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These four forms of propositions bear certain logical relations to each other, as follows:

A close study of these relations, and the symbols expressing them, is necessary for a clear comprehension of the Laws of Opposition stated a little further back, as well as the principles of Conversion which we shall mention a little further on. The following chart, called the Square of Opposition, is also employed by logicians to illustrate the relations between the four classes of propositions:

It will be well for students, at this point, to consider the three following Fundamental Laws of Thought as laid down by the authorities, which are as follows:

Of these laws, Prof. Jevons, a noted authority, says: "Students are seldom able to see at first their full meaning and importance. All arguments may be explained when these self-evident laws are granted; and it is not too much to say that the whole of logic will be plain to those who will constantly use these laws as the key."

INDUCTIVE REASONING

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

REASONING BY INDUCTION

The authorities assume the existence of two kinds of Induction, namely: Perfect Induction; and Imperfect Induction. Other, but similar, terms are employed by different authorities to designate these two classes.

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

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:

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.

THEORY AND HYPOTHESES

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

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.

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

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:

MAKING AND TESTING HYPOTHESES

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. 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."

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."

Jevons says: "In the fourth step , 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 view of the above facts, we shall now proceed to a consideration of that great class of Reasoning known under the term--Deductive Reasoning.

DEDUCTIVE REASONING

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

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.

This axiom has been stated in various terms other than those stated above. For instance: "Whatever may be affirmed or denied of the whole, may be denied or affirmed of the parts;" which form is evidently derived from that used by Hamilton who said: "What belongs, or does not belong, to the containing whole, belongs or does not belong, to each of the contained parts." Aristotle formulated his celebrated Dictum as follows: "Whatever can be predicated affirmatively or negatively of any class or term distributed, can be predicated in like manner of all and singular the classes or individuals contained under it."

Brooks states: "The real reason for the certainty of mathematical reasoning may be stated as follows: First, its ideas are definite, necessary, and exact conceptions of quantity. Second, its definitions, as the description of these ideas are necessary, exact, and indisputable truths. Third, the axioms from which we derive conclusions by comparison are all self-evident and necessary truths. Comparing these exact ideas by the necessary laws of inference, the result must be absolutely true. Or, stated in another way, using these definitions and axioms as the premises of a syllogism, the conclusion follows inevitably. There is no place or opportunity for error to creep in to mar or vitiate our derived truths."

THE SYLLOGISM

This process of reasoning embraces three ideas or objects of thought, in their expression of propositions. It comprises the fundamental or elemental form of reasoning. As Brooks says: "The simplest movement of the reasoning process is the comparing of two objects through their relation to a third." The result of this process is an argument expressed in what is called a Syllogism. Whately says that: "A Syllogism is an argument expressed in strict logical form so that its conclusiveness is manifest from the structure of the expression alone, without any regard to the meaning of the terms." Brooks says: "All reasoning can be and naturally is expressed in the form of the syllogism. It applies to both inductive and deductive reasoning, and is the form in which these processes are presented. Its importance as an instrument of thought requires that it receive special notice."

In order that the nature and use of the Syllogism may be clearly understood, we can do no better than to at once present for your consideration the well-known "Rules of the Syllogism," an understanding of which carries with it a perfect comprehension of the Syllogism itself.

It will be seen that the above Syllogism, whether expressed in words or symbols, is logically valid, because the conclusion must logically follow the premises. And, in this case, the premises being true, it must follow that the conclusion is true. Whately says: "A Syllogism is said to be valid when the conclusion logically follows from the premises; if the conclusion does not so follow, the Syllogism is invalid and constitutes a Fallacy, if the error deceives the reasoner himself; but if it is advanced with the idea of deceiving others it constitutes a Sophism."

The following example will show this arrangement more clearly:

In the Syllogism: "Man is mortal; Socrates is a man; therefore Socrates is mortal," we have the following arrangement: "Mortal," the Major Term; "Socrates," the Minor Term; and "Man," the Middle Term; as follows:

As we have said before, any Syllogism which violates any of the above six syllogisms is invalid and a fallacy.

There are two additional rules which may be called derivative. Any syllogism which violates either of these two derivative rules, also violates one or more of the first six rules as given above in detail.

The principles involved in these two Derivative Rules may be tested by stating Syllogisms violating them. They contain the essence of the other rules, and every syllogism which breaks them will be found to also break one or more of the other rules given.

VARIETIES OF SYLLOGISMS

Then Mrs. Craig made a start for the mountains, taking her household with her, so there were no more opportunities for music. The climate was beginning to tell on Miss Helen and she was so languid and indisposed to effort, that Nan urged her to keep quiet until the rest should be ready to go to the mountains.

So a week passed and then it was decided that all the Corners should go to Myanoshita for a while, and that ended the association with the young men for the time being at least. With the approach of July heat would come the swarms of mosquitoes which started life in rice fields, and with this affliction, added to the humid condition of the atmosphere, the frequent rains and the great dampness, Tokyo promised to be anything but an agreeable summer resort. So Miss Helen and Nan pored over guide-books and decided to make certain journeys by easy stages.

"But," objected Jack who was having a very good time, "we haven't been to Enoshima yet, and I do so want to see those lovely shells."

"Who wants to pick up shells in the pouring rain?" said Jean.

"It doesn't rain every minute," retorted Jack. "There have been some quite pleasant days since we left home."

"But scarcely one since we reached here. I had no idea that Japan was such a moist, unpleasant place."

"Oh, but it is depressing without any sunshine," protested Jean, "and it is so damp all my things are beginning to mould."

"I suppose," remarked Jack who was ready to make capital of any information which came her way, "that is why they wear pongee and crape in these countries; I never thought of it before, but now I see why. Don't you think we might take a day for Enoshima, Aunt Helen, just one day before we go? Even if it rained it wouldn't make so much difference."

"What do you say, Nan?" asked her Aunt Helen.

Nan, who was busy examining a map, traced a line on its surface. "I don't see why we need take a day off to go there specially, when our way leads right past it. Why not stop there over night, or at Kamakura? We always meant to do that, you know, then we could go on the next day. I think it might be the best plan, for it ought to be less tiresome for you and mother."

"Very well, we will decide to do that, for, as you say, Nan, it will be carrying out a former plan and will not be out of our way."

"I shall pray for a pleasant day," said Jack. "I am so glad to find out where it is. If I had known that Myanoshita was in that direction I should have felt easier."

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