Essentials of logic

Preface

This is intended to be a working handbook of logic, and not a history of the subject. No novelty is claimed, except the novelty of simplicity, and freedom from needless technicalities, hereditary nomenclature, and other barnacles ; as, for example, in the treatment of the syllogism and of fallacies. With teachableness and early practical application in view, traditional accretions and controversies have been omitted; and yet, it is believed, without sacrifice of essentials. There is constant endeavor to make evident the relation between different parts of the subject.

Quotation marks have been used freely where they seemed advantageous, but not according to any hard and fast rule.

The book contains a large collection of examples for practice, selected also for their intrinsic value, and illustrating the forms of thought that occur in actual literature, and not merely in logical exercises. Adequate use of these will double the working length of the book.

W. D.

Clarksville, Tennessee, October, 1913.

Chapter I What Logic Is

i. Logic Defined. — Correctness of thought is vital to success in everyday affairs, to progress in knowledge, to science, to literature, to philosophy, to theology, to logic itself. Hence it is of great importance to know the laws of correct thinking, and this knowledge is acquired by the study of logic, which may be defined as the science of the forms and laws of correct thought.

2. Logic not Art but Science. — Science differs from other knowledge in being systematically arranged. Science is knowledge classified or systematized. Such knowledge often forms the basis for the highest skill, or art; but art is concerned with doing, science with knowing. The closeness of this relation of science and art is the reason why the two are often confused or identified. A man, however, may know the science of architecture in minute detail, and yet not have the art to build; a doctor may know the position and use of every tissue in the body, and yet have no skill as a surgeon.

ii

Logic is a science, not an art. Logic does not give us the power to think correctly, but it acquaints us with the laws of correct thought; it gives us the knowledge of what correct thinking must be. The ability to think correctly may be possessed without the study of formal logic, for it is an ability natural to man, but fuller knowledge of What is correct in thought results naturally and increasingly in the use of what is correct.

3. Logic a Distinct Science. — Scientific knowledge is divided into* branches, each called a science, and each distinguished from others by the distinct nature of the facts it investigates and systematizes. Systematized facts as to the human body form the sciences of anatomy and physiology; the plants give us botany; the stars, astronomy; the weather, meteorology; the mind, psychology; thought gives logic. The distinctness of logic from other sciences is due to the distinctness of the facts of thought from the facts treated of in other sciences.

We need to distinguish carefully from each other sciences whose facts are closely related, as physiology and anatomy, physics and chemistry, geology and mineralogy, psychology and logic, or grammar and logic. Grammar, psychology, and logic all have to do with thought. Grammar, however, is concerned not with the correctness of the thought, but merely with the expression of the thought; that is,

WHAT LOGIC IS 13

with the language. Psychology treats of thought solely as an activity of mind, while logic deals with the thought itself; psychology looks at the mind as it thinks, logic at the thought ; psychology considers the mind as it produces thoughts, logic considers the product. Psychology asks: What does the mind do when it thinks? Logic asks: What is the thought ?

As to the nature of thought it is enough to say here, somewhat loosely, that the affirmation or the denial that one of two things, groups of things, qualities, or ideas is related to the other of the two, is a product of thought, or, in brief, a thought. For such affirmation or denial to be warranted, it is evident that its two terms, the subject and the predicate, must first have been compared in mind with each other or with some third term. Thought affirms or denies that one thing is contained in or comprised under another.

4. Logic Related to All Science. — Since science is knowledge classified or systematized, and since it is thought that classifies or systematizes, and logic that treats of thought, it follows that logic is vitally related to every science. The things we think about are many; the correct ways we think about them are few; and as everything we think about must be thought about in some definite way, or form, according to some definite law, if we think correctly ; then it follows that the science of any set of things or facts depends on the forms or laws correct thought must use, in other words, on the facts of logic. Since all science involves thought, every science depends on logic, the science of thought. Logic itself can be a science only by conforming to the laws of thought of which logic treats.

Logic, then, is a science of forms in which a countless variety of thoughts about many things may be cast. Logic treats, not of the things we think about, but of the ways in which we think about things, in trivial everyday affairs or in highest regions of scientific deliberation. Its realm is the universe of thought. One of the forms of thought is the term, or notion. Examples of the notion are: apple, red, sphere, thought, man, each of which, as will appear later, stands for the result of prior comparisons. Other thought-forms are judgments, by means of which we affirm or deny that two notions are related to each other, as : the earth is a sphere, snow is white, justice is not mercy; also, subject and predicate, genera and species, inferences, syllogisms, and many others. The form can be made clear only by filling it with matter, which is a step beyond pure logic, but used freely in pure logic for illustration.

Chapter II

5. What Notions Are. — Grammatically, a notion is a noun or an adjective, as for example, " apple," " red " ; logically, a notion is either a " mark," or quality of a thing, as red is a quality of apple; or the notion stands for a group of qualities, and is called a concept, as apple stands for round, juicy, edible, etc. ; or it stands for a group of things, as apple stands for every member of the group called by that name; psychologically, the notion is the result of abstraction, conception, and generalization. For example, looking at a certain object, we " abstract," or take away in our minds, and consider apart from other qualities, that which we see, and say, the object is red ; handling it, we abstract shape and weight from other qualities and say, it is round, it is heavy; smelling it, we say it is fragrant, and so for each quality perceived by the senses. It is evident that abstraction is purely a mental process in no way affecting the object or its qualities. Each of the abstractions above is expressed in the form of a " judgment," which is thus used in forming no-

iS tions, as well as the notion in forming other judgments. The judgment is discussed in Chapter IV.

The second mental step in forming a notion gathers into one the various marks noted by abstraction, and for convenience gives this bundle of marks a name. This bundle of marks is called " concept," or " gathered together." Thus the red, round, heavy, juicy, edible object of the above illustration of abstraction is called " apple " ; and apple is the concept which includes, or " connotes " the marks red, round, etc.

The third mental step in forming a notion is generalization. The marks red, etc., and the concept apple refer only to the one particular object we have been examining; but we receive similar impressions from other objects, and we say they are red, round, juicy, etc. ; that is, all these objects have these marks, all are apples. Thus we " generalize " the concept apple, we say it includes every object of a certain group, or class, or " genus." In this way we classify our knowledge, we get the genera of science.

The order in which conception and generalization are mentioned above is arbitrary. We may either bundle the marks of one object into a concept and then generalize, or we may observe a number of objects having similar marks, and generalize each of these marks before forming them into a concept.

Marks and concepts are notions. Each form can be changed into the other. The difference is solely in the way the form is used, and may be disregarded in the development of the science of logic.

It is evident that a concept may be viewed as a group of marks or a group of things. " Apple" is said to " connote " the marks juicy, subacid, etc., and to " denote " all apples. To give the marks which a notion includes is to " define " it; to give the classes of things contained under a notion is to " divide " it. Logical definition and division are discussed in sections 7 b and 7 c below.

Many concepts include a practically unlimited number of marks : hence in defining the concept by giving its marks, we select only those essential to the nature of the object or the class, those without which the object would not be what it is. The concept man includes the marks, two-legged, having hair on the head, erect, and many others, the absence of which would still allow the objects denoted by the concept to be men; but the marks animal and rational must be present, else the objects denoted will not be men.

According to what is known as " the law of inverse ratio," the fewer marks a concept contains, the greater the number of things it includes, and the reverse. If to the marks of apple we add " red," the objects included under " red apple " are evidently fewer than those under " apple " ; if from the marks of " rocking-chair " we take the mark " rocking," we have " chair," a class containing more objects than " rocking-chair." All the marks contained in a concept are called the " intension " of the concept; the things contained under a concept are called the " extension " of the concept; and the above law of inverse ratio may be stated thus : The greater the intension, the less the extension, and the reverse.

A notion is also called a " term " (Latin, terminus, end), because in its use as subject or predicate it is the end of the sentence. A notion often consists of a group of words, as, for example: North American Indians, rivers which flow into the Atlantic Ocean.

PRACTICE ON SECTION 5

1. What marks do you find in your notion of gov-

ernment, campus, home, railroad, beauty, food, athletics, miser, education, money?

2. What classes or kinds do you think of as included

under each of the above notions?

3. Name some marks common to two or more of these notions.

4. Name two or more concepts having one of the

following marks in common: free, heavy, fourlegged, juicy, upright, smooth, false, yellow.

5. Does every mark you think of as belonging to one of these concepts, belong to every one of the classes under the concept?

6. Can you name a mark that belongs to some of the classes under a concept, and not to others?

7. Add some mark to one of the above concepts, and show that the resulting concept denotes fewer things; then add other marks successively, noting the effect on the number of things.

8. Illustrate any of the above exercises by using five other concepts.

9. Add two classes of things, such as acute and right

triangle, say what mark has been lost, then add another class to the result, and so on.

10. Give other illustrations of the above exercise.

11. Illustrate abstraction, conception, and generalization by any of the above notions.

12. State the intension and the extension of any of the above notions.

13. Change any of them from marks to concepts, and the reverse.

14. Name some of the marks connoted and some of the kinds denoted by each of the two meanings of the following terms: school, band, mark, notion, bed, log, yard, party.

15. Would a perfect language have words of two meanings ?

6. Kinds of Notions. — The following are the most important of the many ways in which notions may be grouped in classes:

6 a. Notions are " concrete " or " abstract." A concrete notion is the name of a thing or of a quality thought as being in a thing: for example,

" green " is a concrete notion, because it can in this form be thought only as belonging to grass or other green things. It is of course the adjective and not the noun " green," of which this is true. An abstract notion is a quality of a thing thought as being itself a separate thing; for example, greenness, truth, morality, hardness. These are not things, though named as if they were; they are qualities only; respectively, say, of grass, assertions, conduct, iron. All adjectives are concrete. Many nouns originally abstract have by change in usage become concrete; as, "relation" for "relative," "action" for " act."

6 b. Notions are " particular " or " general." A particular, or " singular," or " individual " notion denotes one object; as John, Mr. Brown, this hat, my house. The objects we see, hear, or know through any sense, are particular objects. Abstraction and conception, the first two steps in the formation of notions, described in Section 5, have to do with particular notions; the third step, generalization, with the general notion, or the notion which denotes a " genus," or " kind," or " class " ; as martyr, bean, horse, and all common nouns. The general does not exist except in thought ; real things are individual, singular, particular, and not general. The general notion, however, is often made to denote a particular object by the use of some limiting word or group of words, as " this," " that," " his,"

" the," a relative clause, etc. Sometimes there is ambiguity, and care must be taken, as always, to get at the actual meaning of the words. For example, in the sentence: a man is a complex creature, "a man " is general, and the meaning is the same as : man is a complex creature; but in the sentence: a man is running down the street, " a man " is particular, denoting only one man. Again, in: the ox is patient, and, the ox is dead, we have the double use of " the " ; the first either general or particular, the second particular. The limiting words, phrases, or clauses, which thus reduce the general notion to the particular, are called " particular marks."

It is therefore obvious that all strictly proper names are particular. The difference between the proper name and the common name in its particular use, as above, is that the latter implies marks, the former not. For example, " George " has no meaning whatever, is a mere sign arbitrarily associated with an individual person ; but to designate the same person George as " that boy," attributes to him all the marks of the concept " boy." So-called proper names, however, often show sex, or other marks, and are to that extent not strictly proper: John is of the male sex; the latter part of " Clarksville " means city; so "port," "burg," "ton," and other parts of names.

A proper noun may be used as a general name, some prominent mark possessed by the owner of a well-known proper name being thus attributed to others; as, a Daniel come to judgment; or, he is a Nero, a Herod.

6 c. Notions are " obscure " or " clear." Obscure notions are those not well enough known to be separated in thought from others nearly alike ; as, perhaps, caribou, moose; adhesion, cohesion, attraction; college, university; thought, feeling, consciousness, desire; memory, imagination. On the other hand, we have a clear knowledge of dog, house, ocean, school, etc.

Clear notions may be " indistinct " or " distinct." We have a clear notion of " grass " for example, but our knowledge is not distinct unless we know the marks and the kinds of grass. " Table," " carpet," " bread," " book " are examples of notions distinctly known, for we know their marks and their kinds, the two phases or parts of distinct knowledge.

It appears that clearness is attained by denials, which say what a thing is not; distinctness by affirmations, which say what a thing is : as, he is no fool, he is a man of great ability; a mandolin is not a violin, it is not played with a bow, it has a larger air-chamber, it is played with a plectrum.

Distinct notions may be " inadequate " or " adequate." Inadequate notions are those that do not imply enough marks for them to represent accurately and fully the things in question; while adequate notions are those which imply enough marks for the particular purpose in view.

Distinct notions are also " intuitive " or " symbolic." Intuitive notions are those used in thought by means of a mental image of one of the things included under the notion; as, for example, bed, house, dog, if accompanied by a visual image of a bed, a house, a dog ; or intuitive notions may be accompanied by the actual perception of one object of the class, as when we are looking at a bed, etc., or at a picture of the object. If not so accompanied, the words, bed, house, dog, etc., are mere symbols, and we do not look the thing in the face, as " intuition " means. Many notions, therefore, may be used either intuitively or symbolically ; many others are too complex to be imaged by the mind, and can be used only symbolically; as, education, prosperity, peace, the velocity of light, the government of the United States.

Perfect knowledge is clear, distinct, adequate, intuitive. Most of our knowledge is imperfect ; but the possibility of progress toward perfection is everpresent, and the effort to make the progress is a duty.

Language is the vehicle and the storehouse of thought, and the notion is the fundamental word. The child can make its wants known surprisingly well without other words than the noun. A verb can be resolved always into the verb " be " and a notion. The explanation of the notion is the explanation of language. Without language, the progress of thought would be very narrowly restricted. By imperfect language, the accuracy and the communication of thought are jeopardized. The abstract should be frequently tested by the concrete, the general by the particular, the symbolic by the intuitive.

6 d. The following classifications of marks are important:

ist. Marks are " positive," or " negative " : " wet " is a positive, " dry " a negative mark, meaning merely the absence of moisture. Marks are positive or negative in themselves, without regard to any particular concept to which they may belong.

2nd. Marks are " essential " or " accidental " ; as, " for heating " is an essential, " self-feeding " an accidental mark of stove. Marks are not essential or accidental in themselves, but with regard to the concepts to which they belong.

3rd. Marks are " common " or " peculiar," according as they belong to a concept in common with other concepts, or belong to that concept only : " living " is common to man and horse, " laughing " is peculiar to man, " neighing " is peculiar to horse.

PRACTICE ON SECTION 6

1. Name ten concrete, and ten abstract notions.

2. Name five concrete, and the five corresponding

abstract terms.

3. Change the following notions from general to particular, or the reverse: bread, a storm, the horse, truth, pleasure, fire, charity, king, that tree.

4. Which, if any, of the following are proper names :

Caesar, Czar, Kaiser, Pharaoh, President, Mr. President, Bible, Episcopal Prayer-Book, High School, University of Berlin, Mathematics? What marks are connoted by any which are not proper names?

5. Write a list of ten general terms. Of ten particular terms.

6. Mention several notions not clear to you, naming those not separated clearly from them in your thought.

7. What are some notions you know clearly, but not

distinctly ?

8. Can you think of a notion you know clearly and

distinctly, but not adequately?

9. Give examples of the intuitive and the symbolic representation of several concepts.

10. Mention something your knowledge of which you

deem perfect.

11. Are the following marks positive or negative; that is, do they indicate the presence of a quality, or mere absence: original, negative, essential, complex, sober, protestant, pure, reckless, restful, clean, insane, impartial, silent, dark, weak, hard, soft, paralyzed, oblique?

12. Give the opposite negative or positive mark cor-

responding to each of the above.

13. Which of the marks in list a are essential to any of the concepts in list b? a. round, juicy, living, dead, harmful, peaceable, long-winded, intelligent. b. horse, poison, pencil, orange, bore, corpse, earth, circle.

14. Can you name essential marks of house, kitchen,

stove, study, book, star, plow, spoon, chair, ball, bird, dog, logic, college?

15. Name, if you can, notions of which one of the following is an essential mark : square, blue, loud, sour, smooth, fragrant, animal, logical, mental, conscious, extended, rational, military, strong, scientific, stingy.

16. Use 14 and 15 above, substituting accidental for

essential.

17. What marks can you think of that are peculiar

to any of the notions in 14? What that are common to any two or more of these notions?

7. Relations of Notions. — It was seen in Section 5 that a notion may be viewed either as a group of marks or as a class of things, the marks being the " intension," the things the " extension " of the notion. Intensively, notions may agree, and therefore unite in thought; as, good and great, tall and manly, hardness and weight; or they may disagree in the sense that they do not actually co-exist in one object, which is however a question of matter, not of form, and therefore not in the realm of pure logic; for example, a blue toothache, a happy tree; or again, the two notions may contradict each other, and therefore can not be combined either in thought or in fact, which is the only true logical disagreement; as, theistic and atheistic, wet and dry, learned and ignorant.

Extensively, two notions may represent classes of things in any of five relations:

1st. The two classes may exclude each other and not be immediately contained under some wider class : as desk and walk, spoon and wheelbarrow. This relation may be called " exclusion," and is shown by the fact that no member of either class is a member of the other.

2nd. The two notions may exclude each other, and yet be immediately contained under some wider class; as, Catholic and Protestant under Christian, Greek and Roman under Catholic, right angle and oblique angle under angle. This is the relation of " co-ordination," each notion being of the same rank (Latin, ordo) with respect to the wider class immediately containing it.

3rd. The relation of the notion contained under another to this other is " subordination."

4th. Again, each of two notions may have a part in common with the other, and a part not in common ; as, Europeans and Turks, wholesome food and flesh, men and students. This relation is " intersection."

5th. Lastly, two notions may be equivalent; as negro and darkey, sun and fixed star, leaves and foliage. Such notions are said to be in the relation of " co-extension." The test is that every member of either class is also a member of the other.

Each notion may be represented by a circle, two circles outside of each other illustrating co-ordination, if they are contained in a third circle, or exclusion if there be no third circle; one circle within another illustrates subordination; and two intersecting circles, intersection.

7. a. Classification. — It is evident that a notion which is subordinate to another may have other notions subordinate to it; and that the notion to which others are subordinate may itself be subordinate to some wider notion, so that a system of notions, or classes may be formed. This is classification.

If, as in co-ordination, two classes exclude each other, one of the classes must have at least one mark that the other does not have ; or, to state the matter differently, one class has a positive mark not belonging to the other, while the other has the corresponding negative mark; without this difference in marks, the two classes would of course coincide, and the relation would be co-extension instead of co-ordination. For example, Protestant includes the mark, denying authority of the Pope, while Catholic does not ; right angle has the mark, having a definite magnitude, while oblique angle has not. In fact, each of two co-ordinate classes contains every mark of the wider class to which they are subordinate, and each contains in addition one other mark; as, Catholic and Protestant contain all the marks of Christian, and in addition, respectively, the marks, accepting the authority of the Pope, and, not accepting the authority of the Pope; or, the classes red raspberries and raspberries of other colors have each all the marks of the class raspberry, and in addition, respectively, the marks red and not-red.

It is evident that we are here considering the three processes in the formation of concepts, described in Section 5, from a somewhat different point of view ; for it is by abstraction that we learn the marks, by conception that we form the notion, and when we generalize, we form a class. It is further evident that the law of inverse ratio, of Section 5, applies here, so that each of two co-ordinate classes, as man and brute, having each one mark, as rational and non-rational, respectively, more than the wider class, animal, to which they belong, contains fewer things. To add marks is to diminish the number of things included under the concept, to divide the things under the former concept into two classes, one having the mark in question, the other

So ESSENTIALS OF LOGIC

not; it is to classify, to make classes, or species. On the other hand, to take away marks is to add things, to generalize, to make genera (plural of genus). So genus and species are relative terms; any genus may be divided into species, and these species may in turn be genera of lower species; so any genus except the highest may be in turn a species under a higher genus.

What are the limits to the process? How far can we continue to add marks, to cut off things? Theoretically, until all common marks are included, that is, until we reach a class such that not even any two of its members have a common mark which is not also a mark of all the other members of the class. In such a case there can be no further division into classes, for unless at least two things have such a common mark, division will result not in classes or species, but in separate individuals; and if all the other members of the class have the only marks that are common to any members, any attempt at division would include the whole class, and be no division. For example, a group of men contains one lawyer, one doctor, one merchant, one statesman, and one railroad president, and can be divided on the basis of profession only into individuals. This illustrates the lowest logical species, the species that can not be a genus and be divided into lower species, but only into individuals.

On the other hand, how far can we continue to generalize, to subtract marks, to add things ? Theoretically, until all things are included. Disregarding the metaphysical distinction of matter and form, the widest class, the " summum genus," therefore, is " thing," which has only the one mark, " existing," and denotes everything. For practical scientific ends, classifications do not have to begin with this highest class, but with the class most suitable in the different sciences. Astronomy, for example, starts with heavenly body ; botany, with plant ; and so on. When we do not know the proper class for an object, or do not care to be scientific, we refer it to the " summum genus," and call it a " thing."

The two relations of notions, co-ordination and intersection, are of such importance that they are treated more in detail in Sections 7 b and 7 c immediately below.

PRACTICE ON SECTIONS 7 and 7 a

1. Pick from the following such pairs of notions as

(1) may unite in thought, or (2) do not actually unite in an object, or (3) can not unite in thought or in fact: wide, clear, poor, savage, blue, sweet, arm, pain, wealthy, mean, man, music, sorrow, gentle, short.