Calculating Confidence for Mill’s Methods of Inference

J.S. Mill provides three methods [i] of inductive inference in his 1843 work on the philosophy of science, A System of Logic, Ratiocinative and Inductive. Although this work is today held [ii] to be a naïvely pre-Bayesian theoretical endorsement of simple enumerative induction, it actually innovates on traditional inference by iteratively restricting premises toward higher predictive precision. More importantly, in place of causal relations [iii], the three methods translate inferences into the basic logical relations of implication, material equivalence, and functional predication between phenomena. From these inferences could be derived the theorems, definitions, and models of a sound and complete scientific system independent from any notion of causation.

Below, for each method I give a formula for exactly calculating confidence in any particular inference of that method’s logical relation.