Overview

This book is about the logic of Boolean equations. Such equations were central in the ""algebra of logic"" created in 1847 by Boole [12, 13] and devel­ oped by others, notably Schroder [178], in the remainder of the nineteenth century. Boolean equations are also the language by which digital circuits are described today. Logicians in the twentieth century have abandoned Boole's equation­ based logic in favor of the more powerful predicate calculus. As a result, digital engineers-and others who use Boole's language routinely-remain largely unaware of its utility as a medium for reasoning. The aim of this book, accordingly, is to is to present a systematic outline of the logic of Boolean equations, in the hope that Boole's methods may prove useful in solving present-day problems. Two Logical Languages Logic seeks to reduce reasoning to calculation. Two main languages have been developed to achieve that object: Boole's ""algebra of logic"" and the predicate calculus. Boole's approach was to represent classes (e. g. , happy creatures, things productive of pleasure) by symbols and to represent logical statements as equations to be solved. His formulation proved inadequate, however, to represent ordinary discourse. A number of nineteenth-century logicians, including Jevons [94], Poretsky [159], Schroder [178], Venn [210], and Whitehead [212, 213], sought an improved formulation based on ex­ tensions or modifications of Boole's algebra. These efforts met with only limited success.

Full Product Details

Author:   Frank Markham Brown
Publisher:   Springer
Imprint:   Springer
Edition:   1990 ed.
Dimensions:   Width: 15.50cm , Height: 1.70cm , Length: 23.50cm
Weight:   1.310kg
ISBN:  

9780792391210

ISBN 10:   0792391217
Pages:   276
Publication Date:   01 August 1990
Audience:   College/higher education ,  Professional and scholarly ,  Postgraduate, Research & Scholarly ,  Professional & Vocational
Format:   Hardback
Publisher's Status:   Active
Availability:   In Print  
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Table of Contents

1 Fundamental Concepts.- 1.1 Formulas.- 1.2 Propositions and Predicates.- 1.3 Sets.- 1.4 Operations on Sets.- 1.5 Partitions.- 1.6 Relations.- 1.7 Functions.- 1.8 Operations and Algebraic Systems.- 2 Boolean Algebras.- 2.1 Postulates for a Boolean Algebra.- 2.2 Examples of Boolean Algebras.- 2.3 The Stone Representation Theorem.- 2.4 The Inclusion-Relation.- 2.5 Some Useful Properties.- 2.6 n-Variable Boolean Formulas.- 2.7 n-Variable Boolean Functions.- 2.8 Boole’s Expansion Theorem.- 2.9 The Minterm Canonical Form.- 2.10 The Löwenheim-Müller Verification Theorem.- 2.11 Switching Functions.- 2.12 Incompletely-Specified Boolean Functions.- 2.13 Boolean Algebras of Boolean Functions.- 2.14 Orthonormal Expansions.- 2.15 Boolean Quotient.- 2.16 The Boolean Derivative.- 2.17 Recursive Definition of Boolean Functions.- 2.18 What Good are “Big” Boolean Algebras?.- 3 The Blake Canonical Form.- 3.1 Definitions and Terminology.- 3.2 Syllogistic & Blake Canonical Formulas.- 3.3 Generation of BCF(f).- 3.4 Exhaustion of Implicants.- 3.5 Iterated Consensus.- 3.6 Multiplication.- 4 Boolean Analysis.- 4.1 Review of Elementary Properties.- 4.2 Boolean Systems.- 4.3 Reduction.- 4.4 The Extended Verification Theorem.- 4.5 Poretsky’s Law of Forms.- 4.6 Boolean Constraints.- 4.7 Elimination.- 4.8 Eliminants.- 4.9 Rudundant Variables.- 4.10 Substitution.- 4.11 The Tautology Problem.- 5 Syllogistic Reasoning.- 5.1 The Principle of Assertion.- 5.2 Deduction by Consensus.- 5.3 Syllogistic Formulas.- 5.4 Clausal Form.- 5.5 Producing and Verifying Consequents.- 5.6 Class-Logic.- 5.7 Selective Deduction.- 5.8 Functional Relations.- 5.9 Dependent Sets of Functions.- 5.10 Sum-to-One Subsets.- 5.11 Irredundant Formulas.- 6 Solution of Boolean Equations.- 6.1 Particular Solutions andConsistency.- 6.2 General Solutions.- 6.3 Subsumptive General Solutions.- 6.4 Parametric General Solutions.- 7 Functional Deduction.- 7.1 Functionally Deducible Arguments.- 7.2 Eliminable and Determining Subsets.- 8 Boolean Identification.- 8.1 Parametric and Diagnostic Models.- 8.2 Adaptive Identification.- 9 Recursive Realizations of Combinational Circuits.- 9.1 The Design-Process.- 9.2 Specifications.- 9.3 Tabular Specifications.- 9.4 Strongly Combinational Solutions.- 9.5 Least-Cost Recursive Solutions.- 9.6 Constructing Recursive Solutions.- A Syllogistic Formulas.- A.1 Absorptive Formulas.- A.2 Syllogistic Formulas.- A.3 Prime Implicants.- A.4 The Blake Canonical Form.

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