14. PERFECT INDUCTION.
As has been indicated under simple enumeration, a perfect induction is one in which the premises enumerate all the instances denoted by the conclusion.
ILLUSTRATIONS:
(1) A, B, C, D, and E are all Reactionaries,
(All) The members of the committee are A, B, C, D, and E,
∴ (All) The members of the committee are Reactionaries.
(2) John, James, Albert, and Peter all have perfect eyesight,
John, James, Albert, and Peter are all the boys of my family,
∴ All the boys of my family have perfect eyesight.
(3) The first, second, and third groups are up to grade,
The first, second, and third groups include all of the children in my room,
Hence all the children in my room are up to grade.
Because the conclusion of a perfect induction gives nothing new—nothing but what is found in the premises, some claim that the process is practically valueless. Fromthe viewpoint of the discoverer this position is well taken; yet to universalize particular observations puts the knowledge in compact, usable form, and saves one the trouble of returning each time to the consideration of each particular. Thus as a process which leads to verified universals, perfect induction is a time saver. In the second place it was the method used by Socrates when he desired to lead up to a definition or some other general truth. The Sophists were given to a careless use of the “inductive hazard”; they were prone to generalize from one or two particulars, or what is worse, to establish a generalization and then attempt to fit the particular instances to it. This led to a superficiality which the Great Pagan Educator abhorred. The fact that perfect induction was the method used by Socrates to counteract the teachings of the Sophists, is sufficient vindication for its use in discouraging the indefensible assumptions of to-day, and in inspiring warrantable generalizations based on accurate observation.
In the school room with classes addicted to careless, inaccurate work, to accept nothing but a perfectly induced generalization, when this is feasible, is a most valuable lesson. For example, the teacher may not accept the generalization that all of the “first class” cities of the U. S. are located on navigable waterways, until the pupils have investigated the waterway conditions of every city belonging to the class. On the other hand, there may be individual cases of “cocksureness” which need attention. The teacher can do little for the “know-it-all youngster” until he pricks the bubble of conceit. This may be accomplishedby allowing the youth to draw a generalization, which seems to meet all the requirements of truth arrived at by means of an imperfect induction; then without warning let the teacher give an instance which will show the generalization to be false. This involves what Socrates termed the “torpedo’s shock.” To illustrate: Consider the “prime number” formula given by Jevons. In deriving this, direct the class to add 2 to its square, and to this sum add 41. Give similar directions relative to numbers 3, 4, 7 and 10. Indicating the work as directed, would give the following:
(1) 2 + 2² + 41 = 47
(2) 3 + 3² + 41 = 53
(3) 4 + 4² + 41 = 61
(4) 7 + 7² + 41 = 97
(5) 10 + 10² + 41 = 151
A question or two will make apparent the fact that all the results are prime numbers, and then the generalization may be drawn; namely, X + X² + 41 = prime number. Now without warning, but under the assumption that you desire to test deductively the general formula, let X = 40. This gives (40 + 40² + 41) 1681, which is the square of 41 and is, therefore, not a prime number.