The Art of Logical Thinking; Or, The Laws of Reasoning

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."

CHAPTER XIV.

MAKING AND TESTING HYPOTHESES

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."

The student of The New Psychology sees in the mental operation of the forming of the hypothesis – "the mind thinking the law" – but an instance of the operation of the activities of the Subconscious Mind, or even the Superconscious Mind. (See the volume on the Subconscious Mind in this series.) Not only does this hypothesis give the explanation which the old psychology has failed to do, but it agrees with the ideas of others on the subject as stated in the above quotation from Brooks; and moreover agrees with many recorded instances of the formation of great hypotheses. Sir Wm. Hamilton discovered the very important mathematical law of quaternions while walking one day in the Dublin Observatory. He had pondered long on the subject, but without result. But, finally, on that eventful day he suddenly "felt the galvanic circle of thought" close, and the result was the realization of the fundamental mathematical relations of the problem. Berthelot, the founder of Synthetic Chemistry, has testified that the celebrated experiments which led to his remarkable discoveries were seldom the result of carefully followed lines of conscious thought or pure reasoning processes; but, instead, came to him "of their own accord," so to speak, "as from a clear sky." In these and many other similar instances, the mental operation was undoubtedly purely subjective and subconscious. Dr. Hudson has claimed that the "Subjective Mind" cannot reason inductively, and that its operations are purely and distinctly deductive, but the testimony of many eminent scientists, inventors and philosophers is directly to the contrary.

In this connection the following quotation from Thomson is interesting: "The system of anatomy which has immortalized the name of Oken is the consequence of a flash of anticipation which glanced through his mind when he picked up in a chance walk the skull of a deer, bleached and disintegrated by the weather, and exclaimed after a glance, 'It is part of a vertebral column!' When Newton saw the apple fall, the anticipatory question flashed through his mind, 'Why do not the heavenly bodies fall like this apple?' In neither case had accident any important share; Newton and Oken were prepared by the deepest previous study to seize upon the unimportant fact offered to them, and to show how important it might become; and if the apple and the deer-skull had been wanting, some other falling body, or some other skull, would have touched the string so ready to vibrate. But in each case there was a great step of anticipation; Oken thought he saw a type of the whole skeleton in a single vertebra, while Newton conceived at once that the whole universe was full of bodies tending to fall… The discovery of Goethe, which did for the vegetable kingdom what Oken did for the animal, that the parts of a plant are to be regarded as metamorphosed leaves, is an apparent exception to the necessity of discipline for invention, since it was the discovery of a poet in a region to which he seemed to have paid no especial or laborious attention. But Goethe was himself most anxious to rest the basis of this discovery upon his observation rather than his imagination, and doubtless with good reason… As with other great discoveries, hints had been given already, though not pursued, both of Goethe's and Oken's principles. Goethe left his to be followed up by others, and but for his great fame, perhaps his name would never have been connected with it. Oken had amassed all the materials necessary for the establishment of his theory; he was able at once to discover and conquer the new territory."

It must not be supposed, however, that all hypotheses flashing into the field of consciousness from the Subconsciousness, are necessarily true or correct. On the contrary many of them are incorrect, or at least only partially correct. The Subconsciousness is not infallible or omniscient – it merely produces results according to the material furnished it. But even these faulty hypotheses are often of value in the later formation of a correct one. As Whewell says: "To try wrong guesses is with most persons the only way to hit upon right ones." Kepler is said to have erected at least twenty hypotheses regarding the shape of the earth's orbit before he finally evolved the correct one. As Brooks says: "Even incorrect hypotheses may be of use in scientific research, since they may lead to more correct suppositions." The supposition of the circular motions of the heavenly bodies around the earth as a center, which lead to the conception of epicycles, etc., and at last to the true theory is an illustration of this. So the 'theory of phlogiston' in chemistry, made many facts intelligible, before the true one of 'oxidation' superseded it. And so, as Thomson says, "with the theory that 'Nature abhors a vacuum,' which served to bring together so many cognate facts not previously considered as related. Even an incorrect conception of this kind has its place in science, so long as it is applicable to the facts; when facts occur which it cannot explain, we either correct it or replace it with a new one. The pathway of science, some one remarks, is strewn with the remains of discarded hypotheses."

Halleck says regarding the danger of hasty inference: "Men must constantly employ imperfect induction in order to advance; but great dangers attend inductive inferences made from too narrow experience. A child has experience with one or two dogs at his home. Because of their gentleness, he argues that all dogs are gentle. He does not, perhaps, find out the contrary until he has been severely bitten. His induction was too hasty. He had not tested a sufficiently large number of dogs to form such a conclusion. From one or two experiences with a large crop in a certain latitude, a farmer may argue that the crop will generally be profitable, whereas it may not again prove so for years. A man may have trusted a number of people and found them honest. He concludes that people as a rule are honest, trusts a certain dishonest man, and is ruined. The older people grow, the more cautious they generally become in forming inductive conclusions. Many instances are noted and compared; but even the wisest sometimes make mistakes. It once was a generally accepted fact that all swans were white. Nobody had ever seen a dark swan, and the inference that all swans were white was regarded as certainly true. Black swans were, however, found in Australia."

Brooks says regarding the probability of hypotheses: "The probability of a hypothesis is in proportion to the number of facts and phenomena it will explain. The larger the number of facts and phenomena that it will satisfactorily account for, the greater our faith in the correctness of our supposition… If there is more than one hypothesis in respect to the facts under consideration, that one which accounts for the greatest number of facts is the most probable… In order to verify a hypothesis it must be shown that it will account for all the facts and phenomena. If these facts are numerous and varied, and the subject is so thoroughly investigated that it is quite certain that no important class of facts has been overlooked, the supposition is regarded as true, and the hypothesis is said to be verified. Thus the hypothesis of the 'daily rotation' of the earth on its axis to account for the succession of day and night is accepted as absolutely true. This is the view taken by Dr. Whewell and many other thinkers in respect to the verification of a hypothesis. Some writers, however, as Mill and his school, maintain that in order to verify a hypothesis, we must show not only that it explains all the facts and phenomena, but that there is no other possible hypothesis which will account for them… The former view of verification is regarded as the correct one. By the latter view, it is evident that a hypothesis could never be verified."

Jevons says: "In the fourth step (verification), we proceed to compare these deductions with the facts already collected, or when necessary and practicable, we make new observations and plan new experiments, so as to find out whether the hypothesis agrees with nature. If we meet with several distinct disagreements between our deductions and our observations, it will become likely that the hypothesis is wrong, and we must then invent a new one. In order to produce agreement it will sometimes be enough to change the hypothesis in a small degree. When we get hold of a hypothesis which seems to give results agreeing with a few facts, we must not at once assume that it is certainly correct. We must go on making other deductions from it under various circumstances, and, whenever it is possible, we ought to verify these results, that is, compare them with facts observed through the senses. When a hypothesis is shown in this way to be true in a great many of its results, especially when it enables us to predict what we should never otherwise have believed or discovered, it becomes certain that the hypothesis itself is a true one… Sometimes it will happen that two or even three quite different hypotheses all seem to agree with certain facts, so that we are puzzled which to select… When there are thus two hypotheses, one as good as the other, we need to discover some fact or thing which will agree with one hypothesis and not with the other, because this immediately enables us to decide that the former hypothesis is true and the latter false."

In the above statements regarding the verification of hypotheses we see references made to the testing of the latter upon the "facts" of the case. These facts may be either the observed phenomena or facts apparent to the perception, or else facts obtained by deductive reasoning. The latter may be said to be facts which are held to be true if the hypothesis be true. Thus if we erect the hypothesis that "All men are mortal," we may reason deductively that it will follow that each and every thing that is a man must die sooner or later. Then we test our hypotheses upon each and every man whom we may subject to observation and experiment. If we find a single man who does not die, then the test disproves our hypotheses; if on the contrary all men (the "facts" in the case) prove to be mortal, then is our hypotheses proven or established. The deductive reasoning in this case is as follows: "If so-and-so is true regarding such-and-such a class; and if this particular thing belongs to that class; then it will follow that so-and-so is true regarding this particular thing." This argument is expressed in what is called a Hypothetical Proposition (see Chapter IX), the consideration of which forms a part of the general subject of Deductive Reasoning. Therefore as Jevons has said, "Deductive Reasoning is the Third Step in Inductive Reasoning, and precedes Verification", which we have already considered. Halleck says: "After Induction has classified certain phenomena and thus given us a major premise, we may proceed deductively to apply the inference to any new specimen that can be shown to belong to that class. Induction hands over to deduction a ready-made major premise… Deduction takes that as a fact, making no inquiry about its truth… Only after general laws have been laid down, after objects have been classified, after major premises have been formed, can deduction be employed."

In view of the above facts, we shall now proceed to a consideration of that great class of Reasoning known under the term – Deductive Reasoning.

CHAPTER XV.

DEDUCTIVE REASONING

We have seen that there are two great classes of reasoning, known respectively, as (1) Inductive Reasoning, or the discovery of general truth from particular truths; and (2) Deductive Reasoning, or the discovery of particular truths from general truths.

As we have said, Deductive Reasoning is the process of discovering particular truths from a general truth. Thus from the general truth embodied in the proposition "All horses are animals," when it is considered in connection with the secondary proposition that "Dobbin is a horse," we are able to deduce the particular truth that: "Dobbin is an animal." Or, in the following case we deduce a particular truth from a general truth, as follows: "All mushrooms are good to eat; this fungus is a mushroom; therefore, this fungus is good to eat." A deductive argument is expressed in a deductive syllogism.

Jevons says regarding the last stated illustration: "Here are three sentences which state three different facts; but when we know the two first facts, we learn or gather the third fact from the other two. When we thus learn one fact from other facts, we infer or reason, and we do this in the mind. Reasoning thus enables us to ascertain the nature of a thing without actual trial. If we always needed to taste a thing before we could know whether it was good to eat or not, cases of poisoning would be alarmingly frequent. But the appearance and peculiarities of a mushroom may be safely learned by the eye or the nose, and reasoning upon this information and the fact already well known, that mushrooms are good to eat, we arrive without any danger or trouble at the conclusion that the particular fungus before us is good to eat. To reason, then, is to get some knowledge from other knowledge."

The student will recognize that Deductive Reasoning is essentially an analytic process, because it operates in the direction of analyzing a universal or general truth into its particulars – into the particular parts which are included within it – and asserting of them that "what is true of the general is true of the particular." Thus in the general truth that "All men are mortal," we see included the particular truth that "John Smith is mortal" – John Smith having been discovered to be a man. We deduce the particular truth about John Smith from the general truth about "all men." We analyze "all men" and find John Smith to be one of its particular parts. Therefore, "Deduction is an inference from the whole to its parts; that is, an analytic process."

The student will also recognize that Deductive Reasoning is essentially a descending process, because it operates in the direction of a descent from the universal to the particular; from the higher to the lower; from the broader to the narrower. As Brooks says: "Deduction descends from higher truths to lower truths, from laws to facts, from causes to phenomena, etc. Given the law, we can by deduction descend to the facts that fall under the law, even if we have never before seen the facts; and so from the cause we may pass down to observed and even unknown phenomena."

The general truths which are used as the basis of Deductive Reasoning are discovered in several ways. The majority arise from Inductive Reasoning, based upon experience, observation and experiment. For instance in the examples given above, we could not truthfully assert our belief that: "All horses are animals" unless we had previously studied both the horse and animals in general. Nor without this study could we state that "Dobbin is a horse." Nor could we, without previous study, experience and experiment truthfully assert that: "All mushrooms are good to eat;" or that "this fungus is a mushroom;" and that "therefore, this fungus is good to eat." Even as it is, we must be sure that the fungus really is a mushroom, else we run a risk of poisoning ourselves. General truths of this kind are not intuitive, by any means, but are based upon our own experience or the experience of others.

There is a class of general truths which are called intuitive by some authorities. Halleck says of these: "Some psychologists claim that we have knowledge obtained neither through induction nor deduction; that we recognize certain truths the moment we perceive certain objects, without any process of inference. Under the head of intuitive knowledge are classified such cases as the following: We perceive an object and immediately know that it is a time relation, as existing now and then. We are said to have an intuitive concept of time. When we are told that the whole is greater than a part; that things equal to the same thing are equal to each other; that a straight line cannot enclose space, we immediately, or intuitively, recognize the truth of these statements. Attempts at proof do not make us feel surer of their truth… We say that it is self-evident, or that we know the fact intuitively. The axioms of mathematics and logic are said to be intuitive."

Another class of authorities, however, deny the nature of intuitive knowledge of truth, or intuitive truths. They claim that all our ideas arise from sensation and reflection, and that what we call "intuition" is merely the result of sensation and reflection reproduced by memory or heredity. They hold that the intuitions of animals and men are simply the representation of experiences of the race, or individual, arising from the impressions stored away in the subconsciousness of the individual. Halleck states regarding this: "This school likens intuition to instinct. It grants that the young duck knows water instinctively, plunges into it, and swims without learning. These psychologists believe that there was a time when this was not the case with the progenitors of the duck. They had to gain this knowledge slowly through experience. Those that learned the proper aquatic lesson survived and transmitted this knowledge through a modified structure, to their progeny. Those that failed in the lesson perished in the struggle for existence… This school claims that the intuition of cause and effect arose in the same way. Generations of human beings have seen the cause invariably joined to the effect; hence, through inseparable association came the recognition of their necessary sequence. The tendency to regard all phenomena in these relations was with steadily increasing force transmitted by the laws of heredity to posterity, until the recognition of the relationship has become an intuition."

Another class of general truths is merely hypothetical. Hypothetical means "Founded on or including a hypothesis or supposition; assumed or taken for granted, though not proved, for the purpose of deducing proofs of a point in question." The hypotheses and theories of physical science are used as general truths for deductive reasoning. Hypothetical general truths are in the nature of premises assumed in order to proceed with the process of Deductive Reasoning, and without which such reasoning would be impossible. They are, however, as a rule not mere assumptions, but are rather in the nature of assumptions rendered plausible by experience, experiment and Inductive Reasoning. The Law of Gravitation may be considered hypothetical, and yet it is the result of Inductive Reasoning based upon a vast multitude of facts and phenomena.