Offering a bold new vision on the history of modern logic, Lukas M. Verburgt and Matteo Cosci focus on the lasting impact of Aristotle's syllogism between the 1820s and 1930s. For over two millennia, deductive logic was the syllogism and syllogism was the yardstick of sound human reasoning. During the 19th century, this hegemony fell apart and logicians, including Boole, Frege and Peirce, took deductive logic far beyond its Aristotelian borders. However, contrary to common wisdom, reflections on syllogism were also instrumental to the creation of new logical developments, such as first-order logic and early set theory. This volume presents the period under discussion as one of both tradition and innovation, both continuity and discontinuity. Modern logic broke away from the syllogistic tradition, but without Aristotle's syllogism, modern logic would not have been born. A vital follow up to The Aftermath of Syllogism, this book traces the longue duree history of syllogism from Richard Whately's revival of formal logic in the 1820s through the work of David Hilbert and the Goettingen school up to the 1930s. Bringing together a group of major international experts, it sheds crucial new light on the emergence of modern logic and the roots of analytic philosophy in the 19th and early 20th centuries.

Paris of the year 1900 left two landmarks: the Tour Eiffel, and David Hilbert's celebrated list of twenty-four mathematical problems presented at a conference opening the new century. Kurt Goedel, a logical icon of that time, showed Hilbert's ideal of complete axiomatization of mathematics to be unattainable. The result, of 1931, is called Goedel's incompleteness theorem. Goedel then went on to attack Hilbert's first and second Paris problems, namely Cantor's continuum problem about the type of infinity of the real numbers, and the freedom from contradiction of the theory of real numbers. By 1963, it became clear that Hilbert's first question could not be answered by any known means, half of the credit of this seeming faux pas going to Goedel. The second is a problem still wide open. Goedel worked on it for years, with no definitive results; The best he could offer was a start with the arithmetic of the entire numbers. This book, Goedel's lectures at the famous Princeton Institute for Advanced Study in 1941, shows how far he had come with Hilbert's second problem, namely to a theory of computable functionals of finite type and a proof of the consistency of ordinary arithmetic. It offers indispensable reading for logicians, mathematicians, and computer scientists interested in foundational questions. It will form a basis for further investigations into Goedel's vast Nachlass of unpublished notes on how to extend the results of his lectures to the theory of real numbers. The book also gives insights into the conceptual and formal work that is needed for the solution of profound scientific questions, by one of the central figures of 20th century science and philosophy.

Games, Norms, and Reasons: Logic at the Crossroads provides an overview of modern logic focusing on its relationships with other disciplines, including new interfaces with rational choice theory, epistemology, game theory and informatics. This book continues a series called "Logic at the Crossroads" whose title reflects a view that the deep insights from the classical phase of mathematical logic can form a harmonious mixture with a new, more ambitious research agenda of understanding and enhancing human reasoning and intelligent interaction. The editors have gathered together articles from active authors in this new area that explore dynamic logical aspects of norms, reasons, preferences and beliefs in human agency, human interaction and groups. The book pays a special tribute to Professor Rohit Parikh, a pioneer in this movement.

Contemporary analytic philosophy can generally be characterized by the following tendencies: commitment to first-order predicate logic as the only viable formal logic; rejection of correspondence theories of truth; a view of existence as something expressed by the existential quantifier; a metaphysics that doesn t give the world as a whole its due. This book seeks to offer an alternative analytic theory, one that provides a unified account of what there is, how we speak about it, the underlying logic of our language, how the truth of what we say is determined, and the central role of the real world in all of this. The result is a robust account of reality. The inspiration for many of the ideas that constitute this overall theory comes from such sources as Aristotle, Leibniz, Ryle, and Sommers."

1. The ?rst edition of this book was published in 1977. The text has been well received and is still used, although it has been out of print for some time. In the intervening three decades, a lot of interesting things have happened to mathematical logic: (i) Model theory has shown that insights acquired in the study of formal languages could be used fruitfully in solving old problems of conventional mathematics. (ii) Mathematics has been and is moving with growing acceleration from the set-theoretic language of structures to the language and intuition of (higher) categories, leaving behind old concerns about in?nities: a new view of foundations is now emerging. (iii) Computer science, a no-nonsense child of the abstract computability theory, has been creatively dealing with old challenges and providing new ones, such as the P/NP problem. Planning additional chapters for this second edition, I have decided to focus onmodeltheory, the conspicuousabsenceofwhichinthe ?rsteditionwasnoted in several reviews, and the theory of computation, including its categorical and quantum aspects. The whole Part IV: Model Theory, is new. I am very grateful to Boris I. Zilber, who kindly agreed to write it. It may be read directly after Chapter II. The contents of the ?rst edition are basically reproduced here as Chapters I-VIII. Section IV.7, on the cardinality of the continuum, is completed by Section IV.7.3, discussing H. Woodin's discovery.

The Routledge Handbook of Collective Intentionality provides a wide-ranging survey of topics in a rapidly expanding area of interdisciplinary research. It consists of 36 chapters, written exclusively for this volume, by an international team of experts. What is distinctive about the study of collective intentionality within the broader study of social interactions and structures is its focus on the conceptual and psychological features of joint or shared actions and attitudes, and their implications for the nature of social groups and their functioning. This Handbook fully captures this distinctive nature of the field and how it subsumes the study of collective action, responsibility, reasoning, thought, intention, emotion, phenomenology, decision-making, knowledge, trust, rationality, cooperation, competition, and related issues, as well as how these underpin social practices, organizations, conventions, institutions and social ontology. Like the field, the Handbook is interdisciplinary, drawing on research in philosophy, cognitive science, linguistics, legal theory, anthropology, sociology, computer science, psychology, economics, and political science. Finally, the Handbook promotes several specific goals: (1) it provides an important resource for students and researchers interested in collective intentionality; (2) it integrates work across disciplines and areas of research as it helps to define the shape and scope of an emerging area of research; (3) it advances the study of collective intentionality.

Since the ground-breaking work of Saul Kripke, David Lewis, and others in the 1960s and 70s, one dominant interest of analytic philosophers has been in modal truths, which concerns the questions of what is possible and what is necessary. However, there is considerable controversy over the source and nature of necessity. In Modality and Explanatory Reasoning, Boris Kment takes a novel approach to the study of modality that places special emphasis on understanding the origin of modal notions in everyday thought. Kment argues that the concepts of necessity and possibility originate in a common type of thought experiment-counterfactual reasoning-that allows us to investigate explanatory connections. This procedure is closely related to the controlled experiments of empirical science. Necessity is defined in terms of causation and other forms of explanation such as grounding, the relation that connects metaphysically fundamental facts to non-fundamental ones. Therefore, contrary to a widespread view, explanation is more fundamental than modality. The study of modal facts is important for philosophy, not because these facts are of much metaphysical interest in their own right, but because they provide evidence about explanatory relationships. In the course of developing this position, the book offers new accounts of possible worlds, counterfactual conditionals, essential truths and their role in grounding, and a novel theory of how counterfactuals relate to causation and explanation.

The issue of a logic foundation for African thought connects well with the question of method. Do we need new methods for African philosophy and studies? Or, are the methods of Western thought adequate for African intellectual space? These questions are not some of the easiest to answer because they lead straight to the question of whether or not a logic tradition from African intellectual space is possible. Thus in charting the course of future direction in African philosophy and studies, one must be confronted with this question of logic. The author boldly takes up this challenge and becomes the first to do so in a book by introducing new concepts and formulating a new African culture-inspired system of logic called Ezumezu which he believes would ground new methods in African philosophy and studies. He develops this system to rescue African philosophy and, by extension, sundry fields in African Indigenous Knowledge Systems from the spell of Plato and the hegemony of Aristotle. African philosophers can now ground their discourses in Ezumezu logic which will distinguish their philosophy as a tradition in its own right. On the whole, the book engages with some of the lingering controversies in the idea of (an) African logic before unveiling Ezumezu as a philosophy of logic, methodology and formal system. The book also provides fresh arguments and insights on the themes of decolonisation and Africanisation for the intellectual transformation of scholarship in Africa. It will appeal to philosophers and logicians-undergraduates and post graduate researchers-as well as those in various areas of African studies.

For thousands of years, mathematicians have used the timeless art of logic to see the world more clearly. In The Art of Logic, Royal Society Science Book Prize nominee Eugenia Cheng shows how anyone can think like a mathematician - and see, argue and think better.

Learn how to simplify complex decisions without over-simplifying them. Discover the power of analogies and the dangers of false equivalences. Find out how people construct misleading arguments, and how we can argue back.

Eugenia Cheng teaches us how to find clarity without losing nuance, taking a careful scalpel to the complexities of politics, privilege, sexism and dozens of other real-world situations. Her Art of Logic is a practical and inspiring guide to decoding the modern world.

Alex Oliver and Timothy Smiley provide a natural point of entry to what for most readers will be a new subject. Plural logic deals with plural terms ('Whitehead and Russell', 'Henry VIII's wives', 'the real numbers', 'the square root of -1', 'they'), plural predicates ('surrounded the fort', 'are prime', 'are consistent', 'imply'), and plural quantification ('some things', 'any things'). Current logic is singularist: its terms stand for at most one thing. By contrast, the foundational thesis of this book is that a particular term may legitimately stand for several things at once; in other words, there is such a thing as genuinely plural denotation. The authors argue that plural phenomena need to be taken seriously and that the only viable response is to adopt a plural logic, a logic based on plural denotation. They expound a framework of ideas that includes the distinction between distributive and collective predicates, the theory of plural descriptions, multivalued functions, and lists. A formal system of plural logic is presented in three stages, before being applied to Cantorian set theory as an illustration. Technicalities have been kept to a minimum, and anyone who is familiar with the classical predicate calculus should be able to follow it. The authors' approach is an attractive blend of no-nonsense argumentative directness and open-minded liberalism, and they convey the exciting and unexpected richness of their subject. Mathematicians and linguists, as well as logicians and philosophers, will find surprises in this book. This second edition includes a greatly expanded treatment of the paradigm empty term zilch, a much strengthened treatment of Cantorian set theory, and a new chapter on higher-level plural logic.

This book provides an epistemological study of the great Islamic scholar of Banjarese origin, Syeikh Muhammad Arsyad al-Banjari (1710-1812) who contributed to the development of Islam in Indonesia and, in general, Southeast Asia. The work focuses on Arsyad al-Banjari's dialectical use and understanding of qiyas or correlational inference as a model of parallel reasoning or analogy in Islamic jurisprudence. This constituted the most prominent instrument he applied in his effort of integrating Islamic law into the Banjarese society.This work studies how Arsyad al-Banjari integrates jadal theory or dialectic in Islamic jurisprudence, within his application of qiyas. The author develops a framework for qiyas which acts as the interface between jadal, dialogical logic, and Per Martin-Loef's Constructive Type Theory (CTT). One of the epistemological results emerging from the present study is that the different forms of qiyas applied by Arsyad al-Banjari represent an innovative and sophisticated form of reasoning. The volume is divided into three parts that discuss the types of qiyas as well their dialectical and argumentative aspects, historical background and context of Banjar, and demonstrates how the theory of qiyas comes quite close to the contemporary model of parallel reasoning for sciences and mathematics developed by Paul Bartha (2010). This volume will be of interest to historians and philosophers in general, and logicians and historians of philosophy in particular.

The quality of our lives is determined by the quality of our thinking. The quality of our thinking, in turn, is determined by the quality of our questions, for questions are the engine, the driving force behind thinking. Without questions, we have nothing to think about. Without essential questions, we often fail to focus our thinking on the significant and substantive. When we ask essential questions, we deal with what is necessary, relevant, and indispensable to a matter at hand. We recognize what is at the heart of the matter. Our thinking is grounded and disciplined. We are ready to learn. We are intellectually able to find our way about. To be successful in life, one needs to ask essential questions: essential questions when reading, writing, and speaking; when shopping, working, and parenting; when forming friendships, choosing life-partners, and interacting with the mass media and the Internet. Yet few people are masters of the art of asking essential questions. Most have never thought about why some questions are crucial and others peripheral. Essential questions are rarely studied in school. They are rarely modeled at home. Most people question according to their psychological associations. Their questions are haphazard and scattered. The ideas we provide are useful only to the extent that they are employed daily to ask essential questions. Practice in asking essential questions eventually leads to the habit of asking essential questions. But we can never practice asking essential questions if we have no conception of them. This mini-guide is a starting place for understanding concepts that, when applied, lead to essential questions. We introduce essential questions as indispensable intellectual tools. We focus on principles essential to formulating, analyzing, assessing, and settling primary questions. You will notice that our categories of question types are not exclusive. There is a great deal of overlap

This book applies the formal discipline of logic to everyday discourse. It offers a new analysis of the notion of individual, suggesting that this notion is linguistic, not ontological, and that anything denoted by a proper name in a well-functioning language game is an individual. It further posits that everyday discourse is non-compositional, i.e., its complex expressions are not just the result of putting simpler ones together but react on the latter, modifying their meaning through feedback. The book theorizes that in everyday discourse, there is no algebra of truth values, but the latter can be both input and output of something which has no truth value at all. It suggests that an elementary proposition of everyday discourse (defined as having exactly one predicate) can, in principle, be indefinitely expanded by adding new components, belonging neither to subject nor to predicate, but remain elementary. This book is of interest to logicians and philosophers of language.

It is with great pleasure that we are presenting to the community the second edition of this extraordinary handbook. It has been over 15 years since the publication of the first edition and there have been great changes in the landscape of philosophical logic since then. The first edition has proved invaluable to generations of students and researchers in formal philosophy and language, as well as to consumers of logic in many applied areas. The main logic article in the Encyclopaedia Britannica 1999 has described the first edition as 'the best starting point for exploring any of the topics in logic'. We are confident that the second edition will prove to be just as good. ! The first edition was the second handbook published for the logic commu nity. It followed the North Holland one volume Handbook of Mathematical Logic, published in 1977, edited by the late Jon Barwise, The four volume Handbook of Philosophical Logic, published 1983-1989 came at a fortunate temporal junction at the evolution of logic. This was the time when logic was gaining ground in computer science and artificial intelligence circles. These areas were under increasing commercial pressure to provide devices which help and/or replace the human in his daily activity. This pressure required the use of logic in the modelling of human activity and organisa tion on the one hand and to provide the theoretical basis for the computer program constructs on the other.

Of the four chapters in this book, the first two discuss (albeit in consider ably modified form) matters previously discussed in my papers 'On the Logic of Conditionals' [1] and 'Probability and the Logic of Conditionals' [2], while the last two present essentially new material. Chapter I is relatively informal and roughly parallels the first of the above papers in discussing the basic ideas of a probabilistic approach to the logic of the indicative conditional, according to which these constructions do not have truth values, but they do have probabilities (equal to conditional probabilities), and the appropriate criterion of soundness for inferences involving them is that it should not be possible for all premises of the inference to be probable while the conclusion is improbable. Applying this criterion is shown to have radically different consequences from the orthodox 'material conditional' theory, not only in application to the standard 'fallacies' of the material conditional, but to many forms (e. g. , Contraposition) which have hitherto been regarded as above suspi cion. Many more applications are considered in Chapter I, as well as certain related theoretical matters. The chief of these, which is the most important new topic treated in Chapter I (i. e.

This volume offers English translations of three early works by Ernst Schroeder (1841-1902), a mathematician and logician whose philosophical ruminations and pathbreaking contributions to algebraic logic attracted the admiration and ire of figures such as Dedekind, Frege, Husserl, and C. S. Peirce. Today he still engages the sympathetic interest of logicians and philosophers. The works translated record Schroeder's journey out of algebra into algebraic logic and document his transformation of George Boole's opaque and unwieldy logical calculus into what we now recognize as Boolean algebra. Readers interested in algebraic logic and abstract algebra can look forward to a tour of the early history of those fields with a guide who was exceptionally thorough, unfailingly honest, and deeply reflective.

This volume examines the role of logic in cognitive psychology in light of recent developments. Gonzalo Reyes's new semantic theory has brought the fields of cognitive psychology and logic closer together, and has shed light on how children master proper names and count nouns, and thus acquire knowledge. The chapters highlight the inadequacies of classical logic in its handling of ordinary language and reveal the prospects of applying these new theories to cognitive psychology, cognitive science, linguistics, the philosophy of language and logic.

This volume deals with formal, mechanizable reasoning in modal logics, that is, logics of necessity, possibility, belief, time computations etc. It is therefore of immense interest for various interrelated disciplines such as philosophy, AI, computer science, logic, cognitive science and linguistics. The book consists of 15 original research papers, divided into three parts. The first part contains papers which give a profound description of powerful proof-theoretic methods as applied to the normal modal logic S4. Part II is concerned with a number of generalizations of the standard proof-theoretic formats, while the third part presents new and important results on semantics-based proof systems for modal logic.

"Exposes the vast gap between the actual science underlying AI and the dramatic claims being made for it." -John Horgan "If you want to know about AI, read this book...It shows how a supposedly futuristic reverence for Artificial Intelligence retards progress when it denigrates our most irreplaceable resource for any future progress: our own human intelligence." -Peter Thiel Ever since Alan Turing, AI enthusiasts have equated artificial intelligence with human intelligence. A computer scientist working at the forefront of natural language processing, Erik Larson takes us on a tour of the landscape of AI to reveal why this is a profound mistake. AI works on inductive reasoning, crunching data sets to predict outcomes. But humans don't correlate data sets. We make conjectures, informed by context and experience. And we haven't a clue how to program that kind of intuitive reasoning, which lies at the heart of common sense. Futurists insist AI will soon eclipse the capacities of the most gifted mind, but Larson shows how far we are from superintelligence-and what it would take to get there. "Larson worries that we're making two mistakes at once, defining human intelligence down while overestimating what AI is likely to achieve...Another concern is learned passivity: our tendency to assume that AI will solve problems and our failure, as a result, to cultivate human ingenuity." -David A. Shaywitz, Wall Street Journal "A convincing case that artificial general intelligence-machine-based intelligence that matches our own-is beyond the capacity of algorithmic machine learning because there is a mismatch between how humans and machines know what they know." -Sue Halpern, New York Review of Books

Papers from more than three decades reflect the development of thinkingover the dialogical framework that shapes verbal expression of comprehending experience and that has to be exhibited in responsible argumentations. With dialogical reconstructions of experience owing to the methodical constructivism of the a oeErlangen Schoola it is possible to uncover the origin of many conceptual oppositions in traditional philosophical talk, like natural vs. artificial/cultural, subjective vs. objective, etc., and to solve philosophical riddles connected with them.