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Symbolic Logic and Other Forms of Deductive Reasoning

This text does not presuppose any technical background in math or logic. The first seven chapters cover all the basic components of a first course in symbolic logic, including truth tables, rules for devising formal proofs of validity, multiple quantifiers, properties of relations, enthymemes, and identity. (One exception is that truth trees are not discussed.) The five operator symbols used are: (.) and, (v) or, ( ) not, and also if-then, represented by the sideways U and material equivalence represented by the triple line. There are also four chapters which can be studied without symbolic logic background. Chapter 8 is a study of 7 immediate inferences in Aristotelian logic using A, E, I, O type statements with a detailed proof concerning what existential assumptions are involved. Chapter 9 is a study of classic Boolean syllogism using Venn diagrams to show the validity or invalidity of syllogisms. Chapter 10 is a study of the type of probability problems that are deductive (example: having 2 aces in 5 cards drawn from a randomized deck of cards). Chapter 11 is a study of the types of problems that are often found on standardized tests where certain data are given, and then multiple-choice questions are given where the single correct answer is determined by the data. In the symbolic logic chapters, it is shown many times how putting English statements into symbolic notation reveals the complexity (and sometimes ambiguity) of natural language. Many examples are given of the usage of logic in everyday life, with statements to translate taken from musicals, legal documents, federal tax instructions, etc. Several sections involve arguments given in English, which must be translated into symbolic notation before proof of validity is given. Chapter 7 ends with a careful presentation of Richard's Paradox, challenging those who dismiss the problem because it is not strictly mathematical. The conclusion of this chapter is the most controversial part of the text. Richard's paradox is used to construct a valid symbolic logic proof that Cantor's procedure does not prove there are nondenumerable sets, with a challenge to the reader to identify and prove which premise of the argument is false. There are several uncommon features of the text. For example, there is a section where it is shown how the rules of logic are used in solving Sudoku puzzles. Another section challenges students to devise arguments (premises and conclusion) that can be solved in a certain number of steps (say 3) only by using a certain 3 rules, one time each (for example, Modus Ponens, Simplification, and Conjunction). In proofs of invalidity, if there are 10 simple statements (for example), there are 1024 possible combinations of truth values that the 10 statements can have. But the premises and conclusions are set up so that only 1 of these combinations will make all the premises true and the conclusion false - and this 1 way can be found by forced truth-value assignments, with no need to take options. Another unusual section of the text defines the five operator symbols as relations (for example, Cxy = x conjuncted with y is true), and then statements about the operators are given to determine whether the statements are true or false. To aid in deciding what sections to cover in a given course or time frame, certain sections are labeled "optional" as an indication that understanding these sections is not presupposed by later sections in the text. Although there are a ton of problems with answers in the text, any teacher using this text for a course can receive free of charge an answer book giving answers to all the problems not answered in the text, plus a few cases of additional problems not given in the text, also with answers. Send your request to [email protected], and you will be sent an answer key using your address at the school where you teach.

This text does not presuppose any technical background in math or logic.

Errors of Reasoning. Naturalizing the Logic of Inference

Errors of Reasoning is the long-awaited continuation of the author's investigation of the logic of cognitive systems. The present focus is the individual human reasoner operating under the conditions and pressures of real life with capacities and resources the natural world makes available to him. The ensuing logic is thus agent-centred, goal-directed, and time-and-action oriented. It is also as psychologically real a logic as consistent with lawlike regularities of the better-developed empirical sciences of cognition. A point of departure for the book is that good reasoning is typically reasoning that does not meet the orthodox logician's requirements of either deductive validity or the sort of inductive strength sought for by the statistico-empirical sciences. A central objective here is to fashion a logic for this "third-way" reasoning. In so doing, substantial refinements are proposed for mainline treatments of nonmonotonic, defeasible, autoepistemic and default reasoning. A further departure from orthodox orientations is the eschewal of all idealizations short of those required for the descriptive adequacy of the relevant parts of empirical science. Also banned is any unearned assumption of a logic's normative authority to judge inferential behaviour as it actually occurs on the ground. The logic that emerges is therefore a naturalized logic, a proposed transformation of orthodox logics in the manner of the naturalization, more than forty years ago, of the traditional approaches to analytic epistemology. A byproduct of the transformation is the abandonment of justification as a general condition of knowledge, especially in third-way contexts. A test case for this new approach is an account of erroneous reasoning, including inferences usually judged fallacious, that outperforms its rivals in theoretical depth and empirical sensitivity. Errors of Reasoning is required reading in all research communities that seek a realistic understanding of human inference: Logic, formal and informal, AI and the other branches of cognitive science, argumentation theory, and theories of legal reasoning. Indeed the book is a standing challenge to all normatively idealized theories of assessable human performance. John Woods is Director of The Abductive Systems Group at the University of British Columbia, and was formerly the Charles S. Peirce Professor of Logic in the Group on Logic and Computation in the Department of Computer Science, King's College London. He is author of Paradox and Paraconsistency (2003) and with Dov Gabbay, of Agenda Relevance (2003) and The Reach of Abduction (2005). His pathbreaking The Logic of Fiction appeared in 1974, with a second edition by College Publications, 2009.

A central objective here is to fashion a logic for this "third-way" reasoning. In so doing, substantial refinements are proposed for mainline treatments of nonmonotonic, defeasible, autoepistemic and default reasoning.

The Logic of Law Making in Islam

Women and Prayer in the Legal Tradition

This pioneering study examines the process of reasoning in Islamic law. Some of the key questions addressed here include whether sacred law operates differently from secular law, why laws change or stay the same and how different cultural and historical settings impact the development of legal rulings. In order to explore these questions, the author examines the decisions of thirty jurists from the largest legal tradition in Islam: the Hanafi school of law. He traces their rulings on the question of women and communal prayer across a very broad period of time - from the eighth to the eighteenth century - to demonstrate how jurists interpreted the law and reconciled their decisions with the scripture and the sayings of the Prophet. The result is a fascinating overview of how Islamic law has evolved and the thinking behind individual rulings.

This pioneering study examines the process of reasoning in Islamic law.

Deontic Logic in Computer Science

Normative System Specification

Deontic logic originated from moral philosophy and the philosophy of law as an attempt to formalise normative and legal reasoning. This book draws on the experience of researchers - working in fields as diverse as Artificial Intelligence, linguistics, computer system specification and law - who have discovered the benefits of deontic logic when applied to solving computer science and AI problems. A useful logic in which to specify normative system behaviour, deontic logic has a broad spectrum of possible applications within the field: from legal expert systems to natural language processing, database integrity to electronic contracting and the specification of fault-tolerant software. This book provides a unique and timely assessment of the practical value of deontic logic for computer scientists in AI and law and, more particularly, in such areas as distributed AI and intelligent cooperative information systems.

In The First International Conference on Artificial Intelligence and Law , pages 240-251 . ACM , May 1987 . R.M. Chisholm . ... In C. Walter , editor , Computer Power and Legal reasoning , pages 467-493 . West Publishing Co. , 1985 .

The Social Logic of Taqlid and the Rise of Mukhatasar

The study of the relationship between ijtihamp;ād and Taqlamp;īd has been dominated by an approach that privileges ijtihamp;ād over Taqlamp;īd on the assumption that the former is an intellectually superior mode of legal reasoning. By analyzing the role of Taqlamp;īd in regulating the actions of muftamp;īs and judges as discussed by post-6th/12th century jurists of the Mamp;ālikamp;ī school, I conclude that Taqlamp;īd resulted from the desire to have uniform rules rather than as a result of intellectual stagnation. While ijtihamp;ād was individualistic and solipsistic, Taqlamp;īd was the result of group interpretation that provided an objective basis upon which legal decisions and legal rulings could be described as being either substantively correct or incorrect. Viewed in this light, Taqlamp;īd was originally a desire to limit the discretionary power of legal officials, especially those at the bottom of the legal hierarchy. The desire to possess uniform rules found its logical outcome in the legal genre of the mukhtaamp;ṣar as it emerged in the 7th/13th century. The mukhtaamp;ṣar functioned as the authoritative collection of a legal school's doctrine, and, for that reason, I argue that Islamic law in the age of mukhtaamp;ṣars is best understood as a codified Common Law.

The mukhtaamp;ṣar functioned as the authoritative collection of a legal school's doctrine, and, for that reason, I argue that Islamic law in the age of mukhtaamp;ṣars is best understood as a codified Common Law.

A Practical Logic of Cognitive Systems

The Reach of Abduction: Insight and Trial

The present work is a continuation of the authors' acclaimed multi-volume A Practical Logic of Cognitive Systems. After having investigated the notion of relevance in their previous volume, Gabbay and Woods now turn to abduction. In this highly original approach, abduction is construed as ignorance-preserving inference, in which conjecture plays a pivotal role. Abduction is a response to a cognitive target that cannot be hit on the basis of what the agent currently knows. The abducer selects a hypothesis which were it true would enable the reasoner to attain his target. He concludes from this fact that the hypothesis may be conjectured. In allowing conjecture to stand in for the knowledge he fails to have, the abducer reveals himself to be a satisficer, since an abductive solution is not a solution from knowledge. Key to the authors' analysis is the requirement that a conjectured proposition is not just what a reasoner might allow himself to assume, but a proposition he must defeasibly release as a premiss for further inferences in the domain of enquiry in which the original abduction problem has arisen. The coverage of the book is extensive, from the philosophy of science to computer science and AI, from diagnostics to the law, from historical explanation to linguistic interpretation. One of the volume's strongest contributions is its exploration of the abductive character of criminal trials, with special attention given to the standard of proof beyond a reasonable doubt. Underlying their analysis of abductive reasoning is the authors' conception of practical agency. In this approach, practical agency is dominantly a matter of the comparative modesty of an agent's cognitive agendas, together with comparatively scant resources available for their advancement. Seen in these ways, abduction has a significantly practical character, precisely because it is a form of inference that satisfices rather than maximizes its response to the agent's cognitive target. The Reach of Abduction will be necessary reading for researchers, graduate students and senior undergraduates in logic, computer science, AI, belief dynamics, argumentation theory, cognitive psychology and neuroscience, linguistics, forensic science, legal reasoning and related areas. Key features: - Reach of Abduction is fully integrated with a background logic of cognitive systems. - The most extensive coverage compared to competitive works. - Demonstrates not only that abduction is a form of ignorance preserving inference but that it is a mode of inference that is wholly rational. - Demonstrates the satisficing rather than maximizing character of abduction. - The development of formal models of abduction is considerably more extensive than one finds in existing literature. It is an especially impressive amalgam of sophisticated conceptual analysis and extensive logical modelling. · Reach of Abduction is fully integrated with a background logic of cognitive systems. · The most extensive coverage compared to competitive works · Demonstrates not only that abduction is a form of ignorance preserving inference but that it is a mode of inference that is wholly rational. · Demonstrates the satisficing rather than maximizing character of abduction. · The development of formal models of abduction is considerably more extensive than one finds in existing literature. It is an especially impressive amalgam of sophisticated conceptual analysis and extensive logical modelling.

The present work is a continuation of the authors' acclaimed multi-volume A Practical Logic of Cognitive Systems. After having investigated the notion of relevance in their previous volume, Gabbay and Woods now turn to abduction.

Legal Evidence and Proof

Statistics, Stories, Logic

As a result of recent scandals concerning evidence and proof in the administration of criminal justice - ranging from innocent people on death row in the United States to misuse of statistics leading to wrongful convictions in The Netherlands and elsewhere - inquiries into the logic of evidence and proof have taken on a new urgency both in an academic and practical sense. This study presents a broad perspective on logic by focusing on inference not just in isolation but as embedded in contexts of procedure and investigation. With special attention being paid to recent developments in Artificial Intelligence and the Law, specifically related to evidentiary reasoning, this book provides clarification of problems of logic and argumentation in relation to evidence and proof. As the vast majority of legal conflicts relate to contested facts, rather than contested law, this volume concerning facts as prime determinants of legal decisions presents an important contribution to the field for both scholars and practitioners.

As the vast majority of legal conflicts relate to contested facts, rather than contested law, this volume concerning facts as prime determinants of legal decisions presents an important contribution to the field for both scholars and ...

Truth in Fiction

Rethinking its Logic

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.

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 ...

The Logic of Law Making in Islam South Asia Edition

Women and Prayer in the Legal Tradition

This pioneering study examines the process of reasoning in Islamic law. Some of the key questions addressed here include whether sacred law operates differently from secular law, why laws change or stay the same, and how different cultural and historical settings impact the development of legal rulings. In order to explore these questions, the author examines the decisions of thirty jurists from the largest legal tradition in Islam: the Hanafi school of law. He traces their rulings on the question of women and communal prayer across a very broad period of time from the eighth to the eighteenth century to demonstrate how jurists interpreted the law and reconciled their decisions with the scripture and the sayings of the Prophet. The result is a fascinating overview of how Islamic law has evolved and the thinking behind individual rulings."

This pioneering study examines the process of reasoning in Islamic law.

Logic Programs, Norms and Action

Essays in Honor of Marek J. Sergot on the Occasion of His 60th Birthday

This book is dedicated to Marek Sergot, Professor in Computational Logic at Imperial College London, on the occasion of his 60th birthday. Professor Sergot’s scientific contributions range over many different fields. He has developed a series of novel ideas and formal methods bridging areas including artificial intelligence, computational logic, philosophical logic, legal theory, artificial intelligence and law, multi-agent systems and bioinformatics. By combining his background in logic and computing with his interest in the law, deontic logic, action, and related areas, and applying to all his capacity to understand the subtleties of social interaction and normative reasoning, Professor Sergot has opened up new directions of research, and has been a reference, an inspiration, and a model for many researchers in the fields to which he has contributed. The Festschrift includes several reminiscences and introductory essays describing Professor Sergot's achievements, followed by a series of articles on logic programming, temporal reasoning and action languages, artificial intelligence and law, deontic logic and norm-governed systems, and logical approaches to policies.

This book is dedicated to Marek Sergot, Professor in Computational Logic at Imperial College London, on the occasion of his 60th birthday. Professor Sergot’s scientific contributions range over many different fields.