The elements of logic, a text-book for schools and colleges; being the elementary lessons in logic

Front matter

S^ x*

*W V

THE

Introduction

PAGE

1. Definition op Logic 1

2. Nature op a Law op Thought 2

3. A Science op Thought Possible 3

4. Distinction between Form and Matter . . * 5

5. Logic a General Science 6

6. The Particular Sciences, Special Logics 6

7. Logic both a Science and an Art 7

8. The Usefulness of Logic 8

9. Analysis of an Argument 9

10. Theories of the Real Subject-matter of Logic 10

11. The Three Logical Operations of Mind 12

12. Method of Treatment 16

Section I.— The Various Kinds Of Terms

1. The Meaning of " Term" Explained 17

2. Categorematic and Syncategorematic Words 18

3. Singular Terms 20

4. General Terms 20

5. Collective Terms 21

6. Concrete and Abstract Terms 22

Xll ANALYSIS.

PAGE

7. Positive and Negative Terms 24

8. Privative Terms 26

9. Relative and Absolute Terms 27

10. Summary 28

Section Ii.— The Ambiguity Of Terms

1. Importance op Avoiding Ambiguity 30

2. Univocal and Equivocal Terms 31

3. Kinds and Causes of Ambiguity 33

Section Iii.— Extension And Intension

1. Importance op Understanding this Double Mean-

ing 39

2. Meaning op Extension and Intension 39

3. Various Forms op Expressing Extension and Inten-

sion 41

4. The Variation of Extension and Intension 42

5. The Law of Variation 42

6. connotative and non-connotative terms 43

Section Iv.-the Growth Of Language

1. The Two Principal Processes of Growth 46

2. Generalization 47

3. Specialization 50

4. desynon ymization 51

5. Metaphorical Extension of Meaning 52

6. Origin of the Mental Vocabulary. 53

7. The Fertility of Root words 54

Section V.— The Perfect And The Imperfect Knowledge Of Terms

1. Statement of the Question 56

2. Scheme of Distinctions 56

3. The Intuitive and Symbolic Methods Compared 62

ANALYSIS. Xlll

CHAPTEE II.

PROPOSITIONS.

Section I.— The Kinds Of Propositions. Page

1. Meaning of " Pkoposition " Explained. . . 64

2. Analysis op a Proposition 65

3. Categorical and Conditional Propositions 66

4. The Quantity and Quality op Propositions 67

5. Aristotle's View of Quantity 68

6. Names of the Four Propositions 70

7. Variations from the Logical Form 71

8. The Modality of Propositions 73

Section Ii.— The Opposition Of Propositions

1. The Four Propositions Explained 75

2. The Distribution of Terms 79

3. Table of Results 80

4. Relations of the Four Propositions 80

5. The Scheme of Opposition 83

6. The Laws of Opposition 83

7. The Conditions of Opposition . . 84

8. The Matter of Propositions 85

Section Ii.— The Rules Of The Syllogism

1. The Definition of " Syllogism " 113

2. The Meaning op " Middle Term " . 114

3. The Use of Middle Term in Syllogism 114

4. Statement of the Rules of Syllogism 115

5. Explanation of the Rules 116

Section Iii.— The Moods And Figures Of The Syllogism

1. Explanation op " Moods " 124

2. The Number of Valid Moods 125

3. Explanation op " Figures " 127

ANALYSIS. XV

PAGE

4. The Valid Moods in the Different Figures 128

5. Conclusions Proved in the Different Figures 130

Section Iv.— The Reduction Of Syllogisms

1. The Mnemonic Verses 133

2. Explanation of the Mnemonic Verses 134

3. Conclusions from Particular Premises 139

Section V.— Irregular And Compound Syllogisms

1. The Irregular Mode of Expressing Inferences 141

2. Explanation of " Enthymeme " 142

3. Prosyllogisms and Episyllogisms 144

4 Sorites 145

5. Syllogisms in Extension and in Intension 148

Section Vi.— Conditional Syllogisms

1. Classification of Propositions 149

2. Antecedent and Consequent 150

3. Kinds of Hypothetical Syllogisms 151

4 The Rule for Hypothetical Syllogisms 152

5. The Reduction of Hypothetical to Categorical

6. Fallacies in Hypothetical Syllogisms 155

7. Disjunctive Syllogisms 156

8. The Dilemma 158

CHAPTER IY.

FALLACIES.

Section I.— Logical Fallacies

1. Classification of Logical Fallacies 162

2. The Fallacy of Equivocation 163

3. The Fallacy of Amphibology 164

XVI ANALYSIS.

PAGE

4. The Fallacy op Composition 165

5. The Fallacy of Division 166

6. The Fallacy op Accident 167

7. The Fallacy op the Figure op Speech • , . 168

Section Ii.— Material Fallacies

1. The Classification of Material Fallacies 169

2. The Fallacy of Accident and its Converse 169

3. The Fallacy of Irrelevant Conclusion 171

4. The Fallacy of Petitio Principii 173

5. The Fallacy of the Consequent 175

6. The Fallacy of False Cause 175

7. The Fallacy of Many Questions 176

CHAPTEK V.

INDUCTION.

SECTION I.— THE INDUCTIVE SYLLOGISM.

1. Induction and Deduction Contrasted 178

2. Explanation of Traduction 179

3. Importance of Induction 180

4. Perfect and Imperfect Induction 181

5. The Difference between Perfect and Imperfect

6. The Perfect Inductive Syllogism 183

7. The Perfect Inductive Syllogism Disjunctive 184

8. The Imperfect Inductive Syllogism 184

9. The Fundamental Assumption op Induction 185

SECTION II.— THE FORMS OF INDUCTION.

1. The Character of the Data 187

2, Special Kinds of Induction 195

Ill ANALYSIS. XVii

CHAPTEE VI.

METHOD.

SECTION I.— INDUCTIVE METHOD.

PAGE

1. The Search for Facts 201

2. The Rule for Observation. .' 206

3. The Uses of Hypothesis and Theory 208

4. Definitions of Terms Employed in Investigation. . . 212

5. Canons of Induction 215

SECTION II.— DEDUCTIVE METHOD.

1. The Predicables 227

2. Logical Divisions 234

3. Dichotomy, or Exhaustive Division 236

4. Definition 238

5. Classification 240

6. Requisites of a Good Classification 242

7. Denomination 245

Section Iii.— Complete Method

1. Empirical and Rational Knowledge 249

2. The Elements of Complete Method 251

3. The Nature of Explanation 254

4. Pascal on Method 257

5. Descartes on Method 263

CHAPTEE VII.

RECENT LOGICAL VIEWS.

Section I.— The Quantification Of The Predicate

1. Meaning of the Expression 263

2. Conversion with a Quantified Predicate 264

XV111 . ANALYSIS.

PAGE

3. The Rule for Conversion 266

4 Number of Propositions with a Quantified Predicate 266

5. Number of Syllogisms with Quantified Predicate. 268

6. Hamilton's Notation 268

7. Hamilton's Canon of the Syllogism 270

Section Ii.— Boole's System Of Logic

1. The Difficulty of Dr. Boole's Statement ... 272

2. Application of the Law of Excluded Middle 273

3. Application of the Law of Contradiction 274

4. Universality of the Method 275

5. Comparative Excellence of the System 277

6. The Logical Abacus and the Logical Machine 280

Questions and Exercises 283

1. Definition of Logic.

Logic may be most briefly defined as the Science of Reasoning. It is more commonly defined, however, as the Science of the Laws of Thought, and some logicians think it desirable to specify still more accurately that it is the Science of the Formal, or of the Necessary Laws of Thought. Before these definitions can be of any real use to us we must come to a clear understanding as to the meaning of the expressions ; and it will probably appear that there is no great difference between them.

The name of logic is derived from the common Greek word Tioyog, which usually means word, or the sign and outward manifestation of any inward thought. But the same word was also used to denote the inward thought or reasoning of which words are the expression, and it is thus, probably, that later Greek writers on reasoning, were led to call their science imaTTJfiTj ?i,oyiK7j, or logical science ; also texvt] "koyLK-q, or logical art. The adjective loyturi, being used alone, soon came to be the name of the science, just as Mathematic, Rhetoric, and other names ending in"ic" were originally adjectives, but have been converted into substantives.

INTRODUCTION.

2. Nature of a Law of Thought.

By a Law of Thought we mean a certain uniformity or agreement which exists and must exist in the modes in which all persons think and reason, so long as they do not make what we call mistakes, or fall into self-contradiction and fallacy. The laws of thought are natural laws with which we have no power to interfere, and which are, of course, not to be in any way confused with the artificial laws of a country, which are invented by men and can be altered by them. Every science is occupied in detecting and describing the natural laws which are inflexibly observed by the objects treated in the science.

The science of astronomy investigates the uniform or similar way in which the heavenly bodies, and, in fact, all material substances, tend to fall towards each other, as a stone falls towards the earth, or to move round each other under the influence of this tendency. The universal law of gravitation is thus the natural law of uniformity treated in physical astronomy.

In chemistry the law of equivalent proportions describes the well ascertained fact that each chemical substance enters into combination with every other chemical substance only in certain definite proportions ; as when exactly eight parts by weight of oxygen unite with one part of hydrogen to form water, or sixteen parts of oxygen and six parts of carbon unite to form carbonic acid in the ordinary burning of a flame or fire. Whenever we can detect uniformities, op similarities, we so far create science and arrive at natural laws. But there may be and are, many things so fickle, complicated, and uncertain, that we can never be sure we have detected laws that they will uniformly obey ; in such cases no science, in the proper sense of the word, is possible. There is no such thing, for instance, as a real science of human character, because the human mind is too variable and complicated a subject of investigation. There are no two persons so much alike that you may be sure of one acting in all circumstances as the other would ; it thus becomes impossible to arrange persons in classes so that all who are in the same class shall act uniformly in the same manner in any given circumstances.

3. A Science of Thought Possible.

There is a science of human reason, or thought, apart from the many other acts of mind which belong to human character, because there are modes in which all persons do uniformly think and reason, and must think and reason. Thus, if two things are identical with a third common thing, they are identical with each other. This is a law of thought of a very simple and obvious character, and we may observe concerning it : —

(1) That all people think in accordance with it, and agree that they do so as soon as they understand its meaning.

(2) That they think in accordance with it whatever may be the subject about which they are thinking.

Thus, if the things considered are —

London,

The Metropolis,

The most populous city in Great Britain,

since " the Metropolis is identical with London," and " London is identical with the most populous city in Great Britain," it follows, necessarily, in all minds, that " the Metropolis is identical with the most populous city in Great Britain." Again, if we compare the three following things —

Iron,

The most useful metal,

The cheapest metal, —

and it be allowed that "The most useful metal is Iron," and

"Iron is the cheapest metal," it follows, necessarily, in all minds, that " the most useful metal is the cheapest." We here have two examples of the general truth, that things identical with the same thing are identical with each other; and this, we may say, is a general or necessary form of thought and reasoning. Compare, again, the following three things:—

The earth,

Planets,

Bodies revolving in elliptic orbits.

We cannot say, as before, that "the earth is identical with the planets ; " it is identical only with one of the planets, and we therefore say that "it is a planet." Similarly we may say that " the planets are bodies revolving in elliptic orbits," but only a part of the whole number so revolving. Nevertheless, it follows that if the earth is among the planets, and the planets among bodies revolving in elliptic orbits, then the earth is among the latter.

A very elementary knowledge of chemistry enables us to argue similarly concerning the following : —

Iron,

Metals,

Elementary substances.

Iron is one of the metals, and metals are elements «or simple undecomposable substances, in the sense of being among them or a part of them, but not as composing the whole. It follows, necessarily, that "Iron is one of the elementary substances." We have had, then, two examples of a fixed and necessary form of thought, which is necessary and true, whatever the things may be to which it is applied. The form of argument may be expressed in several different ways, and we shall have to consider it minutely in the lessons on the syllogism. We may express it, for instance, by saying that ' ' part of a part is part of the whole." Iron is part of the class of metals, which is part of the class of elements— hence iron is part of the class of elements.

INTRODUCTION.

4. Distinction between Form and Matter.

In order to apprehend the meaning of the expression, " the necessary forms of thought," we must distinguish between form and matter. A form is something which may remain uniform and unaltered, while the matter thrown into that form may be varied. Medals struck from the same dies have exactly the same form, but they may be of various matter, as bronze, copper, gold, or silver. A building of exactly the same form might be constructed either of stone or bricks ; furniture of exactly similar shape may be made of oak, mahogany, walnut wood, etc. Just as we thus familiarly recognize the difference of form and substance in common tangible things, so we may observe in Logic, that the form of an argument is one thing, quite distinct from the various subjects or matter which may be treated in that form.

We may almost exhibit to the eye the form of reasoning to which belong our two latter arguments, as follows ; —

If within the three pairs of brackets, marked respectively X, r"and Z, we place three names, such that the one in place of X may be said to come under that in T, and that in Y under that in Z, then it necessarily follows that the first (X) comes under the last (Z).

INTRODUCTION.

5. Logic a General Science.

Logic, then, is the science occupied in ascertaining and describing all the general forms of thought which we must employ so long as we reason validly. These forms are very numerous, although the principles on which they are constructed are few and simple. It will hence appear that logic is the most general of all the sciences. Its aid must be more often required than the aid of any other science, because all the particular sciences treat portions only of existing things, and create very different and often unconnected branches of knowledge. But logic treats of those principles and forms of thought which must be employed in every branch of knowledge. It treats of the very origin and foundations of knowledge itself ; and though *it is true that the logical method employed in one science may differ somew r hat from that employed in another science, yet, whatever the particular form may be, it must be logical, and must conform to the laws of thought. There is, in short, something in which all sciences must be similar ; to which they must conform so long as they maintain what is true and self-consistent; and the work of logic is to explain this common basis of all science.

6. The Particular Sciences, Special Logics.

One name which has been given to Logic, namely, the Science of Sciences, very aptly describes the all extensive power of logical principles. The cultivators of special branches of knowledge appear to have been fully aware of the allegiance they owe to the highest of the sciences, for they have usually given names implying this allegiance. The very name of logic occurs as part of nearly all the names recently adopted for the sciences, which are often vulgarly called the "ologies," but are really the " logics," the "o" being only a connecting vowel or part of the previous word. Thus, geology is logic applied to explain the formation of the earth's crust ; biology is logic applied to the phenomena of life ; psychology is logic applied to the nature of the mind ; and the same is the case with physiology, entomology, zoology, teratology, morphology, anthropology, theology, ecclesiology, thalattology, and the rest.* Each science is thus distinctly confessed to be a special logic.

7. Logic both a Science and an Art.

Much discussion of a somewhat trifling character has arisen upon the question whether Logic should be considered a science only, an art only, op both at the same time. Sir W. Hamilton has even taken the trouble to classify almost all the writers on logic according as they held one opinion or the other. But it seems substantially correct and sufficient to say, that logic is a science in so far as it merely investigates the necessary principles and forms of thought, and thus teaches us to understand in what correct thinking consists ; but that it becomes an art when it is occupied in framing rules to assist persons in detecting false reasoning.

* Except Philology, which is differently formed, and means the love or study of words ; the name of this science, if formed upon the same plan, would be logology.

A science teaches us to know and an art to do, and all the

more perfect sciences lead to the creation of corresponding useful arts. Astronomy is the foundation of the art of navigation on the ocean, as well as of the arrangement of the calendar and chronology. Physiology is the basis of the art of medicine, and chemistry is the basis of many useful arts. Logic has similarly been considered as the basis of an art of correct reasoning or investigation which should teach the true method to be observed in all sciences. The celebrated British logician, Duns Scotus, who lived in the 13th century, and called logic the Science of Sciences, called it also the Art of Arts, expressing fully its pre-eminence. Others have thus defined it — " Logic is the art of directing the reason aright in acquiring the knowledge of things, for the instruction both of ourselves and others." Dr. Isaac Watts, adopting this view of logic, called his well-known work " The Art of Thinking."

It may be fairly said, however, that Logic has more the form of a science than of an art, for this reason — all persons necessarily acquire the faculty and habit of reasoning long before they even know the name of logic. This they do by the natural exertion of the powers of mind, or by constant but unconscious imitation of others. They thus observe correctly, but unconsciously, the principles of the science in all very simple cases. But the contradictory opinions and absurd fallacies which are put forth by uneducated persons show that this unaided exercise of mind is not to be trusted when the subject of discussion presents any difficulty or complexity.

8. The Usefulness of Logic.

The study of logic, then, cannot be useless. It not only explains the principles on which every one has often reasoned correctly before, but points out the dangers which exist of erroneous argument. The reasoner thus becomes consciously a correct reasoner and learns consciously to avoid the snares of fallacy. To say that men can reason well without logical science is about as true as to say that they can live healthily without medicine. So they can — as long as they are healthy ; and so can reasoners do without the science of reasoning — as long as they do reason correctly ; but how many are there that can do so ? As well might a man claim to be immortal in his body as infallible in his mind.

And if it be requisite to say a few words in defence of Logic as an art, because circumstances in the past history of the science have given rise to misapprehension, can it be necessary to say anything in its praise as a science? Whatever there is that is great in science or in art or in literature, it is the work of intellect. In bodily form man is kindred with the brutes, and in his perishable part he is but matter. It is the possession of conscious intellect, the power of reasoning by general notions, that raises him above all else upon the earth ; and who can say that the nature and procedure of this intellect is not almost the highest and most interesting subject of study in which we can engage ? In vain would any one deny the truth of the favorite aphorism of Sir W. Hamilton —

In the world there is nothing great but man. In man there is nothing great but mind.

9. Analysis of an Argument.

It has been explained that Logic is the Science of Reasoning, or the Science of those Necessary Laws of Thought which must be observed if we are to argue consistently with ourselves and avoid self-contradiction. Argument or reasoning, therefore, is the strictly proper subject before us. But the most convenient and usual mode of studying logic is to consider first the component parts of which any argument must be made up. Just as an architect must be acquainted with the materials of a building, or a mechanic with the materials of a machine, before he can pretend to be acquainted with its construction, so the materials and instruments with which we must operate in reasoning are suitably described before we proceed to the actual forms of argument.

If we examine a simple argument such as this —

Iron is a metal, Every metal is an element, Therefore Iron is an element, —

we see that it is made up of three statements or assertions, and that each of these contains, besides minor words, two nouns substantive or names of things, and the verb " is/' In short, two names, or terms, when connected by a verb, make up an assertion or proposition ; and three such propositions make up an argument, called in this case a syllogism. Hence it is natural and convenient first to describe terms, as the simplest parts ; next to proceed to the nature and varieties of propositions constructed out of them, and then we shall be in a position to treat of the syllogism as a whole. Such, accordingly, are the three parts of logical doctrine.

10. Theories of the Real Subject-matter of liOgic.

But though we may say that the three parts of logic are concerned with terms, propositions, and syllogisms, it may be said, with equal or greater truth, that the acts of mind indicated by those forms of language are the real subject of our consideration. The opinions, or rather, perhaps, the expressions, of logicians have varied on this point. Archbishop Whately says distinctly that logic is entirely conversant about language ; Sir W. Hamilton, Mr. Manscl, and most other logicians treat it as concerned with the acts or states of mind indicated by the words ; while Mr. J. S. Mill goes back to the things themselves concerning which we argue. Is the subject of logic, then, language, thought, or objects ? The simplest and truest answer is to say that it treats, in a certain sense, of all three. Inasmuch as no reasoning process can be explained or communicated to another person without words, we are practically limited to such reasoning as is reduced to the form of language. Hence we shall always be concerned with words, but only so far as they are the instruments for recording and referring to our thoughts. The grammarian also treats of language, but he treats it as language merely, and his science terminates with the description and explanation of the forms, varieties, and relations of words. Logic also treats of language, but only as the necessary index to the action of mind.

Again, so long as we think correctly, we must think of things as they are. The state of mind within us must correspond to the state of things without us, whenever an opportunity arises for comparing them. It is impossible and inconceivable that iron should prove not to be an elementary substance, if it be a metal, and every metal be an element. We cannot suppose, and there is no reason to suppose, that by the constitution of the mind we are obliged to think of things differently from the manner in which they are. If, then, we may assume that things really agree or differ, according as by correct logical thought we are induced to believe they will, it does not seem that the views of the logicians named are irreconcileable. We treat of things so far as they are the objects of thought, and we treat of language so far as it is the embodiment of thought. If the learner will bear this explanation in mind, he will be saved from some perplexity when he proceeds to read different works on logic, and finds them to vary exceedingly in the mode of treatment, or at least of expression.

11. The Three Logical Operations of Mind.

If, when reduced to language, there be three parts of logic, terms, propositions, and syllogisms, there must be as many different kinds of thought or operations of mind. These are usually called —

(1) Simple apprehension.

(2) Judgment.

(3) Eeasoning, or discourse.

(1) Simple Apprehension is the act of mind by which we merely become aware of something, or have an idea or impression of it brought into the mind. The adjective simple means apart from other things, and apprehension the taking hold by the mind. Thus the name or term iron instantaneously makes the mind think of a strong and very useful metal, but does not tell us anything about it, or compare it with any thing else. The words sun, Jupiter, Sirius, St. PauVs Cathedral, are also terms which call up into the mind certain well-known objects, which dwell in our recollection even when they are not present to our senses. In fact, the use of a term, such as those given as examples, is merely as a substitute for the exhibition of the actual things named.

(2) Judgment is a different action of mind, and consists in comparing together two notions or ideas of objects derived from simple apprehension, so as to ascertain whether they agree or differ. It is evident, therefore, that we cannot judge or compare unless we are conscious of two things, or have the notions of two things in the mind at the same time. Thus, if I compare Jupiter and Sirius, I first simply apprehend each of them ; but, bringing them into comparison, I observe that they agree in being small, bright, shining bodies, which rise and set and move round the heavens with apparently equal speed. By minute examination, however, I notice that Sirius gives a twinkling or intermittent light, whereas Jupiter shines steadily. More prolonged observation shows that Jupiter and Sirius do not really move with equal and regular speed, but that the former changes its position upon the heavens from night to night in no very simple manner. If the comparison be extended to others of the heavenly bodies which are apprehended or seen at the same time, I shall find that there are a multitude of stars which agree with Sirius in giving a twinkling light and in remaining perfectly fixed in relative position to each other, whereas two or three other bodies may be seen which resemble Jupiter in giving a steady light, and also in changing their place from night to night among the fixed stars. I have now by the action of judgment formed in my mind the general notion op concept of fixed stars, by bringing together mentally a number of objects which agree ; while from several other objects I have formed the general notion of planets. Comparing the two general notions together, I find that they do not possess the same qualities or appearances, which I state in the proposition, " Planets are not fixed stars."

The expression " General Notion " is introduced as if the learner were fully acquainted with it. Bat though philosophers have, for more than two thousand years, constantly used the expressions, general notion, idea, conception, concept, etc., they have never succeeded in agreeing exactly as to the meaning of the terms. One class of philosophers, called Nominalists, say that it is all a matter of names, and that when we join together Jupiter, Mars, Saturn, Venus, etc., and call them planets, the common name is the bond between them in our minds. Others, called Realists, have asserted that, besides these particular planets, there is really something which combines the properties common to them all, without any of the differences of size, Color, or motion which distinguish them. Every one allows in the present day, however, that nothing can physically exist corresponding to a general notion, because it must exist here or there, of this size or of that size, and, therefore, it would be one particular planet, and not any planet whatever. The Nominalists, too, seem equally wrong, because language, to be of any use, must denote something, and must correspond, as we have seen, to acts of mind. If then proper names raise up in our minds the images of particular things, like Jupiter, etc., general names should raise up general notions.

The true opinion seems to be that of the philosophers called Conceptualists, who say that the general notion is the knowledge in the mind of the common properties or resemblances of the things embraced under the notion. Thus, the notion planet really means the consciousness in anybody's mind that there are certain heavenly bodies which agree in giving a steady light and in moving about the heavens differently from the fixed stars. It should be added, however, that there are many, including Sir W. Hamilton, who would be counted as Nominalists, and who yet hold that with the general name is associated a consciousness of the resemblance existing between the things denoted by it. Between this form of the doctrine and conceptualism it is not easy to draw a precise distinction, and the subject is of too debatable a character to be pursued in this work.