Chapter XXIII. Of Approximate Generalizations, and Probable Evidence. § 1. The inferences called probable, rest on approximate generalizations 124 2. Approximate generalizations less useful in science than in life 124 3. In what cases they may be resorted to 126 4. In what manner proved 127 5. With what precautions employed 130 6. The two modes of combining probabilities 131 7. How approximate generalizations may be converted into accurate generalizations equivalent to them 136 [Pg viii] Chapter XXIV. Of the Remaining Laws of Nature. § 1. Propositions which assert mere existence 139 2. Resemblance, considered as a subject of science 141 3. The axioms and theorems of mathematics comprise the principal laws of resemblance 143 4. —and those of order in place, and rest on induction by simple enumeration 145 5. The propositions of arithmetic affirm the modes of formation of some given number 146 6. Those of algebra affirm the equivalence of different modes of formation of numbers generally 151 7. The propositions of geometry are laws of outward nature 154 8. Why geometry is almost entirely deductive 156 9. Function of mathematical truths in the other sciences, and limits of that function 158 Chapter XXV. Of the Grounds of Disbelief. § 1. Improbability and impossibility 161 2. Examination of Hume's doctrine of miracles 162 3. The degrees of improbability correspond to differences in the nature of the generalization with which an assertion conflicts 166 4. A fact is not incredible because the chances are against it 170 5. Are coincidences less credible than other facts? 172 6. An opinion of Laplace examined 175 BOOK IV. OF OPERATIONS SUBSIDIARY TO INDUCTION. Chapter I. Of Observation and Description. § 1. Observation, how far a subject of logic 183 2. A great part of what seems observation is really inference 184 3. The description of an observation affirms more than is contained in the observation 187 4. —namely an agreement among phenomena; and the comparison of phenomena to ascertain such agreements is a preliminary to induction 190 [Pg ix] Chapter II. Of Abstraction, or the Formation of Conceptions. § 1. The comparison which is a preliminary to induction implies general conceptions 193 2. —but these need not be pre-existent 194 3. A general conception, originally the result of a comparison, becomes itself the type of comparison 198 4. What is meant by appropriate conceptions 200 5. —and by clear conceptions 203 6. Further illustration of the subject 205 Chapter III. Of Naming, as subsidiary to Induction. § 1. The fundamental property of names as an instrument of thought 209 2. Names are not indispensable to induction 210 3. In what manner subservient to it 211 4. General names not a mere contrivance to economize the use of language 213