An elementary logic

Front matter

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Preface

It is the aim of this book to aid young students in gaining a comprehension of the essential principles of correct thinking.

It aims also to assist those teachers who find the use of a text-book in this subject advantageous. The book contains little that is original. It follows the main tradition of logical doctrine, with such omissions and changes in method as I, during years of experience, have found it desirable to make in an elementary course in logic.

The more important deviations from traditional logic will be found in the treatment of Judgment and the Syllogism and in the part entitled The Logic of Science. I have emphasized the distinction between Judgment and Propositions by giving to the former a separate exposition, preceding the discussion of Propositions.

In the treatment of the Syllogism, some technical matter, particularly that pertaining to moods, has been omitted, and changes have been made in the arrangement of topics which it is hoped will facilitate the comprehension of this subject. I hope I have made some improvement in the treatment of what is customarily called Inductive Logic, or Induction.

6 PREFACE

I have tried to make clearer the nature of Science and the limits of its explanations ; and I have tried to give a more definite statement of the special problems of Science and to explain more clearly the methods of scientific thinking than is done in many books on this subject.

I am glad to acknowledge special indebtedness, both in the growth of my views and in the preparation of this book, to F. H. Bradley, Jevons, Bosanquet, Sigwart, and, preeminently, to John Stuart Mill, whose great work is, I think, still the most valuable contribution to Inductive Logic that has yet been made.

WiLLIAMSTOWN,

Chapter II

The Nature of the Concept. Concept defined and the Two

Significations of the Term Distinguished . . 16, 17

Kinds and Distinctions of Concepts . . . . -17

(1) Simple and Complex Concepts . . . . 17, 18

(2) Universal, Individual, and Collective Concepts . 18, 19

(3) Abstract and Concrete Concepts . . . 20-22

(4) Positive and Negative Concepts ... 22, 23

(5) Absolute and Relative Concepts .... 23

(6) Connotative Names, Extension and Intension of

Concepts 24-26

Chapter III

5. The Meaning of these Terms. Explanation of the So-called

Predicables. Genus, Species, Differentia, Property, and Accident 27-29

6. The Processes of Division, Definition, and Classification 29-34

8 CONTENTS

PAGE

7. Rules for Division and Definition ..... 34, 35

8. Observations upon Division and Classification in Formal Logic 35, 36

Chapter IV

9. The Nature of a Logical Judgment; Analysis of Judgment 37-39 Judgment distinguished —

(i) From Exclamation . . . . . '. .39

(2) From Interrogation ....... 40

10. Judgment and its Verbal Expression. Judgment distin-

guished from Grammatical Sentence. Different Ways

in which Judgment can be Expressed . . . 40-43

11. The Kinds of Judgments —

(1) Categorical Judgment 44

(2) Hypothetical Judgment 44

(3) Disjunctive Judgment ..... 49-51 Distinctive Features of the Hypothetical Judgment. Its Relation to the Categorical Judgment. Characteristics of

the Disjunctive Judgment. Features which distinguish this Judgment from the Categorical and Hypothetical Judgment 44-51

12. The Quality of Judgments. The Meaning of Negation in

Judgment. The Function of Negative Judgments 51-53

13. Other Distinctions in Judgments ...... 54

(1) Analytic and Synthetic Judgments .... 54

True Character of this Distinction .... 55

(2) Modal Judgments 55

The Distinction between Assertorial and Apodictic Judgments 55. 5^

(3) Different Significations of " may," "can," and " must "

in Logical Assertions 5^5^

Chapter V

14. Meaning of the Proposition. Propositions Analyzed. Logi-

cal and Grammatical Analysis .... 59, 60

CONTENTS 9

PAGE

15. Kinds of Propositions. Three Such Kinds corresponding to

the Three Kinds of Judgments. Two Additional Sorts : Exclusive and Exceptive Propositions ... 61, 62

16. Quality of Propositions. Negative Propositions; how^ Dis-

17. Quantity of Propositions. Quantification. Quantity in its

Application to the Subject-term. Universal and Particular Propositions. Quantity in its Application to the Predicate-term. The Method of Quantification of Propositions 65-73

CHAPTER VI Inference. Reasoning

18. The Nature of Inference. Inference Defined. Its Essential

Elements. The Criteria or Tests of Inference. Inference distinguished from Other Mental Operations. Views of Mill and Other Logicians Criticised , 74~^S

19. The Forms of Inference. Deductive and Inductive Inference

Distinguished. Varieties of Deductive and Inductive Inference 85-89

CHAPTER VII The Forms and Methods of Deductive Inference

20. EquipoUence : Its Meaning and the Two Relations on which

it is Based. Various Ways in which EquipoUence is Maintained. Obversion, Conversion, Contraposition, and Added Deterniinants 90-96

21. Inference by Opposition. Contrary, Contradictory, Sub-

contrary, and Subaltern Propositions . . . 96-99

CHAPTER VIII Mediate Inference. The Syllogism

22. Description of the Syllogism. The Categorical, the Hypo-

thetical, and the Disjunctive Syllogism. Irregular Syllogism. Dilemma. Enthymeme. Sorites . . 101-I15 lO CONTENTS

23. Regulative Principles and Rules for the Syllogism. Rule for

the Disjunctive Syllogism, Basis of the Hypothetical Syllogism and the Consequent Rules for its Valid Use. Rules for the Categorical Syllogism ascertained and systematically Stated ...... 1 15-126

Chapter IX

24. Description of Fallacies. Explanation of them and the Main

Division of them into Formal and Material Fallacies. Specific Exposition of Material Fallacies. Formal Fallacies Explained ....... 127-138

25. Classification and Technical Designation of Fallacies . 138-142

26. Value of the Syllogism 142-150

Part Two The Logic Of Science Chapter X

27. The Meaning of Science. The Scientific Conception of the

World. Scientific Explanation. The Special Problems of Science Explained. Ascertainment of Causal Connection by Observation and Experiment. Explanation by Hypothesis. Calculation of Chances and the Method of Statistics 152-159

CHAPTER XI The Ascertainment of Causal Connection by Observation and Experiment

28. Observation and Experiment. Importance and the Difficul-

ties of Observation. Advantages of Experiment. The Aid

it affords in the Ascertainment of Causal Connection 160-165

29. The Regulative Principles for Observation and Experiment.

The So-called Inductive Methods. The Aim of these Methods illustrated by their Use .... 165-174

CONTENTS 1 1

PAGE

30. The Logical Value of the Methods of Observation and

Experiment 175-180

Chapter XII

31. The Essential Features of a Scientific Hypothesis. The

Requisites of a Legitimate Hypothesis . . . 182-184

32. The Method of Explanation by Hypothesis. The Two Essen-

tial Processes. Construction of Hypothesis and Verification. Complete Verification and Proof distinguished from Incomplete Verification and Disproof . . 184-19 1

33. The Value of Rejected Hypotheses .... 191, 192

CHAPTER XIII The Third Special Problem in the Logic of Science

34. Calculation of Chances. The Meaning of Chance and Prob-

ability. The Method of calculating Chances. The Theory

of Probability 193-203

35. The Method of Statistics and the Uses of this Method . 203-207

CHAPTER XIV Generalization from Experience and Analogy

36. Inductive Generalization and its Varieties. Analogical Infer-

ence. Its Scientific and its Practical Value. Principles Vi'hich regulate its Employment .... 208-2H

Chapter XV

37. Fallacies Incident to reasoning upon Matters of Fact. Four

Groups of these Fallacies : Fallacies in Explanation by Observation and Experiment. Fallacies in Explanation by Hypothesis. Fallacies Incident to the Calculation of Chances and the Method of Statistics. Fallacies of Generalization and Analogy . . . . 212-223

APPENDIX. Practical Exercises and Questions . . 224

Section i

the meaning of logic

Logic is the science of thought. Its aim is to ascertain and apply the principles which are regulative for right thinking. By thinking in logic is meant the familiar mental operations of forming ideas, or notions, making assertions, and reasoning. By right or vahd thinking is meant that mode of thinking which attains its end, which is consistency, truth, and knowledge. The subject-matter of logic being thought, this science is not primarily concerned with things or facts of experience ; it is concerned with the knowledge of facts only so far as that knowledge is attained by thinking.

The aim of all serious thought is knowledge; and logic aids the attainment of knowledge in two ways:

(i) It affords a negative criterion of truth or test of knowledge ; for whatever violates the laws of thought cannot be true in any real world.

(2) Logic aids knowledge by defining the principles and laws of thought in accordance with which all knowledge beyond mere sense perception is attained.

If I am to reach knowledge, my thinking must conform to certain principles and laws ; and it is the function of logic, as we have said, to ascertain and apply these constructive principles.

Section 2 divisions in logic

The division of logical doctrines should be based upon a difference between two aims of thought, rather than upon the two forms of inference known as deductive and inductive.

These two kinds of inference are associated with these two distinct fields of logic, but the difference between them is not the basis upon which a division of logical doctrine should be made. Logical thinking aims at two things: consistency and knowledge of fact. By consistency we mean that connection between a given judgment and other judgments which makes a judgment true if the given one is true, and false if the given one is not true.

Consistency means, if I think and assert this, I must also assert that. By truth in matters of fact, I mean agreement or correspondence between thought and fact ; and the certainty of this agreement is knowledge. This difference in the aim of our thinking is the true principle of a division of logic. In accordance with it we have the Logic of Consistency and the Logic of Science. The former is customarily called Formal or Deductive Logic ; the latter, Inductive Logic.

INTRODUCTION 1 5

The function of Formal or Deductive Logic being to establish consistency in our thinking, the function of Inductive Logic the attainment of knowledge, I accordingly divide this study into two parts : —

Part One will deal with the Logic of Consistency; Part Two, with the Logic of Science.

Section 3

the nature of the concept

In formal logic it is customary to present the doctrine of thought under three principal topics : The Concept or Names or Terms, Judgments or Propositions, and Inference.

I shall adhere to this division and order of topic, and proceed first to the Concept.

DEFINITION

A Concept is a mode of thinking in which something is by our thought treated as one thing and distinguished from all other things. In the objective sense of the term, a concept is anything which can be thus thought as one thing, identical with itself and different from other things ; unity and distinctness from other things are consequently the defining characters of the concept in the objective meaning of the term.

THE CONCEFr 1 7

The subject-matter, or content, of a concept is its meaning. The features by which this meaning is distinguished from other meanings constitute the marks, properties, or attributes of a concept. The sum total of these defining marks which are essential to the meaning of a concept are what is meant by its intension, or connotation.

I have chosen the word Concept in preference to Name or Term, which are more commonly used, because concept makes more prominent the mental act or logical process that is being studied; term or name designates properly that which is the product or result of this act. It is regrettable that there is no good word for naming this mode of thinking. Concept in present usage is ambiguous ; it is not always easy to determine whether it is the mental operation itself, or the thing thought about that is meant. The words subjective and objective are the best we have to mark this difference in the meaning of the term ; concept in the subjective sense being the mental act, and concept in the objective sense meaning the product of this act or that which is meant. It is in this objective sense I shall use the term unless I give notice of my intention to use it in the other signification.

Section 4

the kinds and distinctions of concepts

I. Simple and Complex Concepts. — The content of a concept can be extremely simple, and its range very limited ; it can be something that is incapable of analysis or description, as, for example, this particular shade of blue I observe in the sky. This is a concept if, in addition to its being my momentary perception, I think it ; if I judge respecting it that it is Hke the blue I saw in the waters of Lake Geneva ; for by so doing I recognize this bit of color as one thing, distinguishable from other things. The content of a concept may be extremely complex, and its range of meaning illimitable. The universe itself may be a concept. Accordingly, considered in respect to their structure, concepts are Simple or Complex.

Simple concepts are those which do not admit of analysis or separation into other concepts. These concepts are formed at a stage of mental development which precedes the more conscious and purposeful thinking with which logic deals.

Complex concepts are those which can be resolved into other concepts. These concepts are the products of dehberate thinking. Such are the concepts of Science, Philosophy, and those for the most part which are employed in our ordinary intercourse with each other.

2. Concepts are again distinguished as Universal, Individual, and Collective.

A universal concept is formed by uniting in thought those features or marks and those only which are common to a number of individuals.

An individual concept is one which is formed by uniting in thought those marks which distinguish this

THE CONCEPT I9

individual from all other individuals. In traditional logic a general name, answering to a universal concept, is a name which, in the same signification, is applied to all the individuals which constitute a given class or totality. A singular name is one which in the same signification can be applied to but one individual. A collective concept is one which is formed by uniting in thought those marks or properties in virtue of which a number of individuals are considered, not separately or distributively, but as forming a single body or organism. For instance, in forming the concept army, I consider a number of individual men only so far as they are taken together and united in a particular form of organization; they thus form a single body. The individual men who compose an army are not considered distributively, but collectively; this collection implying both plurality and unity. Hence, this concept partakes of the character of both the universal and the singular concept. The collective name of traditional logic, hke the general name, implies a number of individuals; but, unhke the general name, the collective name cannot be applied to these individuals taken distributively, but only as they are taken together. The name army, for instance, is not applicable to the individual soldiers which constitute it; the name horse is applicable to individual animals; but this name army is applicable only to that single organization which these individual soldiers all taken together constitute. The name army apphes only to this unity of all the individuals so united.

3. Abstract and Concrete Concepts. — According to the point of view from which their subject-matter is regarded, concepts are Abstract or Concrete. The true distinction between these is best apprehended by observing two things in the formation of concepts.

The first is, that in forming any concept, particularly in forming the general concept, some abstraction is always involved, since this concept is formed by uniting only the marks which are common to a number of individuals, and there is consequently an abstracting from the other marks which belong as truly to these individuals as do the marks that are included in the concept. The difference between the abstract and the concrete concept does not he in the different ways in which these concepts are formed; not in the fact that abstraction takes place in the formation of one concept and not in the formation of the other; for some amount of abstraction is involved in the formation of any concept.

The second thing to observe is that there are two ways of treating the subject-matter of any concept ; we may either consider this subject-matter as something which possesses properties or marks, in which case we can consider these properties or marks only as they are conceived as belonging to this subject-matter; or, we may single out one or more of these properties, and by abstracting them, so to speak, from that to which they belong, form another concept. For example, in the concept of a Centaur, the various marks which constitute this concept, say a, b, c, d, . . . x, are considered only as they belong to the something called Centaur; and this something is thought of only as possessing these properties. Now let us select one of these marks, say b, and, by abstracting from the other marks, we form a concept, — the concept of the particular property designated by this letter b; this concept is abstract, not because there has been abstraction in forming it, but because its subject-matter is considered apart from the Centaur of which it is a property.

Accordingly a concrete concept is to be defined as one in which the subject-matter is considered as something which has properties, and its properties are considered only as they belong to this subject-matter.

An abstract concept is to be defined as one in which the subject-matter, while it can be the property of something, is considered in abstraction from that something to which this property would belong; thus, whiteness is the property of some conceivable thing, but this property is considered apart from any subject-matter to which it can be attached ; and when it is so considered, the concept is abstract.

This view of the distinction between concrete and abstract concepts explains the fact that many of the abstract names in logic are ambiguous, it being impossible to determine in the lists of such names given in text-books which are abstract and which are concrete. These distinctions as we have explained them are not fixed ; some words doubtless are always names of concrete concepts, as horse, animal, man, state ; others are perhaps as uniformly names of abstract concepts; but very many such words are in some situations names of abstract concepts and in other connections they are names of concrete concepts. For instance, in most situations charity would be rightly called an abstract name or concept; but in the passage, "Charity suffereth long and is kind," the concept is concrete; it is so because it is considered and treated as a something which has properties ; here the attributes are long-suffering and kind.

The same name can therefore be concrete in one usage and abstract in another; only the known intention and use of the author can determine in such cases whether the concept is abstract or concrete. Thus, in the proposition, "The mercy of the Lord is from everlasting to everlasting," mercy is an abstract concept; but in the sentence, "Mercy and peace shall go before him," the same name is concrete.

4. Positive and Negative Concepts. — According as the subject-matter implies affirmation or negation, presence or absence of this subject-matter, concepts are Positive or Negative. What it is to affirm or deny will be explained in the chapter on Judgments. The signs of negation are easy enough to recognize, and there should be no difficulty in distinguishing between positive and negative names. Such prefixes as not, non, invariably denote negative concepts.

Logicians are wont to distinguish a class of concepts which they call Privative. These, it is said, denote the mere absence of some state or quahty which naturally or normally belongs to a subject ; for example, blindness , deafness. These terms, it is said, have no meaning if applied to beings which do not normally possess the powers of sight and hearing. On the other hand, the terms not-seeing, not-hearing, are negative; they are appUcable to inanimate beings as well as to beings which can possess such powers. The difference between negative and privative is, however, one of degree only; a privative name is a negative name which has the additional imphcation that the subject ought normally to possess the specified mark or property.

5. Absolute and Relative Concepts, — In formal logic a name is called Absolute when its meaning is complete without involving a relation to another name. A name is Relative if its complete connotation does involve a relation to some other name. For example, the names husband, parent, brother, king, are relative names; while the names, metal, dog, man, happiness, are absolute names; the meaning of husband, parent, brother, king, is not complete without the relation to wife, other brothers or sisters, subjects; while the meaning of metal, dog, man, happiness, involves no such necessary relation.

The meaning of a relative name must not be confounded with the meaning which a name may have for any mind in consequence of all it suggests to that mind ; in the logical sense of the term a name is not relative because it suggests various other names, every name does that; but because that which is suggested is a part of its meaning. The name home suggests to my mind a thousand things, no one of which is any part of the logical connotation of this name; likewise the name father calls to my mind a hundred things which are not part of the logical meaning of this name; but this name does suggest one thing which is a necessary part of its connotation, viz. offspring, son or daughter ;• and apart from that relation and those two related things, this name has no logical meaning.

6. Connotative and Denotative Names, Extension and Intension of Concepts. — By some logicians names are distinguished as Connotative and Denotative. A connotative name is one which both designates a thing and imphes attributes of this thing; man, for example, is a connotative name, since it means individual beings and implies certain attributes belonging to these individuals. A denotative name is one which points out or distinguishes some individual, and does not imply in its meaning attributes.

I have given the customary distinction ; but it is not, I think, the right one. The real distinction is that of function or use of the name, or rather the intention of the thinker. In using the name man, the purpose of the thinker is to designate a certain group of qualities or attributes which are possessed in common by an indefinite number of individuals; these individuals are of importance, and are meant, only so far as they possess these attributes; it is the attributes and not the individuals that the name signifies, which constitute its connotation. The name John Smith is used to point out this individual, and to distinguish him from other individuals.

This distinction of connotation and denotation is not, properly speaking, a distinction in names themselves; it is rather a distinction in the purpose or intention of the one who is dealing with that which the names mean. The distinction which these logicians make is identical with that between universal or general, and individual or singular names.

We have seen that every name has a connotation; there are no denotative names in the sense in which these logicians use the term denotative, viz. a name which connotes no attributes. Even such individual names as John Smith must have some connotation ; for it is only by virtue of some properties that I can distinguish this individual named John Smith, my John Smith, from other individuals, or from the John Smith that somebody else may mean.

The distinctions of Extension and Intension apply to the subject-matter of every concept, and it is better to make extension synonymous with denotation and intension synonymous with connotation. Accordingly, by the intension or connotation of a concept or name is meant the marks which are common to all the individuals to which the name appHes; and by the extension or denotation of a concept or name is meant the individuals which possess these marks.

Every name has accordingly both extension and intension; this extension may be very great in the cases of names of classes or it may be reduced to a single individual in the case of a singular name. So with intension : it ranges from very few marks to an innumerable number of marks. A law of relation between extension and intension is sometimes stated in this way; extension and intension vary in inverse ratio, or the greater the extension of a name the less is its intension. A glance at the way in which general concepts are formed makes^ the truth of this statement obvious.

Section 5

the meaning of these terms

By Division in logic is meant the separation of a given class into the lesser classes which are contained within it. By Definition is meant the specification or statement of the marks which distinguish these lesser classes within some larger containing class. By Classification is meant the systematic arrangement and distribution of individual objects so as to form classes.

Division and classification are complementary processes; and these processes cannot in reahty be separated from each other ; every division involves a classification, and every classification is at the same time a process of division.

A very intimate relation also exists between division and definition. A definition is involved in every process of division; and division is possible only as there is at the same time definition. I cannot systematically and completely divide a class without at the same time defining each of the successively formed classes;

nor can I define any one of these classes without at the same time dividing the larger class in which it is contained. It should also be observed that division and definition are closely related to extension and intension of names, division being the systematic statement or unfolding of the extension of a name, definition the systematic statement of the intension of a name.

Before proceeding to the exposition of these processes I have defined, it is necessary to explain a group of terms which occur in formal logic; they are the so-called predicables. Genus, Species, Differentia, Property, and Accident.

A genus is the larger class in the process of logical division; species are the lesser classes into which the genus is divided. Difference or differentia are the marks which distinguish the species from each other within a given genus. Property or proprium is a mark which belongs to every individual of a class, but which is not a part of the connotation or meaning of the name ; thus, equahty of its three angles to two right angles is a property of every triangle, but this mark is not essential to the meaning of a triangle, not necessary to its logical definition. By accident is meant a mark which belongs to some individual or to a part of the individual members of a class, but never to all the members of a class ; for example, a diameter of six inches may be a mark of a certain circle, or of a number of circles; but it is not a mark of all circles.

There is a twofold distinction between property and accident : —

(i) A property is supposed to depend upon the essential nature or essence of the species or genus; while the accident does not. Thus, the equality of the radii of a circle depends upon the essential nature of the circle; hence it is a property of all circles; but a given length of a radius, say six feet, does not depend upon the nature of all circles; it is only an accident of certain circles or of one circle.

(2) The property mark must belong to every individual of the class; the accident cannot belong to every individual, and may belong to only one individual. The student must not understand that genus and species in logic designate fixed classes ; they are, on the contrary, only relative distinctions. The same class can be a species relative to a larger class of vi^hich it is a part, and a genus relative to a lesser class into which it can be subdivided; thus, horse is a species relative to the genus quadrupeds, and a genus to the various kinds of horses into which we can divide this class.

Section 6

the processes of division, definition, and classification

From this explanation of the technical terms, we pass to the processes of division, definition, etc. We can best understand these processes if we study a concrete case. Accordingly, let us suppose we are to divide the books in a hbrary. Our problem is both one of division and classification; for, at the outset, these books are a class of things in virtue of certain marks common to them all. Now, it is obvious that different divisions and different classifications of these books are possible. The first step, therefore, in our task is the selection of a principle, or basis, of this proposed division and classification ; in technical phrase a fundamentum divisionis. We will accordingly make the subject-matter of which these books treat the basis of the division, the fundamentum divisionis. We first divide the genus books into the following species ; Histories, Books of Science, Literature, and Philosophy. This division is neither exact nor exhaustive, but it will answer our purpose. We note that the species are coordinate; the following diagram will show this : —

BOOKS

History Science Literature Philosophy

We note also that a peculiarity of the subject-matter treated in each of these species of books constitutes the differentia of each species: thus, the differentia of the history books is, that they treat of the actions of human beings in society.

Now let our division be carried one step farther, and we obtain the following new classes: History gives Ancient, Mediaeval, and Modern History; Physical Science gives Physics, Chemistry, and Biology ; Literature gives Fiction, Essays, and Poetry; Philosophy gives Logic, Theory of Knowledge, and Metaphysics. There results from this second step in our division the following things : —