The Art of Logical Thinking Part 5

It will be seen that the process of Inductive Reasoning is essentially _a synthetic process_, because it operates in the direction of combining and uniting particular facts or truths into general truths or laws which comprehend, embrace and include them all. As Brooks says: "The particular facts are united by the mind into the general law; the general law embraces the particular facts and binds them together into a unity of principle and thought. Induction is thus a process of thought from the parts to the whole--a synthetic process." It will also be seen that the process of Inductive Reasoning is essentially _an ascending process_, because it ascends from particular facts to general laws; particular truths to universal truths; from the lower to the higher, the narrower to the broader, the smaller to the greater.

Brooks says of Inductive Reasoning: "The relation of induction to deduction will be clearly seen. Induction and Deduction are the converse, the opposites of each other. Deduction derives a particular truth from a general truth; Induction derives a general truth from particular truths. This ant.i.thesis appears in every particular.

Deduction goes from generals to particulars; Induction goes from particulars to generals. Deduction is an a.n.a.lytic process; Induction is a synthetic process. Deduction is a descending process--it goes from the higher truth to the lower truth; Induction is an ascending process--it goes from the lower truth to the higher. They differ also in that Deduction may be applied to necessary truths, while Induction is mainly restricted to contingent truths." Hyslop says: "There have been several ways of defining this process. It has been usual to contrast it with Deduction. Now, deduction is often said to be reasoning from general to particular truths, from the containing to the contained truth, or from cause to effect. Induction, therefore, by contrast is defined as reasoning from the particular to the general, from the contained to the containing, or from effect to cause. Sometimes induction is said to be reasoning from the known to the unknown. This would make deduction, by contrast, reasoning from the unknown to the known, which is absurd. The former ways of representing it are much the better. But there is still a better way of comparing them. Deduction _is reasoning in which the conclusion is contained in the premises_. This is a ground for its cert.i.tude and we commit a fallacy whenever we go beyond the premises as shown by the laws of the distribution of terms. In contrast with this, then, we may call inductive reasoning _the process by which we go beyond the premises in the conclusion_.... The process here is to start from given facts and to infer some other probable facts more general or connected with them. In this we see the process of going beyond the premises. There are, of course, certain conditions which regulate the legitimacy of the procedure, just as there are conditions determining deduction. They are _that the conclusion shall represent the same general kind as the premises_, with a possibility of accidental differences. But it goes beyond the premises in so far as _known_ facts are concerned."

The following example may give you a clearer idea of the processes of Inductive Reasoning:

_First Step._ Preliminary Observation. _Example_: We notice that all the particular _magnets_ which have come under our observation _attract iron_. Our mental record of the phenomena may be stated as: "A, B, C, D, E, F, G, etc., and also X, Y, and Z, all of which are _magnets_, in all observed instances, and at all observed times, _attract iron_."

_Second Step._ The Making of Hypotheses. _Example_: Upon the basis of the observations and experiments, as above stated, and applying the axiom of Inductive Reasoning, that: "What is true of the many, is true of the whole," we feel justified in forming a hypothesis or inference of a general law or truth, applying the facts of the particulars to the general, whole or universal, thus: "_All_ magnets attract iron."

_Third Step._ Deductive Reasoning. _Example_: Picking up a magnet regarding which we have had no experience and upon which we have made no experiments, we reason by the syllogism, as follows: (1) _All_ magnets attract iron; (2) _This thing_ is a magnet; therefore (3) _This thing_ will attract iron. In this we apply the axiom of Deductive Reasoning: "Whatever is true of the whole is true of the parts."

_Fourth Step._ Verification. _Example_: We then proceed to test the hypothesis upon the particular magnet, so as to ascertain whether or not it agrees with the particular facts. If the magnet does not attract iron we know that either our hypothesis is wrong and that _some_ magnets do _not_ attract iron; or else that our _judgment_ regarding that particular "thing" being a magnet is at fault and that it is _not_ a magnet. In either case, further examination, observation and experiment is necessary. In case the particular magnet _does_ attract iron, we feel that we have verified our hypothesis and our judgment.

CHAPTER XII.

REASONING BY INDUCTION

The term "Induction," in its logical usage, is defined as follows: "(a) The process of investigating and collecting facts; and (b) the deducing of an inference from these facts; also (c) sometimes loosely used in the sense of an inference from observed facts." Mill says: "_Induction_, then, is that operation of the mind, by which we infer that what we know to be true in a particular case or cases, will be true in all cases which resemble the former in certain a.s.signable respects. In other words, _Induction_ is the process by which we conclude that what is true of certain individuals of a cla.s.s, is true of the whole cla.s.s, or that what is true at certain times will be true in similar circ.u.mstances at all times."

The _Basis of Induction_ is the axiom that: "_What is true of the many is true of the whole_." Esser, a well known authority, states this axiom in rather more complicated form, as follows: "That which belongs or does not belong to many things of the same kind, belongs or does not belong to all things of the same kind."

This basic axiom of Induction rests upon the conviction that Nature's laws and manifestations are regular, orderly and _uniform_. If we a.s.sume that Nature does not manifest these qualities, then the axiom must fall, and all inductive reason must be fallacious. As Brooks well says: "Induction has been compared to a ladder upon which we ascend from facts to laws. This ladder cannot stand unless it has something to rest upon; and this something is our faith in the constancy of Nature's laws." Some authorities have held that this perception of the uniformity of Nature's laws is in the nature of an _intuitive_ truth, or an inherent law of our intelligence. Others hold that it is in itself an _inductive_ truth, arrived at by experience and observation at a very early age. We are held to have noticed the uniformity in natural phenomena, and almost instinctively infer that this uniformity is continuous and universal.

The authorities a.s.sume the existence of two kinds of Induction, namely: (1) Perfect Induction; and (2) Imperfect Induction. Other, but similar, terms are employed by different authorities to designate these two cla.s.ses.

_Perfect Induction_ necessitates a knowledge of _all_ the particulars forming a cla.s.s; that is, _all_ the individual objects, persons, things or facts comprising a cla.s.s must be known and enumerated in this form of Induction. For instance, if we _knew positively_ all of Brown's children, and that their names were John, Peter, Mark, Luke, Charles, William, Mary and Susan, respectively; and that each and every one of them were freckled and had red hair; then, in that case, instead of simply _generalizing_ and stating that: "John, Peter, Mark, Luke, Charles, William, Mary and Susan, who are _all_ of Brown's children, are freckled and have red hair," we would save words, and state the inductive conclusion: "All Brown's children are freckled and have red hair." It will be noticed that in this case _we include in the process only what is stated in the premise itself_, and we do not extend our inductive process beyond the actual data upon which it is based. This form of Induction is sometimes called "Logical Induction," because the inference is a logical necessity, without the possibility of error or exception. By some authorities it is held not to be Induction at all, in the strict sense, but little more than a simplified form of enumeration.

In actual practice it is seldom available, for it is almost impossible for us to know all the particulars in inferring a general law or truth.

In view of this difficulty, we fall back upon the more practical form of induction known as:

_Imperfect Induction_, or as it is sometimes called "Practical Induction," by which is meant the inductive process of reasoning in which we a.s.sume that the particulars or facts actually known to us correctly represent those which are not actually known, and hence the whole cla.s.s to which they belong. In this process it will be seen that _the conclusion extends_ beyond the data upon which it is based. In this form of Induction we must actually employ the principle of the axiom: "What is true of the many is true of the whole"--that is, must _a.s.sume_ it to be a fact, not because we _know_ it by actual experience, but because we infer it from the axiom which also agrees with past experience. The conclusion arrived at may not always be true in its fullest sense, as in the case of the conclusion of Perfect Induction, but is the result of an inference based upon a principle which gives us a reasonable right to a.s.sume its truth in absence of better knowledge.

In considering the actual steps in the process of Inductive Reasoning we can do no better than to follow the cla.s.sification of Jevons, mentioned in the preceding chapter, the same being simple and readily comprehended, and therefore preferable in this case to the more technical cla.s.sification favored by some other authorities. Let us now consider these four steps.

_First Step._ Preliminary observation. It follows that without the experience of oneself or of others in the direction of observing and remembering particular facts, objects, persons and things, we cannot hope to acquire the preliminary facts for the generalization and inductive inference necessary in Inductive Reasoning. It is necessary for us to form a variety of clear Concepts or ideas of facts, objects, persons and things, before we may hope to generalize from these particulars. In the chapters of this book devoted to the consideration of Concepts, we may see the fundamental importance of the formation and acquirement of correct Concepts. Concepts are the fundamental material for correct reasoning. In order to produce a perfect finished product, we must have perfect materials, and a sufficient quant.i.ty of them. The greater the knowledge one possesses of the facts and objects of the outside world, the better able is he to reason therefrom. Concepts are the raw material which must feed the machinery of reasoning, and from which the final product of perfected thought is produced. As Halleck says: "There must first be a presentation of materials. Suppose that we wish to form the concept _fruit_. We must first perceive the different kinds of fruit--cherry, pear, quince, plum, currant, apple, fig, orange, etc. Before we can take the next step, we must be able to form distinct and accurate images of the various kinds of fruit. If the concept is to be absolutely accurate, not one kind of fruit must be overlooked.

Practically this is impossible; but many kinds should be examined. Where perception is inaccurate and stinted, the products of thought cannot be trustworthy. No building is firm if reared on insecure foundations."

In the process of Preliminary Observation, we find that there are two ways of obtaining a knowledge of the facts and things around us. These two ways are as follows:

I. By _Simple Observation_, or the perception of the happenings which are manifested without our interference. In this way we perceive the motion of the tides; the movement of the planets; the phenomena of the weather; the pa.s.sing of animals, etc.

II. By the _Observation of Experiment_, or the perception of happenings in which we interfere with things and then observe the result. An _experiment_ is: "A trial, proof, or test of anything; an act, operation, or process designed to discover some unknown truth, principle or effect, or to test some received or reputed truth or principle."

Hobbes says: "To have had many _experiments_ is what we call _experience_." Jevons says: "Experimentation is observation with something more; namely, regulation of the things whose behavior is to be observed. The advantages of experiment over mere observation are of two kinds. In the first place, we shall generally know much more certainly and accurately with what we are dealing, when we make experiments than when we simply observe natural events.... It is a further advantage of artificial experiments, that they enable us to discover entirely new substances and to learn their properties.... It would be a mistake to suppose that the making of an experiment is inductive reasoning, and gives us without further trouble the laws of nature. _Experiments only give us the facts upon which we may afterward reason...._ Experiments then merely give facts, and it is only by careful reasoning that we can learn when the same facts will be observed again. _The general rule is that the same causes will produce the same effects._ Whatever happens in one case will happen in all like cases, provided that they are really like, and not merely apparently so.... When we have by repeated experiments tried the effect which all the surrounding things might have on the result, we can then reason with much confidence as to similar results in similar circ.u.mstances.... In order that we may, from our observations and experiments, learn the law of nature and become able to foresee the future, we must perform the process of generalization. To generalize is to draw a general law from particular cases, and to infer that what we see to be true of a few things is true of the whole genus or cla.s.s to which these things belong. It requires much judgment and skill to generalize correctly, because everything depends upon the number and character of the instances about which we reason."

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.

CHAPTER XIII.

THEORY AND HYPOTHESES

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

A _Hypothesis_ is: "A supposition, proposition or principle _a.s.sumed or taken for granted_ in order to draw a conclusion or inference in proof of the point or question; a proposition a.s.sumed 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 a.s.sumption_, whose truth must be tested by observed facts. The a.s.sumption 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 a.s.sumes 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 a.s.sociated 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 At.w.a.ter 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."

At.w.a.ter 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 cla.s.s, and warrant a valid inductive conclusion."

Jevons gives us the following valuable rules:

I. "Whenever we can alter the quant.i.ty 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 quant.i.ty 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 cla.s.s 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 circ.u.mstances, 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 a.s.sist 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_."

CHAPTER XIV.