Survey 1979: Equational Logic
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How Peircean Was the “'Fregean' Revolution” in Logic?
HOW PEIRCEAN WAS THE “‘FREGEAN’ REVOLUTION” IN LOGIC? Irving H. Anellis Peirce Edition, Institute for American Thought Indiana University – Purdue University at Indianapolis Indianapolis, IN, USA [email protected] Abstract. The historiography of logic conceives of a Fregean revolution in which modern mathematical logic (also called symbolic logic) has replaced Aristotelian logic. The preeminent expositors of this conception are Jean van Heijenoort (1912–1986) and Don- ald Angus Gillies. The innovations and characteristics that comprise mathematical logic and distinguish it from Aristotelian logic, according to this conception, created ex nihlo by Gottlob Frege (1848–1925) in his Begriffsschrift of 1879, and with Bertrand Rus- sell (1872–1970) as its chief This position likewise understands the algebraic logic of Augustus De Morgan (1806–1871), George Boole (1815–1864), Charles Sanders Peirce (1838–1914), and Ernst Schröder (1841–1902) as belonging to the Aristotelian tradi- tion. The “Booleans” are understood, from this vantage point, to merely have rewritten Aristotelian syllogistic in algebraic guise. The most detailed listing and elaboration of Frege’s innovations, and the characteristics that distinguish mathematical logic from Aristotelian logic, were set forth by van Heijenoort. I consider each of the elements of van Heijenoort’s list and note the extent to which Peirce had also developed each of these aspects of logic. I also consider the extent to which Peirce and Frege were aware of, and may have influenced, one another’s logical writings. AMS (MOS) 2010 subject classifications: Primary: 03-03, 03A05, 03C05, 03C10, 03G27, 01A55; secondary: 03B05, 03B10, 03E30, 08A20; Key words and phrases: Peirce, abstract algebraic logic; propositional logic; first-order logic; quantifier elimina- tion, equational classes, relational systems §0. -
Logic Machines and Diagrams
LOGIC M ACH IN ES AND DIAGRAMS Martin Gardner McGRAW-HILL BOOK COMPANY, INC. New York Toronto London 1958 LOGIC MACHINES AND DIAGRAMS Copyright 1958 by the McGraw-Hill Book Company, Inc. Printed in the United States of America. All rights reserved. This book, or parts thereof, may not be reproduced in any form without permission of the publishers. Library of Congress Catalog Card Number: 58-6683 for C. G. Who thinks in a multivalued system all her own. Preface A logic machine is a device, electrical or mechanical, designed specifically for solving problems in formal logic. A logic diagram is a geometrical method for doing the same thing. The two fields are closely intertwined, and this book is the first attempt in any language to trace their curious, fascinating histories. Let no reader imagine that logic machines are merely the play- of who to have a recreational interest in things engineers happen , symbolic logic. As we move with terrifying speed into an age of automation, the engineers and mathematicians who design our automata constantly encounter problems that are less mathemati- cal in form than logical. It has been discovered, for example, that symbolic logic can be applied fruitfully to the design and simplification of switching circuits. It has been found that electronic calculators often require elaborate logic units to tell them what steps to follow in tackling certain problems. And in the new field of operations research, annoying situations are constantly arising for which techniques of symbolic logic are surprisingly appropriate. The last chaptej: of this book suggests some of the ways in which logic machines may play essential roles in coping with the stagger- ing complexities of an automated technology. -
Equational Logic and Abstract Algebra Taje I. Ramsamujh, Florida
EQUATIONAL LOGIC AND ABSTRACT ALGEBRA* Taje I. Ramsamujh Florida International University Mathematics Department ABSTRACT Equational logic is a formalization of the deductive methods encountered in studying the set of all equations that can be derived from a given fixed set of equations. So it is naturally associated with abstract algebraic structures. The equations involved are interpreted as being true for all the variables involved and so are best thought of as identities. In complexity, equational logic sits somewhere between propositional and first-order logic. And even though it may appear simple at first sight, many of the problems are very interesting and non-trivial. Several are actually quite difficult and some are still open. In this exposition our goal is to introduce equational logic in an informal way as a bridge from the propositional logic to the first-order logic and contrast it with them. ---------------------------------------------- * This is an expanded version of a talk presented at the MAA Florida Section Annual Meeting at FGCU, March 2nd-3rd, 2001. 1 1. Introduction. Equational logic is often referred to as universal algebra because of its natural association with abstract algebraic structures but this is not the view we shall take here. We shall view it as a logic and accordingly begin by giving a general outline of what is a logic. There is a large body of work on equational logic and we refer the reader to the very comprehensive survey by Taylor [1979], even though it may be a little dated. Our goal is to show how equational logic can be use to motivate the study of first-order logic and model theory. -
A Field Guide to Equational Logic
J. Symbolic Computation (1992) 14, 371-397 A Field Guide to Equational Logic GEORGE F . McNULTY Department of Mathematics, University of South Carolina Columbia, SC 29208, USA (mcnultyOmilo . math . sarolina . edu) (Received 12 May 1992) 1. Introduction The concepts that can be expressed by means of equations and the kinds of proofs that may be devised using equations are central concerns of equational logic . The concept of a ring is ordinarily presented by saying that a ring is a system (R, +, -, , 0,1) in which the following equations are true : x+(y+z) ;Z~ (x+y)+z x •(y .z) :s(x .y) .z x •(y+z)%rx •y +x •z x+y ;Z1 y+x x •1 --x (x+y) •z ;Z~ x •z +y •z -x+x . 0 1 •x sz~ x x+0 :x A ring is an algebra-meaning here a nonempty set endowed with a system of finitary operations. Equations, on the other hand, are certain strings of formal symbols. The concept of truth establishes a binary relation between equations and algebras: an equation s ;t; t is true in an algebra A . This relationship underlies virtually all work in equational logic. By way of this relation each equational theory-that is, each set of equations closed under logical consequence-is associated with a variety of algebras : the class of all algebras in which each equation of the theory is true . Through this connection syntactical and computational tools developed for dealing with equations can be brought to bear on algebraic questions about varieties . Conversely, algebraic techniques and concepts from the theory of varieties can be employed on the syntactical side . -
Tutorial on Universal Algebra
Tutorial on Universal Algebra Peter Jipsen Chapman University August 13, 2009 Peter Jipsen (Chapman University) Tutorial on Universal Algebra BLAST August 13, 2009 1 Outline Aim: cover the main concepts of universal algebra This is a tutorial Slides give definitions and results, few proofs Learning requires doing, so participants get individual exercises They vary in difficulty Part I: basic universal algebra and lattice theory Part II: some advances of the last 3 decades Peter Jipsen (Chapman University) Tutorial on Universal Algebra BLAST August 13, 2009 2 Survey How many of the following books are freely available for downloading? 01234567 = Not sure Stan Burris and H. P. Sankapannavar, “A Course in Universal Algebra”, Springer-Verlag, 1981 David Hobby and Ralph McKenzie, “The Structure of Finite Algebras”, Contemporary Mathematics v. 76, American Mathematical Society, 1988 Ralph Freese and Ralph McKenzie, “Commutator theory for congruence modular varieties”, Cambridge University Press, 1987 Jarda Jezek, “Universal Algebra”, 2008 Peter Jipsen and Henry Rose, “Varieties of Lattices”, Lecture Notes in Mathematics 1533, Springer-Verlag, 1992 Keith Kearnes and Emil Kiss, “The shape of congruence lattices”, 2006 Peter Jipsen (Chapman University) Tutorial on Universal Algebra BLAST August 13, 2009 3 ALL of them Stan Burris and H. P. Sankapannavar, “A Course in Universal Algebra”, Springer-Verlag, 1981, online at www.math.uwaterloo.ca/∼snburris David Hobby and Ralph McKenzie, “The Structure of Finite Algebras”, Contemporary Mathematics v. 76, -
Polynomial Ring Calculus for Modal Logics: a New Semantics and Proof Method for Modalities Juan C
THE REVIEW OF SYMBOLIC LOGIC Volume 4, Number 1, March 2011 POLYNOMIAL RING CALCULUS FOR MODAL LOGICS: A NEW SEMANTICS AND PROOF METHOD FOR MODALITIES JUAN C. AGUDELO State University of Campinas—UNICAMP and Eafit University WALTER CARNIELLI State University of Campinas—UNICAMP and SQIG—IT Abstract. A new (sound and complete) proof style adequate for modal logics is defined from the polynomial ring calculus (PRC). The new semantics not only expresses truth conditions of modal formulas by means of polynomials, but also permits to perform deductions through polynomial handling. This paper also investigates relationships among the PRC here defined, the algebraic semantics for modal logics, equational logics, the Dijkstra–Scholten equational-proof style, and rewriting systems. The method proposed is throughly exemplified for S5, and can be easily extended to other modal logics. §1. Yet another semantics for modal logic? Besides its indisputable success, there may be reasons to doubt about the universal acceptability of the possible-worlds semantics (often called relational semantics or Kripke semantics) for modal logics, basically because they reduce (or equate) meaning to extensions of worlds. Indeed, criticisms against possible-worlds semantics abound. For instance, Kuczynski (2007) purports to show that the very assumptions of possible-worlds semantics lead to the absurd conclusion that all propositions are necessarily true. Even if this critique would be perhaps circumscribed to uses of possible-worlds semantics in quantified modal logic, it is not easy to rebut the well-known Quine’s criticisms about the difficulties in separating the notion of possibility within a world from the notion of consistency of the world’s description (cf., Section 1 of Quine, 1953). -
A Course in Universal Algebra
The Millennium Edition Stanley Burris H. P. Sankappanavar A Course in Universal Algebra With 36 Illustrations c S. Burris and H.P. Sankappanavar All Rights Reserved This Edition may be copied for personal use This book is dedicated to our children Kurosh Phillip Burris Veena and Geeta and Sanjay Sankappanavar Preface to the Millennium Edition The original 1981 edition of A Course in Universal Algebra has now been LaTeXed so the authors could make the out-of-print Springer-Verlag Gradu- ate Texts in Mathematics edition available once again, with corrections. The subject of Universal Algebra has flourished mightily since 1981, and we still believe that A Course in Universal Algebra offers an excellent introduction to the subject. Special thanks go to Lis D’Alessio for the superb job of LaTeXing this edition, and to NSERC for their support which has made this work possible. v Acknowledgments First we would like to express gratitude to our colleagues who have added so much vital- ity to the subject of Universal Algebra during the past twenty years. One of the original reasons for writing this book was to make readily available the beautiful work on sheaves and discriminator varieties which we had learned from, and later developed with H. Werner. Recent work of, and with, R. McKenzie on structure and decidability theory has added to our excitement, and conviction, concerning the directions emphasized in this book. In the late stages of polishing the manuscript we received valuable suggestions from M. Valeriote, W. Taylor, and the reviewer for Springer-Verlag. For help in proof-reading we also thank A. -
Polynomial Ring Calculus for Modal Logics: a New Semantics and Proof Method for Modalities
Polynomial ring calculus for modal logics: a new semantics and proof method for modalities Juan C. Agudelo∗ Walter Carnielli† Abstract A new (sound and complete) adequate semantics for the modal logic S5 is defined from the polynomial ring calculus (PRC), which permit to perform modal deductions through polynomial handling. This paper also investigates relationships among the PRC here defined, the algebraic semantics for modal logics, equational logics and the Dijkstra-Scholten equational-proof style. The method proposed can be easily extended to other modal logics. Yet another semantics for modal logic? Besides its indisputable success, there may be reasons to doubt about the uni- versal acceptability of the possible worlds semantics (often called relational se- mantics or Kripke semantics) for modal logics, basically because they reduce (or equate) meaning with extensions of worlds. Indeed, criticisms against possible-worlds semantics abound. For instance, [16] intends to show that the very assumptions of possible-worlds semantics lead to the absurd conclusion that all propositions are necessarily true. Even if this critique would be perhaps circumscribed to uses of possible-worlds semantics in quantified modal logic, it is not easy to rebut to the well-known Quine’s criticisms about the difficulties in separating the notion of possibility within a world from the notion of consistency of the world’s description (cf. Section 1 of [19]). But there are other issues with regard to the possible-worlds account of modal notions. As argued in [12], the mathematics associated with such seman- tics may be quite complicated, besides the inappropriateness of the possible- worlds to formalizing knowledge, belief and other non-alethic modal concepts.