Boole's Logic and Probability (eBook)

A Critical Exposition from the Standpoint of Contemporary Algebra, Logic and Probability Theory

T. Hailperin (Autor)

eBook Download: PDF

1986 | 2. Auflage
427 Seiten
Elsevier Science (Verlag)
978-0-08-088005-1 (ISBN)

Infos

Since the publication of the first edition in 1976, there has been a notable increase of interest in the development of logic. This is evidenced by the several conferences on the history of logic, by a journal devoted to the subject, and by an accumulation of new results. This increased activity and the new results - the chief one being that Boole's work in probability is best viewed as a probability logic - were influential circumstances conducive to a new edition.

Chapter 1, presenting Boole's ideas on a mathematical treatment of logic, from their emergence in his early 1847 work on through to his immediate successors, has been considerably enlarged. Chapter 2 includes additional discussion of the ``uninterpretable'' notion, both semantically and syntactically. Chapter 3 now includes a revival of Boole's abandoned propositional logic and, also, a discussion of his hitherto unnoticed brush with ancient formal logic. Chapter 5 has an improved explanation of why Boole's probability method works. Chapter 6, Applications and Probability Logic, is a new addition. Changes from the first edition have brought about a three-fold increase in the bibliography.

Inhalt

Front Cover 1
Boole's Logic and Probability 4
Copyright Page 5
Contents 11
Preface to the First Edition 8
Preface to the Second Edition 10
Introduction 14
Chapter 0. Requisites From Algebra, Logic and Probability 22
0.1. Preliminaries 22
0.2. Algebraic structures 23
0.3. First order theories, models, extensions 28
0.4. Semirings, Commutative rings with unit 33
0.5. Boolean algebras and Boolean rings. Propositional logic 36
0.6. Rings of quotients. Boolean quotients 44
0.7. Fourier elimination. Solvability of linear systems 49
0.8. Linear programming 52
0.9. Probability theory 57
0.10. Miscellaneous 64
PART I . LOGIC 74
Chapter 1. Boole's Logic of Class Terms 76
1.0. Symbolical Algebra 76
1.1. Boole's first essay 79
1.2. The basic principle 83
I.3. Symbols of Logic and "operations of the mind". The fundamental law 90
1.4. Boole's algebra of +, -, X, 0, 1 96
1.5. Primary propositions and class terms 97
1.6. Principles of symbolical reasoning. Development 102
1.7. Interpretation 106
1.8. Elimination. Reduction 112
1.9. Abbreviation. Perfection of method 117
1.10. Treatment of "some". Aristotelian logic 121
1.11. De Morgan. Jevons, Peirce, Macfarlane, Venn 126
1.12. Propositions numerically definite 142
1.13. Notes to Chapter 1 145
Chapter 2. Formalization of Boole's Logic 148
2.1. The calculus of multisets. Axioms for multiset algebra 148
2.2. Boole's Algebra (SM algebras) 152
2.3. Idempotents. Boolean multiset terms 154
2.4. Boole's notion of "uninterpretable" 162
2.5. The indefinite class symbol . 165
2.6. The solution of Boolean multiset equations for an unknown 168
2.7. Boolean equations and division 173
2.8. Additional remarks on 0/0 and 1/0' 179
2.9. The partial algebra of Boolean quotients 181
2.10. Notes to Chapter 2 183
Chapter 3. Boole's Propositional Logic 186
3.0. The two theories 186
3.1. The calculus of elective symbols (operators) 187
3.2. The logic of hypotheticals 192
3.3. Secondary propositions and "time" 197
3.4. Boole's illustrative examples 202
3.5. Justification of the logic of secondary propositions 215
3.6. A two- to four-valued connective 217
3.7. Notes to Chapter 3 222
PART II. PROBABILITY 226
Chapter 4. Probability From Boole's Viewpoint 228
4.1. Critique of the standard theory 228
4.2. An additional principle 232
4.3. A general method 236
4.4. The problem of "absolute" probabilities I 250
4.5. Boole's method without the mutual independence 254
4.6. Elementary illustrations of Boole's method 256
4.7. Conditions of possible experience. Bounds on the probability of events 269
4.8. Wilbraham's and Peirce's criticisms 279
4.9. Notes to Chapter 4 292
Chapter 5. Boole'S Probability Made Rigorous 298
5.1. Simple and Boole probability algebras, calculi, models 299
5.2. Conditioned-events probability realm 306
5.3. Reprise of Boole's General Problem in Probability 309
5.4. Justification for Boole's solution of the General Problem 314
5.5. Conditions of possible experience -a consistency (solvability) problem 323
5.6. The problem of absolute probabilities II 328
5.7. Boole's General Problem linearly programmed 351
5.8. Notes to Chapter 5 363
Chapter 6. Applications. Probability Logic 368
6.1. Michell's argument and inverse probability 368
6.2. Boole's Challenge Problem 372
6.3. Further problems on causes 381
6.4. Probability of judgements 389
6.5. Combination of testimonies 396
6.6. Probability logic 402
6.7. An extension of probability logic 412
6.8. Notes to Chapter 6 418
Bibliography 424
Index 438

Produkt-Details

• Erscheint lt. Verlag: 1.10.1986
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Allgemeines / Lexika
• Themenwelt: Mathematik / Informatik ► Mathematik ► Logik / Mengenlehre
• Themenwelt: Mathematik / Informatik ► Mathematik ► Statistik
• Themenwelt: Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik
• Themenwelt: Technik
• ISBN-10: 0-08-088005-3 / 0080880053
• ISBN-13: 978-0-08-088005-1 / 9780080880051