set theory. "The Evolution of Laws" attempts to formulate the question "do laws of nature evolve?" Two papers concern space and time, two others, the electrostatic and quantum theories of matter. The collection concludes with a pair (...) of papers on the ethical import of science.—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)

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 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 rigorous treatment of elementary logic can best be characterized by noting that it relies heavily on semantical analyses of systems of logic running from the propositional calculus right through to a system of second-order arithmetic. The first chapter covers a multiplicity of topics: the concept of consequence, proofs and calculi, the symbolization of mathematical propositions. Hermes then painstakingly constructs quantification theory: first, the language itself, then its semantics; he then presents a completely set up predicate calculus, giving special attention (...) to derivability and decidability problems; a long, well-worked out completeness proof along the lines of Henkin is given, and some of its consequences are drawn out. Hermes then constructs a system of arithmetic in second-order logic, where he is especially concerned with categoricity of its interpretations. The last chapter provides further material on the functional calculus of first order: extended predicate calculi, normal forms for expressions. Excepting for a very few passages, this book should be accessible to everyone interested in formal logic.—P. J. M. (shrink)

Chomsky is widely mentioned in those philosophical circles whose interest centers on the analysis of language, but until now he has really been little read; this new work will remedy that situation. Here Chomsky, building on a presupposed acquaintance with linguistics, provides a stimulating examination of four major areas of linguistic theory: first, generative grammars are studied in their relation to language learning and understanding, then they are further considered as theories of linguistic use and competence; Chomsky here sets out (...) the criteria required for a grammar to be considered adequate—the justification of grammars. The second chapter examines syntactic theory with reference to depth structure and the problem of the relation of syntactic description to the traditional, correct account of how language is used. Depth structures and their import with reference to transformations within grammars are the subject matter of the third chapter. The last section considers, among other problems, the interface between syntax and semantics, especially the problems of degrees of grammaticality. A number of notes which go further into certain topics appear after the main text, and there is a useful bibliography. The main thrust of the book is a syntactical treatment of natural language by using transformational generative grammars as descriptive. Although he suggests empirical tests for the models he has constructed, he does not provide an attempt to fully verify the proposed system. This book will certainly be essential for an understanding of the structure of language viewed from the syntactic point of view.—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)

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 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 volume contains papers and abstracts of papers delivered at the colloquium at Tihany, Hungary in 1962. There were seven sections; mathematical logic, computers and automata theory, circuit theory, mathematical linguistics, computers and programming, applications of computers in economics, artificial intelligence. Among the more interesting—to the reviewer—were these papers: one by Church concerning an independence problem in recursive arithmetic; Muller—characterizing classes of recursive functions; a long and philosophically stimulating study by Watanabe on a formalization of inductive logic; Kiefer—applications of set (...) theoretic techniques in linguistics; Abraham and Salapina—recognition of synonymity by machine; Gelernter—problem-solving graphs and heuristic programming. A number of papers which were read are not included in this volume.—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)

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)

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)

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)

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)

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)

This is a much-revised edition of the 1941 text; the most significant changes are as follows: sections ## 21-30 are revised into seven and the discussions on validity, truth-functional implication, truth, etc., are revised in form to bring them in line with more modern approaches; the third chapter, except for the terminological changes which pervade the text is essentially unchanged from the original; the last chapter on quantificational inference is much-revised in all but the first few sections. The tone of (...) the book is now very much closer to his Methods of Logic, but the last chapter of the latter covers far more of glimpses ahead than does the corresponding one here.—P. J. M. (shrink)

Novikov is one of Russia's leading logicians and the appearance of this fine textbook is a good indicator of increasing American interest in Soviet logic. The book contains some new material, including a new independence proof of the rule of complete induction from the remaining axioms of first-order arithmetic. The first third of this work consists in chapters on propositional algebra and the propositional calculus. The first-order predicate calculus comes next under discussion: here a number of important classical results—Gödel's incompleteness (...) theorem, the Compactness theorem, Skolem-Löwenheim theorem—are proved rigorously. The author throughout the book leans fairly heavily on model-theoretic techniques, hence knowledge of some algebra, especially elementary field theory, will be useful. The last two chapters are concerned with first order arithmetic and the elements of proof theory. The only shortcoming is the lack of a bibliography and related scholarly apparatus; the student using this text may have difficulty locating material in other publications relevant to that in the book without outside assistance. The translation is very smooth and clear. Altogether, a first-class job.—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)

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)

This is a collection of papers read at an international logic colloquium held at Oxford in 1963. The first half contains articles on intuitionistic and modal logics, the propositional calculus, and languages with infinitely long expressions by such logicians as Kripke, Bull, Harrop, and Tait. The second part is primarily concerned with recursive functions and features a monograph by Crossley on constructive order types, as well as contributions by Goodstein, Schütte, and Wang, among others. Especially noteworthy is Kripke's paper which (...) applies Cohen's methods to the semantics of intuitionistic logic.—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)

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 is the first English translation of the work of Eugenio Garin, one of the foremost modern historians of the Italian Renaissance. The present text, translated so intelligently, is based on the revised Italian edition of 1958.. Garin treats the growth of Italian humanism from Petrarch in the fourteenth century to its point of radical transformation with Tommaso Campanella at the beginning of the seventeenth century. The commentary on Giordano Bruno is especially clear, concise, and penetrating. For Garin, the elements (...) of Renaissance humanism are multiple. It sought its own rhetoric and shrank from the dry syllogistic reasoning of scholastic philosophy. It used philology as an essential tool and basis of a revolutionary historiography. As well, Renaissance humanism was engage in civic life. In seeing Renaissance humanism as the link with the modern world, Garin opposes those historians of science who posit medieval science and philosophy as the vital bridge. Garin minimizes the importance of Aristotelianism and finds Renaissance humanism severely hostile to scholasticism. Nevertheless, the book is a rich mine of facts, summarization, and syntheses of texts; the very index of the book, restricted to Italian Renaissance figures, occupies five pages of double columns. In addition, Peter Munz has written an introduction which is a full-blown critique—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 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)

In this textbook on elementary logic the authors present a rigorous treatment of first the propositional, and then the predicate calculi. The first two chapters deal with the former topic exclusively: there is much emphasis on translation of ordinary-language sentences into logic and testing their validity; also a proof notation consisting of nested boxes, similar to the Fitch subproof technique, is introduced and used. The third and fourth chapters are concerned with quantification theory in application to language analysis; the next (...) chapter discusses more theoretical topics including a proof procedure for formulae in prenex normal form and decision procedures for restricted classes of such formulae. Identity is treated next, with the following chapter on definite descriptions. The eighth chapter is more formal in approach and has as its subject the abstract approach to logical systems: the notion of the extension of a theory is introduced at this stage; two examples of formal theories—the theory of commutative ordered fields, and that of real numbers—are presented and come under extensive elaboration. The last chapter discusses variable-binding operations and presents the theory of convergence of series as an example of such a theory with these operations. Each chapter has numerous exercises, and most have historical remarks and lists of theorems of that chapter.—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)

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)

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)

We have here a systematic examination of systems of logic cast in "natural deduction" form, in the widest sense of that word. Naturally enough, the author begins with Gentzen-style systems, moves by means of the inversion principle in int-elim logics to deductions of classical and intuitionistic logic which have a canonical "normal form"; more briefly, but always with clarity, Prawitz examines natural deduction for second-order and modal logics, and also systems with even less usual forms of implication. Three appendices cover (...) the Sequenzen-kalkül of Gentzen, a system of set theory of Fitch, and some alternative formulations of natural deductions due to Jáskowski and others. There is a short bibliography. There are also some hilarious misprints, few serious.—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)

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)

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)

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'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)

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)

In this monograph the author presents the special and general theories of relativity from a geochronometrical viewpoint. The amount of mathematics demanded is not too great, and one can get quite far along on the material on vectors presented early in the book. The first three chapters especially derive from the work of A. A. Robb several decades ago: they treat foundations of geochronometry, one-plus-one geochronometry and its generalization to a one-plus-three system. Later chapters cover such staples as the Lorentz (...) transformation, the metric and dynamical tensors, dynamical relationships of particles, accelerating and curving events, geodesics, and applications to the red-shift of spectral lines and bending of light-rays. The best introduction to the study of the foundation of physics is physics itself; so those philosophers interested in the foundations of relativity theory will find here a lucid and logically respectable exposition of that theory, and in that way an understanding of its fundamentals and problems will be reached.—P. J. M. (shrink)

In this lecture Evans studies one of the basic presuppositions of empiricism: the central doctrine of the theory of meaning of propositions, with special reference to the cleavage between verbal definition—the defining of words by reference to other words—and ostensive definition—the defining of a word by "pointing" to its object. The author shows that ostensive definition is wholly inadequate to the task of defining such words as nouns and adjectives—one can point only to their instances in the world and not (...) to what the noun or adjective actually refers. Thus with the breakdown of this distinction in definition goes the weakening of the division between mathematical and empirical sentences which together were believed to exhaust all possible meaningful expressions.—P. J. M. (shrink)