(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.
15. TRADUCTION.
It may have been noted by the student that “perfect induction” is not induction at all according to the definition; viz.: Inductive reasoning is reasoning from less general premises to a more general conclusion. Referring to the first illustration of the previous section it is apparentthat the conclusion is no broader than the premises. Ostensibly, the conclusion is a mere summary, or a generalization of the facts mentioned in the premises. Moreover perfect induction does not readily conform to the definition of deductive reasoning, as in this the movement must be from the more general to the less. We are thus forced to the conclusion that perfect induction is a form of a third type of reasoning which is known under the cognomen of traduction. This is from the Latin trans, and ducere meaning to lead across. Definition: Traductive reasoning is reasoning to a conclusion which is neither less general nor more general than the premises.
Aside from the case of perfect induction there are other types which well illustrate traduction. These are: First. Reasoning from particular (or individuals) to particular (or individuals).
ILLUSTRATION:
Highland Street is the longest street in Jamaica,
Highland Street is not so long as Broadway of New York City,
∴ The longest street of Jamaica is not so long as Broadway of New York City.