Originally published in 1962. This book gives an account of the concepts and methods of a basic part of logic. In chapter I elementary ideas, including those of truth-functional argument and truth-functional validity, are explained. Chapter II begins with a more comprehensive account of truth-functionality; the leading characteristics of the most important monadic and dyadic truth-functions are described, and the different notations in use are set forth. The main part of the book describes and explains three different methods of testing truth-functional aguments and agument forms for validity: the truthtable method, the deductive method and the method of normal forms; for the benefit mainly of readers who have not acquired in one way or another a general facility in the manipulation of symbols some of the procedures have been described in rather more detail than is common in texts of this kind. In the final chapter the author discusses and rejects the view, based largely on the so called paradoxes of material implication, that truth-functional logic is not applicable in any really important way to arguments of ordinary discourse.

Originally published in 1966. Professor Rescher's aim is to develop a "logic of commands" in exactly the same general way which standard logic has already developed a "logic of truth-functional statement compounds" or a "logic of quantifiers". The object is to present a tolerably accurate and precise account of the logically relevant facets of a command, to study the nature of "inference" in reasonings involving commands, and above all to establish a viable concept of validity in command inference, so that the logical relationships among commands can be studied with something of the rigour to which one is accustomed in other branches of logic.

Originally published in 1988. This text gives a lucid account of the most distinctive and influential responses by twentieth century philosophers to the problem of the unity of the proposition. The problem first became central to twentieth-century philosophy as a result of the depsychoiogising of logic brought about by Bradley and Frege who, responding to the 'Psychologism' of Mill and Hume, drew a sharp distinction between the province of psychology and the province of logic. This author argues that while Russell, Ryle and Davidson, each in different ways, attempted a theoretical solution, Frege and Wittgenstein (both in the Tractatus and the Investigations) rightly maintained that no theoretical solution is possible. It is this which explains the importance Wittgenstein attached in his later work to the idea of agreement in judgments. The two final chapters illustrate the way in which a response to the problem affects the way in which we think about the nature of the mind. They contain a discussion of Strawson's concept of a person and provide a striking critique of the philosophical claims made by devotees of artificial intelligence, in particular those made by Daniel Dennett.

Originally published in 1941. Professor Ushenko treats of current problems in technical Logic, involving Symbolic Logic to a marked extent. He deprecates the tendency, in influential quarters, to regard Logic as a branch of Mathematics and advances the intuitionalist theory of Logic. This involves criticism of Carnap, Russell,Wittgenstein, Broad and Whitehead, with additional discussions on Kant and Hegel. The author believes that the union of Philosophy and Logic is a natural one, and that an exclusively mathematical treatment cannot give an adequate account of Logic. A fundamental characteristic of Logic is comprehensiveness, which brings out the affinity between logic and philosophy, for to be comprehensive is the aim of philosophical ambition.

Written by one of the preeminent researchers in the field, this book provides a comprehensive exposition of modern analysis of causation. It shows how causality has grown from a nebulous concept into a mathematical theory with significant applications in the fields of statistics, artificial intelligence, economics, philosophy, cognitive science, and the health and social sciences. Judea Pearl presents and unifies the probabilistic, manipulative, counterfactual, and structural approaches to causation and devises simple mathematical tools for studying the relationships between causal connections and statistical associations. The book will open the way for including causal analysis in the standard curricula of statistics, artificial intelligence, business, epidemiology, social sciences, and economics. Students in these fields will find natural models, simple inferential procedures, and precise mathematical definitions of causal concepts that traditional texts have evaded or made unduly complicated. The first edition of Causality has led to a paradigmatic change in the way that causality is treated in statistics, philosophy, computer science, social science, and economics. Cited in more than 5,000 scientific publications, it continues to liberate scientists from the traditional molds of statistical thinking. In this revised edition, Judea Pearl elucidates thorny issues, answers readers questions, and offers a panoramic view of recent advances in this field of research. Causality will be of interests to students and professionals in a wide variety of fields. Anyone who wishes to elucidate meaningful relationships from data, predict effects of actions and policies, assess explanations of reported events, or form theories of causal understanding and causal speech will find this book stimulating and invaluable."

Originally published in 1973. This book is directed to the student of philosophy whose background in mathematics is very limited. The author strikes a balance between material of a philosophical and a formal kind, and does this in a way that will bring out the intricate connections between the two. On the formal side, he gives particular care to provide the basic tools from set theory and arithmetic that are needed to study systems of logic, setting out completeness results for two, three, and four valued logic, explaining concepts such as freedom and bondage in quantificational logic, describing the intuitionistic conception of the logical operators, and setting out Zermelo's axiom system for set theory. On the philosophical side, he gives particular attention to such topics as the problem of entailment, the import of the Loewenheim-Skolem theorem, the expressive powers of quantificational logic, the ideas underlying intuitionistic logic, the nature of set theory, and the relationship between logic and set theory. There are exercises within the text, set out alongside the theoretical ideas that they involve.

Originally published in 1962. A clear and simple account of the growth and structure of Mathematical Logic, no earlier knowledge of logic being required. After outlining the four lines of thought that have been its roots - the logic of Aristotle, the idea of all the parts of mathematics as systems to be designed on the same sort of plan as that used by Euclid and his Elements, and the discoveries in algebra and geometry in 1800-1860 - the book goes on to give some of the main ideas and theories of the chief writers on Mathematical Logic: De Morgan, Boole, Jevons, Pierce, Frege, Peano, Whitehead, Russell, Post, Hilbert and Goebel. Written to assist readers who require a general picture of current logic, it will also be a guide for those who will later be going more deeply into the expert details of this field.

Originally published in 1937. A short account of the traditional logic, intended to provide the student with the fundamentals necessary for the specialized study. Suitable for working through individualy, it will provide sufficient knowledge of the elements of the subject to understand materials on more advanced and specialized topics. This is an interesting historic perspective on this area of philosophy and mathematics.

Originally published in 1934. This fourth edition originally published 1954., revised by C. W. K. Mundle. "It must be the desire of every reasonable person to know how to justify a contention which is of sufficient importance to be seriously questioned. The explicit formulation of the principles of sound reasoning is the concern of Logic". This book discusses the habit of sound reasoning which is acquired by consciously attending to the logical principles of sound reasoning, in order to apply them to test the soundness of arguments. It isn't an introduction to logic but it encourages the practice of logic, of deciding whether reasons in argument are sound or unsound. Stress is laid upon the importance of considering language, which is a key instrument of our thinking and is imperfect.

This volume tells the story of the legacy and impact of the great German polymath Gottfried Wilhelm Leibniz (1646-1716). Leibniz made significant contributions to many areas, including philosophy, mathematics, political and social theory, theology, and various sciences. The essays in this volume explores the effects of Leibniz's profound insights on subsequent generations of thinkers by tracing the ways in which his ideas have been defended and developed in the three centuries since his death. Each of the 11 essays is concerned with Leibniz's legacy and impact in a particular area, and between them they show not just the depth of Leibniz's talents but also the extent to which he shaped the various domains to which he contributed, and in some cases continues to shape them today. With essays written by experts such as Nicholas Jolley, Pauline Phemister, and Philip Beeley, this volume is essential reading not just for students of Leibniz but also for those who wish to understand the game-changing impact made by one of history's true universal geniuses.

Originally published in 1923 Chance and Error examines the vagaries of chance, and how this is the result of the interference of yes and no. The book basis its examination of chance on the idea of a two-sided coin. The book stipulates that contradictories are head and tail, or yes and no. When the coin is flipped in the air yes normally wins half of the trials, but this includes half of the half that normally go to no. Thus, normally in one quarter of the trials there is an interference of yes and no. From this the chance of any number of heads or tails can be easily calculated, and all results that are attained by more difficult mathematics are secured. The book uses this idea to examine interference of yes and no in everyday life and argues that this causes the variations in everything that goes on around us in nature and in our daily life. This book will be of interest to philosophers of logic, as well as mathematicians.

A New World of Geometry Awaits Your Discovery! The last stone falls out ... a rush of ancient air ... the glint of gold ... the tingle of discovery ... When explorers first opened the tombs of the ancient pharaohs, they knew that they had discovered something wonderful. That feeling, that same passionate sense of discovery, is one of the most powerful educational tools a text can deliver. Geometry by Discovery is an exciting new approach to geometry. This ground-breaking text taps the pedagogical value of discovery to help students stretch their geometric perspective and hone their geometric intuition. It actively engages students in solving mathematical problems, and empowers them to be successful problem-solvers and discoverers of mathematical ideas.

Spherical Geometry and Its Applications introduces spherical geometry and its practical applications in a mathematically rigorous form. The text can serve as a course in spherical geometry for mathematics majors. Readers from various academic backgrounds can comprehend various approaches to the subject. The book introduces an axiomatic system for spherical geometry and uses it to prove the main theorems of the subject. It also provides an alternate approach using quaternions. The author illustrates how a traditional axiomatic system for plane geometry can be modified to produce a different geometric world - but a geometric world that is no less real than the geometric world of the plane. Features: A well-rounded introduction to spherical geometry Provides several proofs of some theorems to appeal to larger audiences Presents principal applications: the study of the surface of the earth, the study of stars and planets in the sky, the study of three- and four-dimensional polyhedra, mappings of the sphere, and crystallography Many problems are based on propositions from the ancient text Sphaerica of Menelaus

Originally published in 1994, The Incommensurability Thesis is a critical study of the Incommensurability Thesis of Thomas Kuhn and Paul Feyerabend. The book examines the theory that different scientific theories may be incommensurable because of conceptual variance. The book presents a critique of the thesis and examines and discusses the arguments for the theory, acknowledging and debating the opposing views of other theorists. The book provides a comprehensive and detailed discussion of the incommensurability thesis.

Contents:
Introduction. 1. Philosophy of logic 2. Philosophy of mathematics in the 20th century. 3. Frege 4. Wittgenstein's Tractatus 5. Logical postivism 6. The philosophy of physics 7. The philosophy of science 8. Chance, cause and conduct: probability theory and the explanation of human action 9. Cybernetics 10. Descartes' legacy: the mechanist/vitalist debates.

During the first few decades of the twentieth century, philosophers and mathematicians mounted a sustained effort to clarify the nature of mathematics. This led to considerable discord, even enmity, and yielded fascinating and fruitful work of both a mathematical and a philosophical nature. It was one of the most exhilarating intellectual adventures of the century, pursued at an extraordinarily high level of acuity and imagination. Its legacy principally consists of three original and finely articulated programs that seek to view mathematics in the proper light: logicism, intuitionism, and finitism. Each is notable for its symbiotic melding together of philosophical vision and mathematical work: the philosophical ideas are given their substance by specific mathematical developments, which are in turn given their point by philosophical reflection.

This book provides an accessible, critical introduction to these three projects as it describes and investigates both their philosophical and their mathematical components.

Solutions manual is available upon request.

This edited collection is the first of its kind to explore the view called perspectivism in philosophy of science. The book brings together an array of essays that reflect on the methodological promises and scientific challenges of perspectivism in a variety of fields such as physics, biology, cognitive neuroscience, and cancer research, just as a few examples. What are the advantages of using a plurality of perspectives in a given scientific field and for interdisciplinary research? Can different perspectives be integrated? What is the relation between perspectivism, pluralism, and pragmatism? These ten new essays by top scholars in the field offer a polyphonic journey towards understanding the view called 'perspectivism' and its relevance to science.

This book explores and articulates the concepts of the continuous and the infinitesimal from two points of view: the philosophical and the mathematical. The first section covers the history of these ideas in philosophy. Chapter one, entitled 'The continuous and the discrete in Ancient Greece, the Orient and the European Middle Ages,' reviews the work of Plato, Aristotle, Epicurus, and other Ancient Greeks; the elements of early Chinese, Indian and Islamic thought; and early Europeans including Henry of Harclay, Nicholas of Autrecourt, Duns Scotus, William of Ockham, Thomas Bradwardine and Nicolas Oreme. The second chapter of the book covers European thinkers of the sixteenth and seventeenth centuries: Galileo, Newton, Leibniz, Descartes, Arnauld, Fermat, and more. Chapter three, 'The age of continuity,' discusses eighteenth century mathematicians including Euler and Carnot, and philosophers, among them Hume, Kant and Hegel. Examining the nineteenth and early twentieth centuries, the fourth chapter describes the reduction of the continuous to the discrete, citing the contributions of Bolzano, Cauchy and Reimann. Part one of the book concludes with a chapter on divergent conceptions of the continuum, with the work of nineteenth and early twentieth century philosophers and mathematicians, including Veronese, Poincare, Brouwer, and Weyl. Part two of this book covers contemporary mathematics, discussing topology and manifolds, categories, and functors, Grothendieck topologies, sheaves, and elementary topoi. Among the theories presented in detail are non-standard analysis, constructive and intuitionist analysis, and smooth infinitesimal analysis/synthetic differential geometry. No other book so thoroughly covers the history and development of the concepts of the continuous and the infinitesimal.

One of the only volumes that brings the humanities, social sciences and even the natural sciences under one remit to look at how they can be researched in an integrated and useful way, with policy and real world implications in terms of how we relate in and to the world. Interdisciplinarity and Transdisciplinarity have been around for a long time, but as as we move through a digital age they are becoming more and more important and interesting to the scholarly community and beyond. There is nothing on the market that pulls all of these subjects across disciplines together and works out a framework to construct the analysis in a way that asks and answers useful questions.

This book is an attempt to change our thinking about thinking. Anna Sfard undertakes this task convinced that many long-standing, seemingly irresolvable quandaries regarding human development originate in ambiguities of the existing discourses on thinking. Standing on the shoulders of Vygotsky and Wittgenstein, the author defines thinking as a form of communication. The disappearance of the time-honoured thinking-communicating dichotomy is epitomised by Sfard's term, commognition, which combines communication with cognition. The commognitive tenet implies that verbal communication with its distinctive property of recursive self-reference may be the primary source of humans' unique ability to accumulate the complexity of their action from one generation to another. The explanatory power of the commognitive framework and the manner in which it contributes to our understanding of human development is illustrated through commognitive analysis of mathematical discourse accompanied by vignettes from mathematics classrooms.

The Twentieth Century has seen a dramatic rise in the use of probability and statistics in almost all fields of research. This has stimulated many new philosophical ideas on probability.
Philosophical Theories of Probability is the first book to present a clear, comprehensive and systematic account of these various theories and to explain how they relate to one another. Gillies also offers a distinctive version of the propensity theory of probability, and the intersubjective interpretation, which develops the subjective theory.

Phanes (fa-nays) means "manifester" or "revealer", and is related to the Greek words "light" and "to shine forth".

Phanes Press was founded in 1985 to publish quality books on the spiritual, philosophical, and cosmological traditions of the Western world. Since that time, we have published 45 books, including five volumes of Alexandria, a book-length journal of cosmology, philosophy, myth, and culture.

The year 2000 marks our fifteen-year anniversary, and we are working to bring out more interdisciplinary works, including books on creativity, psychology, literature, and the intersections between science, spirituality, and culture.

The longest work on number symbolism to survive from the ancient world. Contains helpful footnotes, an extensive glossary, bibliography, & foreword by Keith Critchlow.

Reissuing five works originally published between 1937 and 1991, this collection contains books addressing the subject of time, from a mostly philosophic point of view but also of interest to those in the science and mathematics worlds. These texts are brought back into print in this small set of works addressing how we think about time, the history of the philosophy of time, the measurement of time, theories of relativity and discussions of the wider thinking about time and space, among other aspects. One volume is a thorough bibliography collating references on the subject of time across many disciplines.

Originally published in 1976. This comprehensive study discusses in detail the philosophical, mathematical, physical, logical and theological aspects of our understanding of time and space. The text examines first the many different definitions of time that have been offered, beginning with some of the puzzles arising from our awareness of the passage of time and shows how time can be understood as the concomitant of consciousness. In considering time as the dimension of change, the author obtains a transcendental derivation of the concept of space, and shows why there has to be only one dimension of time and three of space, and why Kant was not altogether misguided in believing the space of our ordinary experience to be Euclidean. The concept of space-time is then discussed, including Lorentz transformations, and in an examination of the applications of tense logic the author discusses the traditional difficulties encountered in arguments for fatalism. In the final sections he discusses eternity and the beginning and end of the universe. The book includes sections on the continuity of space and time, on the directedness of time, on the differences between classical mechanics and the Special and General theories of relativity, on the measurement of time, on the apparent slowing down of moving clocks, and on time and probability.

Originally published in 1980. What is time? How is its structure determined? The enduring controversy about the nature and structure of time has traditionally been a diametrical argument between those who see time as a container into which events are placed, and those for whom time cannot exist without events. This controversy between the absolutist and the relativist theories of time is a central theme of this study. The author's impressive arguments provide grounds for rejecting both these theories, firstly by establishing that 'empty' time is possible, and secondly by showing, through a discussion of the structure of time which involves considering whether time might be cyclical, branching, beginning or non-beginning, that the absolutist theory of time is untenable. This book then advances two new theories, and succeeds in shifting the traditional debate about time to a consideration of time as a theoretical structure and as a theoretical framework.