Kitabı oku: «Thought-Culture; Or, Practical Mental Training», sayfa 6

If the student will select some familiar term and endeavor to define it correctly, writing down the result, and will then compare the latter with the definition given in some standard dictionary, he will see a new light regarding logical definition. Practice in definition, conducted along these lines, will cultivate the powers of analysis and conception and will, at the same time, tend toward the acquiring of correct and scientific methods of thought and clear expression.

Hyslop gives the following excellent Rules of Logical Definition, which should be followed by the student in his exercises:

"I. A definition should state the essential attributes of the species defined.

"II. A definition must not contain the name or word defined. Otherwise the definition is called a circulus in definiendo (defining in a circle).

"III. The definition must be exactly equivalent to the species defined.

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

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

Logical Synthesis is the exact opposite of Logical Analysis. In the latter we strive to separate and take apart; in the former we strive to bind together and combine the particulars into the general. Beginning with individual things and comparing them with each other according to observed points of resemblance, we proceed to group them into species or narrow classes. These classes, or species, we then combine with similar ones, into a larger class or genus; and then, according to the same process, into broader classes as we have shown in the first part of this chapter.

The process of Synthesis is calculated to develop and cultivate the mind in several directions and exercises along these lines will give a new habit and sense of orderly arrangement, which will be most useful to the student in his every-day life. Halleck says: "Whenever a person is comparing a specimen to see whether it may be put in the same class with other specimens, he is thinking. Comparison is an absolutely essential factor of thought, and classification demands comparison. The man who has not properly classified the myriad individual objects with which he has to deal, must advance like a cripple. He, only, can travel with seven-league boots, who has thought out the relations existing between these stray individuals and put them into their proper classes. In a minute a business man may put his hand on any one of ten thousand letters if they are properly classified. In the same way, the student of history, sociology or any other branch, can, if he studies the subjects aright, have all his knowledge classified and speedily available for use… In this way, we may make our knowledge of the world more minutely exact. We cannot classify without seeing things under a new aspect."

The study of Natural History, in any or all of its branches, will do much to cultivate the power of Classification. But one may practice classification with the objects around him in his every-day life. Arranging things mentally, into small classes, and these into larger, one will soon be able to form a logical connection between particular ideas and general ideas; particular objects and general classes. The practice of classification gives to the mind a constructive turn – a "building-up" tendency, which is most desirable in these days of construction and development. Regarding some of the pitfalls of classification, Jevons says:

"In classifying things, we must take great care not to be misled by outward resemblances. Things may seem to be very much like each other which are not so. Whales, porpoises, seals and several other animals live in the sea exactly like fish; they have a similar shape and are usually classed among fish. People are said to go whale-fishing. Yet these animals are not really fish at all, but are much more like dogs and horses and other quadrupeds than they are like fish. They cannot live entirely under water and breathe the air contained in the water like fish, but they have to come up to the surface at intervals to take breath. Similarly, we must not class bats with birds because they fly about, although they have what would be called wings; these wings are not like those of birds and in truth bats are much more like rats and mice than they are like birds. Botanists used at one time to classify plants according to their size, as trees, shrubs or herbs, but we now know that a great tree is often more similar in its character to a tiny herb than it is to other great trees. A daisy has little resemblance to a great Scotch thistle; yet the botanist regards them as very similar. The lofty growing bamboo is a kind of grass, and the sugarcane also belongs to the same class with wheat and oats."

Remember that analysis of a genus into its component species is accomplished by a separation according to differences; and species are built up by synthesis into a genus because of resemblances. The same is true regarding individual and species, building up in accordance to points of resemblance, while analysis or separation is according to points of difference.

The use of a good dictionary will be advantageous to the student in developing the power of Generalization or Conception. Starting with a species, he may build up to higher and still higher classes by consulting the dictionary; likewise, starting with a large class, he may work down to the several species composing it. An encyclopedia, of course, is still better for the purpose in many cases. Remember that Generalization is a prime requisite for clear, logical thinking. Moreover, it is a great developer of Thought.

CHAPTER XI
JUDGMENT

We have seen that in the several mental processes which are grouped together under the general head of Understanding, the stage or step of Abstraction is first; following which is the second step or phase, called Generalization or Conception. The third step or phase is that which is called Judgment. In the exercise of the faculty of Judgment, we determine the agreement or disagreement between two concepts, ideas, or objects of thought, by comparing them one with another. From this process of comparison arises the Judgment, which is expressed in the shape of a logical Proposition. A certain form of Judgment must be used, however, in the actual formation of a Concept, for we must first compare qualities, and make a judgment thereon, in order to form a general idea. In this place, however, we shall confine ourselves to the consideration of the faculty of Judgment in the strictly logical usage of the term, as previously stated.

We have seen that the expression of a concept is called a Term, which is the name of the concept. In the same way when we compare two terms (expressions of concepts) and pass Judgment thereon, the expression of that Judgment is called a Proposition. In every Judgment and Proposition there must be two Terms or Concepts, connected by a little word "is" or "are," or some form of the verb "to be," in the present tense indicative. This connecting word is called the Copula. For instance, we may compare the two terms horse and animal, as follows: "A horse is an animal," the word is being the Copula or symbol of the affirmative Judgment, which connects the two terms. In the same way we may form a negative Judgment as follows: "A horse is not a cow." In a Proposition, the term of which something is affirmed is called the Subject; and the term expressing that which is affirmed of the subject is called the Predicate.

Besides the distinction between affirmative Judgments, or Propositions, there is a distinction arising from quantity, which separates them into the respective classes of particular and universal. Thus, "all horses are animals," is a universal Judgment; while "some horses are black" is a particular Judgment. Thus all Judgments must be either affirmative or negative; and also either particular or universal. This gives us four possible classes of Judgments, as follows, and illustrated symbolically:

1. Universal Affirmative, as "All A is B."

2. Universal Negative, as "No A is B."

3. Particular Affirmative, as "Some A is B."

4. Particular Negative, as "Some A is not B."

The Term or Judgment is said to be "distributed" (that is, extended universally) when it is used in its fullest sense, in which it is used in the sense of "each and every" of its kind or class. Thus in the proposition "Horses are animals" the meaning is that "each and every" horse is an animal – in this case the subject is "distributed" or made universal. But the predicate is not "distributed" or made universal, but remains particular or restricted and implies merely "some." For the proposition does not mean that the class "horses" includes all animals. For we may say that: "Some animals are not horses." So you see we have several instances in which the "distribution" varies, both as regards the subject and also the predicate. The rule of logic applying in this case is as follows:

1. In universal propositions, the subject is distributed.

2. In particular propositions, the subject is not distributed.

3. In negative propositions, the predicate is distributed.

4. In affirmative propositions, the predicate is not distributed.

A little time devoted to the analysis and understanding of the above rules will repay the student for his trouble, inasmuch as it will train his mind in the direction of logical distinction and judgment. The importance of these rules will appear later.

Halleck says: "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. The progress of the next hundred years will join many concepts, which now seem to have no common qualities. If the vast amount of energy latent in the sunbeams, in the rays of the stars, in the winds, in the rising and falling of the tides, is treasured up and applied to human purposes, it will be a fresh triumph for judgment. This world is rolling around in a universe of energy, of which judgment has as yet harnessed only the smallest appreciable fraction. Fortunately, judgment is ever working and silently comparing things that, to past ages, have seemed dissimilar; and it is constantly abstracting and leaving out of the field of view those qualities which have simply served to obscure the point at issue." Brooks says: "The power of judgment is of great value to its products. It is involved in or accompanies every act of the intellect, and thus lies at the foundation of all intellectual activity. It operates directly in every act of the understanding; and even aids the other faculties of the mind in completing their activities and products."

The best method of cultivating the power of Judgment is the exercise of the faculty in the direction of making comparisons, of weighing differences and resemblances, and in generally training the mind along the lines of Logical Thinking. Another volume of this series is devoted to the latter subject, and should aid the student who wishes to cultivate the habit of logical and scientific thought. The study of mathematics is calculated to develop the faculty of Judgment, because it necessitates the use of the powers of comparison and decision. Mental arithmetic, especially, will tend to strengthen, and exercise this faculty of the mind.

Geometry and Logic will give the very best exercise along these lines to those who care to devote the time, attention and work to the task. Games, such as chess, and checkers or draughts, tend to develop the powers of Judgment. The study of the definitions of words in a good dictionary will also tend to give excellent exercise along the same lines. The exercises given in this book for the cultivation and development of the several faculties, will tend to develop this particular faculty in a general way, for the exercise of Judgment is required at each step of the way, and in each exercise.

Brooks says: "It should be one of the leading objects of the culture of young people to lead them to acquire the habit of forming judgments. They should not only be led to see things, but to have opinions about things. They should be trained to see things in their relations, and to put these relations into definite propositions. Their ideas of objects should be worked up into thoughts concerning the objects. Those methods of teaching are best which tend to excite a thoughtful habit of mind that notices the similitudes and diversities of objects, and endeavors to read the thoughts which they embody and of which they are the symbols."

The exercises given at the close of the next chapter, entitled "Derived Judgments," will give to the mind a decided trend in the direction of logical judgment. We heartily recommend them to the student.

The student will find that he will tend to acquire the habit of clear logical comparison and judgment, if he will memorize and apply in his thinking the following excellent Primary Rules of Thought, stated by Jevons:

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

"II. Law of Contradiction: Nothing can at the same time and place both be and not be.

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

Jevons says of these laws: "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 their key."

CHAPTER XII
DERIVED JUDGMENTS

As we have seen, a Judgment is obtained by comparing two objects of thought according to their agreement or difference. The next higher step, that of logical Reasoning, consists of the comparing of two ideas through their relation to a third. This form of reasoning is called mediate, because it is effected through the medium of the third idea. There is, however, a certain process of Understanding which comes in between this mediate reasoning on the one hand, and the formation of a plain judgment on the other. Some authorities treat it as a form of reasoning, calling it Immediate Reasoning or Immediate Inference, while others treat it as a higher form of Judgment, calling it Derived Judgment. We shall follow the latter classification, as best adapted for the particular purposes of this book.

The fundamental principle of Derived Judgment is that ordinary Judgments are often so related to each other that one Judgment may be derived directly and immediately from another. The two particular forms of the general method of Derived Judgment are known as those of (1) Opposition; and (2) Conversion; respectively.

In order to more clearly understand the logical processes involved in Derived Judgment, we should acquaint ourselves with the general relations of Judgments, and with the symbolic letters used by logicians as a means of simplifying the processes of thought. Logicians denote each of the four classes of Judgments or Propositions by a certain letter, the first four vowels – A, E, I and O, being used for the purpose. It has been found very convenient to use these symbols in denoting the various forms of Propositions and Judgments. The following table should be memorized for this purpose:

Universal Affirmative, symbolized by "A."

Universal Negative, symbolized by "E."

Particular Affirmative, symbolized by "I."

Particular Negative, symbolized by "O."

It will be seen that these four forms of Judgments bear certain relations to each other, from which arises what is called opposition. This may be better understood by reference to the following table called the Square of Opposition:

Thus, A and E are contraries; I and O are sub-contraries; A and I, and also E and O are subalterns; A and O, and also E and I are contradictories.

The following will give a symbolic table of each of the four Judgments or Propositions with the logical symbols attached:

(A) "All A is B."

(E) "No A is B."

(I) "Some A is B."

(O) "Some A is not B."

The following are the rules governing and expressing the relations above indicated:

I. Of the Contradictories: One must be true, and the other must be false. As for instance, (A) "All A is B;" and (O) "Some A is not B;" cannot both be true at the same time. Neither can (E) "No A is B;" and (I) "Some A is B;" both be true at the same time. They are contradictory by nature, – and if one is true, the other must be false; if one is false, the other must be true.

II. Of the Contraries: If one is true the other must be false; but, both may be false. As for instance, (A) "All A is B;" and (E) "No A is B;" cannot both be true at the same time. If one is true the other must be false. But, both may be false, as we may see when we find we may state that (I) "Some A is B." So while these two propositions are contrary, they are not contradictory. While, if one of them is true the other must be false, it does not follow that if one is false the other must be true, for both may be false, leaving the truth to be found in a third proposition.

III. Of the Subcontraries: If one is false the other must be true; but both may be true. As for instance, (I) "Some A is B;" and (O) "Some A is not B;" may both be true, for they do not contradict each other. But one or the other must be true – they can not both be false.

IV. Of the Subalterns: If the Universal (A or E) be true the Particular (I or O) must be true. As for instance, if (A) "All A is B" is true, then (I) "Some A is B" must also be true; also, if (E) "No A is B" is true, then "Some A is not B" must also be true. The Universal carries the particular within its truth and meaning. But; If the Universal is false, the particular may be true or it may be false. As for instance (A) "All A is B" may be false, and yet (I) "Some A is B" may be either true or false, without being determined by the (A) proposition. And, likewise, (E) "No A is B" may be false without determining the truth or falsity of (O) "Some A is not B."

But: If the Particular be false, the Universal also must be false. As for instance, if (I) "Some A is B" is false, then it must follow that (A) "All A is B" must also be false; or if (O) "Some A is not B" is false, then (E) "No A is B" must also be false. But: The Particular may be true, without rendering the Universal true. As for instance: (I) "Some A is B" may be true without making true (A) "All A is B;" or (O) "Some A is not B" may be true without making true (E) "No A is B."

The above rules may be worked out not only with the symbols, as "All A is B," but also with any Judgments or Propositions, such as "All horses are animals;" "All men are mortal;" "Some men are artists;" etc. The principle involved is identical in each and every case. The "All A is B" symbology is merely adopted for simplicity, and for the purpose of rendering the logical process akin to that of mathematics. The letters play the same part that the numerals or figures do in arithmetic or the a, b, c; x, y, z, in algebra. Thinking in symbols tends toward clearness of thought and reasoning.

Exercise: Let the student apply the principles of Opposition by using any of the above judgments mentioned in the preceding paragraph, in the direction of erecting a Square of Opposition of them, after having attached the symbolic letters A, E, I and O, to the appropriate forms of the propositions.

Then let him work out the following problems from the Tables and Square given in this chapter.

1. If "A" is true; show what follows for E, I and O. Also what follows if "A" be false.

2. If "E" is true; show what follows for A, I and O. Also what follows if "E" be false.

3. If "I" is true; show what follows for A, E and O. Also what follows if "I" be false.

4. If "O" is true; show what follows for A, E and I. Also what happens if "O" be false.

CONVERSION OF JUDGMENTS

Judgments are capable of the process of Conversion, or the change of place of subject and predicate. Hyslop says: "Conversion is the transposition of subject and predicate, or the process of immediate inference by which we can infer from a given preposition another having the predicate of the original for its subject, and the subject of the original for its predicate." The process of converting a proposition seems simple at first thought but a little consideration will show that there are many difficulties in the way. For instance, while it is a true judgment that "All horses are animals," it is not a correct Derived Judgment or Inference that "All animals are horses." The same is true of the possible conversion of the judgment "All biscuit is bread" into that of "All bread is biscuit." There are certain rules to be observed in Conversion, as we shall see in a moment.

The Subject of a judgment is, of course, the term of which something is affirmed; and the Predicate is the term expressing that which is affirmed of the Subject. The Predicate is really an expression of an attribute of the Subject. Thus when we say "All horses are animals" we express the idea that all horses possess the attribute of "animality;" or when we say that "Some men are artists," we express the idea that some men possess the attributes or qualities included in the concept "artist." In Conversion, the original judgment is called the Convertend; and the new form of judgment, resulting from the conversion, is called the Converse. Remember these terms, please.

The two Rules of Conversion, stated in simple form, are as follows:

I. Do not change the quality of a judgment. The quality of the converse must remain the same as that of the convertend.

II. Do not distribute an undistributed term. No term must be distributed in the converse which is not distributed in the convertend.

The reason of these rules is that it would be contrary to truth and logic to give to a converted judgment a higher degree of quality and quantity than is found in the original judgment. To do so would be to attempt to make "twice 2" more than "2 plus 2."

There are three methods or kinds of Conversion, as follows: (1) Simple Conversion; (2) Limited Conversion; and (3) Conversion by Contraposition.

In Simple Conversion, there is no change in either quality or quantity. For instance, by Simple Conversion we may convert a proposition by changing the places of its subject and predicate, respectively. But as Jevons says: "It does not follow that the new one will always be true if the old one was true. Sometimes this is the case, and sometimes it is not. If I say, 'some churches are wooden-buildings,' I may turn it around and get 'some wooden-buildings are churches;' the meaning is exactly the same as before. This kind of change is called Simple Conversion, because we need do nothing but simply change the subjects and predicates in order to get a new proposition. We see that the Particular Affirmative proposition can be simply converted. Such is the case also with the Universal Negative proposition. 'No large flowers are green things' may be converted simply into 'no green things are large flowers.'"

In Limited Conversion, the quantity is changed from Universal to Particular. Of this, Jevons continues: "But it is a more troublesome matter, however, to convert a Universal Affirmative proposition. The statement that 'all jelly fish are animals,' is true; but, if we convert it, getting 'all animals are jelly fish,' the result is absurd. This is because the predicate of a universal proposition is really particular. We do not mean that jelly fish are 'all' the animals which exist, but only 'some' of the animals. The proposition ought really to be 'all jelly fish are some animals,' and if we converted this simply, we should get, 'some animals are all jelly fish.' But we almost always leave out the little adjectives some and all when they would occur in the predicate, so that the proposition, when converted, becomes 'some animals are jelly fish.' This kind of change is called Limited Conversion, and we see that a Universal Affirmative proposition, when so converted, gives a Particular Affirmative one."

In Conversion by Contraposition, there is a change in the position of the negative copula, which shifts the expression of the quality. As for instance, in the Particular Negative "Some animals are not horses," we cannot say "Some horses are not animals," for that would be a violation of the rule that "no term must be distributed in the converse which is not distributed in the convertend," for as we have seen in the preceding chapter: "In Particular propositions the subject is not distributed." And in the original proposition, or convertend, "animals" is the subject of a Particular proposition. Avoiding this, and proceeding by Conversion by Contraposition, we convert the Convertend (O) into a Particular Affirmative (I), saying: "Some animals are not-horses;" or "Some animals are things not horses;" and then proceeding by Simple Conversion we get the converse, "Some things not horses are animals," or "Some not-horses are animals."

The following gives the application of the appropriate form of Conversion to each of the several four kind of Judgments or Propositions:

(A) Universal Affirmative: This form of proposition is converted by Limited Conversion. The predicate not being distributed in the convertend, it cannot be distributed in the converse, by saying "all." ("In affirmative propositions the predicate is not distributed.") Thus by this form of Conversion, we convert "All horses are animals" into "Some animals are horses." The Universal Affirmative (A) is converted by limitation into a Particular Affirmative (I).

(E) Universal Negative: This form of proposition is converted by Simple Conversion. In a Universal Negative both terms are distributed. ("In universal propositions, the subject is distributed;" "In negative propositions, the predicate is distributed.") So we may say "No cows are horses," and then convert the proposition into "No horses are cows." We simply convert one Universal Negative (E) into another Universal Negative (E).

(I) Particular Affirmative: This form of proposition is converted by Simple Conversion. For neither term is distributed in a Particular Affirmative. ("In particular propositions, the subject is not distributed. In affirmative propositions, the predicate is not distributed.") And neither term being distributed in the convertend, it must not be distributed in the converse. So from "Some horses are males" we may by Simple Conversion derive "Some males are horses." We simply convert one Particular Affirmative (I), into another Particular Affirmative (I).

(O) Particular Negative: This form of proposition is converted by Contraposition or Negation. We have given examples and illustrations in the paragraph describing Conversion by Contraposition. The Particular Negative (I) is converted by contraposition into a Particular Affirmative (I) which is then simply converted into another Particular Affirmative (I).

There are several minor processes or methods of deriving judgments from each other, or of making immediate inferences, but the above will give the student a very fair idea of the minor or more complete methods.

Exercise: The following will give the student good practice and exercise in the methods of Conversion. It affords a valuable mental drill, and tends to develop the logical faculties, particularly that of Judgment. The student should convert the following propositions, according to the rules and examples given in this chapter:

1. All men are reasoning beings.

2. Some men are blacksmiths.

3. No men are quadrupeds.

4. Some birds are sparrows.

5. Some horses are vicious.

6. No brute is rational.

7. Some men are not sane.

8. All biscuit is bread.

9. Some bread is biscuit.

10. Not all bread is biscuit.