Feature logics are the logical basis for so-called unification grammars studied in computational linguistics. We investigate the expressivity of feature terms with negation and the functional uncertainty construct needed for the description of long-distance dependencies and obtain the following results: satisfiability of feature terms is undecidable, sort equations can be internalized, consistency of sort equations is decidable if there is at least one atom, and consistency of sort equations is undecidable if there is no atom.

This paper gives a simple method for providing categorial brands of feature-based unification grammars with a model-theoretic semantics. The key idea is to apply the paradigm of fibred semantics (or layered logics, see Gabbay (1990)) in order to combine the two components of a feature-based grammar logic. We demonstrate the method for the augmentation of Lambek categorial grammar with Kasper/Rounds-style feature logic. These are combined by replacing (or annotating) atomic formulas of the first logic, i.e. the (...) basic syntactic types, by formulas of the second. Modelling such a combined logic is less trivial than one might expect. The direct application of the fibred semantics method where a combined atomic formula like np (num: sg & pers: 3rd) denotes those strings which have the indicated property and the categorial operators denote the usual left- and right-residuals of these string sets, does not match the intuitive, unification-based proof theory. Unification implements a global bookkeeping with respect to a proof whereas the direct fibring method restricts its view to the atoms of the grammar logic. The solution is to interpret the (embedded) feature terms as global feature constraints while maintaining the same kind of fibred structures. For this adjusted semantics, the anticipated proof system is sound and complete. (shrink)

While monadic second-order logic (MSO) has played a prominent role in model theoretic syntax, modal logics have been used in this context since its inception. When comparing propositional dynamic logic (PDL) to MSO over trees, Kracht (1997) noted that there are tree languages that can be defined in MSO that can only be defined in PDL by adding new features whose distribution is predictable. He named such features “inessential features”. We show that Kracht’s observation can be extended to other (...) modal logics of trees in two ways. First, we demonstrate that for each stronger logic, there exists a tree language that can only be defined in a weaker logic with inessential features. Second, we show that any tree language that can be defined in a stronger logic, but not in some weaker logic, can be defined with inessential features. Additionally, we consider Kracht’s definition of inessential features more closely. It turns out that there are features whose distribution can be predicted, but who fail to be inessential in Kracht’s sense. We will look at ways to modify his definition. (shrink)

One of the biggest challenges in understanding perception is to understand how the nervous system manages to integrate the multiple codes it uses to represent features in multiple sensory modalities. From different cortical areas, which might separately register the sight of something red and the touch of something smooth, one effortlessly generates the perception of one thing that is both red and smooth. This process has been variously called "feature integration", "binding", or "synthesis". Citing some current models and some (...) historical precursors, this paper makes some simple observations about the logic of feature integration. I suggest that "feature conjunction" is not strictly speaking conjunction at all, but rather joint predication; and that the critical task in "binding" is not simply grouping scattered representations together, or providing them a common label, but rather identifying those that have a common subject matter-those that are. (shrink)

Featuring an exceptionally clear writing style and a wealth of real-world examples and exercises, Logic, Second Edition, shows how logic relates to everyday life, demonstrating its applications in such areas as the workplace, media and entertainment, politics, science and technology, student life, and elsewhere.Thoroughly revised and expanded in this second edition, the text now features 2600 exercises, more than 1000 of them new; three new chapters on legal arguments, moral arguments, and analyzing a long essay; enhanced pedagogy; and much more.FEATURES* (...) 2600 exercises--more than 1000 of them new--breathe new life into logic* Real-world examples help bring logic down to earth for students* A unique, extended explanation or model of the answer to the first question of each "Check Your Understanding" section shows students what is expected of their answers* An additional 25% of the exercises are answered at the back of the book* "Profiles in Logic" provide short sketches of logicians, philosophers, mathematicians, and others associated with logic* "Logic Challenge" problems present puzzles and paradoxes that end each chapter on a fun note* Additional pedagogical elements--marginal definitions, key terms, a glossary, reference boxes, and bulleted chapter summaries--make the material even more accessible* Detailed guides help students learn to create "truth tables" and Venn diagramsSUPPORT PACKAGEAn Instructor's Manual with Computerized Test Bank on CD includes: Solutions to all exercises in the book, enhanced by explanations and answers that elucidate the details of the correct answersKey Terms and a summary for each chapterA customizable Computerized Test Bank--with multiple-choice, true/false, and fill-in-the-blank questions--that allows instructors to give exams or homework problems that can be auto-gradedA traditional "pencil-and-paper" Test Bank and answer key containing the same questions as the Computerized Test BankPowerPoint-based lecture outlinesThe Instructor's Manual with Computerized Test Bank and the traditional Test Bank are also available in printed format.A Companion Website offers:Instructor Resources :A downloadable version of the Instructor's Manual PowerPoint-based lecture outlinesStudent Resources:Brief chapter summariesInteractive Flash Cards with key terms and definitionsWeb links and other media resourcesPractice quizzes with answers and explanations The Learning Management System Cartridges include, in a fully downloadable format:Instructor's Manual and Computerized Test Bank Student material from the companion websiteThe Perfect Text for Your Course... No Matter How You Teach ItYour course is unique; you have your own teaching philosophy and style. Whether you teach traditional Introduction to Logic, Critical Thinking/Informal Logic, or Formal Logic, Baronett: Logic, Second Edition, can be tailored to fulfill your course needs.We'll make it easy for you; choose one of our Alternate Editions or build the book you want, chapter by chapter.Option 1. Author Stan Baronett has suggested four Alternate Editions that may work for you. Each Alternate Edition comes in full color, with answers to problems, a full glossary, and an index. The books are in stock and available for ordering. Please see the ISBN information below.Logic: Concise EditionChapters 1, 3, 4, 5, 6, 7, 8Order using ISBN: 978-0-19-994129-2Logic: An Emphasis on Critical Thinking and Informal LogicChapters 1, 2, 3, 4, 10, 11, 12, 13 A-E, 14, 15Order using ISBN: 978-0-19-994128-5Logic: An Emphasis on Formal LogicChapters 1, 5, 6, 7, 8, 9Order using ISBN: 978-0-19-994126-1Logic: With Diagramming used in Chapter 4. Informal FallaciesFull textOrder using ISBN: 978-0-19-997081-0Option 2. Create your own customized textbook by choosing the specific chapters that you need for your course. Please contact your Oxford University Press Sales Representative or call 800.280.0280 for details. (shrink)

This robust, clear, and well-researched textbook for classes in logic introduces students to both formal logic and to the virtues of intellectual inquiry. Part 1 challenges students to develop the analytical skills of deductive and inductive reasoning, showing them how to identify and evaluate arguments. Part 2 helps students develop the intellectual virtues of the wise inquirer. The book includes helpful pedagogical features such as practice exercises and a concluding summary with definitions of key concepts for each chapter. Resources for (...) professors and students are available through Baker Academic's Textbook eSources. (shrink)

The major reason given in the philosophical literature for dissatisfaction with theories of vagueness based on fuzzy logic is that such theories give rise to a problem of higherorder vagueness or artificial precision. In this paper I first outline the problem and survey suggested solutions: fuzzy epistemicism; measuring truth on an ordinal scale; logic as modelling; fuzzy metalanguages; blurry sets; and fuzzy plurivaluationism. I then argue that in order to decide upon a solution, we need to understand the true nature (...) and source of the problem. Two possible sources are discussed: the problem stems from the very nature of vagueness—from the defining features of vague predicates; or the problem stems from the way in which the meanings of predicates are determined—by the usage of speakers together with facts about their environment and so on. I argue that the latter is the true source of the problem, and on this basis that fuzzy plurivaluationism is the correct solution. (shrink)

The purpose of this paper is to explore the question of how truthmaker theorists ought to think about their subject in relation to logic. Regarding logic and truthmaking, I defend the view that considerations drawn from advances in modal logic have little bearing on the legitimacy of truthmaker theory. To do so, I respond to objections Timothy Williamson has lodged against truthmaker theory. As for the logic of truthmaking, I show how the project of understanding the logical features of the (...) truthmaking relation has led to an apparent impasse. I offer a new perspective on the logic of truthmaking that both explains the problem and offers a way out. (shrink)

A concise yet rigorous introduction to logic and discrete mathematics. This book features a unique combination of comprehensive coverage of logic with a solid exposition of the most important fields of discrete mathematics, presenting material that has been tested and refined by the authors in university courses taught over more than a decade. The chapters on logic - propositional and first-order - provide a robust toolkit for logical reasoning, emphasizing the conceptual understanding of the language and the semantics of classical (...) logic as well as practical applications through the easy to understand and use deductive systems of Semantic Tableaux and Resolution. The chapters on set theory, number theory, combinatorics and graph theory combine the necessary minimum of theory with numerous examples and selected applications. Written in a clear and reader-friendly style, each section ends with an extensive set of exercises, most of them provided with complete solutions which are available in the accompanying solutions manual. Key Features: Suitable for a variety of courses for students in both Mathematics and Computer Science. Extensive, in-depth coverage of classical logic, combined with a solid exposition of a selection of the most important fields of discrete mathematics Concise, clear and uncluttered presentation with numerous examples. Covers some applications including cryptographic systems, discrete probability and network algorithms. Logic and Discrete Mathematics: A Concise Introduction is aimed mainly at undergraduate courses for students in mathematics and computer science, but the book will also be a valuable resource for graduate modules and for self-study. (shrink)

Every branch of mathematics has its subject matter, and one of the distinguishing features of logic is that so many of its fundamental objects of study are rooted in language. The subject deals with terms, expressions, formulas, theorems, and proofs. When we speak about these notions informally, we are talking about things that can be written down and communicated with symbols. One of the goals of mathematical logic is to introduce formal definitions that capture our intuitions about such objects and (...) enable us to reason about them precisely. At the most basic level, syntactic objects can be viewed as strings of symbols. For concreteness, we can identify symbols with particular set-theoretic objects, but for most purposes, it doesn't matter what they are; all that is needed is that they are distinct from one another. (shrink)

The Fourth Edition of this long-established text retains all the key features of the previous editions, covering the basic topics of a solid first course in ...

An account of the logic of bivalent languages with truth-value gaps is given. This account is keyed to the use of tables introduced by S. C. Kleene. The account has two guiding ideas. First, that the bivalence property insures that the language satisfies classical logic. Second, that the general concepts of a valid sentence and an inconsistent sentence are, respectively, as sentences which are not false in any model and sentences which are not true in any model. What recommends this (...) approach is (1) its relative simplicity, and (2) the fact that it leaves the fundamental features of classical logic intact. (shrink)

One of the central figures in the Kyoto School, Miki Kiyoshi wrote Logic of Imagination as a series of articles between 1937 and 1943. Translating this seminal work into English for the first time, with contextual notes throughout, this book features an introduction and biographical information about the author. Miki's thinking about the imagination illuminates our contemporary understanding of technology and how we behave in the world.

This comprehensive volume features the proceedings of the Third International Workshop on Logics for New-Generation Artificial Intelligence and the International Workshop on Logic, AI and Law, held in Hangzhou, China on September 8-9 and 11-12, 2023. The collection offers a diverse range of papers that explore the intersection of logic, artificial intelligence, and law. With contributions from some of the leading experts in the field, this volume provides insights into the latest research and developments in the applications of logic (...) in these areas. It is an essential resource for researchers, practitioners, and students interested in the latest advancements in logic and its applications to artificial intelligence and law. (shrink)

The paper discusses a four-valued propositional logic FOUR≤, similar to Belnap's logic, which can be used to describe incomplete or inconsistent knowledge. In addition to the two classical logical values tt, ff, FOUR≤ features also two nonclassical values: ⊥, representing incomplete information, and ⊤, representing inconsistency. The nonclassical values are incomparable, and together with the classical ones they form a diamond-shaped lattice L4 known from Belnap's logic, which underlies the semantics of FOUR≤. The set of connectives contains those of Belnap's (...) logic, together with an additional operator representing the lattice order, which decisively increases the expressive power of the logic. In the paper we develop a complete decompositional deduction system in Rasiowa-Sikorski style for FOUR≤. (shrink)

An entertaining series of logic problems and puzzles of increasing difficulty, and all relating important mathematical and logical concepts, includes mind-benders, paradoxes, metapuzzles, number exercises, and a mathematical novel.

Throughout history, a believer did not need logical proof to believe in Almighty God. This is because the spiritual proof was always enough to admit the existence of Almighty God and to submit to Him. Finding Almighty God is not a mathematical equation that needs to be proven. Rather, it is a spiritual feeling due to a call from inside a human being. The relationship between Almighty God and humans is spiritual rather logical.However, with the advances in science and technology, (...) this spiritual relationship decreases. Nowadays, modern man is looking for logic and scientific answers to many questions relating to Almighty God and religions. In fact, nonbelievers or atheists are not the only ones looking for answers to these types of questions; believers also want to increase their faith and remove doubts from their hearts. The main objective of this book is to provide logical answers to questions relating to belief in Almighty God, creations and Islam as the last heavenly religion. These questions are grouped under several topics, namely: 1) Almighty God, 2) the creation, 3) the soul, 4) the hereafter, 5) the destiny and freedom in choices, 6) the holy Quran and 7) the Islam.0The main feature of this book is that the author provides logic flowcharts with each presented question relating to the different topics. Each logic flowchart has three phases: a start, a process and an end.0Prof. Dr. Magd Abdel Wahab is a full-time Professor of Applied Mechanics in the Faculty of Engineering and Architecture at Ghent University, Belgium.Beside his interest in engineering, Prof. Wahab has a strong interest in research in Islamic religion. During the last 15 years, he has served as an Associate Imam, and has delivered Friday ceremony speeches and Islamic talks in several mosques in the UK and Belgium. This book, Logic and Islam, summarises his experiences and provides logical answers to questions related to Almighty God, creations and Islam. (shrink)

This book reports on cutting-edge research concerning social practices. Merging perspectives from various disciplines, including philosophy, biology, psychology and cognitive science, and economy, it discusses theoretical aspects of social behavior along with models to investigate them, and presenting key case studies as well. Further, it describes concepts related to habits, routines, and rituals and examines important features of human action, such as intentionality and choice, exploring the influence of specific social practices in different situations. Based on a workshop held on (...) April 2022 at the World Congress on Universal Logic (UNILOG 22), in Crete, and including additional invited chapters, the book offers fresh insights into the fields of social practice and the cognitive, computational, and philosophical tools to understand them. (shrink)

This book is the first to present a comprehensive investigation of the technical features of the metainferential logics developed in the last years, with their most relevant results and applications. It provides some new paths to define and investigate metainferential logics and offers a thorough study of the semantics and the proof-theories of this new and exciting variety of families of logics. This volume examines the hierarchies of metainferential logics and gives a general and systematic theory (...) of them, and of the truth theories based on these logics. This book puts forward the prospects for truth-theories based on the metainferential logics of the TS/ST hierarchy and argues for its promise noting that each of these logics can be safely expanded with a transparent truth predicate. It also goes onto to explore new developments in three fields related to logics – namely metainferential logics built by means of the Weak Kleene schema and combining them with logics defined through the Strong Kleene schema, proof-theoretic presentations, and those with a with a global or an absolutely global validity standard, instead of a local one. This book is of interest to scholars in formal logic. (shrink)

The paper examines the interrelationship between mathematics and logic, arguing that a central characteristic of each has an essential role within the other. The first part is a reconstruction of and elaboration on Paul Bernays’ argument, that mathematics and logic are based on different directions of abstraction from content, and that mathematics, at its core it is a study of formal structures. The notion of a study of structure is clarified by the examples of Hilbert’s work on the axiomatization of (...) geometry and Hilbert et al.’s formalist proof theory. It is further argued that the structural aspect of logic puts it under the purview of the mathematical, analogously to how the deductive nature of mathematics puts it under the purview of logic. This is then linked, in the second part, to certain aspects of Gödel’s critique of Carnap’s conventionalism, that ‘mere syntax’ cannot capture the full content of mathematics, which is revealed to be closely related to the characteristic of mathematics argued for by Bernays. Finally, this is connected with Gödel’s latter-day views about two kinds of formality, intensional and extensional, and the relationship between them. (shrink)

This book introduces the reader to relevant logic and provides the subject with a philosophical interpretation. The defining feature of relevant logic is that it forces the premises of an argument to be really used in deriving its conclusion. The logic is placed in the context of possible world semantics and situation semantics, which are then applied to provide an understanding of the various logical particles and natural language conditionals. The book ends by examining various applications of relevant logic (...) and presenting some interesting open problems. It will be of interest to a range of readers including advanced students of logic, philosophical and mathematical logicians, and computer scientists. (shrink)

Logic is essential to correct reasoning and also has important theoretical applications in philosophy, computer science, linguistics, and mathematics. This book provides an exceptionally clear introduction to classical logic, with a unique approach that emphasizes both the hows and whys of logic. Here Nicholas Smith thoroughly covers the formal tools and techniques of logic while also imparting a deeper understanding of their underlying rationales and broader philosophical significance. In addition, this is the only introduction to logic available today that presents (...) all the major forms of proof--trees, natural deduction in all its major variants, axiomatic proofs, and sequent calculus. The book also features numerous exercises, with solutions available on an accompanying website. -/- Logic is the ideal textbook for undergraduates and graduate students seeking a comprehensive and accessible introduction to the subject. -/- Provides an essential introduction to classical logic Emphasizes the how and why of logic Covers both formal and philosophical issues Presents all the major forms of proof--from trees to sequent calculus Features numerous exercises, with solutions available online The ideal textbook for undergraduates and graduate students . (shrink)

본 논문은『十門和諍論』〈佛性有無和諍門〉에 나타난 불성 유 · 무(佛性有無)에 대한 논쟁 및 이에 대한 원효(617∼686) 해석의 특성을 검토하는 것을 목적으로 한다. 기존 연구에서는 유성 · 무성론의 논법보다 원효의 화쟁 내용이나 일심과의 관련에 초점을 두었다. 하지만『十門和諍論』의 불성 논의에는 원효의 회통 부분이 남아 있지 않다. 따라서 유성론과 무성론의 논쟁에 대한 원효 해석을 통해 화쟁 방식을 추론할 수밖에 없다는 점에서, 원효가 불성 유 · 무(佛性有無) 논의를 정리하는 방식에 대한 검토가 필요하다고 보았다.BR 이러한 시각에서 본 논문에서는 불성 유 · 무에 대한 두 입장의 타당성과 이에 대한 (...) 원효 해석의 특성을 검토하고자 한다. 이를 통해 귀류법(prasaṅga) 등 전통적 논법을 사용하면서도 오류논법 등을 포섭하여 부정을 통한 긍정으로 이쟁(異諍)을 화(和)하는 방식을 분석할 것이다. 또한 법성(dharmatā)과 도리(yukti)의 차원에서 다양한 경증과 이증을 통해 해당 논의를 회통하는 방식을 검토할 것이다.BR 『十門和諍論』잔간에서는 유성론과 무성론의 양 입장을 서술한 후, 관련 오류를 지적하는 방식으로 논의가 진행되고 있다. 단, 무성론 비판에서 경증 자체가 아니라 경증에 대한 잘못된 해석이나 논리적 오류만을 드러내는 점을 고려해 보면, 유성론 비판에서도 이와 유사한 형태를 취했을 것으로 추정된다. 특히『釋華嚴敎分記圓通鈔』의 인용내용 등에 의거할 때, 산실부에서 이어졌을 화쟁은 대승적 견지에서 양론을 동등하게 포섭했을 것으로 생각된다. 다만 부정 이후, 긍정을 유도한다는 점에서『涅槃宗要』의 긍정적 종합과 달리, ‘선별적 부정 및 관점에 따른 긍정’ 방식이 활용되었을 것으로 보인다. (shrink)

_Framing effects_ concern the having of different attitudes towards logically or necessarily equivalent contents. Framing is of crucial importance for cognitive science, behavioral economics, decision theory, and the social sciences at large. We model a typical kind of framing, grounded in (i) the structural distinction between beliefs activated in working memory and beliefs left inactive in long term memory, and (ii) the topic- or subject matter-sensitivity of belief: a feature of propositional attitudes which is attracting growing research attention. We (...) introduce a class of models featuring (i) and (ii) to represent, and reason about, agents whose belief states can be subject to framing effects. We axiomatize a logic which we prove to be sound and complete with respect to the class. (shrink)

With the advent of classical logic we are continuing to observe an adherence to the laws of logic. Moreover, the system of classical logic exhibits a prominent role within analytic philosophy. Given that the laws of logic have persistently endured in actively defining classical logic and its preceding system of logic, it begs the question as to whether it actually proves to be consistent with Islam. To consider this inquiry in a broader manner; it would be an investigation into the (...) consistency between Islam and the logic which has been the predominant driving force of analytic philosophy. Despite the well documented engagement and novel contributions made in the field of logic by Arab and Islamic theologians/logicians, I think this question deserves examination not just in terms of classical logic but also from perspectives which go beyond classical logic, namely, non-classical logic. Doing so, would I believe, retain this inquiry within the purview of analytic philosophy despite the reference to non-classical logic. To be more specific, this question would be directed toward the Islamic theologian who espouses the system of classical logic in attempting to make sense of an absolute ineffable God of Islam. The inquiry would seek to determine if classical logic is consistent (amenable) in making sense of an absolute ineffable God of Islam. This would principally involve an analysis which determines whether the metaphysical assumptions of the laws of logic (more specifically the law of non-contradiction) are consistent in making sense of an absolute ineffable God of Islam. I shall argue that it is inconsistent. I shall establish my position on this matter by demonstrating why classical logic is inconsistent (not amenable) with an absolute ineffable God of Islam. Although, I am principally concerned with classical logic, my argument is as applicable to all earlier systems of logic as much as it is to classical logic. This is on the basis that both systems of logic, namely, all preceding systems and classical logic, consider the laws of logic as defining features. (shrink)

We consider two formalisations of the notion of a compositionalsemantics for a language, and find some equivalent statements in termsof substitutions. We prove a theorem stating necessary and sufficientconditions for the existence of a canonical compositional semanticsextending a given partial semantics, after discussing what features onewould want such an extension to have. The theorem involves someassumptions about semantical categories in the spirit of Husserl andTarski.

This two-volume work, first published in 1843, was John Stuart Mill's first major book. It reinvented the modern study of logic and laid the foundations for his later work in the areas of political economy, women's rights and representative government. In clear, systematic prose, Mill disentangles syllogistic logic from its origins in Aristotle and scholasticism and grounds it instead in processes of inductive reasoning. An important attempt at integrating empiricism within a more general theory of human knowledge, the work constitutes (...) essential reading for anyone seeking a full understanding of Mill's thought. Volume 1 contains Mill's introduction, which elaborates upon his definition of logic as 'not the science of Belief, but the science of Proof, or Evidence'. It also features discussions of the central components of logical reasoning - propositions and syllogisms - in relation to Mill's theories of inductive reasoning and experimental method. (shrink)

_Logic: The Basics_ is an accessible introduction to several core areas of logic. The first part of the book features a self-contained introduction to the standard topics in classical logic, such as: · mathematical preliminaries · propositional logic · quantified logic · English and standard ‘symbolic translations’ · tableau procedures. Alongside comprehensive coverage of the standard topics, this thoroughly revised second edition also introduces several philosophically important nonclassical logics, free logics, and modal logics, and gives the reader (...) an idea of how they can take their knowledge further. With its wealth of exercises, _Logic: The Basics _is a useful textbook for courses ranging from the introductory level to the early graduate level, and also as a reference for students and researchers in philosophical logic. (shrink)

Designed for use by philosophy students, this book provides an accessible, yet technically sound treatment of modal logic and its philosophical applications. Every effort has been made to simplify the presentation by using diagrams in place of more complex mathematical apparatus. These and other innovations provide philosophers with easy access to a rich variety of topics in modal logic, including a full coverage of quantified modal logic, non-rigid designators, definite descriptions, and the de-re de-dictio distinction. Discussion of philosophical issues concerning (...) the development of modal logic is woven into the text. The book uses natural deduction systems and also includes a diagram technique that extends the method of truth trees to modal logic. This feature provides a foundation for a novel method for showing completeness, one that is easy to extend to systems that include quantifiers. (shrink)

Not only Greek philosophers listened to prophets. René Descartes also long pondered over the familiar words of Ausonius: "What path shall I choose for myself in life?" And-what an amusing play on words-the idea of methodos entered philosophy with Descartes. The very path of his metaphysical thought was the last possible one within the confines of Aristotelian metaphysics, when, having posited certain principles of discourse, in the end they themselves are conclusively reached and the entire system becomes a manifest certainty. (...) Descartes provides a glimpse of the possibility of a new type of metaphysical reflection in which the correctness of the chosen principles is grounded in the completeness and noncontradictoriness of the system built on their basis but not confined to them. (shrink)

Well-Structured Mathematical Logic does for logic what Structured Programming did for computation: make large-scale work possible. From the work of George Boole onward, traditional logic was made to look like a form of symbolic algebra. In this work, the logic undergirding conventional mathematics resembles well-structured computer programs. A very important feature of the new system is that it structures the expression of mathematics in much the same way that people already do informally. In this way, the new system is (...) simultaneously machine-parsable and user-friendly, just as Structured Programming is for algorithms. Unlike traditional logic, the new system works with you, not against you, as you use it to structure--and understand--the mathematics you work with on a daily basis. The book provides a complete guide to its subject matter. It presents the major results and theorems one needs to know in order to use the new system effectively. Two chapters provide tutorials for the reader in the new way that symbols move when logical calculations are performed in the well-structured system. Numerous examples and discussions are provided to illustrate the system's many results and features. Well-Structured Mathematical Logic is accessible to anyone who has at least some knowledge of traditional logic to serve as a foundation, and is of interest to all who need a system of pliant, user-friendly mathematical logic to use in their work in mathematics and computer science. (shrink)

This chapter contains sections titled: Introduction Mixed Inferences and Mixed Compounds Alethic Pluralism as Functionalism More than One Logic? Conclusion.

In this paper we will consider the existing notions of bilateralism in the context of proof-theoretic semantics and propose, based on our understanding of bilateralism, an extension to logical multilateralism. This approach differs from what has been proposed under this name before in that we do not consider multiple speech acts as the core of such a theory but rather multiple consequence relations. We will argue that for this aim the most beneficial proof-theoretical realization is to use sequent calculi with (...) multiple sequent arrows satisfying some specific conditions, which we will lay out in this paper. We will unfold our ideas with the help of a case study in logical tetralateralism and present an extension of Almukdad and Nelson’s propositional constructive four-valued logic by unary operations of meaningfulness and nonsensicality. We will argue that in sequent calculi with multiple sequent arrows it is possible to maintain certain features that are desirable if we assume an understanding of the meaning of connectives in the spirit of proof-theoretic semantics. The use of multiple sequent arrows will be justified by the presence of congruentiality-breaking unary connectives. (shrink)

Originally published in 1973, this book shows that methods developed for the semantics of systems of formal logic can be successfully applied to problems about the semantics of natural languages; and, moreover, that such methods can take account of features of natural language which have often been thought incapable of formal treatment, such as vagueness, context dependence and metaphorical meaning. Parts 1 and 2 set out a class of formal languages and their semantics. Parts 3 and 4 show that these (...) formal languages are rich enought to be used in the precise description of natural languages. Appendices describe some of the concepts discussed in the text. (shrink)

This brief paperback is designed for symbolic/formal logic courses. It features the tree method proof system developed by Jeffrey. The new edition contains many more examples and exercises and is reorganized for greater accessibility.

According to Douglas Walton, the concept of presumption relates to both logical and dialogical components. Logically, a presumption is the conclusion of a presumptive defeasible inference. Dialogically, the function of a presumptions to shift the burden of proof to the respondent in order to move the dialogue forward when the proponent, due to an objective lack of evidence, cannot present a sufficiently persuasive proposition. Presumptive status, assigned only at the argumentation stage of dialogue, is provisional: a particular presumption stands until (...) further counter-indications are found. This paper points out some possible difficulties pertaining to Walton’s dialogical model of presumption. Firstly, once a presumption's typical practical function is made precise, Walton’s idea of presumption starts to lose its conceptual cohesion, thereby becoming applicable in the opening stage of dialogue, as well. Secondly, cohesion is lost also in practical contexts, since Walton's "seat belts" example might indicate that presumptions are not evidentially defeasible if the practical stakes are raised high enough. Most importantly, contextual evaluation may reveal that presumptions do not shift the burden of proof in any specific or genuine way. (shrink)

Logic is usually thought to concern itself only with features that sentences and arguments possess in virtue of their logical structures or forms. The logical form of a sentence or argument is determined by its syntactic or semantic structure and by the placement of certain expressions called “logical constants.”[1] Thus, for example, the sentences Every boy loves some girl. and Some boy loves every girl. are thought to differ in logical form, even though they share a common syntactic and semantic (...) structure, because they differ in the placement of the logical constants “every” and “some”. By contrast, the sentences Every girl loves some boy. and Every boy loves some girl. are thought to have the same logical form, because “girl” and “boy” are not logical constants. Thus, in order to settle questions about logical form, and ultimately about which arguments are logically valid and which sentences logically true, we must distinguish the “logical constants” of a language from its nonlogical expressions. (shrink)

This chapter contains sections titled: Introduction Logical Consequence Logical Pluralism Applications of Logic Conclusion References.