The Principles of Science

CHAPTER I.. Introduction.

INTRODUCTION.

SECTION PAGE

1. Introduction 1

2. The Powers of Mind concerned in the Creation of Science 4

3. Laws of Identity and Difference 5

4. The Nature of the Laws of Identity and Difference 6

5. The Process of Inference 9

6. Deduction and Induction 11

7. Symbolic Expression of Logical Inference 13

8. Expression of Identity and Difference 14

9. General Formula of Logical Inference 17

10. The Propagating Power of Similarity 20

11. Anticipations of the Principle of Substitution 21

12. The Logic of Relatives 22

CHAPTER II.. Terms.

TERMS.

1. Terms 24

2. Twofold meaning of General Names 25

3. Abstract Terms 27

4. Substantial Terms 28

5. Collective Terms 29

6. Synthesis of Terms 30

7. Symbolic Expression of the Law of Contradiction 31

8. Certain Special Conditions of Logical Symbols 32

CHAPTER III.. Propositions.

PROPOSITIONS.

1. Propositions 36

2. Simple Identities 37

3. Partial Identities 40

4. Limited Identities 42

5. Negative Propositions 43

6. Conversion of Propositions 46

7. Twofold Interpretation of Propositions 47

CHAPTER IV.. Deductive Reasoning.

DEDUCTIVE REASONING.

1. Deductive Reasoning 49

2. Immediate Inference 50

3. Inference with Two Simple Identities 51

4. Inference with a Simple and a Partial Identity 53

5. Inference of a Partial from Two Partial Identities 55

6. On the Ellipsis of Terms in Partial Identities 57

7. Inference of a Simple from Two Partial Identities 58

8. Inference of a Limited from Two Partial Identities 59

9. Miscellaneous Forms of Deductive Inference 60

10. Fallacies 62

CHAPTER V.. Disjunctive Propositions.

DISJUNCTIVE PROPOSITIONS.

1. Disjunctive Propositions 66

2. Expression of the Alternative Relation 67

3. Nature of the Alternative Relation 68

4. Laws of the Disjunctive Relation 71

5. Symbolic Expression of the Law of Duality 73

6. Various Forms of the Disjunctive Proposition 74

7. Inference by Disjunctive Propositions 76

CHAPTER VI.. The Indirect Method of Inference.

THE INDIRECT METHOD OF INFERENCE.

1. The Indirect Method of Inference 81

2. Simple Illustrations 83

3. Employment of the Contrapositive Proposition 84

4. Contrapositive of a Simple Identity 86

5. Miscellaneous Examples of the Method 88

6. Mr. Venn’s Problem 90

7. Abbreviation of the Process 91

8. The Logical Alphabet 94

9. The Logical Slate 95

10. Abstraction of Indifferent Circumstances 97

11. Illustrations of the Indirect Method 98

12. Second Example 99

13. Third Example 100

14. Fourth Example 101

15. Fifth Example 101

16. Fallacies Analysed by the Indirect Method 102

17. The Logical Abacus 104

18. The Logical Machine 107

19. The Order of Premises 114

20. The Equivalence of Propositions 115

21. The Nature of Inference 118

CHAPTER VII.. Induction.

INDUCTION.

1. Induction 121

2. Induction an Inverse Operation 122

3. Inductive Problems for Solution by the Reader 126

4. Induction of Simple Identities 127

5. Induction of Partial Identities 130

6. Solution of the Inverse or Inductive Problem, involving Two Classes 134

7. The Inverse Logical Problem, involving Three Classes 137

8. Professor Clifford on the Types of Compound Statement involving Four Classes 143

9. Distinction between Perfect and Imperfect Induction 146

10. Transition from Perfect to Imperfect Induction 149

BOOK II.. Number, Variety, and Probability.

NUMBER, VARIETY, AND PROBABILITY.

CHAPTER VIII.

PRINCIPLES OF NUMBER.

1. Principles of Number 153

2. The Nature of Numbe 156

3. Of Numerical Abstraction 158

4. Concrete and Abstract Number 159

5. Analogy of Logical and Numerical Terms 160

6. Principle of Mathematical Inference 162

7. Reasoning by Inequalities 165

8. Arithmetical Reasoning 167

9. Numerically Definite Reasoning 168

10. Numerical meaning of Logical Conditions 171

CHAPTER IX.. The Variety of Nature, or the Doctrine of Combinations

THE VARIETY OF NATURE, OR THE DOCTRINE OF COMBINATIONS AND PERMUTATIONS.

1. The Variety of Nature 173

2. Distinction of Combinations and Permutations 177

3. Calculation of Number of Combinations 180

4. The Arithmetical Triangle 182

5. Connexion between the Arithmetical Triangle and the Logical Alphabet 189

6. Possible Variety of Nature and Art 190

7. Higher Orders of Variety 192

CHAPTER X.. Theory of Probability.

THEORY OF PROBABILITY.

1. Theory of Probability 197

2. Fundamental Principles of the Theory 200

3. Rules for the Calculation of Probabilities 203

4. The Logical Alphabet in questions of Probability 205

5. Comparison of the Theory with Experience 206

6. Probable Deductive Arguments 209

7. Difficulties of the Theory 213

CHAPTER XI.. Philosophy of Inductive Inference.

PHILOSOPHY OF INDUCTIVE INFERENCE.

1. Philosophy of Inductive Inference 218

2. Various Classes of Inductive Truths 219

3. The Relation of Cause and Effect 220

4. Fallacious Use of the Term Cause 221

5. Confusion of Two Questions 222

6. Definition of the Term Cause 224

7. Distinction of Inductive and Deductive Results 226

8. The Grounds of Inductive Inference 228

9. Illustrations of the Inductive Process 229

10. Geometrical Reasoning 233

11. Discrimination of Certainty and Probability 235

CHAPTER XII.. The Inductive or Inverse Application of the Theory

THE INDUCTIVE OR INVERSE APPLICATION OF THE THEORY OF PROBABILITY.

1. The Inductive or Inverse Application of the Theory 240

2. Principle of the Inverse Method 242

3. Simple Applications of the Inverse Method 244

4. The Theory of Probability in Astronomy 247

5. The General Inverse Problem 250

6. Simple Illustration of the Inverse Problem 253

7. General Solution of the Inverse Problem 255

8. Rules of the Inverse Method 257

9. Fortuitous Coincidences 261

10. Summary of the Theory of Inductive Inference 265

BOOK III.. Methods of Measurement.

METHODS OF MEASUREMENT.

CHAPTER XIII.

THE EXACT MEASUREMENT OF PHENOMENA.

1. The Exact Measurement of Phenomena 270

2. Division of the Subject 274

3. Continuous quantity 274

4. The Fallacious Indications of the Senses 276

5. Complexity of Quantitative Questions 278

6. The Methods of Accurate Measurement 282

7. Conditions of Accurate Measurement 282

8. Measuring Instruments 284

9. The Method of Repetition 288

10. Measurements by Natural Coincidence 292

11. Modes of Indirect Measurement 296

12. Comparative Use of Measuring Instruments 299

13. Systematic Performance of Measurements 300

14. The Pendulum 302

15. Attainable Accuracy of Measurement 303

CHAPTER XIV.. Units and Standards of Measurement.

UNITS AND STANDARDS OF MEASUREMENT.

1. Units and Standards of Measurement 305

2. Standard Unit of Time 307

3. The Unit of Space and the Bar Standard 312

4. The Terrestrial Standard 314

5. The Pendulum Standard 315

6. Unit of Density 316

7. Unit of Mass 317

8. Natural System of Standards 319

9. Subsidiary Units 320

10. Derived Units 321

11. Provisional Units 323

12. Theory of Dimensions 325

13. Natural Constants 328

14. Mathematical Constants 330

15. Physical Constants 331

16. Astronomical Constants 332

17. Terrestrial Numbers 333

18. Organic Numbers 333

19. Social Numbers 334

CHAPTER XV.. Analysis of Quantitative Phenomena.

ANALYSIS OF QUANTITATIVE PHENOMENA.

1. Analysis of Quantitative Phenomena 335

2. Illustrations of the Complication of Effects 336

3. Methods of Eliminating Error 339

4. Method of Avoidance of Error 340

5. Differential Method 344

6. Method of Correction 346

7. Method of Compensation 350

8. Method of Reversal 354

CHAPTER XVI.. The Method of Means.

THE METHOD OF MEANS.

1. The Method of Means 357

2. Several Uses of the Mean Result 359

3. The Mean and the Average 360

4. On the Average or Fictitious Mean 363

5. The Precise Mean Result 365

6. Determination of the Zero Point 368

7. Determination of Maximum Points 371

CHAPTER XVII.. The Law of Error.

THE LAW OF ERROR.

1. The Law of Error 374

2. Establishment of the Law of Error 375

3. Herschel’s Geometrical Proof 377

4. Laplace’s and Quetelet’s Proof of the Law 378

5. Logical Origin of the Law of Error 383

6. Verification of the Law of Error 383

7. The Probable Mean Result 385

8. The Probable Error of Results 386

9. Rejection of the Mean Result 389

10. Method of Least Squares 393

11. Works upon the Theory of Probability 394

12. Detection of Constant Errors 396

BOOK IV.. Inductive Investigation.

INDUCTIVE INVESTIGATION.

CHAPTER XVIII.

OBSERVATION.

1. Observation 399

2. Distinction of Observation and Experiment 400

3. Mental Conditions of Correct Observation 402

4. Instrumental and Sensual Conditions of Correct Observation 404

5. External Conditions of Correct Observation 407

6. Apparent Sequence of Events 409

7. Negative Arguments from Non-Observation 411

CHAPTER XIX.. Experiment.

EXPERIMENT.

1. Experiment 416

2. Exclusion of Indifferent Circumstances 419

3. Simplification of Experiments 422

4. Failure in the Simplification of Experiments 424

5. Removal of Usual Conditions 426

6. Interference of Unsuspected Conditions 428

7. Blind or Test Experiments 433

8. Negative Results of Experiment 434

9. Limits of Experiment 437

CHAPTER XX.. Method of Variations.

METHOD OF VARIATIONS.

1. Method of Variations 439

2. The Variable and the Variant 440

3. Measurement of the Variable 441

4. Maintenance of Similar Conditions 443

5. Collective Experiments 445

6. Periodic Variations 447

7. Combined Periodic Changes 450

8. Principle of Forced Vibrations 451

9. Integrated Variations 452

CHAPTER XXI.. Theory of Approximation.

THEORY OF APPROXIMATION.

1. Theory of Approximation 456

2. Substitution of Simple Hypotheses 458

3. Approximation to Exact Laws 462

4. Successive Approximations to Natural Conditions 465

5. Discovery of Hypothetically Simple Laws 470

6. Mathematical Principles of Approximation 471

7. Approximate Independence of Small Effects 475

8. Four Meanings of Equality 479

9. Arithmetic of Approximate Quantities 481

CHAPTER XXII.. Quantitative Induction.

QUANTITATIVE INDUCTION.

1. Quantitative Induction 483

2. Probable Connexion of Varying Quantities 484

3. Empirical Mathematical Laws 487

4. Discovery of Rational Formulæ 489

5. The Graphical Method 492

6. Interpolation and Extrapolation 495

7. Illustrations of Empirical Quantitative Laws 499

8. Simple Proportional Variation 501

CHAPTER XXIII.. The Use of Hypothesis.

THE USE OF HYPOTHESIS.

1. The Use of Hypothesis 504

2. Requisites of a good Hypothesis 510

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