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Proof Theory: Second Edition (Dover Books on Mathematics #Volume 81)

by Gaisi Takeuti

Focusing on Gentzen-type proof theory, this volume presents a detailed overview of creative works by author Gaisi Takeuti and other twentieth-century logicians. The text explores applications of proof theory to logic as well as other areas of mathematics. Suitable for advanced undergraduates and graduate students of mathematics, this long-out-of-print monograph forms a cornerstone for any library in mathematical logic and related topics.The three-part treatment begins with an exploration of first order systems, including a treatment of predicate calculus involving Gentzen's cut-elimination theorem and the theory of natural numbers in terms of Gödel's incompleteness theorem and Gentzen's consistency proof. The second part, which considers second order and finite order systems, covers simple type theory and infinitary logic. The final chapters address consistency problems with an examination of consistency proofs and their applications.

Proof Theory and Algebra in Logic (Short Textbooks in Logic)

by Hiroakira Ono

This book offers a concise introduction to both proof-theory and algebraic methods, the core of the syntactic and semantic study of logic respectively. The importance of combining these two has been increasingly recognized in recent years. It highlights the contrasts between the deep, concrete results using the former and the general, abstract ones using the latter. Covering modal logics, many-valued logics, superintuitionistic and substructural logics, together with their algebraic semantics, the book also provides an introduction to nonclassical logic for undergraduate or graduate level courses.The book is divided into two parts: Proof Theory in Part I and Algebra in Logic in Part II. Part I presents sequent systems and discusses cut elimination and its applications in detail. It also provides simplified proof of cut elimination, making the topic more accessible. The last chapter of Part I is devoted to clarification of the classes of logics that are discussed in the second part. Part II focuses on algebraic semantics for these logics. At the same time, it is a gentle introduction to the basics of algebraic logic and universal algebra with many examples of their applications in logic. Part II can be read independently of Part I, with only minimum knowledge required, and as such is suitable as a textbook for short introductory courses on algebra in logic.

A Proof Theory for Description Logics (SpringerBriefs in Computer Science)

by Alexandre Rademaker

Description Logics (DLs) is a family of formalisms used to represent knowledge of a domain. They are equipped with a formal logic-based semantics. Knowledge representation systems based on description logics provide various inference capabilities that deduce implicit knowledge from the explicitly represented knowledge. A Proof Theory for Description Logics introduces Sequent Calculi and Natural Deduction for some DLs (ALC, ALCQ). Cut-elimination and Normalization are proved for the calculi. The author argues that such systems can improve the extraction of computational content from DLs proofs for explanation purposes.

Proofiness

by Charles Seife

The bestselling author of Zero shows how mathematical misinformation pervades-and shapes-our daily lives. According to MSNBC, having a child makes you stupid. You actually lose IQ points. Good Morning America has announced that natural blondes will be extinct within two hundred years. Pundits estimated that there were more than a million demonstrators at a tea party rally in Washington, D.C., even though roughly sixty thousand were there. Numbers have peculiar powers-they can disarm skeptics, befuddle journalists, and hoodwink the public into believing almost anything. "Proofiness," as Charles Seife explains in this eye-opening book, is the art of using pure mathematics for impure ends, and he reminds readers that bad mathematics has a dark side. It is used to bring down beloved government officials and to appoint undeserving ones (both Democratic and Republican), to convict the innocent and acquit the guilty, to ruin our economy, and to fix the outcomes of future elections. This penetrating look at the intersection of math and society will appeal to readers of Freakonomics and the books of Malcolm Gladwell.

Proofiness

by Charles Seife

The bestselling author of Zero shows how mathematical misinformation pervades-and shapes-our daily lives. According to MSNBC, having a child makes you stupid. You actually lose IQ points. Good Morning America has announced that natural blondes will be extinct within two hundred years. Pundits estimated that there were more than a million demonstrators at a tea party rally in Washington, D. C. , even though roughly sixty thousand were there. Numbers have peculiar powers-they can disarm skeptics, befuddle journalists, and hoodwink the public into believing almost anything. "Proofiness," as Charles Seife explains in this eye-opening book, is the art of using pure mathematics for impure ends, and he reminds readers that bad mathematics has a dark side. It is used to bring down beloved government officials and to appoint undeserving ones (both Democratic and Republican), to convict the innocent and acquit the guilty, to ruin our economy, and to fix the outcomes of future elections. This penetrating look at the intersection of math and society will appeal to readers of Freakonomics and the books of Malcolm Gladwell. .

Proofs 101: An Introduction to Formal Mathematics

by Joseph Kirtland

Proofs 101: An Introduction to Formal Mathematics serves as an introduction to proofs for mathematics majors who have completed the calculus sequence (at least Calculus I and II) and a first course in linear algebra. The book prepares students for the proofs they will need to analyze and write the axiomatic nature of mathematics and the rigors of upper-level mathematics courses. Basic number theory, relations, functions, cardinality, and set theory will provide the material for the proofs and lay the foundation for a deeper understanding of mathematics, which students will need to carry with them throughout their future studies. Features Designed to be teachable across a single semester Suitable as an undergraduate textbook for Introduction to Proofs or Transition to Advanced Mathematics courses Offers a balanced variety of easy, moderate, and difficult exercises

Proofs and Algorithms

by Gilles Dowek

Logic is a branch of philosophy, mathematics and computer science. It studies the required methods to determine whether a statement is true, such as reasoning and computation. Proofs and Algorithms: Introduction to Logic and Computability is an introduction to the fundamental concepts of contemporary logic - those of a proof, a computable function, a model and a set. It presents a series of results, both positive and negative, - Church's undecidability theorem, Gödel's incompleteness theorem, the theorem asserting the semi-decidability of provability - that have profoundly changed our vision of reasoning, computation, and finally truth itself. Designed for undergraduate students, this book presents all that philosophers, mathematicians and computer scientists should know about logic.

Proofs and Models in Philosophical Logic (Elements in Philosophy and Logic)

by Greg Restall

This Element is an introduction to recent work proofs and models in philosophical logic, with a focus on the semantic paradoxes the sorites paradox. It introduces and motivates different proof systems and different kinds of models for a range of logics, including classical logic, intuitionistic logic, a range of three-valued and four-valued logics, and substructural logics. It also compares and contrasts the different approaches to substructural treatments of the paradox, showing how the structural rules of contraction, cut and identity feature in paradoxical derivations. It then introduces model theoretic treatments of the paradoxes, including a simple fixed-point model construction which generates three-valued models for theories of truth, which can provide models for a range of different non-classical logics. The Element closes with a discussion of the relationship between proofs and models, arguing that both have their place in the philosophers' and logicians' toolkits.

Proofs and Refutations

by John Worrall Lakatos Imre Zahar Elie Imre Lakatos Elie Zahar

Imre Lakatos's Proofs and Refutations is an enduring classic, which has never lost its relevance. Taking the form of a dialogue between a teacher and some students, the book considers various solutions to mathematical problems and, in the process, raises important questions about the nature of mathematical discovery and methodology. Lakatos shows that mathematics grows through a process of improvement by attempts at proofs and critiques of these attempts, and his work continues to inspire mathematicians and philosophers aspiring to develop a philosophy of mathematics that accounts for both the static and the dynamic complexity of mathematical practice. With a specially commissioned Preface written by Paolo Mancosu, this book has been revived for a new generation of readers.

Proofs and Refutations

by John Worrall Elie Zahar

Proofs and Refutations is essential reading for all those interested in the methodology, the philosophy and the history of mathematics. Much of the book takes the form of a discussion between a teacher and his students. They propose various solutions to some mathematical problems and investigate the strengths and weaknesses of these solutions. Their discussion (which mirrors certain real developments in the history of mathematics) raises some philosophical problems and some problems about the nature of mathematical discovery or creativity. Imre Lakatos is concerned throughout to combat the classical picture of mathematical development as a steady accumulation of established truths. He shows that mathematics grows instead through a richer, more dramatic process of the successive improvement of creative hypotheses by attempts to 'prove' them and by criticism of these attempts: the logic of proofs and refutations.

Proofs of the Cantor-Bernstein Theorem

by Arie Hinkis

This book offers an excursion through the developmental area of research mathematics. It presents some 40 papers, published between the 1870s and the 1970s, on proofs of the Cantor-Bernstein theorem and the related Bernstein division theorem. While the emphasis is placed on providing accurate proofs, similar to the originals, the discussion is broadened to include aspects that pertain to the methodology of the development of mathematics and to the philosophy of mathematics. Works of prominent mathematicians and logicians are reviewed, including Cantor, Dedekind, Schröder, Bernstein, Borel, Zermelo, Poincaré, Russell, Peano, the Königs, Hausdorff, Sierpinski, Tarski, Banach, Brouwer and several others mainly of the Polish and the Dutch schools. In its attempt to present a diachronic narrative of one mathematical topic, the book resembles Lakatos' celebrated book Proofs and Refutations. Indeed, some of the observations made by Lakatos are corroborated herein. The analogy between the two books is clearly anything but superficial, as the present book also offers new theoretical insights into the methodology of the development of mathematics (proof-processing), with implications for the historiography of mathematics.

Propaganda and International Criminal Law: From Cognition to Criminality

by Predrag Dojčinović

This book addresses the conceptual and evidentiary issues relating to the treatment of propaganda in international criminal law. Bringing together an interdisciplinary range of scholars, researchers and legal practitioners from Africa, Australia, Europe and the United States, the book provides an in-depth analysis of the nature, position and role of the concept of propaganda in mass atrocity crimes trials. A sequel to the earlier Propaganda, War Crimes Trials and International Law: From Speakers’ Corner to War Crimes (Routledge, 2011) this book is the first to synthesize the knowledge, procedures and methods of international criminal law with the social cognitive sciences. Including a comprehensive overview of the most relevant case law, jurisprudence and scientific studies, the book also offers a series of practical insights and strategies for both academics and legal professionals. An invaluable resource for those working in the area of international criminal law, this book will also be of interest to academics, practitioners and students with relevant interests in legal theory, politics, linguistics and psychology.

Propaganda and Nation Building: Selling the Irish Free State (Routledge New Directions in PR & Communication Research)

by Kevin Hora

This book examines the origins of Ireland in its first independent incarnation, the Irish Free State (1922-1937). It explores how contemporary public relations and propaganda techniques were used to construct an identity for this new state – a state which after enduring seven years of insurrection and civil war, became one of the most stable democracies in Europe. This stability, the book argues, was constructed not solely through policies enacted by governments, but through the construction of a Gaelic, Catholic and Celtic national identity. By shifting the perspective to how nation building was communicated, it weaves an interdisciplinary narrative that initiates a new understanding of nation building - providing insights of increasing relevance in current world events. Avoiding a simplistic cause and effect history of public relations, the book examines the uses and effects of early public relations from a political and societal perspective and suggests that while governments were only modestly successful in their varied propaganda efforts, cumulatively they facilitated a transition from violence to peace. This will be of interest to researchers and advanced students with an interest in public relations, propaganda studies, nation building and Irish studies.

A Proper Sense of Honor: Service and Sacrifice in George Washington's Army

by Caroline Cox

Starting with the decision by patriot leaders to create a corps of officers who were gentlemen and a body of soldiers who were not, Caroline Cox examines the great gap that existed in the conditions of service of soldiers and officers in the Continental army.

The Proper Study Of Mankind: An Anthology of Essays

by Isaiah Berlin

‘He becomes everyman’s guide to everything exciting in the history of ideas’ New York Review of BooksIsaiah Berlin was one of the leading thinkers of the twentieth century, and one of the finest writers. The Proper Study Of Mankind selects some of his best essays in which his insights both illuminate the past and offer a key to the burning issues of today.The full (and enormous) range of his work is represented here, from the exposition of his most distinctive doctrine - pluralism - to studies of Machiavelli, Tolstoy, Churchill and Roosevelt. In these pages he encapsulates the principal movements that characterise the modern age: romanticism, historicism, Fascism, relativism, irrationalism and nationalism. His ideas are always tied to the people who conceived them, so that abstractions are brought alive. EDITED BY HENRY HARDY AND ROGER HAUSHEER AND WITH A NEW FOREWORD BY ANDREW MARR

Properties (Elements in Metaphysics)

by Anna-Sofia Maurin

Although the subject matter of this Element is properties, do not expect in-depth introductions to the various views on properties 'on the market'. Instead, here that subject matter is treated meta-philosophically. Rather than ask and try to answer a question like do properties exist? this Element asks what reasons one might have for thinking that properties exist (what problem properties, if they exist, are there to solve), what counts as solving that (or those) problems (including what counts as 'a property'), as well as how we ought to proceed when trying to find out if properties exist (by which method this ought to be decided). As it turns out, these questions and their answers are all intricately intertwined. Theory comparison and theory evaluation is in other words (and perhaps not that surprisingly) tricky. Do properties exist? After reading this Element all we can say is therefore this: that depends.

Properties and Propositions: The Metaphysics of Higher-Order Logic

by Robert Trueman

This book articulates and defends Fregean realism, a theory of properties based on Frege's insight that properties are not objects, but rather the satisfaction conditions of predicates. Robert Trueman argues that this approach is the key not only to dissolving a host of longstanding metaphysical puzzles, such as Bradley's Regress and the Problem of Universals, but also to understanding the relationship between states of affairs, propositions, and the truth conditions of sentences. Fregean realism, Trueman suggests, ultimately leads to a version of the identity theory of truth, the theory that true propositions are identical to obtaining states of affairs. In other words, the identity theory collapses the gap between mind and world. This book will be of interest to anyone working in logic, metaphysics, the philosophy of language or the philosophy of mind.

Properties of Life: Toward a Theory of Organismic Biology (Vienna Series in Theoretical Biology)

by Bernd Rosslenbroich

A coherent and comprehensive theory of life that synthesizes the specific properties of living organisms.Despite continued advances, science has until now struggled to describe the specific properties that define a living being. By synthesizing several aspects of organismic biology and contemporary science, Properties of Life by Bernd Rosslenbroich generates a coherent concept of the singular quality of being alive—a concept that provides a crucial foundation for scientists, farmers, and medical practitioners and helps explain how we all interact with the world around us and within ourselves.Is an organism an aggregate of parts or an integrated system with agency? Is it a passive stimulus-response machine or a being equipped with subjectivity and consciousness? Rosslenbroich argues that the way people in different fields understand life determines their assumptions about organic function and behavior. In medicine, this extends to the human organism, which influences prevention, diagnosis, and treatment. Drawing attention to a long-standing but underappreciated line of thought in organismic biology, Rosslenbroich&’s original idea emphasizes the autonomy of living processes, their network characteristics, and their self-determined organization in time and structure. A timely and revelatory book, Properties of Life formulates an integrated, unified theory that remains flexible enough to accommodate future developments and resilient enough to withstand the challenges of different theoretical and disciplinary backgrounds.

Properties, Powers and Structures: Issues in the Metaphysics of Realism (Routledge Studies in Metaphysics)

by Brian Ellis Alexander Bird Howard Sankey

While the phrase "metaphysics of science" has been used from time to time, it has only recently begun to denote a specific research area where metaphysics meets philosophy of science—and the sciences themselves. The essays in this volume demonstrate that metaphysics of science is an innovative field of research in its own right. The principle areas covered are: The modal metaphysics of properties: What is the essential nature of natural properties? Are all properties essentially categorical? Are they all essentially dispositions, or are some categorical and others dispositional? Realism in mathematics and its relation to science: What does a naturalistic commitment of scientific realism tell us about our commitments to mathematical entities? Can this question be framed in something other than a Quinean philosophy? Dispositions and their relation to causation: Can we generate an account of causation that takes dispositionality as fundamental? And if we take dispositions as fundamental (and hence not having a categorical causal basis), what is the ontological ground of dispositions? Pandispositionalism: Could all properties be dispositional in nature? Natural kinds: Are there natural kinds, and if so what account of their nature should we give? For example, do they have essences? Here we consider how these issues may be illuminated by considering examples from reals science, in particular biochemistry and neurobiology.

Property (Key Concepts in Political Theory #9)

by Robert Lamb

Few political ideas are as divisive and controversial for some – and yet taken for granted by others – as the ownership of private property. For its defenders, private ownership is a fundamental right that protects individual freedom and ensures wider economic benefits for the community; for its critics, by contrast, property is institutionalised theft, responsible for lamentable levels of inequality and poverty. In this book, Robert Lamb explores philosophical arguments deployed to conceptualise, justify, and criticise private property ownership. He introduces the radical case against property advanced by anarchist and socialist writers, before analysing some of the most important and influential arguments in its favour. Lamb explains and assesses the various defences of property rights advanced by Locke, Hume, Hegel, J. S. Mill, and Nozick. He then shows how theorists such as John Rawls and his followers encourage us to rethink the very nature of ownership in a democratic society. This engaging synthesis of historical and contemporary theories of property will be essential reading for students and scholars of political philosophy.

Property and its Forms in Classical German Philosophy

by David James

The theme of property is directly relevant to some of the most divisive social and political issues today, such as wealth inequality and the question of whether governments should limit it by introducing measures that restrict the right to property. Yet what is property? And when seeking to answer this question, do we tend to identify the concept with just one dominant historical form of property? In this book, David James reconstructs the theories of property developed by four key figures in classical German philosophy - Kant, Fichte, Hegel and Marx. He argues that although their theories of property are different, the concept of social recognition plays a crucial role in all of them, and assesses these philosophers' arguments for the specific forms of property they claim should exist in a society that is genuinely committed to the idea of freedom.

Property and Justice: A Liberal Theory of Natural Rights (Political Philosophy for the Real World)

by Billy Christmas

This book gives an account of a full spectrum of property rights and their relationship to individual liberty. It shows that a purely deontological approach to justice can deal with the most complex questions regarding the property system. Moreover, the author considers the economic, ecological, and technological complexities of our real-world property systems. The result is a more conceptually sound account of natural rights and the property system they demand. If we think that liberty should be at the centre of justice, what does that mean for the property system? Economists and lawyers widely agree that a property system must be composed of many different types of property: the kind of private ownership one has over one’s person and immediate possessions, as well as the kinds of common ownership we each have in our local streets, as well as many more. However, theories of property and justice have not given anything approaching an adequate account of the relationship between liberty and any other form of property other than private ownership. It is often thought that a basic commitment to liberty cannot really tell us how to arrange the major complexities of the property system, which diverge from simple private ownership. Property and Justice demonstrates how philosophical rigour coupled with interdisciplinary engagement enables us to think clearly about how to deal with real-world problems. It will be of interest to political philosophers, political theorists, and legal theorists working on property rights and justice.

Property and Justice: A Trend Towards Marxist Political Philosophy (China Perspectives)

by Zhang Wenxi

This book attempts to answer the question “what on earth is the Marxist political philosophy?” The author discusses Karl Marx's vision of the good life and the good society, focusing in particular on the interrelationship between property and justice. Strictly speaking, the subject of the present book should not be placed within an a priori conceptual framework, which essentially focuses on academic notions of the good society, good government and justice. However, for the purposes of this book, Marxist political philosophy, namely communism, is already embodied in two aspects. This book argues that it is the noisy and confusing modern political philosophy that obscures the actualisation of human nature unfolded by the Marxist political philosophy before us. The book will be essential reading for students and scholars of philosophy, Marxism, Chinese studies and Chinese Marxist philosophy.

Property and Political Order in Africa

by Catherine Boone

In sub-Saharan Africa, property relationships around land and access to natural resources vary across localities, districts and farming regions. These differences produce patterned variations in relationships between individuals, communities and the state. This book captures these patterns in an analysis of structure and variation in rural land tenure regimes. In most farming areas, state authority is deeply embedded in land regimes, drawing farmers, ethnic insiders and outsiders, lineages, villages and communities into direct and indirect relationships with political authorities at different levels of the state apparatus. The analysis shows how property institutions - institutions that define political authority and hierarchy around land - shape dynamics of great interest to scholars of politics, including the dynamics of land-related competition and conflict, territorial conflict, patron-client relations, electoral cleavage and mobilization, ethnic politics, rural rebellion, and the localization and 'nationalization' of political competition.

Property Is Theft!

by Iain Mckay Pierre-Joseph Proudhon

"An indispensable source book for anyone interested in Proudhon's ideas and the origins of the socialist and anarchist movements in nineteenth-century Europe."-Robert Graham, editor of Anarchism: A Documentary History of Libertarian Ideas"Iain McKay's introduction offers a sure-footed guide through the misconceptions surrounding Proudhon's thought."-Mark Leier, author of Bakunin: The Creative PassionMore influential than Karl Marx during his lifetime, Pierre-Joseph Proudon's work has long been out of print or unavailable in English. Iain McKay's comprehensive collection is a much-needed and timely historical corrective.Iain McKay is the editor of An Anarchist FAQ.

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