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Author: J. Michael Dunn Publisher: ISBN: 9781848903180 Category : Mathematics Languages : en Pages : 144
Book Description
J. Michael Dunn's PhD dissertation occupies a unique place in the development of the algebraic approach to logic. In The Algebra of Intensional Logics, Dunn introduced De Morgan monoids, a class of algebras in which the algebra of R (the logic of relevant implication) is free. This is an example where a logic's algebra is neither a Boolean algebra with further operations, nor a residuated distributive lattice. De Morgan monoids served as a paradigm example for the algebraization of other relevance logics, including E, the logic of entailment and R-Mingle (RM), the extension of R with the mingle axiom. De Morgan monoids extend De Morgan lattices, which algebraize the logic of first-degree entailments that is a common fragment of R and E. Dunn studied the role of the four-element De Morgan algebra D in the representation of De Morgan lattices, and from this he derived a completeness theorem for first-degree entailments. He also showed that every De Morgan lattice can be embedded into a 2-product of Boolean algebras, and proved related results about De Morgan lattices in which negation has no fixed point. Dunn also developed an informal interpretation for first-degree entailments utilizing the notion of aboutness, which was motivated by the representation of De Morgan lattices by sets. Dunn made preeminent contributions to several areas of relevance logic in his career spanning more than half a century. In proof theory, he developed sequent calculuses for positive relevance logics and a tableaux system for first-degree entailments; in semantics, he developed a binary relational semantics for the logic RM. The use of algebras remained a central theme in Dunn's work from the proof of the admissibility of the rule called γ to his theory of generalized Galois logics (or ``gaggles''), in which the residuals of arbitrary operations are considered. The representation of gaggles---utilizing relational structures---gave a new framework for relational semantics for relevance and for so-called substructural logics, and led to an information-based interpretation of them.
Author: J. Michael Dunn Publisher: ISBN: 9781848903180 Category : Mathematics Languages : en Pages : 144
Book Description
J. Michael Dunn's PhD dissertation occupies a unique place in the development of the algebraic approach to logic. In The Algebra of Intensional Logics, Dunn introduced De Morgan monoids, a class of algebras in which the algebra of R (the logic of relevant implication) is free. This is an example where a logic's algebra is neither a Boolean algebra with further operations, nor a residuated distributive lattice. De Morgan monoids served as a paradigm example for the algebraization of other relevance logics, including E, the logic of entailment and R-Mingle (RM), the extension of R with the mingle axiom. De Morgan monoids extend De Morgan lattices, which algebraize the logic of first-degree entailments that is a common fragment of R and E. Dunn studied the role of the four-element De Morgan algebra D in the representation of De Morgan lattices, and from this he derived a completeness theorem for first-degree entailments. He also showed that every De Morgan lattice can be embedded into a 2-product of Boolean algebras, and proved related results about De Morgan lattices in which negation has no fixed point. Dunn also developed an informal interpretation for first-degree entailments utilizing the notion of aboutness, which was motivated by the representation of De Morgan lattices by sets. Dunn made preeminent contributions to several areas of relevance logic in his career spanning more than half a century. In proof theory, he developed sequent calculuses for positive relevance logics and a tableaux system for first-degree entailments; in semantics, he developed a binary relational semantics for the logic RM. The use of algebras remained a central theme in Dunn's work from the proof of the admissibility of the rule called γ to his theory of generalized Galois logics (or ``gaggles''), in which the residuals of arbitrary operations are considered. The representation of gaggles---utilizing relational structures---gave a new framework for relational semantics for relevance and for so-called substructural logics, and led to an information-based interpretation of them.
Author: Daniel Gallin Publisher: Elsevier ISBN: 148327473X Category : Mathematics Languages : en Pages : 159
Book Description
North-Holland Mathematics Studies, 19: Intensional and Higher-Order Modal Logic: With Applications to Montague Semantics focuses on an approach to the problem of providing a precise account of natural language syntax and semantics, including the set-theoretic semantical methods, Boolean models, and two-sorted type theory. The book first offers information on intensional logic and alternative formulations of intensional logic. Topics include two-sorted type theory, normal forms, extensions and intensional logic, modal T-logic, persistence in intensional logic, generalized completeness of intensional logic, and natural language and intensional logic. The text then examines higher-order modal logic and algebraic semantics. Discussions focus on Cohen's independence results, topological models of MLp, modal independence results, Boolean models of MLp, relative strength of intensional logic and MLp, propositional operators, modal predicate logic, and propositions in MLp. The monograph is a valuable reference for mathematicians and researchers interested in intensional and higher-order modal logic.
Author: Maarten de Rijke Publisher: Springer Science & Business Media ISBN: 9401582424 Category : Philosophy Languages : en Pages : 390
Book Description
This volume contains a selection of papers presented at a Seminar on Intensional Logic held at the University of Amsterdam during the period September 1990-May 1991. Modal logic, either as a topic or as a tool, is common to most of the papers in this volume. A number of the papers are con cerned with what may be called well-known or traditional modal systems, but, as a quick glance through this volume will reveal, this by no means implies that they walk the beaten tracks. In deed, such contributions display new directions, new results, and new techniques to obtain familiar results. Other papers in this volume are representative examples of a current trend in modal logic: the study of extensions or adaptations of the standard sys tems that have been introduced to overcome various shortcomings of the latter, especially their limited expressive power. Finally, there is another major theme that can be discerned in the vol ume, a theme that may be described by the slogan 'representing changing information. ' Papers falling under this heading address long-standing issues in the area, or present a systematic approach, while a critical survey and a report contributing new techniques are also included. The bulk of the papers on pure modal logic deal with theoreti calor even foundational aspects of modal systems.
Author: Zoran Majkic Publisher: Walter de Gruyter GmbH & Co KG ISBN: 3110981432 Category : Computers Languages : en Pages : 542
Book Description
This book introduces the properties of conservative extensions of First Order Logic (FOL) to new Intensional First Order Logic (IFOL). This extension allows for intensional semantics to be used for concepts, thus affording new and more intelligent IT systems. Insofar as it is conservative, it preserves software applications and constitutes a fundamental advance relative to the current RDB databases, Big Data with NewSQL, Constraint databases, P2P systems and Semantic Web applications. Moreover, the many-valued version of IFOL can support the AI applications based on many-valued logics.
Author: J. Michael Dunn Publisher: OUP Oxford ISBN: 0191589225 Category : Languages : en Pages : 490
Book Description
This comprehensive text demonstrates how various notions of logic can be viewed as notions of universal algebra. It is aimed primarily for logisticians in mathematics, philosophy, computer science and linguistics with an interest in algebraic logic, but is also accessible to those from a non-logistics background. It is suitable for researchers, graduates and advanced undergraduates who have an introductory knowledge of algebraic logic providing more advanced concepts, as well as more theoretical aspects. The main theme is that standard algebraic results (representations) translate into standard logical results (completeness). Other themes involve identification of a class of algebras appropriate for classical and non-classical logic studies, including: gaggles, distributoids, partial- gaggles, and tonoids. An imporatant sub title is that logic is fundamentally information based, with its main elements being propositions, that can be understood as sets of information states. Logics are considered in various senses e.g. systems of theorems, consequence relations and, symmetric consequence relations.
Author: S. Shapiro Publisher: Elsevier ISBN: 0080880045 Category : Science Languages : en Pages : 237
Book Description
``Platonism and intuitionism are rival philosophies of Mathematics, the former holding that the subject matter of mathematics consists of abstract objects whose existence is independent of the mathematician, the latter that the subject matter consists of mental construction... both views are implicitly opposed to materialistic accounts of mathematics which take the subject matter of mathematics to consist (in a direct way) of material objects...'' FROM THE INTRODUCTIONAmong the aims of this book are: - The discussion of some important philosophical issues using the precision of mathematics. - The development of formal systems that contain both classical and constructive components. This allows the study of constructivity in otherwise classical contexts and represents the formalization of important intensional aspects of mathematical practice. - The direct formalization of intensional concepts (such as computability) in a mixed constructive/classical context.
Author: Dov M. Gabbay Publisher: Springer Science & Business Media ISBN: 9401593094 Category : Philosophy Languages : en Pages : 337
Book Description
The notion of negation is one of the central logical notions. It has been studied since antiquity and has been subjected to thorough investigations in the development of philosophical logic, linguistics, artificial intelligence and logic programming. The properties of negation-in combination with those of other logical operations and structural features of the deducibility relation-serve as gateways among logical systems. Therefore negation plays an important role in selecting logical systems for particular applications. At the moment negation is a 'hot topic', and there is an urgent need for a comprehensive account of this logical key concept. We therefore have asked leading scholars in various branches of logic to contribute to a volume on "What is Negation?". The result is the present neatly focused collection of re search papers bringing together different approaches toward a general characteri zation of kinds of negation and classifications thereof. The volume is structured into four interrelated thematic parts. Part I is centered around the themes of Models, Relevance and Impossibility. In Chapter 1 (Negation: Two Points of View), Arnon Avron develops two characteri zations of negation, one semantic the other proof-theoretic. Interestingly and maybe provokingly, under neither of these accounts intuitionistic negation emerges as a genuine negation. J. Michael Dunn in Chapter 2 (A Comparative Study of Various Model-theoretic Treatments of Negation: A History of Formal Negation) surveys a detailed correspondence-theoretic classifcation of various notions of negation in terms of properties of a binary relation interpreted as incompatibility.
Author: Hajnal Andréka Publisher: Springer Nature ISBN: 3031148878 Category : Mathematics Languages : en Pages : 337
Book Description
This book gives a comprehensive introduction to Universal Algebraic Logic. The three main themes are (i) universal logic and the question of what logic is, (ii) duality theories between the world of logics and the world of algebra, and (iii) Tarskian algebraic logic proper including algebras of relations of various ranks, cylindric algebras, relation algebras, polyadic algebras and other kinds of algebras of logic. One of the strengths of our approach is that it is directly applicable to a wide range of logics including not only propositional logics but also e.g. classical first order logic and other quantifier logics. Following the Tarskian tradition, besides the connections between logic and algebra, related logical connections with geometry and eventually spacetime geometry leading up to relativity are also part of the perspective of the book. Besides Tarskian algebraizations of logics, category theoretical perspectives are also touched upon. This book, apart from being a monograph containing state of the art results in algebraic logic, can be used as the basis for a number of different courses intended for both novices and more experienced students of logic, mathematics, or philosophy. For instance, the first two chapters can be used in their own right as a crash course in Universal Algebra.
Author: J. Michael Dunn
Author: J. Michael Dunn
Author: Jon Michael Dunn
Author: Daniel Gallin
Author: Maarten de Rijke
Author: Zoran Majkic
Author: J. Michael Dunn
Author: S. Shapiro
Author: Louis Couturat
Author: Dov M. Gabbay
Author: Hajnal Andréka