CHAPTER XVII

COMBINATION OF INDUCTION WITH DEDUCTION

Sec. 1.  We have now reviewed Mill’s five Canons of Inductive Proof.  At bottom, as he observes, there are only two, namely, Agreement and Difference:  since the Double Method, Variations and Residues are only special forms of the other two.  Indeed, in their function of proof, they are all reducible to one, namely, Difference; for the cogency of the method of Agreement (as distinguished from a simple enumeration of instances agreeing in the coincidence of a supposed cause and its effect), depends upon the omission, in one instance after another, of all other circumstances; which omission is a point of difference.

The Canons are an analysis of the conditions of proving directly (where possible), by means of observation or experiment, any proposition that predicates causation.  But if we say ’by means of observation or experiment,’ it is not to be understood that these are the only means and that nothing else is involved; for it has been shown that the Law of Causation is itself an indispensable foundation of the evidence.  In fact Inductive Logic may be considered as having a purely formal character.  It consists (1) in a statement of the Law of Cause and Effect; (2) in certain immediate inferences from this Law, expanded into the Canons; (3) in the syllogistic application of the Canons to special predications of causation by means of minor premises, showing that certain instances satisfy the Canons.

At the risk of some pedantry, we may exhibit the process as follows (cf. Prof.  Ray’s Logic:  Appendix D): 

Whatever relation of events has certain marks is a case of causation;

    The relation A:  p has some or all of these marks (as shown
    by observation and by the conformity of instances to such or
    such a Canon): 

Therefore, the relation A:  p is a case of causation.  Now, the parenthesis, “as shown by the conformity, etc.,” is an adscititious member of an Epicheirema, which may be stated, as a Prosyllogism, thus: 

If an instance, etc. (Canon of Difference);

The instances A B C B C are of the kind required: 
p q r’ q r

Therefore, A, present where p occurs and absent where it
does not occur, is an indispensable antecedent of p.

Such is the bare Logic of Induction:  so that, strictly speaking, observation or experiment is no part of the logic, but a means of applying the logic to actual, that is, not merely symbolical, propositions.  The Formal Logic of Induction is essentially deductive; and it has been much questioned whether any transition from the formal to the material conditions of proof is possible.  As long as we are content to illustrate the Canons with symbols, such as A and p, all goes well; but can we in any actual investigation show that the relevant facts or ‘instances’ correspond with those symbols?