The rules for valid induction and hypothesis deducible from this theory are as follows:—
1. The explaining syllogism, that is to say, the deductive syllogism one of whose premises is inductively or hypothetically inferred from the other and from its conclusion, must be valid.
2. The conclusion is not to be held as absolutely true, but only until it can be shown that, in the case of induction, S′ was taken from some narrower class than M, or, in the case of hypothesis, that P′ was taken from some higher class than M.
3. From the last rule it follows as a corollary that in the case of induction the subject of the premises must be a sum of subjects, and that in the case of hypothesis the predicate of the premises must be a conjunction of predicates.
4. Also, that this aggregate must be of different objects or qualities and not of mere names.
5. Also, that the only principle upon which the instanced subjects or predicates can be selected is that of belonging to M.8
Hence the formulæ are
Induction
S′ S″ S‴, &c. are taken at random as M’s,
S′ S″ S‴, &c. are P;
∴ Any M is probably P.
Hypothesis
Any M is, for instance, P′ P″ P‴, &c.,
S is P′ P″ P‴, &c.;
∴ S is probably M.
§2. Moods and Figures of Probable Inference
It is obvious that the explaining syllogism of an induction or hypothesis may be of any mood or figure.
It would also seem that the conclusion of an induction or hypothesis may be contraposed with one of the premises.
§3. Analogy
The formula of analogy is as follows:—
S’, S”, and S’” are taken at random from such a class that their characters at random are such as P’, P”, P’”.
t is P′, P″, and P’”.
S’, S”, and S’” are q.
∴ t is q.
Such an argument is double. It combines the two following:—
1
S’, S”, S’” are taken as being P’, P”, P’”.
S’, S”, S’” are q.
∴ (By induction) P’, P”, P’” is q.
t is P’, P”, P’”.
∴ (Deductively) t is q.
2
S’, S”, S”’ are, for instance, P’, P”, P’”.
t is P’, P”, P’”.
∴ (By hypothesis) t has the common characters of S’, S”, S’”.
S’, S”, S”‘ are q.
∴ (Deductively) t is q.
Owing to its double character, analogy is very strong with only a moderate number of instances.
§4. Formal Relations of the above Forms of Argument
If we take an identical proposition as the fact to be explained by induction and hypothesis, we obtain the following formulæ.
By Induction
S, S′, S″ are taken at random as being M,
S, S′, S″ have the characters common to S, S′, S″.
∴ Any M has the characters common to S, S′, S″.
By Hypothesis
M is, for instance, P, P′, P″
Whatever is at once P, P′, and P″ is P, P′, P″.
∴ Whatever is at once P, P′, and P” is M.
By means of the substitution thus justified, Induction and Hypothesis can be reduced to the general type of syllogism, thus:—
Induction
S, S′, S″ are taken as M,
S, S′, S″ are P;
∴ Any M is P.
Reduction
S, S′, S″ are P;
Almost any M has the common characters of S, S′, S″.
∴Almost any M is P.
Hypothesis
M is, for instance, P′, P″, P‴,
S is P’, P″, P‴;
∴ S is M.
Reduction
Whatever is, at once, P′, P″, P‴ is like M,
S is P’, P″, P‴;
∴ S is like M.
Induction may, therefore, be defined as argument which assumes that a whole collection, from which a number of instances have been taken at random, has all the common characters of those instances; hypothesis, as an argument which assumes that a term which necessarily involves a certain number of characters, which have been lighted upon as they occurred, and have not been picked out, may be predicated of any object which has all these characters.
There is a resemblance between the transposition of propositions by which the forms of probable inference are derived and the contraposition by which the indirect figures are derived; in the latter case there is a denial or change of modal quality; while in the former there is reduction from certainty to probability, and from the sum of all results to some only, or a change in modal quantity. Thus probable inference is related to apagogical proof, somewhat as the third figure is to the second. Among probable inferences, it is obvious that hypothesis corresponds to the second figure, induction to the third, and analogy to the second-third.
1. Neither of these terms is quite satisfactory. Enthymeme is usually defined as a syllogism with a premise suppressed. This seems to determine the same sphere as the definition I have given; but the doctrine of a suppressed premise is objectionable. The sense of a premise which is said to be suppressed is either conveyed in some way, or it is not. If it is, the premise is not suppressed in any sense which concerns the logician; if it is not, it ceases to be a premise altogether. What I mean by the distinction is this. He who is convinced that Sortes is mortal because he is a man (the latter belief not only being the cause of the former, but also being felt to be so) necessarily says to himself that all such arguments are valid. This genus of argument is either clearly or obscurely recognized.