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

Accordingly, the feature of including objects in a concept or term is called its extension; while the feature of including attributes or qualities is called its intension. It follows as a natural consequence that the greater the extension of a term, the less its intension; the greater its intension, the less its extension. We will understand this more clearly when we consider that the more individuals contained in a term, the fewer common properties or qualities it can contain; and the more common properties, the fewer individuals. As Brooks says: "The concept man has more extension than poet, orator or statesman, since it embraces more individuals; and less intension, since we must lay aside the distinctive attributes of poet, orator and statesman in order to unite them in a common class man." In the same way the general term animal is quite extended for it includes a large number of individual varieties of very different and varied characteristics and qualities; as for instance, the lion, camel, dog, oyster, elephant, snail, worm, snake, etc. Accordingly its intension must be small for it can include only the qualities common to all animals, which are very few indeed. The definition of the term shows how small is its intension, as: "Animal. An organic being, rising above a vegetable in various respects, especially in possessing sensibility, will and the power of voluntary motion." Another narrows the intension still further when he defines animal as: "a creature which possesses, or has possessed, life." Halleck says: "Animal is very narrow in intension, very broad in extension. There are few qualities common to all animals, but there is a vast number of animals. To give the full meaning of the term in extension, we should have to name every animal, from the microscopic infusoria to the tiger, from the angleworm to the whale. When we decrease the extension to one species of animal, horse, the individuals are fewer, the qualities more numerous."

The importance of forming clear and distinct concepts and of grouping, classifying and generalizing these into larger and broader concepts and terms is recognized by all authorities and is generally regarded as forming the real basis of all constructive thought. As Brooks says: "Generalization lies at the basis of language: only as man can form general conceptions is it possible for him to form a language.... Nearly all the ordinary words in our language are general rather than particular.... This power of generalization lies also at the basis of science. Had we no power of forming general ideas, each particular object would be a study by itself, and we should thus never pass beyond the very alphabet of knowledge. Judgments, except in the simplest form, would be impossible; and it is difficult to see how even the simplest form of the syllogism could be constructed. No general conclusion could be drawn from particulars, nor particular conclusions from generals; and thus neither inductive nor deductive reasoning would be possible. The classifications of science could not be made; and knowledge would end at the very threshold of science."

CHAPTER VII.
THE MEANING OF TERMS

Each concrete general term has two meanings, (1) the actual concrete thing, person or object to which the term is applied; and (2) the qualities, attributes or properties of those objects, persons or things in consequence of which the term is applied. For instance, in the case of the concrete term book, the first meaning consists of the general idea of the thing which we think of as a book, and the second meaning consists of the various qualities which go to make that thing a book, as the printed pages, the binding, the form, the cover, etc. Not only is that particular thing a book, but every other thing having the same or similar properties also must be a book. And so, whenever I call a thing a book it must possess the said qualities. And, whenever I combine the ideas of these qualities in thought, I must think of a book. As Jevons says: "In reality, every ordinary general term has a double meaning: it means the things to which it is applied, ... it also means, in a totally different way, the qualities and peculiarities implied as being in the things. Logicians say that the number of things to which a term applies is the extension of the term; while the number of qualities or peculiarities implied is the intension."

The extension and intension of terms has been referred to in the previous chapter. The general classification of the degrees of extension of a general term is expressed by the two terms, Genus and Species, respectively. The classification of the character of the intension of a term is expressed by the term, Difference, Property and Accident, respectively.

Genus is a term indicating: "a class of objects containing several species; a class more extensive than a species; a universal which is predicable of several things of different species."

Species is a term denoting: "a smaller class of objects than a genus, and of two or more of which a genus is composed; a predicable that expresses the whole essence of its subject in so far as any common term can express it."

An authority says: "The names species and genus are merely relative and the same common term may, in one case, be the species which is predicated of an individual, and in another case the individual of which a species is predicated. Thus the individual, George, belongs to the logical species Man, while Man is an individual of the logical species Animal." Jevons says: "It is desirable to have names by which to show that one class is contained in another, and accordingly we call the class which is divided into two or more smaller ones, the genus, and the smaller ones into which it is divided, the species." Animal is a genus of which man is a species; while man, in turn, is a genus of which Caucasian is a species; and Caucasian, in turn, becomes a genus of which Socrates becomes a species. The student must avoid confusing the logical meaning of the terms genus and species with the use of the same terms in Natural History. Each class is a "genus" to the class below it in extension; and each class is a "species" to the class above it in extension. At the lowest extreme of the scale we reach what is called the infima species, which cannot be further subdivided, as for instance "Socrates"—this lowest species must always be an individual object, person or thing. At the highest extreme of the scale we reach what is summum genus, or highest genus, which is never a species of anything, for there is no class higher than it, as for instance, "being, existence, reality, truth, the absolute, the infinite, the ultimate," etc. Hyslop says: "In reality there is but one summum genus, while there may be an indefinite number of infimae species. All intermediate terms between these extremes are sometimes called subalterns, as being either genera or species, according to the relation in which they are viewed."

Passing on to the classification of the character of the intension of terms, we find:

Difference, a term denoting: "The mark or marks by which the species is distinguished from the rest of the genus; the specific characteristic." Thus the color of the skin is a difference between the Negro and the Caucasian; the number of feet the difference between the biped and the quadruped; the form and shape of leaves the difference between the oak and the elm trees, etc. Hyslop says: "Whatever distinguishes one object from another can be called the differentia. It is some characteristic in addition to the common qualities and determines the species or individual under the genus."

Property, a term denoting: "A peculiar quality of anything; that which is inherent in or naturally essential to anything." Thus a property is a distinguishing mark of a class. Thus black skin is a property of the Negro race; four feet a property of quadrupeds; a certain form of leaf a property of the oak tree. Thus a difference between two species may be a property of one of the species.

Accident, a term denoting: "Any quality or circumstance which may or may not belong to a class, accidentally as it were; or, whatever does not really constitute an essential part of an object, person or thing." As, for instance, the redness of a rose, for a rose might part with its redness and still be a rose—the color is the accident of the rose. Or, a brick may be white and still be a brick, although the majority of bricks are red—the redness or whiteness of the brick are its accidents and not its essential properties. Whately says: "Accidents in Logic are of two kinds—separable and inseparable. If walking be the accident of a particular man, it is a separable one, for he would not cease to be that man though he stood still; while, on the contrary, if Spaniard is the accident connected with him, it is an inseparable one, since he never can cease to be, ethnologically considered, what he was born."

Arising from the classification of the meaning or content of terms, we find the process termed "Definition."

Definition is a term denoting: "An explanation of a word or term." In Logic the term is used to denote the process of analysis in which the properties and differences of a term are clearly stated. There are of course several kinds of definitions. For instance, there is what is called a Real Definition, which Whately defines as: "A definition which explains the nature of the thing by a particular name." There is also what is called a Physical Definition, which is: "A definition made by enumerating such parts as are actually separable, such as the hull, masts, etc., of a ship." Also a Logical Definition, which is: "A definition consisting of the genus and the difference. Thus if a planet be defined as 'a wandering star,' star is the genus, and wandering points out the difference between a planet and an ordinary star." An Accidental Definition is: "A definition of the accidental qualities of a thing." An Essential Definition is: "a definition of the essential properties and differences of an object, person or thing."

Crabbe discriminates between a Definition and an Explanation, as follows: "A definition is correct or precise; an explanation is general or ample. The definition of a word defines or limits the extent of its signification; it is the rule for the scholar in the use of any word; the explanation of a word may include both definition and illustration; the former admits of no more words than will include the leading features in the meaning of any term; the latter admits of an unlimited scope for diffuseness on the part of the explainer."

Hyslop gives the following excellent explanation of the Logical Definition, which as he states is the proper meaning of the term in Logic. He states:

"The rules which regulate Logical Definition are as follows:

1. A definition should state the essential attributes of the species defined.

2. A definition must not contain the name of word defined. Otherwise the definition is called a circulus in definiendo.

3. The definition must be exactly equivalent to the species defined.

4. A definition should not be expressed in obscure, figurative, or ambiguous language.

5. A definition must not be negative when it can be affirmative."

A correct definition necessarily requires the manifestation of the two respective processes of Analysis and Synthesis.

Analysis is a term denoting: "The separation of anything into its constituent elements, qualities, properties and attributes." It is seen at once that in order to correctly define an object, person or thing, it is first necessary to analyze the latter in order to perceive its essential and accidental properties or differences. Unless the qualities, properties and attributes are clearly and fully perceived, we cannot properly define the object itself.

Synthesis is a term denoting: "The act of joining or putting two or more things together; in Logic: the method by composition, in opposition to the method of resolution or analysis." In stating a definition we must necessarily join together the various essential qualities, properties and attributes, which we have discovered by the process of analysis; and the synthesized combination, considered as a whole, is the definition of the object expressed by the term.

CHAPTER VIII.
JUDGMENTS

In every act of Judgment there must be at least two concepts to be examined and compared. This comparison must lead to a Judgment regarding their agreement or disagreement. For instance, we have the two concepts, horse and animal. We examine and compare the two concepts, and find that there is an agreement between them. We find that the concept horse is included in the higher concept of animal and therefore, we assert that: "The horse is an animal." This is a statement of agreement and is, therefore, a Positive Judgment. We then compare the concepts horse and cow and find a disagreement between them, which we express in the statement of the Judgment that: "The horse is not a cow." This Judgment, stating a disagreement is what is called a Negative Judgment.

In the above illustration of the comparison between the concepts horse and animal we find that the second concept animal is broader than the first, horse, so broad in fact that it includes the latter. The terms are not equal, for we cannot say, in truth, that "an animal is the horse." We may, however, include a part of the broader conception with the narrower and say: "some animals are horses." Sometimes both concepts are of equal rank, as when we state that: "Man is a rational animal."

In the process of Judgment there is always the necessity of the choice between the Positive and the Negative. When we compare the concepts horse and animal, we must of necessity decide either that the horse is an animal, or else that it is not an animal.

The importance of the process of Judgment is ably stated by Halleck, as follows: "Were isolated concepts possible, they would be of very little use. Isolated facts are of no more service than unspun wool. We might have a concept of a certain class of three-leaved ivy, as we might also of poisons. Unless judgment linked these two concepts and decided that this species of ivy is poisonous, we might take hold of it and be poisoned. We might have a concept of bread and also one of meat, fruit and vegetables. If we also had a concept of food, unrelated to these, we should starve to death, for we should not think of them as foods. A vessel, supposing itself to be far out at sea, signaled another vessel that the crew were dying of thirst. That crew certainly had a concept of drinkable things and also of water. To the surprise of the first, the second vessel signaled back, 'Draw from the sea and drink. You are at the mouth of the Amazon.' The thirsty crew had not joined the concept drinkable to the concept of water over the ship's side. A man having taken an overdose of laudanum, his wife lost much valuable time in sending out for antidotes, because certain of her concepts had not been connected by judgment. She had good concepts of coffee and of mustard; she also knew that an antidote to opium was needed; but she had never linked these concepts and judged that coffee and mustard were antidotes to opium. The moment she formed that judgment she was a wiser woman for her knowledge was related and usable.... Judgment is the power revolutionizing the world. The revolution is slow because nature's forces are so complex, so hard to be reduced to their simplest forms and so disguised and neutralized by the presence of other forces.... Fortunately judgment is ever silently working and comparing things that, to past ages, have seemed dissimilar; and it is continually abstracting and leaving out of the field of view those qualities which have simply served to obscure the point at issue."

Judgment may be both analytic or synthetic in its processes; and it may be neither. When we compare a narrow concept with a broader one, as a part with a whole, the process is synthetic or an act of combination. When we compare a part of a concept with another concept, the process is analytic. When we compare concepts equal in rank or extent, the process is neither synthetic nor analytic. Thus in the statement that: "A horse is an animal," the judgment is synthetic; in the statement that: "some animals are horses," the judgement is analytic; in the statement that: "a man is a rational animal," the judgment is neither analytic nor synthetic.

Brooks says: "In one sense all judgments are synthetic. A judgment consists of the union of two ideas and this uniting is a process of synthesis. This, however, is a superficial view of the process. Such a synthesis is a mere mechanical synthesis; below this is a thought-process which is sometimes analytic, sometimes synthetic and sometimes neither analytic nor synthetic."

The same authority states: "The act of mind described is what is known as logical judgment. Strictly speaking, however, every intelligent act of the mind is accompanied with a judgment. To know is to discriminate and, therefore, to judge. Every sensation or cognition involves a knowledge and so a judgment that it exists. The mind cannot think at all without judging; to think is to judge. Even in forming the notions which judgment compares, the mind judges. Every notion or concept implies a previous act of judgment to form it: in forming a concept, we compare the common attributes before we unite them; and comparison is judgment. It is thus true that 'Every concept is a contracted judgment; every judgment an expanded concept.' This kind of judgment, by which we affirm the existence of states of consciousness, discriminate qualities, distinguish percepts and form concepts, is called primitive or psychological judgment."

In Logical Judgment there are two aspects; i.e., Judgment by Extension and Judgment by Intension. When we compare the two concepts horse and animal we find that the concept horse is contained in the concept animal and the judgment that "a horse is an animal" may be considered as a Judgment by Extension. In the same comparison we see that the concept horse contains the quality of animality, and in attributing this quality to the horse, we may also say "the horse is an animal," which judgment may be considered as a Judgment by Intension. Brooks says: "Both views of Judgment are correct; the mind may reach its judgment either by extension or by intension. The method by extension is usually the more natural."

When a Judgment is expressed in words it is called a Proposition. There is some confusion regarding the two terms, some holding that a Judgment and a proposition are identical, and that the term "proposition" may be properly used to indicate the judgment itself. But the authorities who seek for clearness of expression and thought now generally hold that: "A Proposition is a Judgment expressed in words." In the next chapter, in which we consider Propositions, we shall enter into a more extended consideration of the subject of Judgments as expressed in Propositions, which consideration we omit at this point in order to avoid repetition. Just as the respective subjects of Concepts and Terms necessarily blend into each other, so do the respective subjects of Judgments and Propositions. In each case, too, there is the element of the mental process on the one hand and the verbal expression of it on the other hand. It will be well to keep this fact in mind.

CHAPTER IX.
PROPOSITIONS

The Parts of a Proposition are: (1) the Subject, or that of which something is affirmed or denied; (2) the Predicate, or the something which is affirmed or denied regarding the Subject; and (3) the Copula, or the verb serving as a link between the Subject and the Predicate.

In the Proposition: "Man is an animal," the term man is the Subject; the term an animal is the Predicate; and the word is, is the Copula. The Copula is always some form of the verb to be, in the present tense indicative, in an affirmative Proposition; and the same with the negative particle affixed, in a negative Proposition. The Copula is not always directly expressed by the word is or is not, etc., but is instead expressed in some phrase which implies them. For instance, we say "he runs," which implies "he is running." In the same way, it may appear at times as if the Predicate was missing, as in: "God is," by which is meant "God is existing." In some cases, the Proposition is inverted, the Predicate appearing first in order, and the Subject last, as in: "Blessed are the peacemakers;" or "Strong is Truth." In such cases judgment must be used in determining the matter, in accordance with the character and meaning of the terms.

An Affirmative Proposition is one in which the Predicate is affirmed to agree with the Subject. A Negative Proposition is one in which the agreement of the Predicate and Subject is denied. Examples of both of these classes have been given in this chapter.

Another classification of Propositions divides them in three classes, as follows (1) Categorical; (2) Hypothetical; (3) Disjunctive.

A Categorical Proposition is one in which the affirmation or denial is made without reservation or qualification, as for instance: "Man is an animal;" "the rose is a flower," etc. The fact asserted may not be true, but the statement is made positively as a statement of reality.

A Hypothetical Proposition is one in which the affirmation or denial is made to depend upon certain conditions, circumstances or suppositions, as for instance: "If the water is boiling-hot, it will scald;" or "if the powder be damp, it will not explode," etc. Jevons says: "Hypothetical Propositions may generally be recognized by containing the little word 'if;' but it is doubtful whether they really differ much from the ordinary propositions.... We may easily say that 'boiling water will scald,' and 'damp gunpowder will not explode,' thus avoiding the use of the word 'if.'"

A Disjunctive Proposition is one "implying or asserting an alternative," and usually containing the conjunction "or," sometimes together with "either," as for instance: "Lightning is sheet or forked;" "Arches are either round or pointed;" "Angles are either obtuse, right angled or acute."

Another classification of Propositions divides them in two classes as follows: (1) Universal; (2) Particular.

A Universal Proposition is one in which the whole quantity of the Subject is involved in the assertion or denial of the Predicate. For instance: "All men are liars," by which is affirmed that all of the entire race of men are in the category of liars, not some men but all the men that are in existence. In the same way the Proposition: "No men are immortal" is Universal, for it is a universal denial.

A Particular Proposition is one in which the affirmation or denial of the Predicate involves only a part or portion of the whole of the Subject, as for instance: "Some men are atheists," or "Some women are not vain," in which cases the affirmation or denial does not involve all or the whole of the Subject. Other examples are: "A few men," etc.; "many people," etc.; "certain books," etc.; "most people," etc.

Hyslop says: "The signs of the Universal Proposition, when formally expressed, are all, every, each, any, and whole or words with equivalent import." The signs of Particular Propositions are also certain adjectives of quantity, such as some, certain, a few, many, most or such others as denote at least a part of a class.

The subject of the Distribution of Terms in Propositions is considered very important by Logicians, and as Hyslop says: "has much importance in determining the legitimacy, or at least the intelligibility, of our reasoning and the assurance that it will be accepted by others." Some authorities favor the term, "Qualification of the Terms of Propositions," but the established usage favors the term "Distribution."

The definition of the Logical term, "Distribution," is: "The distinguishing of a universal whole into its several kinds of species; the employment of a term to its fullest extent; the application of a term to its fullest extent, so as to include all significations or applications." A Term of a Proposition is distributed when it is employed in its fullest sense; that is to say, when it is employed so as to apply to each and every object, person or thing included under it. Thus in the proposition, "All horses are animals," the term horses is distributed; and in the proposition, "Some horses are thoroughbreds," the term horses is not distributed. Both of these examples relate to the distribution of the subject of the proposition. But the predicate of a proposition also may or may not be distributed. For instance, in the proposition, "All horses are animals," the predicate, animals, is not distributed, that is, not used in its fullest sense, for all animals are not horses—there are some animals which are not horses and, therefore, the predicate, animals, not being used in its fullest sense is said to be "not distributed." The proposition really means: "All horses are some animals."

There is however another point to be remembered in the consideration of Distribution of Terms of Propositions, which Brooks expresses as follows: "Distribution generally shows itself in the form of the expression, but sometimes it may be determined by the thought. Thus if we say, 'Men are mortal,' we mean all men, and the term men is distributed. But if we say 'Books are necessary to a library,' we mean, not 'all books' but 'some books.' The test of distribution is whether the term applies to 'each and every.' Thus when we say 'men are mortal,' it is true of each and every man that he is mortal."

The Rules of Distribution of the Terms of Proposition are as follows:

1. All universals distribute the subject.

2. All particulars do not distribute the subject.

3. All negatives distribute the predicate.

4. All affirmatives do not distribute the predicate.

The above rules are based upon logical reasoning. The reason for the first two rules is quite obvious, for when the subject is universal, it follows that the whole subject is involved; when the subject is particular it follows that only a part of the subject is involved. In the case of the third rule, it will be seen that in every negative proposition the whole of the predicate must be denied the subject, as for instance, when we say: "Some animals are not horses," the whole class of horses is cut off from the subject, and is thus distributed. In the case of the fourth rule, we may readily see that in the affirmative proposition the whole of the predicate is not denied the subject, as for instance, when we say that: "Horses are animals," we do not mean that horses are all the animals, but that they are merely a part or portion of the class animal—therefore, the predicate, animals, is not distributed.

In addition to the forms of Propositions given there is another class of Propositions known as Definitive or Substitutive Propositions, in which the Subject and the Predicate are exactly alike in extent and rank. For instance, in the proposition, "A triangle is a polygon of three sides" the two terms are interchangeable; that is, may be substituted for each other. Hence the term "substitutive." The term "definitive" arises from the fact that the respective terms of this kind of a proposition necessarily define each other. All logical definitions are expressed in this last mentioned form of proposition, for in such cases the subject and the predicate are precisely equal to each other.

CHAPTER X.
IMMEDIATE REASONING

Opposition exists between propositions having the same subject and predicate, but differing in quality or quantity, or both. The Laws of Opposition are as follows:

I. (1) If the universal is true, the particular is true. (2) If the particular is false, the universal is false. (3) If the universal is false, nothing follows. (4) If the particular is true, nothing follows.

II. (1) If one of two contraries is true, the other is false. (2) If one of two contraries is false, nothing can be inferred. (3) Contraries are never both true, but both may be false.

III. (1) If one of two sub-contraries is false, the other is true. (2) If one of two sub-contraries is true, nothing can be inferred concerning the other. (3) Sub-contraries can never be both false, but both may be true.

IV. (1) If one of two contradictories is true, the other is false. (2) If one of two contradictories is false, the other is true. (3) Contradictories can never be both true or both false, but always one is true and the other is false.

In order to comprehend the above laws, the student should familiarize himself with the following arrangement, adopted by logicians as a convenience:

Examples of the above: Universal Affirmative (A): "All men are mortal;" Universal Negative (E): "No man is mortal;" Particular Affirmative (I): "Some men are mortal;" Particular Negative (O): "Some men are not mortal."

The following examples of abstract propositions are often used by logicians as tending toward a clearer conception than examples such as given above:

(A) "All A is B."

(I) "Some A is B."

(E) "No A is B."

(O) "Some A is not B."

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

A and E are styled contraries. I and O are sub-contraries; A and I and also E and O are called subalterns; A and O and also I and E are styled contradictories.

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:

Conversion is the process of immediate reasoning by which we infer from a given proposition another proposition having the predicate of the original for its subject and the subject of the original for its predicate; or stated in a few words: Conversion is the transposition of the subject and predicate of a proposition. As Brooks states it: "Propositions or judgments are converted when the subject and predicate change places in such a manner that the resulting judgment is an inference from the given judgment." The new proposition, resulting from the operation or Conversion, is called the Converse; the original proposition is called the Convertend.

The Law of Conversion is that: "No term must be distributed in the Converse that is not distributed in the Convertend." This arises from the obvious fact that nothing should be affirmed in the derived proposition than there is in the original proposition.

There are three kinds of Conversion; viz: (1) Simple Conversion; (2) Conversion by Limitation; (3) Conversion by Contraposition.

In Simple Conversion there is no change in either quality or quantity. In Conversion by Limitation the quality is changed from universal to particular. In Conversion by Negation the quality is changed but not the quantity. Referring to the classification tables and symbols given in the preceding pages of this chapter, we may now proceed to consider the application of these methods of Conversion to each of the four kinds of propositions; as follows:

The Universal Affirmative (symbol A) proposition is converted by Limitation, or by a change of quality from universal to particular. The predicate not being "distributed" in the convertend, we must not distribute it in the converse by saying "all." Thus in this case we must convert the proposition, "all men are mortal" (A), into "some mortals are men" (I).

The Universal Negative (symbol E) is converted by Simple Conversion, in which there is no change in either quality or quantity. For since both terms of "E" are distributed, they may both be distributed in the converse without violating the law of conversion. Thus "No man is mortal" is converted into: "No mortals are men." "E" is converted into "E."

The Particular Affirmative (symbol I) is also converted by Simple Conversion in which there is no change in either quality or quantity. For since neither term is distributed in "I," neither term may be distributed in the converse, and the latter must remain "I." For instance; the proposition: "Some men are mortal" is converted into the proposition, "Some mortals are men."

The Particular Negative (symbol O) is converted by Conversion by Negation, in which the quality is changed but not the quantity. Thus in converting the proposition: "Some men are not mortal," we must not say "some mortals are not men," for in so doing we would distribute men in the predicate, where it is not distributed in the convertend. Avoiding this, we transfer the negative particle from the copula to the predicate so that the convertend becomes "I" which is converted by Simple Conversion. Thus we transfer "Some men are not mortal" into "Some men are not-mortal" from which we easily convert (by simple Conversion) the proposition: "Some not-mortals are men."

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:

The Law of Identity, which states that: "The same quality or thing is always the same quality or thing, no matter how different the conditions in which it occurs."

The Law of Contradiction, which states that: "No thing can at the same time and place both be and not be."

The Law of Excluded Middle, which states that: "Everything must either be or not be; there is no other alternative or middle course."

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."

CHAPTER XI.
INDUCTIVE REASONING

Inductive Reasoning, as we have said, is the process of discovering general truth from particular truths, or inferring general laws from particular facts. Thus, from the experience of the individual and the race regarding the particular truth that each and every man under observation has been observed to die sooner or later, it is inferred that all men die, and hence, the induction of the general truth that "All men must die." Or, as from experience we know that the various kinds of metals expand when subjected to heat, we infer that all metals are subject to this law, and that consequently we may arrive by inductive reasoning at the conclusion that: "All metals expand when subjected to heat." It will be noticed that the conclusion arrived at in this way by Inductive Reasoning forms the fundamental premise in the process of Deductive Reasoning. As we have seen elsewhere, the two processes, Inductive and Deductive Reasoning, respectively are interdependent—resting upon one another.

Jevons says of Inductive Reasoning: "In Deductive Reasoning we inquire how we may gather the truth contained in some propositions called Premises, and put into another proposition called the Conclusion. We have not yet undertaken to find out how we can learn what propositions really are true, but only what propositions are true when other ones are true. All the acts of reasoning yet considered would be called deductive because we deduce, or lead down the truth from premises to conclusion. It is an exceedingly important thing to understand deductive inference correctly, but it might seem to be still more important to understand inductive inference, by which we gather the truth of general propositions from facts observed as happening in the world around us." Halleck says: "Man has to find out through his own experience, or that of others, the major premises from which he argues or draws his conclusions. By induction we examine what seems to us a sufficient number of individual cases. We then conclude that the rest of these cases, which we have not examined, will obey the same general law.... Only after general laws have been laid down, after objects have been classified, after major premises have been formed, can deduction be employed."

Strange as may now appear, it is a fact that until a comparatively recent period in the history of man, it was held by philosophers that the only way to arrive at all knowledge was by means of Deductive Reasoning, by the use of the Syllogism. The influence of Aristotle was great and men preferred to pursue artificial and complicated methods of Deductive Reasoning, rather than to reach the truth by obtaining the facts from Nature herself, at first hand, and then inferring general principle from the facts so gathered. The rise of modern scientific methods of reasoning, along the lines of Inductive Inference, dates from about 1225-1300. Roger Bacon was one of the first to teach that we must arrive at scientific truth by a process of observation and experimentation on the natural objects to be found on all sides. He made many discoveries by following this process. He was ably seconded by Galileo who lived some three hundred years later, and who also taught that many great general truths might be gained by careful observation and intelligent inference. Lord Francis Bacon, who lived about the same time as Galileo, presented in his Novum Organum many excellent observations and facts regarding the process of Inductive Reasoning and scientific thought. As Jevons says: "Inductive logic inquires by what manner of reasoning we can gather the laws of nature from the facts and events observed. Such reasoning is called induction, or inductive inquiry, and, as it has actually been practiced by all the great discoverers in science, it consists in four steps."

The Four Steps in Inductive Reasoning, as stated by Jevons, are as follows:

First Step.—Preliminary observation.

Second Step.—The making of hypotheses.

Third Step.—Deductive reasoning.

Fourth Step.—Verification.

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 antithesis appears in every particular. Deduction goes from generals to particulars; Induction goes from particulars to generals. Deduction is an analytic 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 certitude 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 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 assume 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 assume 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 classes.

Perfect Induction necessitates a knowledge of all the particulars forming a class; that is, all the individual objects, persons, things or facts comprising a class 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 assume that the particulars or facts actually known to us correctly represent those which are not actually known, and hence the whole class 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 assume 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 assume 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 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.

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 quantity 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 passing 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 circumstances.... 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 class 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

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."