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Having seen that the first step in Inductive Reasoning is Preliminary Observation, let us now consider the next steps in which we may see what we do with the facts and ideas which we have acquired by this Observation and Experiment.

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Following Jevons' classification, we find that the Second Step in Inductive Reasoning is that called "The Making of Hypotheses."

A Hypothesis is: "A supposition, proposition or principle assumed or taken for granted in order to draw a conclusion or inference in proof of the point or question; a proposition assumed or taken for granted, though not proved, for the purpose of deducing proof of a point in question." It will be seen that a Hypothesis is merely held to be possibly or probably true , and not certainly true ; it is in the nature of a working assumption , whose truth must be tested by observed facts. The assumption may apply either to the cause of things, or to the laws which govern things. Akin to a hypothesis, and by many people confused in meaning with the latter, is what is called a Theory.

A Theory is: "A verified hypothesis; a hypothesis which has been established as, apparently, the true one." An authority says " Theory is a stronger word than hypothesis . A theory is founded on principles which have been established on independent evidence. A hypothesis merely assumes the operation of a cause which would account for the phenomena, but has not evidence that such cause was actually at work. Metaphysically, a theory is nothing but a hypothesis supported by a large amount of probable evidence." Brooks says: "When a hypothesis is shown to explain all the facts that are known, these facts being varied and extensive, it is said to be verified, and becomes a theory. Thus we have the theory of universal gravitation, the Copernican theory of the solar system, the undulatory theory of light, etc., all of which were originally mere hypotheses. This is the manner in which the term is usually employed in the inductive philosophy; though it must be admitted that it is not always used in this strict sense. Discarded hypotheses are often referred to as theories; and that which is actually a theory is sometimes called a hypothesis."

The steps by which we build up a hypothesis are numerous and varied. In the first place we may erect a hypothesis by the methods of what we have described as Perfect Induction, or Logical Induction. In this case we proceed by simple generalization or simple enumeration. The example of the freckled, red-haired children of Brown, mentioned in a previous chapter, explains this method. It requires the examination and knowledge of every object or fact of which the statement or hypothesis is made. Hamilton states that it is the only induction which is absolutely necessitated by the laws of thought. It does not extend further than the plane of experience. It is akin to mathematical reasoning.

Far more important is the process by which hypotheses are erected by means of inferences from Imperfect Induction, by which we reason from the known to the unknown, transcending experience, and making true inductive inferences from the axiom of Inductive Reasoning. This process involves the subject of Causes. Jevons says: "The cause of an event is that antecedent, or set of antecedents, from which the event always follows. People often make much difficulty about understanding what the cause of an event means, but it really means nothing beyond the things that must exist before in order that the event shall happen afterward ."

Causes are often obscure and difficult to determine. The following five difficulties are likely to arise: I. The cause may be out of our experience, and is therefore not to be understood; II. Causes often act conjointly, so that it is difficult to discover the one predominant cause by reason of its associated causes; III. Often the presence of a counteracting, or modifying cause may confuse us; IV. Often a certain effect may be caused by either of several possible causes; V. That which appears as a cause of a certain effect may be but a co-effect of an original cause.

Mill formulated several tests for ascertaining the causal agency in particular cases, in view of the above-stated difficulties. These tests are as follows: (1) The Method of Agreement; (2) The Method of Difference; (3) The Method of Residues; and (4) The Method of Concomitant Variations. The following definitions of these various tests are given by Atwater as follows:

Method of Agreement : "If, whenever a given object or agency is present without counteracting forces, a given effect is produced, there is a strong evidence that the object or agency is the cause of the effect."

Method of Difference : "If, when the supposed cause is present the effect is present, and when the supposed cause is absent the effect is wanting, there being in neither case any other agents present to effect the result, we may reasonably infer that the supposed cause is the real one."

Method of Residue : "When in any phenomena we find a result remaining after the effects of all known causes are estimated, we may attribute it to a residual agent not yet reckoned."

Method of Concomitant Variations : "When a variation in a given antecedent is accompanied by a variation of a given consequent, they are in some manner related as cause and effect."

Atwater adds: "Whenever either of these criteria is found free from conflicting evidence, and especially when several of them concur, the evidence is clear that the cases observed are fair representatives of the whole class, and warrant a valid inductive conclusion."

Jevons gives us the following valuable rules:

I. "Whenever we can alter the quantity of the things experimented on, we can apply a rule for discovering which are causes and which are effects , as follows: We must vary the quantity of one thing, making it at one time greater and at another time less, and if we observe any other thing which varies just at the same times, it will in all probability be an effect ."

II. "When things vary regularly and frequently, there is a simple rule, by following which we can judge whether changes are connected together as causes and effects , as follows: Those things which change in exactly equal times are in all likelihood connected together."

III. "It is very difficult to explain how it is that we can ever reason from one thing to a class of things by generalization, when we cannot be sure that the things resemble each other in the important points .... Upon what grounds do we argue? We have to get a general law from particular facts. This can only be done by going through all the steps of inductive reasoning. Having made certain observations, we must frame hypotheses as to the circumstances, or laws from which they proceed. Then we must reason deductively; and after verifying the deductions in as many cases as possible, we shall know how far we can trust similar deductions concerning future events.... It is difficult to judge when we may, and when we may not, safely infer from some things to others in this simple way, without making a complete theory of the matter. The only rule that can be given to assist us is that if things resemble each other in a few properties only, we must observe many instances before inferring that these properties will always be joined together in other cases ."

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The older philosophers and logicians were often at a loss how to reasonably account for the origin of hypotheses. It will be seen, after giving the matter a little thought, that the actual formation of the hypothesis is more than a mere grouping together or synthesis of facts or ideas—there is another mental process which actually evolves the hypothesis or theory—which gives a possible reason . What is this mental process? Let us consider the matter. Brooks well says: "The hypotheses of science originate in what is called anticipation. They are not the result of a mere synthesis of facts, for no combination of facts can give the law or cause. We do not see the law; we see the facts and the mind thinks the law . By the power of anticipation, the mind often leaps from a few facts to the cause which produces them or the law which governs them. Many hypotheses were but a happy intuition of the mind . They were the result of what La Place calls 'a great guess,' or what Plato so beautifully designates as 'a sacred suspicion of truth.' The forming of hypotheses requires a suggestive mind, a lively fancy, a philosophic imagination, that catches a glimpse of the idea through the form, or sees the law standing behind the fact."