Mill's Methods of Induction

The Method of Agreement holds that if two or more instances of a phenomenon share only one antecedent circumstance, that circumstance is the cause or part of the cause. The Method of Difference is more powerful: if a case where the phenomenon occurs and a case where it does not occur differ only in one circumstance — present in the first case, absent in the second — that circumstance is the cause. Between them these two methods underwrite the structure of the controlled experiment, the most reliable instrument of scientific inquiry.

Where perfect agreement and difference are impossible — as in much of medicine, economics, and social inquiry — the Method of Concomitant Variations applies: a phenomenon that varies whenever another phenomenon varies is causally related to it, even if neither can be wholly absent or present. The Method of Residues deducts from any phenomenon the effects already known to be produced by certain antecedents; what remains must be the effect of the remaining antecedents. This method is particularly valuable in astronomy and chemistry, where multiple causes operate simultaneously.

For Mill, these five methods are not mechanical algorithms but the formulation of what careful scientists have always implicitly done. Their value is to make the logic of inductive inference explicit, to reveal its structure, and to identify the conditions under which it succeeds or fails. All knowledge of matters of fact — including the laws of physics, chemistry, and the human mind — rests ultimately on inductive inference of this kind. The methods are the grammar of empirical science.

The five methods are set out in Book III of A System of Logic (1843) under "Of Induction," Chapters 8–10. They have been extensively debated in philosophy of science ever since.