George Boole (1815–1864)
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George Boole?
iCompute For more fun computing lessons and resources visit: Who was George Boole? 8 He was an English mathematician 8 He believed that human thought could be George Boole written down as ‘rules’ 8 His ideas led to boolean logic which is Biography for children used by computers today The story of important figures in the history of computing George Boole (1815 – 1864) © iCompute 2015 www.icompute -uk.com iCompute Why is George Boole important? 8 He invented a set of rules for thinking that are used by computers today 8 The rules were that some statements can only ever be ‘true’ or ‘false’ 8 His idea was first used in computers as switches either being ‘on’ or ‘off’ 8 Today this logic is used in almost every device and almost every line of computer code His early years nd 8 Born 2 November 1815 8 His father was a struggling shoemaker 8 George had had very little education and taught himself maths, French, German and Latin © iCompute 2015 www.icompute -uk.com iCompute 8 He also taught himself Greek and published a translation of a Greek poem in 1828 at the age of 14! 8 Aged 16, the family business collapsed and George began working as a teacher to support the family 8 At 19 he started his own school 8 In 1840 he began having his books about mathematics published 8 In 1844, he was given the first gold medal for Mathematics by the Royal Society 8 Despite never having been to University himself, in 1849 he became professor of Mathematics at Queens College Cork in Ireland 8 He married Mary Everett in 1855 8 They had four daughters between 1956-1864 -
Laws of Thought and Laws of Logic After Kant”1
“Laws of Thought and Laws of Logic after Kant”1 Published in Logic from Kant to Russell, ed. S. Lapointe (Routledge) This is the author’s version. Published version: https://www.routledge.com/Logic-from-Kant-to- Russell-Laying-the-Foundations-for-Analytic-Philosophy/Lapointe/p/book/9781351182249 Lydia Patton Virginia Tech [email protected] Abstract George Boole emerged from the British tradition of the “New Analytic”, known for the view that the laws of logic are laws of thought. Logicians in the New Analytic tradition were influenced by the work of Immanuel Kant, and by the German logicians Wilhelm Traugott Krug and Wilhelm Esser, among others. In his 1854 work An Investigation of the Laws of Thought on Which are Founded the Mathematical Theories of Logic and Probabilities, Boole argues that the laws of thought acquire normative force when constrained to mathematical reasoning. Boole’s motivation is, first, to address issues in the foundations of mathematics, including the relationship between arithmetic and algebra, and the study and application of differential equations (Durand-Richard, van Evra, Panteki). Second, Boole intended to derive the laws of logic from the laws of the operation of the human mind, and to show that these laws were valid of algebra and of logic both, when applied to a restricted domain. Boole’s thorough and flexible work in these areas influenced the development of model theory (see Hodges, forthcoming), and has much in common with contemporary inferentialist approaches to logic (found in, e.g., Peregrin and Resnik). 1 I would like to thank Sandra Lapointe for providing the intellectual framework and leadership for this project, for organizing excellent workshops that were the site for substantive collaboration with others working on this project, and for comments on a draft. -
AUGUSTUS DE MORGAN and the LOGIC of RELATIONS the New Synthese Historical Library Texts and Studies in the History of Philosophy
AUGUSTUS DE MORGAN AND THE LOGIC OF RELATIONS The New Synthese Historical Library Texts and Studies in the History of Philosophy VOLUME 38 Series Editor: NORMAN KRETZMANN, Cornell University Associate Editors: DANIEL ELLIOT GARBER, University of Chicago SIMO KNUUTTILA, University of Helsinki RICHARD SORABJI, University of London Editorial Consultants: JAN A. AERTSEN, Free University, Amsterdam ROGER ARIEW, Virginia Polytechnic Institute E. JENNIFER ASHWORTH, University of Waterloo MICHAEL AYERS, Wadham College, Oxford GAIL FINE, Cornell University R. J. HANKINSON, University of Texas JAAKKO HINTIKKA, Boston University, Finnish Academy PAUL HOFFMAN, Massachusetts Institute of Technology DAVID KONSTAN, Brown University RICHARD H. KRAUT, University of Illinois, Chicago ALAIN DE LIBERA, Ecole Pratique des Hautes Etudes, Sorbonne DAVID FATE NORTON, McGill University LUCA OBERTELLO, Universita degli Studi di Genova ELEONORE STUMP, Virginia Polytechnic Institute ALLEN WOOD, Cornell University The titles published in this series are listed at the end of this volume. DANIEL D. MERRILL Department of Philosophy, Oberlin College, USA AUGUSTUS DE MORGAN AND THE LOGIC OF RELATIONS KLUWER ACADEMIC PUBLISHERS DORDRECHT I BOSTON I LONDON Library of Congress Cataloging. in·Publication Data Merrill, Daniel D. (Daniel Davy) Augustus De Morgan and the lOglC of relations / Daniel D. Merril. p. cm. -- <The New synthese historical library; 38) 1. De Morgan, Augustus, 1806-1871--Contributions in logic of relations. 2. Inference. 3. Syllogism. 4. Logic, Symbolic and mathematical--History--19th cenTury. I. Title. II. Series. BC185.M47 1990 160' .92--dc20 90-34935 ISBN·13: 978·94·010·7418·6 e-ISBN -13: 978-94-009-2047-7 DOl: 10.1007/978-94-009-2047-7 Published by Kluwer Academic Publishers, P.O. -
Boolean Algebra.Pdf
Section 3 Boolean Algebra The Building Blocks of Digital Logic Design Section Overview Binary Operations (AND, OR, NOT), Basic laws, Proof by Perfect Induction, De Morgan’s Theorem, Canonical and Standard Forms (SOP, POS), Gates as SSI Building Blocks (Buffer, NAND, NOR, XOR) Source: UCI Lecture Series on Computer Design — Gates as Building Blocks, Digital Design Sections 1-9 and 2-1 to 2-7, Digital Logic Design CECS 201 Lecture Notes by Professor Allison Section II — Boolean Algebra and Logic Gates, Digital Computer Fundamentals Chapter 4 — Boolean Algebra and Gate Networks, Principles of Digital Computer Design Chapter 5 — Switching Algebra and Logic Gates, Computer Hardware Theory Section 6.3 — Remarks about Boolean Algebra, An Introduction To Microcomputers pp. 2-7 to 2-10 — Boolean Algebra and Computer Logic. Sessions: Four(4) Topics: 1) Binary Operations and Their Representation 2) Basic Laws and Theorems of Boolean Algebra 3) Derivation of Boolean Expressions (Sum-of-products and Product-of-sums) 4) Reducing Algebraic Expressions 5) Converting an Algebraic Expression into Logic Gates 6) From Logic Gates to SSI Circuits Binary Operations and Their Representation The Problem Modern digital computers are designed using techniques and symbology from a field of mathematics called modern algebra. Algebraists have studied for a period of over a hundred years mathematical systems called Boolean algebras. Nothing could be more simple and normal to human reasoning than the rules of a Boolean algebra, for these originated in studies of how we reason, what lines of reasoning are valid, what constitutes proof, and other allied subjects. The name Boolean algebra honors a fascinating English mathematician, George Boole, who in 1854 published a classic book, "An Investigation of the Laws of Thought, on Which Are Founded the Mathematical Theories of Logic and Probabilities." Boole's stated intention was to perform a mathematical analysis of logic. -
Vasiliev and the Foundations of Logic
Chapter 4 Vasiliev and the Foundations of Logic Otávio Bueno Abstract Nikolai Vasiliev offered a systematic approach to the development of a class of non-classical logics, which he called “Imaginary Logics”. In this paper, IexaminecriticallysomeofthecentralfeaturesofVasiliev’sapproachtological theory, suggesting its relevance to contemporary debates in the philosophy of logic. IarguethatthereismuchofsignificantvalueinVasiliev’swork,whichdeserves close philosophical engagement. Keywords Vasiliev • logical pluralism • Revisability • a priori • negation 4.1 Introduction: Six Central Features of Vasiliev’s Approach to Logical Theory Nikolai Vasiliev’s approach to logical theory has a number of features. Six of them, in particular, are worth highlighting: (a) logical pluralism (there is a plurality of logics, depending on the subject matter under consideration); (b) logical revisability (certain logical laws can be revised depending on the subject matter); (c) logical non-a priorism (certain logical laws are empirically based); (d) logical contingency (given the empirical nature of some logical laws, they are ultimately contingent; in this context, issues regarding the scope of logic are also examined, with the accompanying distinction between laws of objects and laws of thought); (e) the nature of negation (negation is characterized via incompatibility; it is not just difference, nor is it grounded on absence, and it is inferred rather than perceived); and (f) logical commitment (why Vasiliev is not a dialetheist, after all). In this paper, I will examine each of these features, and suggest the relevance of many of Vasiliev’s proposals to contemporary philosophical reflection about the foundations of logic. Although the terms I use to describe some of the views O. Bueno (!) Department of Philosophy, University of Miami, Coral Gables, FL 33124, USA e-mail: [email protected] ©SpringerInternationalPublishingAG2017 43 V. -
Abstract Consequence and Logics
Abstract Consequence and Logics Essays in honor of Edelcio G. de Souza edited by Alexandre Costa-Leite Contents Introduction Alexandre Costa-Leite On Edelcio G. de Souza PART 1 Abstraction, unity and logic 3 Jean-Yves Beziau Logical structures from a model-theoretical viewpoint 17 Gerhard Schurz Universal translatability: optimality-based justification of (not necessarily) classical logic 37 Roderick Batchelor Abstract logic with vocables 67 Juliano Maranh~ao An abstract definition of normative system 79 Newton C. A. da Costa and Decio Krause Suppes predicate for classes of structures and the notion of transportability 99 Patr´ıciaDel Nero Velasco On a reconstruction of the valuation concept PART 2 Categories, logics and arithmetic 115 Vladimir L. Vasyukov Internal logic of the H − B topos 135 Marcelo E. Coniglio On categorial combination of logics 173 Walter Carnielli and David Fuenmayor Godel's¨ incompleteness theorems from a paraconsistent perspective 199 Edgar L. B. Almeida and Rodrigo A. Freire On existence in arithmetic PART 3 Non-classical inferences 221 Arnon Avron A note on semi-implication with negation 227 Diana Costa and Manuel A. Martins A roadmap of paraconsistent hybrid logics 243 H´erculesde Araujo Feitosa, Angela Pereira Rodrigues Moreira and Marcelo Reicher Soares A relational model for the logic of deduction 251 Andrew Schumann From pragmatic truths to emotional truths 263 Hilan Bensusan and Gregory Carneiro Paraconsistentization through antimonotonicity: towards a logic of supplement PART 4 Philosophy and history of logic 277 Diogo H. B. Dias Hans Hahn and the foundations of mathematics 289 Cassiano Terra Rodrigues A first survey of Charles S. -
Paradox and Foundation Zach Weber Submitted in Total Fulfilment of The
Paradox and Foundation Zach Weber Submitted in total fulfilment of the requirements of the degree of Doctor of Philosophy May 2009 School of Philosophy, Anthropology and Social Inquiry The University of Melbourne This is to certify that - the thesis comprises only my original work towards the PhD, - due acknowledgement has been made in the text to all other material used, - the thesis is less than 100,000 words in length. Preface Dialethic paraconsistency is an approach to formal and philosophical theories in which some but not all contradictions are true. Advancing that program, this thesis is about paradoxes and the foundations of mathematics, and is divided accordingly into two main parts. The first part concerns the history and philosophy of set theory from Cantor through the independence proofs, focusing on the set concept. A set is any col- lection of objects that is itself an object, with identity completely determined by membership. The set concept is called naive because it is inconsistent. I argue that the set concept is inherently and rightly paradoxical, because sets are both intensional and extensional objects: Sets are predicates in extension. All consistent characterizations of sets are either not explanatory or not coherent. To understand sets, we need to reason about them with an appropriate logic; paraconsistent naive set theory is situated as a continuation of the original foundational project. The second part produces a set theory deduced from an unrestricted compre- hension principle using the weak relevant logic DLQ, dialethic logic with quantifiers. I discuss some of the problems involved with embedding in DLQ, especially related to identity and substitution. -
Aristotle on Non-Contradiction: Philosophers Vs
Journal of Ancient Philosophy J. anc. philos. (Engl. ed.), São Paulo, v.7, n.2. p. 51-74, 2013. ISSN 1981-9471 - FFLCH/USP DOI: http://dx.doi.org/10.11606/issn.1981-9471.v7i2p51-74 www.revistas.usp.br/filosofiaantiga Aristotle on Non-Contradiction: Philosophers vs. Non-Philosophers Jean-Louis Hudry USP, Brazil Abstract: Aristotle’s principle of non-contradiction (PNC) has been interpreted by Łukasiewicz through three distinct formulations, namely ontological, logical, and psychological. Many have criticized Łukasiewicz’s position, but they still maintain that Aristotle defends distinct formulations. In contrast, this paper shows that Aristotle suggests only one formulation of the PNC. This unique formulation belongs to philosophy as the first science, so that the philosophers think of the PNC as a necessarily true principle, owing to their meta-physical cognition of the nature of things. Yet, there is another way to understand this formulation. Indeed, the non-philosophers believe in the PNC, without being able to understand its necessary truth, due to their ignorance of philosophy. Thus, Aristotle has to convince them that the PNC is the most certain opinion of all, and his dialectical justifications are purposely weak, as they are only concerned with the defense of a common opinion. In Chapter 3 of Metaphysics Gamma, Aristotle introduces the so-called principle of non-contradiction (hereafter PNC).1 It is important to put this principle into context in order to understand why and how Aristotle introduces it: It is proper for the one who best cognizes (gnôrizonta) each genus to be able to state the most certain principles (archas) of an actual thing, so that the one who cognizes beings, qua beings (tôn ontôn hêi onta), will also be able to state the most certain principles of all things. -
George Boole. Selected Manuscripts on Logic and Its Philosophy (Science Networks
George Boole. Selected Manuscripts on Logic and its Philosophy (Science Networks. Historical Studies, vol. 20). Ivor Grattan-Guinness/G´erard Bor- net, eds. Birkh¨auser Verlag: Basel/Boston/Berlin, 1997, e 37.30, ISBN 0- 8176-5456-9, lxiv + 236 pp. Reviewed by Volker Peckhaus Universit¨at Paderborn Kulturwissenschaftliche Fakult¨at Warburger Str. 100, D – 33098 Paderborn E-mail: [email protected] George Boole, the founder of the algebra of logic, did not stop working on logic after having published his seminal An Investigation of the Laws of Thought (1854 ). Among his aims was to write a philosophical comple- ment to his writings in the algebra of logic, but he died before he was able to finish it. All this was known already to his contemporaries and although leading authorities like Augustus De Morgan, Bertrand Russell, Philip E. B. Jourdain and Louis Couturat were involved in attempts to publish Boole’s posthumous logical papers, only bits and pieces from his logical manuscripts have been published up to now. This insufficient situation comes now to an end with this scholarly edition by Ivor Grattan-Guinness and G´erard Bornet. The editors have compiled a selection of about 40 % of Boole’s unpublished manuscripts on logic “from a scattered and chronologically unclear textual universe” (p. xxv). The bad condition of the manuscripts is due to Boole’s working style. He usually started each time from scratch producing not really substantial texts but notes and further work-plans. This is explained by Boole himself when he reported his plans to Augustus De Morgan in March 1859 (quoted p. -
The Logical Analysis of Key Arguments in Leibniz and Kant
University of Mississippi eGrove Electronic Theses and Dissertations Graduate School 2016 The Logical Analysis Of Key Arguments In Leibniz And Kant Richard Simpson Martin University of Mississippi Follow this and additional works at: https://egrove.olemiss.edu/etd Part of the Philosophy Commons Recommended Citation Martin, Richard Simpson, "The Logical Analysis Of Key Arguments In Leibniz And Kant" (2016). Electronic Theses and Dissertations. 755. https://egrove.olemiss.edu/etd/755 This Thesis is brought to you for free and open access by the Graduate School at eGrove. It has been accepted for inclusion in Electronic Theses and Dissertations by an authorized administrator of eGrove. For more information, please contact [email protected]. THE LOGICAL ANALYSIS OF KEY ARGUMENTS IN LEIBNIZ AND KANT A Thesis presented in partial fulfillment of requirements for the degree of Master of Arts in the Department of Philosophy and Religion The University of Mississippi By RICHARD S. MARTIN August 2016 Copyright Richard S. Martin 2016 ALL RIGHTS RESERVED ABSTRACT This paper addresses two related issues of logic in the philosophy of Gottfried Leibniz. The first problem revolves around Leibniz’s struggle, throughout the period of his mature philosophy, to reconcile his metaphysics and epistemology with his antecedent theological commitments. Leibniz believes that for everything that happens there is a reason, and that the reason God does things is because they are the best that can be done. But if God must, by nature, do what is best, and if what is best is predetermined, then it seems that there may be no room for divine freedom, much less the human freedom Leibniz wished to prove. -
Leibniz and the Principle of Sufficient Reason
Leibniz and the Principle of Sufficient Reason by Owen Pikkert A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Department of Philosophy University of Toronto © Copyright by Owen Pikkert 2018 Abstract Leibniz and the Principle of Sufficient Reason Owen Pikkert Doctor of Philosophy Department of Philosophy University of Toronto 2018 Leibniz’s principle of sufficient reason (PSR) is the claim that everything has an explanation. It rules out brute facts, inexplicable primitives, and purely random events. On the usual view, Leibniz grounds the PSR in purely descriptive truths. On my view, Leibniz grounds the PSR in this being the best of all possible worlds. God only creates the best, and a world in which the PSR is true is better than a world in which it is false. For the PSR ensures that the world has an explanatory structure, the investigation of which facilitates human happiness. This way of grounding the PSR faces at least two problems. The first problem is that it presupposes that the PSR is a contingent principle, even though most commentators take it to be necessary. But I argue that Leibniz is indeed committed to the contingency of the PSR. I demonstrate this by showing how, for Leibniz, PSR-violating entities such as vacua, atoms, and indiscernible bodies are possible but not actual. I also argue that the contingency of the ii PSR does not conflict with Leibniz’s other modal commitments. In particular, it does not conflict with the modal status of his principle of the identity of indiscernibles, nor with the modal status of his theory of truth. -
What Does It Mean to Say That Logic Is Formal?
WHAT DOES IT MEAN TO SAY THAT LOGIC IS FORMAL? by John Gordon MacFarlane A.B., Philosophy, Harvard College, 1991 M.A., Philosophy, University of Pittsburgh, 1994 M.A., Classics, University of Pittsburgh, 1997 Submitted to the Graduate Faculty of Arts and Sciences in partial fulfillment of the requirements for the degree of Doctor of Philosophy University of Pittsburgh 2000 i Robert Brandom, Distinguished Service Professor of Philosophy (Director) Nuel Belnap, Alan Ross Anderson Distinguished Professor of Philosophy (Second Reader) Joseph Camp, Professor of Philosophy Danielle Macbeth, Associate Professor of Philosophy, Haverford College (Outside Reader) Kenneth Manders, Associate Professor of Philosophy Gerald Massey, Distinguished Service Professor of Philosophy ii WHAT DOES IT MEAN TO SAY THAT LOGIC IS FORMAL? John Gordon MacFarlane, PhD University of Pittsburgh, 2000 Much philosophy of logic is shaped, explicitly or implicitly, by the thought that logic is distinctively formal and abstracts from material content. The distinction between formal and material does not appear to coincide with the more familiar contrasts between a pri- ori and empirical, necessary and contingent, analytic and synthetic—indeed, it is often invoked to explain these. Nor, it turns out, can it be explained by appeal to schematic inference patterns, syntactic rules, or grammar. What does it mean, then, to say that logic is distinctively formal? Three things: logic is said to be formal (or “topic-neutral”) (1) in the sense that it provides constitutive norms for thought as such, (2) in the sense that it is indifferent to the particular identities of objects, and (3) in the sense that it abstracts entirely from the semantic content of thought.