A proof is a successful demonstration that a conclusion necessarily follows by logical reasoning from axioms which are considered evident for the given context and agreed upon by the community. It is this concept that sets mathematics apart from other disciplines and distinguishes it as the prototype of a deductive science. Proofs thus are utterly relevant for research, teaching and communication in mathematics and of particular interest for the philosophy of mathematics. In computer science, moreover, proofs have proved to be a rich source for already certified algorithms. This book provides the reader with a collection of articles covering relevant current research topics circled around the concept 'proof'. It tries to give due consideration to the depth and breadth of the subject by discussing its philosophical and methodological aspects, addressing foundational issues induced by Hilbert's Programme and the benefits of the arising formal notions of proof, without neglecting reasoning in natural language proofs and applications in computer science such as program extraction.

This book presents adaptive logics as an intuitive and powerful framework for modeling defeasible reasoning. It examines various contexts in which defeasible reasoning is useful and offers a compact introduction into adaptive logics. The author first familiarizes readers with defeasible reasoning, the adaptive logics framework, combinations of adaptive logics, and a range of useful meta-theoretic properties. He then offers a systematic study of adaptive logics based on various applications. The book presents formal models for defeasible reasoning stemming from different contexts, such as default reasoning, argumentation, and normative reasoning. It highlights various meta-theoretic advantages of adaptive logics over other logics or logical frameworks that model defeasible reasoning. In this way the book substantiates the status of adaptive logics as a generic formal framework for defeasible reasoning.

This volume provides a comprehensive collection of classic and contemporary readings in the philosophy of logic. The selections include some of the most important, technically exact, yet lucid and readable expositions of central concepts and controversies in philosophical logic. Areas of coverage include classical logic, truth, propositions and meaning, quantifiers and quantificational theory, validity, inference and entailment, and modality, intensionality and propositional attitude.

Since the articles are written from a variety of perspectives, the reader is able to critically assess the background and current trends in philosophical applications of symbolic logic. The volume also examines the limitations of classical and nonstandard logics.

The book complements "Philosophy of Mathematics: An Anthology" and "A Companion to Philosophical Logic, "also edited by Dale Jacquette.

Forms of Truth and the Unity of Knowledge addresses a philosophical subject-the nature of truth and knowledge-but treats it in a way that draws on insights beyond the usual confines of modern philosophy. This ambitious collection includes contributions from established scholars in philosophy, theology, mathematics, chemistry, biology, psychology, literary criticism, history, and architecture. It represents an attempt to integrate the insights of these disciplines and to help them probe their own basic presuppositions and methods. The essays in Forms of Truth and the Unity of Knowledge are collected into five parts, the first dealing with division of knowledge into multiple disciplines in Western intellectual history; the second with the foundational disciplines of epistemology, logic, and mathematics; the third with explanation in the natural sciences; the fourth with truth and understanding in disciplines of the humanities; and the fifth with art and theology. Contributors: Vittorio Hoesle, Keith Lehrer, Robert Hanna, Laurent Lafforgue, Thomas Nowak, Francisco J. Ayala, Zygmunt Pizlo, Osborne Wiggins, Allan Gibbard, Carsten Dutt, Aviezer Tucker, Nicola Di Cosmo, Michael Lykoudis, and Celia Deane-Drummond.

This new digital edition of The Trial and Death of Socrates: Euthyphro, Apology, Crito and Phaedo presents Benjamin Jowett's classic translations, as revised by Enhanced Media Publishing. A number of new or expanded annotations are also included.

Every thoughtful person must ask, "What do I know?" The two most explosive fields, religion and politics, are notably filled with strident and conflicting claims. Close analysis in clear language reveals that no one knows what he or she is talking about. Because of the challenge of unexamined assumptions, of unclear cause-and-effect relationships, and of the rarity of reliable sources, a person who wants to be open-minded cannot avoid adopting skepticism as the least embarrassing philosophy. Some discoveries made in this book: *Reason appears to prove nothing *Intuition is probably a delusion *Facts are slippery *Religious people yearn for suicide *Why socialism cannot work *Where conservatives screwed up badly (as they admit) *The equation STAR+2R+R3=GPS explains the cultural history of the world *Shakespeare was a skeptic *Dante's curious insight into love *Passing the Magic Johnson test *Tom De Lay does not realize that relativism is as American as apple pie *Hamlet, who never existed, is more real than you or I. Here is a sample observation: "People believe in God because the Bible tells all about him, and they believe in the Bible because God wrote or inspired it. This is a classic case of the Fallacy of Circular Reasoning."

"Introductory Logic for College Students" teaches students logical principles and argument forms in an accessible and understandable way. The book effectively uses explanations, illustrations, and examples that are particularly appropriate for those who are new to the discipline.
This high-interest and student-friendly text engages readers, and challenges them to apply what they are learning through exercises and problems that are both interesting and academically rigorous. Dialogue boxes throughout the book address typical challenges students face when studying certain principles of logic, and explain how to use these skills in argument analysis and construction.
Clear and concise, "Introductory Logic for College Students" doesn't weigh the reader down with extraneous material not typically covered in a one-semester introductory course. Written as a stand-alone text, this book is ideal for courses in logic and critical thinking courses that focus on formal arguments.
Corinne Painter earned her Ph.D. in philosophy at Loyola University in Chicago, Illinois. Dr. Painter is a member of the professional faculty of the philosophy department at Washtenaw Community College in Ann Arbor, where her course offerings include logic, critical thinking, ethics, social-political philosophy, and philosophy of religion. She is the co-editor of and a contributor to "Phenomenology and the Non-Human Animal: At the Limits of Experience," and has published numerous articles on animal ethics, ancient philosophy, and Continental philosophy.

Is critical argumentation an effective way to overcome disagreement? And does the exchange of arguments bring opponents in a controversy closer to the truth? This study provides a new perspective on these pivotal questions. By means of multi-agent simulations, it investigates the truth and consensus-conduciveness of controversial debates. The book brings together research in formal epistemology and argumentation theory. Aside from its consequences for discursive practice, the work may have important implications for philosophy of science and the way we construe scientific rationality as well.

This monograph examines truth in fiction by applying the techniques of a naturalized logic of human cognitive practices. The author structures his project around two focal questions. What would it take to write a book about truth in literary discourse with reasonable promise of getting it right? What would it take to write a book about truth in fiction as true to the facts of lived literary experience as objectivity allows? It is argued that the most semantically distinctive feature of the sentences of fiction is that they areunambiguously true and false together. It is true that Sherlock Holmes lived at 221B Baker Street and also concurrently false that he did. A second distinctive feature of fiction is that the reader at large knows of this inconsistency and isn't in the least cognitively molested by it. Why, it is asked, would this be so? What would explain it? Two answers are developed. According to the no-contradiction thesis, the semantically tangled sentences of fiction are indeed logically inconsistent but not logically contradictory. According to the no-bother thesis, if the inconsistencies of fiction were contradictory, a properly contrived logic for the rational management of inconsistency would explain why readers at large are not thrown off cognitive stride by their embrace of those contradictions. As developed here, the account of fiction suggests the presence of an underlying three - or four-valued dialethic logic. The author shows this to be a mistaken impression. There are only two truth-values in his logic of fiction. The naturalized logic of Truth in Fiction jettisons some of the standard assumptions and analytical tools of contemporary philosophy, chiefly because the neurotypical linguistic and cognitive behaviour of humanity at large is at variance with them. Using the resources of a causal response epistemology in tandem with the naturalized logic, the theory produced here is data-driven, empirically sensitive, and open to a circumspect collaboration with the empirical sciences of language and cognition.

How science can convey a profound sense of wonder, connectedness, and optimism about the human condition.
This book makes a compelling case that now more than ever the public at large needs to appreciate the critical-thinking tools that science has to offer and be educated in basic science literacy. The author emphasizes that the methods and facts of science are accessible to everyone, and that, contrary to popular belief, understanding science does not require extraordinary intelligence. He also notes that scientific rationality and critical thinking are not only good for our physical well-being but also are fully in sync with our highest moral codes. He illustrates the many ways in which the scientific worldview offers a profound sense of wonder, connectedness, and optimism about the human condition, an inspiring perspective that satisfies age-old spiritual aspirations.
At a time of daunting environmental challenges and rampant misinformation, this book provides a welcome corrective and reason to hope for the future.

One hundred years ago, Russell and Whitehead published their epoch-making Principia Mathematica (PM), which was initially conceived as the second volume of Russell's Principles of Mathematics (PoM) that had appeared ten years before. No other works can be credited to have had such an impact on the development of logic and on philosophy in the twentieth century. However, until now, scholars only focused on the first parts of the books - that is, on Russell's and Whitehead's theory of logic, set-theory and arithmetic.
Sebastien Gandon aims at reversing the perspective. His goal is to give a picture of Russell's logicism based on a detailed reading of the developments dealing with advanced mathematics - namely projective geometry and the theory of quantity. This book is not only the first study ever made of the 'later' portions of PoM and PM, it also shows how this shift of perspective compels us to change our view of the logicist program taken as a whole.

In Hinge Epistemology, eminent epistemologists investigate Wittgenstein's concept of basic certainty or 'hinge certainty'. The volume begins by examining the salient features of 'hinges': Are they propositions that enjoy a special kind of non-evidential justification? Are they objects of knowledge or ways of acting mistaken for known propositions? Various attempts are then made to integrate hinges in the development of a viable epistemology: Can they shed light on the conditions of satisfaction for knowledge and justification? Do they offer a solution to scepticism? Finally, the application of hinges is explored in such areas as common knowledge and intellectual loyalty. The volume attests to the importance of hinge certainty and Wittgenstein's On Certainty for mainstream epistemology.

Frege's puzzle concerning belief reports has been in the middle of the discussion on semantics and pragmatics of attitude reports: The intuition behind the opacity does not seem to be consistent with the thesis of semantic innocence according to which the semantic value of proper names is nothing but their referent. Main tasks of this book include providing truth-conditional content of belief reports. Especially, the focus is on semantic values of proper names. The key aim is to extend Crimmins's basic idea of semantic pretense and the introduction of pleonastic entities proposed by Schiffer. They enable us to capture Frege's puzzle in the analysis without giving up semantic innocence. To reach this conclusion, two issues are established. First, based on linguistic evidence, the frame of belief reports functions adverbially rather than relationally. Second, the belief ascriptions, on which each belief report is made, must be analyzed in terms of the measurement-theoretic analogy.

Truth Through Proof defends an anti-platonist philosophy of mathematics derived from game formalism. Classic formalists claimed implausibly that mathematical utterances are truth-valueless moves in a game. Alan Weir aims to develop a more satisfactory successor to game formalism utilising a widely accepted, broadly neo-Fregean framework, in which the proposition expressed by an utterance is a function of both sense and background circumstance. This framework allows for sentences whose truth-conditions are not representational, which are made true or false by conditions residing in the circumstances of utterances but not transparently in the sense.
Applications to projectivism and fiction pave the way for the claim that mathematical utterances are made true or false by the existence of concrete proofs or refutations, though these truth-making conditions form no part of their sense or informational content.
The position is compared with rivals, an account of the applicability of mathematics developed, and a new account of the nature of idealisation proffered in which it is argued that the finitistic limitations Godel placed on proofs are without rational justification. Finally a non-classical logical system is provided in which excluded middle fails, yet enough logical power remains to recapture the results of standard mathematics.

This book presents the current state of the art regarding the application of logical tools to the problems of theory and practice of lawmaking. It shows how contemporary logic may be useful in the analysis of legislation, legislative drafting and legal reasoning concerning different contexts of law making. Elaborations of the process of law making have variously emphasised its political, social or economic aspects. Yet despite strong interest in logical analyses of law, questions remains about the role of logical tools in law making. This volume attempts to bridge that gap, or at least to narrow it, drawing together some important research problems-and some possible solutions-as seen through the work of leading contemporary academics. The volume encompasses 20 chapters written by authors from 16 countries and it presents diversified views on the understanding of logic (from strict mathematical approaches to the informal, argumentative ones) and differentiated choices concerning the aspects of law making taken into account. The book presents a broad set of perspectives, insights and results into the emerging field of research devoted to the logical analysis of the area of creation of law. How does logic inform lawmaking? Are legal systems consistent and complete? How can legal rules be represented by means of formal calculi and visualization techniques? Does the structure of statutes or of legal systems resemble the structure of deductive systems? What are the logical relations between the basic concepts of jurisprudence that constitute the system of law? How are theories of legal interpretation relevant to the process of legislation? How might the statutory text be analysed by means of contemporary computer programs? These and other questions, ranging from the theoretical to the immediately practical, are addressed in this definitive collection.

In a fragment entitled Elementa Nova Matheseos Universalis (1683?) Leibniz writes "the mathesis [...] shall deliver the method through which things that are conceivable can be exactly determined"; in another fragment he takes the mathesis to be "the science of all things that are conceivable." Leibniz considers all mathematical disciplines as branches of the mathesis and conceives the mathesis as a general science of forms applicable not only to magnitudes but to every object that exists in our imagination, i.e. that is possible at least in principle. As a general science of forms the mathesis investigates possible relations between "arbitrary objects" ("objets quelconques"). It is an abstract theory of combinations and relations among objects whatsoever. In 1810 the mathematician and philosopher Bernard Bolzano published a booklet entitled Contributions to a Better-Grounded Presentation of Mathematics. There is, according to him, a certain objective connection among the truths that are germane to a certain homogeneous field of objects: some truths are the "reasons" ("Grunde") of others, and the latter are "consequences" ("Folgen") of the former. The reason-consequence relation seems to be the counterpart of causality at the level of a relation between true propositions. Arigorous proof is characterized in this context as a proof that shows the reason of the proposition that is to be proven. Requirements imposed on rigorous proofs seem to anticipate normalization results in current proof theory. The contributors of Mathesis Universalis, Computability and Proof, leading experts in the fields of computer science, mathematics, logic and philosophy, show the evolution of these and related ideas exploring topics in proof theory, computability theory, intuitionistic logic, constructivism and reverse mathematics, delving deeply into a contextual examination of the relationship between mathematical rigor and demands for simplification.

This book contains 20 essays tracing the work of David Zarefsky, a leading North American scholar of argumentation from a rhetorical perspective.The essays cohere around 4 general themes: objectives for studying argumentation rhetorically, approaches to rhetorical study of argumentation, patterns and schemes of rhetorical argumentation, and case studies illustrating the potential of studying argumentation rhetorically.These articles are drawn from across Zarefsky's 45-year career. Many of these articles originally appeared in publications that are difficult to access today, and this collection brings the reader up to date on the topic.

Zarefsky's scholarship focuses on the role of language in political argumentation, the ways in which argumentation creates public knowledge and belief, the influence of framing and context on what is said and understood, the deployment of particular patterns and schemes of argumentation in public reasoning, and the influence of debate on politics and governance. All these topics are addressed in this book.

Each of the conceptual essays includes brief application to specific cases, and five extended case studies are also presented in this volume. The case studies cover different themes: two explore famous political debates, the third focuses on presidential rhetoric across the course of United States history, the fourth on the arguments for liberalism at a time of political polarization, and the fifth on the contemporary effort to engage the United States with the Muslim world.

This bookis ofinterest to scholars in the fields of philosophy, logic, law, philosophy of law, and legal history. The range of topics and concepts addressed, the interplay of concepts and cases and the unifying perspective of rhetorical argumentation make this book a valuable read for students of argumentative practice, whether rhetorically or otherwise."

This book provides a critical examination of how the choice of what to believe is represented in the standard model of belief change. In particular the use of possible worlds and infinite remainders as objects of choice is critically examined. Descriptors are introduced as a versatile tool for expressing the success conditions of belief change, addressing both local and global descriptor revision. The book presents dynamic descriptors such as Ramsey descriptors that convey how an agent's beliefs tend to be changed in response to different inputs. It also explores sentential revision and demonstrates how local and global operations of revision by a sentence can be derived as a special case of descriptor revision. Lastly, the book examines revocation, a generalization of contraction in which a specified sentence is removed in a process that may possibly also involve the addition of some new information to the belief set.

A Chance for Possibility defends the view that the objective modal realm is tripartite: truths about possible worlds supervene on modal truths, which in turn supervene on truths about objective chances. An understanding of supervenience in terms of grounding is developed which - unlike the standard modal characterization - allows the question of what modal truths supervene on to have a non-trivial answer. Relying on this understanding, a negative result is established: modal truths do not supervene on truths about possible worlds, whether possible worlds are conceived of as Lewisian concreta or as abstract objects of some kind. Instead, a conception of pleonastic possible worlds is developed that reverses the direction of supervenience. On the basis of linguistic considerations concerning our use of natural language 'might' and 'might have' sentences, Steinberg finally argues that truths about objective chances are able to provide a supervenience base for modal truths. A Chance for Possibility is an investigation in analytic metaphysics, drawing on related work in the philosophy of logic and language as well as linguistics. It provides a detailed case study for the fruitful use of a notion of grounding in the clarification and evaluation of longstanding philosophical issues.

This skit of Bertrand Russell's philosophy was originally published in 1918 by Russell's correspondent friend Jourdain. The introduction explains that the contents purport to be lost papers written by Mr. B*rtr*nd R*ss*ll, a contemporary of Bertrand Russell. This politically humorous volume from the early 20th Century parodies the writing style of Russell as well as his theories.