Propositional Logic As A Boolean Algebra A New Perspective

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Propositional Logic As a Boolean Algebra - a New Perspective

Author : William Veatch
Publisher : Createspace Independent Publishing Platform
ISBN 13 : 9781979069236
Total Pages : 252 pages
Book Rating : 4.0/5 (692 download)

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Book Synopsis Propositional Logic As a Boolean Algebra - a New Perspective by : William Veatch

Download or read book Propositional Logic As a Boolean Algebra - a New Perspective written by William Veatch and published by Createspace Independent Publishing Platform. This book was released on 2017-12-02 with total page 252 pages. Available in PDF, EPUB and Kindle. Book excerpt: William S. Veatch Propositional Logic as a Boolean Algebra - A New Perspective Vol. 1 This Volume 1 considers the question of whether we can interpret Traditional Propositional Logic using the Logic Operations OR, AND, and NOT as a Boolean Algebra when viewed in the broader context of the Mathematics of Ideas as developed in the author's book: "Math Without Numbers - The Mathematics of Ideas - Vol.1 Foundations." The answer is "yes," provided, that we make some changes to how OR, AND, and NOT are defined and implemented. Basically, we equate OR, AND, and NOT to Union, Intersection, and Complementation for purposes of combining Propositions to form sets, but we develop a new methodology for assigning Truth Values. To implement our new style of Propositional Logic in Math Without Numbers, or MWN for short, the author creates three separate but related Universes of Discourse, each of which constitutes a Boolean Algebra using Union, Intersection, and Complementation: Ideas (Order 1), Propositions (Order 2), and Logic Formulas (Order 3). We see that the Truth Values of Propositions and Logic Formulas are inextricably linked to the set relationships of the Ideas comprising the subjects and predicates of the Propositions. In the end, we see that we can view Traditional Propositional Logic as a subset of a larger system of MWN Propositional Logic. Traditional Propositional Logic is a special case concerning an Order 2 Domain with a single Atom, whereas MWN Propositional Logic goes on to examine Order 2 Domains with multiple Atoms. In developing this new theory of Propositional Logic, the author proposes a new methodology for assigning Truth Values. The underlying premise is that every Idea is either an Atom or a Compound made up of Atoms, but only Atoms have a binary Truth/False Truth Value. Compounds, if homogeneous, may have a clear Truth Value, but unlike Atoms, Compounds may consist of a heterogeneous mix of True and False Atoms, such that there is no clear Truth Value for such "Mixed Sets" of Atoms. Depending upon the context, we may be able to create a rule for assigning a Truth Value to a Mixed Set, but it requires some exercise of discretion. This is consistent with the premise that mathematics can tell us how to think, but not what to think. This book is intended for anyone interested in Logic.

Logic as Algebra

Author : Paul Halmos
Publisher : American Mathematical Soc.
ISBN 13 : 1470451662
Total Pages : 141 pages
Book Rating : 4.4/5 (74 download)

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Book Synopsis Logic as Algebra by : Paul Halmos

Download or read book Logic as Algebra written by Paul Halmos and published by American Mathematical Soc.. This book was released on 2019-01-30 with total page 141 pages. Available in PDF, EPUB and Kindle. Book excerpt: Here is an introduction to modern logic that differs from others by treating logic from an algebraic perspective. What this means is that notions and results from logic become much easier to understand when seen from a familiar standpoint of algebra. The presentation, written in the engaging and provocative style that is the hallmark of Paul Halmos, from whose course the book is taken, is aimed at a broad audience, students, teachers and amateurs in mathematics, philosophy, computer science, linguistics and engineering; they all have to get to grips with logic at some stage. All that is needed.

Logic, Epistemology, and the Unity of Science

Author : Shahid Rahman
Publisher : Springer Science & Business Media
ISBN 13 : 9048124867
Total Pages : 617 pages
Book Rating : 4.0/5 (481 download)

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Book Synopsis Logic, Epistemology, and the Unity of Science by : Shahid Rahman

Download or read book Logic, Epistemology, and the Unity of Science written by Shahid Rahman and published by Springer Science & Business Media. This book was released on 2009-03-16 with total page 617 pages. Available in PDF, EPUB and Kindle. Book excerpt: The first volume in this new series explores, through extensive co-operation, new ways of achieving the integration of science in all its diversity. The book offers essays from important and influential philosophers in contemporary philosophy, discussing a range of topics from philosophy of science to epistemology, philosophy of logic and game theoretical approaches. It will be of interest to philosophers, computer scientists and all others interested in the scientific rationality.

Set Theoretical Logic-The Algebra of Models

Author : W Felscher
Publisher : CRC Press
ISBN 13 : 9789056992668
Total Pages : 298 pages
Book Rating : 4.9/5 (926 download)

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Book Synopsis Set Theoretical Logic-The Algebra of Models by : W Felscher

Download or read book Set Theoretical Logic-The Algebra of Models written by W Felscher and published by CRC Press. This book was released on 2000-05-30 with total page 298 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is an introduction to mathematical logic in which all the usual topics are presented: compactness and axiomatizability of semantical consequence, Löwenheim-Skolem-Tarski theorems, prenex and other normal forms, and characterizations of elementary classes with the help of ultraproducts. Logic is based exclusively on semantics: truth and satisfiability of formulas in structures are the basic notions. The methods are algebraic in the sense that notions such as homomorphisms and congruence relations are applied throughout in order to gain new insights. These concepts are developed and can be viewed as a first course on universal algebra. The approach to algorithms generating semantical consequences is algebraic as well: for equations in algebras, for propositional formulas, for open formulas of predicate logic, and for the formulas of quantifier logic. The structural description of logical consequence is a straightforward extension of that of equational consequence, as long as Boolean valued propositions and Boolean valued structures are considered; the reduction of the classical 2-valued case then depends on the Boolean prime ideal theorem.

Early Analytic Philosophy - New Perspectives on the Tradition

Author : Sorin Costreie
Publisher : Springer
ISBN 13 : 3319242148
Total Pages : 455 pages
Book Rating : 4.3/5 (192 download)

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Book Synopsis Early Analytic Philosophy - New Perspectives on the Tradition by : Sorin Costreie

Download or read book Early Analytic Philosophy - New Perspectives on the Tradition written by Sorin Costreie and published by Springer. This book was released on 2016-01-21 with total page 455 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume discusses some crucial ideas of the founders of the analytic philosophy: Gottlob Frege, Bertrand Russell and Ludwig Wittgenstein, or the ‘golden trio’. The book shows how these ‘old’ ideas are still present and influential in the current philosophical debates and to what extent these debates echo the original ideas. The collection aim is twofold: to better understand these fruitful ideas by placing them in the original setting, and to systematically examine these ideas in the context of the current debates animating philosophical discussions today. Divided into five sections, the book first sets the stage and offers a general introduction to the background influences, as well as delimitations of the initial foundational positions. This first section contains two papers dedicated to the discussion of realism and the status of science at that time, followed by two papers that tackle the epistemic status of logical laws. The next three sections constitute the core of the volume, each being dedicated to the most important figures in the early analytic tradition: Frege, Russell, and Wittgenstein. The last section gathers several essays that discuss either the relation between two or more analytic thinkers, or various important concepts such as ‘predicativism’ and ‘arbitrary function’, or the principles of abstraction and non-contradiction.

Completeness Theory for Propositional Logics

Author : Witold A. Pogorzelski
Publisher : Springer Science & Business Media
ISBN 13 : 3764385189
Total Pages : 178 pages
Book Rating : 4.7/5 (643 download)

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Book Synopsis Completeness Theory for Propositional Logics by : Witold A. Pogorzelski

Download or read book Completeness Theory for Propositional Logics written by Witold A. Pogorzelski and published by Springer Science & Business Media. This book was released on 2008-05-25 with total page 178 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book develops the theory of one of the most important notions in the methodology of formal systems. Particularly, completeness plays an important role in propositional logic where many variants of the notion have been defined. This approach allows also for a more profound view upon some essential properties of propositional systems. For these purposes, the theory of logical matrices, and the theory of consequence operations is exploited.

Fuzzy Logic and Mathematics

Author : Radim Belohlavek
Publisher : Oxford University Press
ISBN 13 : 0190200022
Total Pages : 480 pages
Book Rating : 4.1/5 (92 download)

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Book Synopsis Fuzzy Logic and Mathematics by : Radim Belohlavek

Download or read book Fuzzy Logic and Mathematics written by Radim Belohlavek and published by Oxford University Press. This book was released on 2017-05-03 with total page 480 pages. Available in PDF, EPUB and Kindle. Book excerpt: The term "fuzzy logic," as it is understood in this book, stands for all aspects of representing and manipulating knowledge based on the rejection of the most fundamental principle of classical logic---the principle of bivalence. According to this principle, each declarative sentence is required to be either true or false. In fuzzy logic, these classical truth values are not abandoned. However, additional, intermediate truth values between true and false are allowed, which are interpreted as degrees of truth. This opens a new way of thinking---thinking in terms of degrees rather than absolutes. For example, it leads to the definition of a new kind of sets, referred to as fuzzy sets, in which membership is a matter of degree. The book examines the genesis and development of fuzzy logic. It surveys the prehistory of fuzzy logic and inspects circumstances that eventually lead to the emergence of fuzzy logic. The book explores in detail the development of propositional, predicate, and other calculi that admit degrees of truth, which are known as fuzzy logic in the narrow sense. Fuzzy logic in the broad sense, whose primary aim is to utilize degrees of truth for emulating common-sense human reasoning in natural language, is scrutinized as well. The book also examines principles for developing mathematics based on fuzzy logic and provides overviews of areas in which this has been done most effectively. It also presents a detailed survey of established and prospective applications of fuzzy logic in various areas of human affairs, and provides an assessment of the significance of fuzzy logic as a new paradigm.

New Perspectives on Cybernetics

Author : G. Vijver
Publisher : Springer Science & Business Media
ISBN 13 : 9401580626
Total Pages : 262 pages
Book Rating : 4.4/5 (15 download)

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Book Synopsis New Perspectives on Cybernetics by : G. Vijver

Download or read book New Perspectives on Cybernetics written by G. Vijver and published by Springer Science & Business Media. This book was released on 2013-06-29 with total page 262 pages. Available in PDF, EPUB and Kindle. Book excerpt: Gertrudis Van de Vijver· Seminar of Logic and Epistemology University of Ghent Before being classified under the fashionable denominators of complexity and chaos, self-organization and autonomy were intensely inquired into in the cybernetic tradition. Despite all rejections that cybernetics has gone through in the second half of this century, today its importance is more and more recognized. Its decisive influence for connectionist theories, autopoietic and constructivist theories, for different forms of applied or experimental epistemology, is being more and more understood and generally accepted. It is mainly due to the success of connectionist models that we observe today a revival of interest for cybernetics. The 1943 article by McCulloch and Pitts is evidently a founding article. Cybernetics has however a much broader interest than the one linked to technical-mathematical details relevant to the construction of networks. For instance, the evolution from first to second order cybernetics, the ways of approaching biological and cognitive phenomena in the latter and the limits that were formulated there, are particularly meaningful to understand current developments and divergences in connectionism. A nuanced picture of cybernetic's history and its present state is therefore clearly epistemologically essential.

Sheaves, Games, and Model Completions

Author : Silvio Ghilardi
Publisher : Springer Science & Business Media
ISBN 13 : 940159936X
Total Pages : 246 pages
Book Rating : 4.4/5 (15 download)

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Book Synopsis Sheaves, Games, and Model Completions by : Silvio Ghilardi

Download or read book Sheaves, Games, and Model Completions written by Silvio Ghilardi and published by Springer Science & Business Media. This book was released on 2013-04-17 with total page 246 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is an example of fruitful interaction between (non-classical) propo sitionallogics and (classical) model theory which was made possible due to categorical logic. Its main aim consists in investigating the existence of model completions for equational theories arising from propositional logics (such as the theory of Heyting algebras and various kinds of theories related to proposi tional modal logic ). The existence of model-completions turns out to be related to proof-theoretic facts concerning interpretability of second order propositional logic into ordinary propositional logic through the so-called 'Pitts' quantifiers' or 'bisimulation quantifiers'. On the other hand, the book develops a large number of topics concerning the categorical structure of finitely presented al gebras, with related applications to propositional logics, both standard (like Beth's theorems) and new (like effectiveness of internal equivalence relations, projectivity and definability of dual connectives such as difference). A special emphasis is put on sheaf representation, showing that much of the nice categor ical structure of finitely presented algebras is in fact only a restriction of natural structure in sheaves. Applications to the theory of classifying toposes are also covered, yielding new examples. The book has to be considered mainly as a research book, reporting recent and often completely new results in the field; we believe it can also be fruitfully used as a complementary book for graduate courses in categorical and algebraic logic, universal algebra, model theory, and non-classical logics. 1.

Boolean Reasoning

Author : Frank Markham Brown
Publisher : Courier Corporation
ISBN 13 : 0486164594
Total Pages : 304 pages
Book Rating : 4.4/5 (861 download)

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Book Synopsis Boolean Reasoning by : Frank Markham Brown

Download or read book Boolean Reasoning written by Frank Markham Brown and published by Courier Corporation. This book was released on 2012-02-10 with total page 304 pages. Available in PDF, EPUB and Kindle. Book excerpt: Concise text begins with overview of elementary mathematical concepts and outlines theory of Boolean algebras; defines operators for elimination, division, and expansion; covers syllogistic reasoning, solution of Boolean equations, functional deduction. 1990 edition.

Mathematical Logic

Author : René Cori
Publisher :
ISBN 13 : 9780198500483
Total Pages : 338 pages
Book Rating : 4.5/5 (4 download)

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Book Synopsis Mathematical Logic by : René Cori

Download or read book Mathematical Logic written by René Cori and published by . This book was released on 2000 with total page 338 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Perspectives on Universal Logic

Author : J. Y. Beziau
Publisher : Polimetrica s.a.s.
ISBN 13 : 8876990771
Total Pages : 442 pages
Book Rating : 4.8/5 (769 download)

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Book Synopsis Perspectives on Universal Logic by : J. Y. Beziau

Download or read book Perspectives on Universal Logic written by J. Y. Beziau and published by Polimetrica s.a.s.. This book was released on 2007 with total page 442 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Universal Algebraic Logic

Author : Hajnal Andréka
Publisher : Springer Nature
ISBN 13 : 3031148878
Total Pages : 337 pages
Book Rating : 4.0/5 (311 download)

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Book Synopsis Universal Algebraic Logic by : Hajnal Andréka

Download or read book Universal Algebraic Logic written by Hajnal Andréka and published by Springer Nature. This book was released on 2022-11-01 with total page 337 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Proof-theoretical Coherence

Author : Kosta Došen
Publisher : College Publications
ISBN 13 :
Total Pages : 404 pages
Book Rating : 4.:/5 (1 download)

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Book Synopsis Proof-theoretical Coherence by : Kosta Došen

Download or read book Proof-theoretical Coherence written by Kosta Došen and published by College Publications. This book was released on 2004 with total page 404 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book in categorial proof theory formulates in terms of category theory a generalization close to linear algebra of the notions of distributive lattice and Boolean algebra. These notions of distributive lattice category and Boolean category codify a plausible nontrivial notion of identity of proofs in classical propositional logic, which is in accordance with Gentzen's cut-elimination procedure for multiple-conclusion sequents modified by admitting new principles called union of proofs and zero proofs. It is proved that these notions of category are coherent in the sense that there is a faithful structure-preserving functor from freely generated distributive lattice categories and Boolean categories into the category whose arrows are relations between finite ordinals-a category related to generality of proofs and to the notion of natural transformation. These coherence results yield a simple decision procedure for equality of proofs. Coherence in the same sense is also proved for various more general notions of category that enter into the notions of distributive lattice category and Boolean category. Some of these coherence results, like those for monoidal and symmetric monoidal categories are well known, but are here presented in a new light. The key to this categorification of the proof theory of classical propositional logic is distribution of conjunction over disjunction that is not an isomorphism as in cartesian closed categories.

The Logico-Algebraic Approach to Quantum Mechanics

Author : C.A. Hooker
Publisher : Springer Science & Business Media
ISBN 13 : 9401017956
Total Pages : 611 pages
Book Rating : 4.4/5 (1 download)

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Book Synopsis The Logico-Algebraic Approach to Quantum Mechanics by : C.A. Hooker

Download or read book The Logico-Algebraic Approach to Quantum Mechanics written by C.A. Hooker and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 611 pages. Available in PDF, EPUB and Kindle. Book excerpt: The twentieth century has witnessed a striking transformation in the un derstanding of the theories of mathematical physics. There has emerged clearly the idea that physical theories are significantly characterized by their abstract mathematical structure. This is in opposition to the tradi tional opinion that one should look to the specific applications of a theory in order to understand it. One might with reason now espouse the view that to understand the deeper character of a theory one must know its abstract structure and understand the significance of that struc ture, while to understand how a theory might be modified in light of its experimental inadequacies one must be intimately acquainted with how it is applied. Quantum theory itself has gone through a development this century which illustrates strikingly the shifting perspective. From a collection of intuitive physical maneuvers under Bohr, through a formative stage in which the mathematical framework was bifurcated (between Schrödinger and Heisenberg) to an elegant culmination in von Neumann's Hilbert space formulation the elementary theory moved, flanked even at the later stage by the ill-understood formalisms for the relativistic version and for the field-theoretic altemative; after that we have a gradual, but constant, elaboration of all these quantal theories as abstract mathematical struc tures (their point of departure being von Neumann's formalism) until at the present time theoretical work is heavily preoccupied with the manip ulation of purely abstract structures.

Mathematical Logic and Computation

Author : Jeremy Avigad
Publisher : Cambridge University Press
ISBN 13 : 1108478751
Total Pages : 527 pages
Book Rating : 4.1/5 (84 download)

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Book Synopsis Mathematical Logic and Computation by : Jeremy Avigad

Download or read book Mathematical Logic and Computation written by Jeremy Avigad and published by Cambridge University Press. This book was released on 2022-09-30 with total page 527 pages. Available in PDF, EPUB and Kindle. Book excerpt: A thorough introduction to the fundamental methods and results in mathematical logic, and its foundational role in computer science.

A Logical Approach to Discrete Math

Author : David Gries
Publisher : Springer Science & Business Media
ISBN 13 : 1475738374
Total Pages : 517 pages
Book Rating : 4.4/5 (757 download)

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Book Synopsis A Logical Approach to Discrete Math by : David Gries

Download or read book A Logical Approach to Discrete Math written by David Gries and published by Springer Science & Business Media. This book was released on 2013-03-14 with total page 517 pages. Available in PDF, EPUB and Kindle. Book excerpt: Here, the authors strive to change the way logic and discrete math are taught in computer science and mathematics: while many books treat logic simply as another topic of study, this one is unique in its willingness to go one step further. The book traets logic as a basic tool which may be applied in essentially every other area.