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This institution reserves the right to refuse to accept a copy order if, in its judgement, fulfillment of the order would involve violation of the copyright law. A UTHOR : LIGHTFOOT, JOHN REV TITLE: LOGIC AND EDUCATION AN ELEMENTARY TEXT PLACE: LONDON DA TE : 1899 Restrictions on Use: COLUMBIA UNIVERSITY LIBRARIES PRESERVATION DEPARTMENT BIBLIOGRAPHIC MICROFORM TARGET Master Negative # ^B- 61167-3. Original Material as Filmed - Existing Bibliographic Record . g n I-ieJjtfoot, P.ev. John. Logic and education; an elementary t«xt-book Of deductive and inductive logic. Iondonl899. D. 112 + 2p. (Roj:^ standard series.) I. 10l7fi(j FILM SIZE: 3S^,^n. IMAGE PLACEMENT: lA Ql DATE FILMED: ^//r/93 TECHNICAL MICROFORM DATA iARC IB IIB REDUCTION RATIO:___/Z^ INITIALS_^^^^_ FILMED BY: RESEARCH PUBLICATIONf^. TMC WOnnRRTnCP PT Association for information and Image IManagement 1100 Wayne Avenue, Suite 1100 Silver Spring, Maryland 20910 301/587-8202 Centimeter 12 3 4 lllllllllll LUJ 5 6 liiiiliiiili 7 8 9 10 11 iliiiiliiiiliiiiliiiiliiiilimliiiiliiiil iiiiliiiiliiiiliiiiliiiiliiiiliiiiliiiiliiiiliiiiliiiiliiiiliiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii^ 12 13 14 J 15 mm T TTT 1 Inch es T NT IT I I nil I I 1.0 Ki 2.8 ^ 1 3.2 ■ 6.3 If «^ 1.4 2.5 22 I.I 2.0 1.8 1.6 1.25 I I I ^'^^^ '/ MflNUFflCTURED TO flllM STRNDflRDS BY APPLIED IMAGE, INC. \^ ■<^>BM|«MM*)i -•** ? ~> ttmmmmmimmmmm rtH ««iMaii immtm lOiwi # AND M&tmOOT, M.A., D*S«, &.- ,v & G9. rtfc'^s"/, ■ lUJwri tmtim^ i n »(i C''' iti < I i . i > ii ;.l. i> H i 'l>a iiiii ' i. V i >. ■feHBiMmil im^titmtmtn •*t>hMi«anMipi /- --r^jE LOGIC AND EDUCATION: AN ELEMENTARY TEXT-BOOK OF DEDUCTIVE AND INDUCTIVE LOGIC. BY Rev. JOHN LIGHTFOOT, M.A., D.Sc, (Vicar of Cross Stone), Author of " Studies in Philosophy^' " Text-Book of the Thirty-nine Articles," etc. I London : EALPH, HOLLAND & CO., Temple Chambers, E.G. 1899. [ALL RIGHTS RESERVED.] . 1 1 , vi:;'' 1 1 s C5 <^^ PREFACE. This book is intended for students commencing the study of Logic. It is an elementary exposition of the subject, but will be found sufficiently comprehensive for ordinary purposes. Tlie liile of the book indicates ihe primary object oi tkd writer. Both in the text and especially in the exercises, the bearing of Logic upon the Science of Education has been constantly kept in view. J. LiaHTFOOT. Cross Stone Vicarage, 310515 AUTHOR'S NOTE. -o Students making their first acquaintance with the subject are advised to adopt the following plan on their first reading of the book :— Omit Chapter I. Read Chapter II., omitting the section "Divisions of the Subject." Omit Chapter III- Chapter IV. should be thoroughly mastered. Read Chapter V., omitting the " Predicables." Omit Chapter VI. Chapter VII. is very important, and the exercises on page 50 should be carefullj- worked. Read Chapter VIII., but " Obver- sion " and " Contraposition " may be omitted on first reading. Chap- ters IX. and X. are very important. Chapters XI. and XII. may be neglected on first reading. Chapter XIII. is important (to bottom of page 84). Chapter XIV. is easy and interesting. Pay special attention to the "petitio principii" and "ignoratio elenchi" (pages 92-94), Cliaptei- XV. to end of book is very important. On second reading no part of the book should be neglected. Notice that the exercises contain only few questions involving mere repro- duction of the text. It has been assumed that students can and will construct such questions for themselves. J.L CONTENTS. ■o- PAGE. Chapter I.- Chapter II.- Chapter III.- Chapter IV.- -The Relation of Logic to other Branches of Philosophy '. -Definition of Logic and Divisions of the Subject . . . ... -The Axioms of Logic . -Terms : Their Definition and Classifica tioii . . . , . Chapter Y.— The Denotation and Connotation of Terms . . . . , -Definition and Division of Terms . -Propositions ..... -Imraediate Liference -Mediate Inference — The Syllogism -The Figures of the Syllogism -The Reduction of Syllogisms -Irregular and Compound SyUogisms -Conditional SyUogisms . . , -Fallacies of Deduction . , , -Inductive Logic .... -The Preliminaries to Induction . -The Inductive Canons . -Arguments Similar to Induction , Chapter VI. Chapter VII.- Chapter VIII.- Chapter IX.- Chapter X. Chapter XL- Chapter XII.- Chapter XIII. Chapter YIV.- Chapter XV. Chapter XVI.. Chapter XVII. Chapter XVIII. 12 19 22 27 34 41 51 61 69 74 79 82 80 97 101 104 108 CHAPTER I. i The Reiation of Logic to other Branches of Philosophy. > ^-], Logic is usually considered the proper introduction to the Blud^' of Philosophy, It is well, therefore, to got a preliiuLiarv view of the subjects which are included under the general term ** Philosophy." There are certain questions which must always be of great importance to those whose profession it is to train the minds of children. For instance, the question *'What am I?" is obviously as important as the question ♦' What is the sun ? "' Now, there are many similar questions to ** What am I ? " that wiU suggest themselves as of great importance in this respect ; e.g. : — How did I become what I am ? What are the fixed rules, or laws, which govern the development of mind in man ? To what laws must all my conscious thinking conform, so that error and self-contradiction may be avoided ? Philosophy is the name given to that branch of study which attempts to give an answer to these and similar questions. A coursG in Philosophy is usually divided into three sections : — Logic, psychology, metaphysios ; and the object of .' I \ h * h • 8 THE DIVISIONS OF PHILOSOPHY. these three departments may be thus briefly stated :— 1. Logic— Here we investigate the laws to which all our " thinking " must conform, in order that we may avoid error and self-contradiction in our thinking process. 2. Psychology. — Here is considered our xmwer of think- ing, its growth, and the laws by which " mind " in the individual is governed. I 3. Metaphysics. — Under thi^ name wd oi^cuss a number of very difficult and speculative questions about the ultimate grounds of our beUefs and opinions. This very brief statement must, for the sake of clearness, be considered more in detail. Philosophy, we have said, exists in three forms, and first we have : — I. PHILOSOPHY IN THE FORM OP LOGIC. Knowledge in its simplest form and from our earliest days, comes to us through sensation. But this elementary knowledge is from the first extended by reflection {i.e., by thinking) and by reasoning. The earhest efforts of a teacher are devoted to making the scholar reason correctly. And this effort is continued throughout the pupil's school-hfe. Evidently, then, it is of first importance that all who teach should themselves clearly recognise the laws to which ** correct reasoning " must conform. Putting the matter in its briefest form, we may say that "correct reasoning" must have two qualities, viz. : — {a) Self-consistency ; (6) Consistency with known facts. If our ** reasoning " is wanting in either of these qualities, it is incorrect, or fallacious. In our efforts, then, to extend and to improve our know- ledge. Logic comes to our aid by showing how fallacies of 1 THE DIVISIONS OF PHILOSOPHY. 9 thought and fallacies of expression may be avoided. Logic is thus : — A systematised body of tests and rules, by the aid of which we may determine whether (a) Our thinking is correct thinking {i.e., in accordance with the laws of thought) ; and % Whether our thinking is in agreement with factS) and with the known laws of nature* Logic and Rhetoric both aim at the formation of conclusions. The Logician seeks to convince that the conclusion mmt be ; The Ithetorician seeks to persuade that the conclusion ought to be. II. PHILOSOPHY IN THE FORM OF PSYCHOLOGY. The simplest reflection suggests to us that all our knowing, reasoning and beheving presupposes that we have a "mind" which knows, reasons and believes. "What the *' mind " really is we do not know. But we do know^ how^ the raind raanifests itself. If we do not know the inind itself, we know its phenomena. Sensation, knowledge, memory, imagination, reasoning, all these are phenomena of the mind. All these, too, have their laws of growth and development, and Psycho- logy is the orderly investigation of the phenomena of mind and the laws by which they are governed. Hereafter it wiU be seen that all man's conscious thinking manifests itself in mental judgments. Logic investigates the laws whicli these judgments, when formed, must obey. Thus we see the relation between Logic and Psychology. Psychology investigates the process by which the mind forms judgments. Logic studies the result, i.e., the juclgments when formed. in. PHILOSOPHY IN THE FORM OF METAPHYSICS. We have remarked that we do not know what " mind " really is. But all the same, men have felt themselves bound to hold some theory about it. So, too, we do not know precisely what '* matter " really is, but chemists and others have 10 ETHICS, OR MORAL PHILOSOPHY, I a theory about it. So too in every department of human study there are certain things which have to be taken for granted ; things which cannot be proved, but which we are compelled to assume. Now, Philosophy in the form of Metaphysics attempts to give some account of these subjects. It attempts to explain the ultimate nature of mind and matter, and to give a reasoned account of those truths which ordinary science takes for granted. The word "Metaphysic" suggests the subjects with which its study is concerned. Tlie word means that wbich is •' after," or beyond physical or ordinary scientific investigation. It only remains in this general sketch of the province of Philosophy that we should see precisely the place of Moral Philosophy, or Etliics. Ethical study is the considera- tiou of a certain definite set of facts and opinions which regulate the behaviour of men as individuals, and as members of the community. It is because of the great practical importance of these that they are reserved for special and separate ireatnient. In Ethics, or Moral Philo- sophy, we learn how the knowledge of moral distinctions (i.e., of right and wrong) are obtained, and we investi- gate the laws which govern the moral hfe. But it must be observed that when we are studying the growth of the know- ledge of moral distinctions we are really engaged in the study of Ethical Psychology. So, too, when we further consider questions like the ultimate nature and destiny of the soul, and its relation to the Supreme Being, we are then in the province ol Ethical Metaphysicsi Having now got a general view of the whole province of PhUosophy, students preparing for the Certificate Examinations of the Education Department should consider how the various subjects are distributed over their course. In the First Year, *'The general principles of Logic'* are required. EXEBCISE8. 11 i» In {}ie Second Year^ *' The elements of Psychology ** applied to teaching, and "The elements of Ethics," with special reference to the government and dis- cipUne of children. In the Third Year, more advanced Ethics and Psycho- logy in relation to teaching.* EXERCISES ON CHAPTER I. 1. Should Logic precede Psychology as a subject of study ? Give reasons for your answer. 2. State briefly tJie fundamental relation in which Logic stand: to Psychology and Metaphysics, * The higher branch of Piiilosophy (Metaphysics) is not required for the Certificate Examinations, but always forms part of University Examinations when " Mental Science " is selected as a subject for a degree. ■as CHAPTER II. -o- Definition of Logic and Divisions of the Subject. Logic is usuaUy defined as " The Science of Reasoning or Inference." This definition, though sufficient for general purposes, is not sufficiently precise. The object of Logic is to unfold to us the ideal, or perfect conditions to which all our thinldng must conform in order to be correct thinking. Its •object is to show us how we must arrange the matter about which we think, in order that our thought shall be coherent and non-contradictory. A better definition, therefore, will be: — Logic is the science of the laws which regulate valid thought. It is of the utmost importance to fix clearly the meaning attached to each word in this definition. (a) A Science is a systematised body of knowledge about some particular subject-matter. (&) A Law is a statement of a general truth, i.e., a truth which holds good universally in that science, (c) The word "Thought" is used both for the proceu of thinking, and for the product of thinking. N.B.— Knowledge of isolated facts is not Science ; nor is a truth Which holds good only in certain instances a " Law." In Logic we consider every simple complete thought as an assertion or a denial. I ■ i W\ 1 DEFINITION OP LOGIC. 13 Every assertion or denial we call a ** judgment.*' Every judgment when expressed in words we call a " ppoposiiion." Every assertion or denial which it is possible to make is of such a character that when it is made certain other assertions or denials follow from it as a necessary consequence. These latter are called " inferences." An inference is thus a judgment which follows as a neces- sary consequence from some previous assertion, and one which the mind is obUged to make on pain of self-contradiction. If I assert the general fact that "All men are mortal," I am obliged, on pain of self-contradiction, to infer that this or that pai'ticular man is mortal. A child has had its attention called to the fact that a piece of cork thrown into the water alw^ays floats, and is also taught how to recognise a piece of cork. The child is now on any occasion able to make two assertions : — {a) Cork always floats ; (Jb) This is a piece of cork ; and from these two assertions, the further one that we call an " inference " follows, and must follow, viz.^ this piece of cork thrown into the Avater will float. From our earliest days, our conscious life is largely occupied in drawing inferences. This will appear if we reflect how in teaching and in ordinary conversation we are constantly using such words as therefore, for, because, since, etc. Every such word marks the drawing of some inference. Education at school, and experience in after life is really little else than the development of the inferential connections between pro- positions. Now this work of drawing inferences has certain definite rules and laws which must be observed. And it is by these laws that all our inferences may be tested. If the laws have been broken then the inference drawn is an mvalid one. An 14 LOGIC, A SCIENCE OR ART. invalid inference we call a fallacy. * No one knows better than a teacher how prone children are to draw wrong inferences. And indeed all through life men are liable to draw inferences which are fallacious, the main object of Logic being to show how these wrong conclusions may be avoided. It does this by educating man's power of distinguishing the consistent and the conclusive from that which is incondstent and inconclusive. The principles of Logic find their application in every walk of life, and in every branch of science. This is recognised by the names given to tlie various sciences. Thus in the name *' Geology " the last four letters are only another form of the word "logic," and the term "geology" means "Logic applied to explain the crust of the earth." So, too, theology means "Logic applied to explain Divine matters," and so on. Since then the rules of Logic find their application iri the processes of every special science, Logic has been very properly called tlie Science of Sciences. Some writers have considered it needful to discuss at great length whether Ijogic shoxild be called a Science or an Art. As a matter of fact it may be considered as either or both. Science is sound know- ledge, an Art is the instrument by which science works. In studying a science we are gathering knowledge, in learning an art we are pre- paiing to do something. Logic is a Science in so far as it nnfolds the conditions of valid thought. Logic is an Art in so far as it devises rules for enabling men to apply their thought to things consistently and coherently. Of course it is not iniphed that a man is unable to think or reason correctly unless he has learnt Logic. Plenty of people speak correct English who have never learnt the rules of Grammar. From our earliest childhood we have been accustomed to draw conclusions, and no doubt we have generally obeyed the laws of Logic in doing so, without being in the least aware what those laws were. In such cases we were thinking logically without being conscious of the logical DIVISIONS OF LOGIC AND PSYCHOLOGY. 15 principles which our thinking exempHfied. On the other hand, a course of logical study must bring into prominence the laws which constitute vaHd, consistent thought. The student who has patiently worked through a course of Logic, is much more hkely hereafter to think consistently and coherently than one who is ignorant of Logic. And this will be found to be especially true in those cases where, owing to the complexity of thought and the ambiguity of language, there is serious danger of fallacy even to the cleverest intellects. DIVISIONS OF THE SUBJECT. There are two main divisions of Logic, viz. : — (a) Deductive and (6) Inductive Logic. "We must get a preliminary view of the scope of these : — (a) Deductive Logic (sometimes called pure or Formal Logic). This division of the subject is the orderly, scientific unfolding of those forms and conditions to wliicli our "think- ing " must conform in order to be valid thinking. Our " thinking " manifests itself in three distinct stages, and each stage has its pecuHar product or result : — STAGE 1. PROCESS. Formation of ideas or con- cepts of things. STAGE 2. Forming judgments about these ideas— i.^., making men- tal asBertions or denials about them. STAGE 3. Drawing mental inferences firom these judgments. RESULT. Signifying these concepts or ideas by " Names." Expressing these judgments in " Propositions.** Signifying this mental pro- cess by the '^ Syllogism." stage 1, is often described as " simple apprehension," and stands for the action of the mind in being aware of anything, having an idea, 16 INDUCTIVE LOGIC. or concept Of it. The psychological analysis of "simple apprehen- sion," however, shows it to be a complex and not a simple process. N.B.— Simple apprehension, judgment and reasoning (inference) are the psychological processes. Names (terms), propositions and syllogisms are the corresponding results which are the subject matter of Logic. Consequently Deductive Logic has three sub-divisions answering to these three stages. The laws of thought : — 1. Concerning our Concepts of things ; 2. Concerning our Judgment of things ; 3. Concerning our Reasoning about things. And since our thinking is liable to error, we shall require to supplement the above by an account of the Fallacies to which we are Uable if the laws of pure logic are violated. (6) Inductive Logic (sometimes called Applied or Mixed Logic). The scope of this second main division is best seen from an account of its sub-divisions. 1. Definition.— In all our thinking about nature we are obliged to use language. But language, at the best, is only an imperfect instrument for expressing all that is in the thinker's mind. The Logical doctrine of delinition aims at improving the relations of language to thoU'^ht, and especially to man's thought about things. 2. Inductive and Analogical Proof.— This section first investigates the ineanhig and value of the central presupposition of all science, viz: "The uniformity of nature." It reveals the fact that nature is not a chaos, but an orderly coherent system of cause and effect. Next it deals with the obvious fact that untrained minds are liable to confuse mere accidental coinc i dence with trueconse- quence and r eal scient ific connection. To guard against this, Logic provides certain canons or rules, and by these unfolds the standard of scientific proof. It thus enables men to distinguish between evidence which is properly / logic and grammar. IT (a) Inductive and therefore reliable, and evidence which is only (6) Analogical or probable. This section is thus really the logic of all the physic al sciences. 8. Historical Proof.— If the language of nature is hable to be misunderstood, much more so is the spoken and written language of men. Therefore Logic has certain rules to lay down respecting human testimony which form the criteria of Historical Proof. These divisions and sub-divisions of the whole domain of Logic ixiay be shown thus : — Logic. (The Science of the laws which regulate valid thought.) Deductive Logic. Inductive Logic. Thought as Thought as Thought as Definition Concept. Judgment. Inference. I Testimony and Authority. Proof : (a) Inductive. (6) Analogical. Note that Logic is concerned primarily with Thought, Grammar with Language. Logic considers Language only as the instrument of Thought. The " Parts of Speech " which are the mam feature in the grammatical analysis of language are not recognised by Logic. Only those words which can properly express a concept are within its range. Words that can express a concept are grouped together as Terras, and it is quite immaterial whether, in grammatical language, they are nouns, pronouns, adjectives, or verbs. In Grammar and Logic the simplest expression of a complete thought is a Simple Sentence. But the logical analysis of a sentence differs from the grammatical. The predicate of a logical sentence is always the complete assertion made of the subject. 18 EXERCISES. EXERCISES ON CHAPTER II. 1. Define " Science*^ and *^ Art." Discuss tJie q^uestion ivhstlier Logic is a Science or an Art. 2. Wfuit is Logic ? What are ths chief uses of its study ? Why should teacJiers especially make a study of it ? 3. Many people think quite correctly who have never studied Lofjic ; tvJiy, then, tuaste time in studying it? 4. Logic has been defined as *' the science of the laics of thought." In this definition what is meant by the terms " Science," '* Law," and *' Thotight " ? 5. Discuss the relation of Logic to Grammar and Rhetoric. 6. What practical value may be attributed to Logic (a) in tJie detection of error, (b) in the discovery of truth I 7. Explain the logical icords term, proposition and syllogism, and give tJie psychological ivords for the corresponding mental act of each. J s CHAPTER III. — The Axioms of Logic. Logic has thus far been shown to be the practical science which unfolds to us the ideal of self-consistent thought. In its later sections it supplies the student with certain canons or rules for applying our thought to things. Now, there are certain principles or axioms which form the essence of self- consistency. When these are drawn out they will appear to the student as self-evident truths. All the same they require consideration, and after we have examined them they must be regarded as. axioms. Speakmg quite generally we may say that these several axioms imply one general truth, viz. : — ** Thought which is evidently self-contradictory is ivi- possible.*' Every description of fallacy is really a thought that is self- contradictory. Eal; wo may so express ourselves that ike contradiction is not obvious to those to whom we are speaking, nor even to om'solves. Thus, in such an argument as the following : — *• He who is most hungry eats most ; He who eats least is most hungry : Therefore, he who eats least eats most," we feel there is contradiction somewhere, but it would require some consideration to point out where the contradiction really lay. It might be better to gay that a fallacy is really the absence of thought, i.e., of logical thought ; the absence usually being concealed under a veil of words. \. wmmm mm 20 THE LAWS OF THOUGHT. The general truth stated above— m i CliASSIPICATION OP TERMS. 25 The student must carefully consider this distinction of terms. Think, for example, why " mountain" is called a connotative term, but Snowdon, the name of a particular mountain is not. Now the name " Snowdon " might suggest to anyone with sufficient geo- graphical knowledge, all the attributes implied in the term " mountain." But a word is not connotative because it may suggest facts or attributes which are otherwise knotcn, but only when it actually implies them. Many logicians have overlooked this, and have considered proper names connotative. In answering a question in an examination it would be wise to give your reason for consider- ing a proper name as " non-connotative." 5. (a) An Absolute Term is a name which is complete in itself, i.e. J which in its meaning impUes no reference to anything else ; as gas, sound, tree, etc. (h) A Relative Term Is a name which not only denotes some object, but also implies in its signification the existence of some other object called the correlative. Thus when we use the term friend or father for some man, we imply the existence of some other person or persons to which the man stands in the relation of friendship or fatherhood. These definitions of the various hinds of terms must he tlioroughly understood, and the student must be well exercised in the classification of terms. When the appended examples are attempted it will be found a more difficult task than might be supposed. The main difficulty will be found in decidin" whether an abstract term is general or singular. Some logicians argue that all abstract names are singular. Thus the adjective " red " is the name of red objects, but it implies the possession by them of the quality '* redness," and this quality has one single meaning. It is much simpler, however, to consider some abstracts general on the ground that they are names of attributes ot which there are various kinds or subdivisions; e.^., the word colour which is a name common to whiteness, red- ness, etc., or the term whiteness in respect of the various shades of whiteness to which it is applied in common. But 26 EXEBCISES. just because the point is a disputed one, you should give your reason for classifying abstract terms as general or singular, A further difficulty arises in dealing with terms that are equivocal, i.e., capable of being used in several senses. Indeed, some writers make a further classification of terms, as Univocal (terms which can only suggest one meaning) and Equivocal or Ambiguous (terms which may have two or more meanings). An equivocal term is really two or more terms with identical spelling, and should be so treated. Thus the term *' force " is equivocal, as it might mean an army or that which causes motion, etc., and each meaning demands a distinct classifica- tion of the word. It is better, therefore, to say at once if a term is equivocal or univocal, and then proceed. CHAPTER V. EXERCISES ON CHAPTER IV. 1. Discuss tlie grammatical parts of speech from a logical point of view. 2. May terms he classified as catcgoremattc and syncategore- matic ? Give reasons for your answer. 3. Describe a ''collective term.'' Illustrate tJie di^culty of distinguishing tJiese from general or abstract terms. 4. Classify the following terms : donkey, reagent, red, redness, London, sugar, Mihado of Japan, miensiiy, also, vexation, hlind, emotion, darkness, foot, Westminster Abbey, uncle. 5. Point out tJw ambiguity, if any, of the following terms : vice, hydrogen, peer, paper, sense, minister, tea-cup, interest. 6. Distinguish betiveen the meaning of tJie terms abstract and concrete, and show the applicability of these terms (1) to parts of speech, and (2) to arithmetic. Say what is the use of the distinction. ^"4 o The Denotation and Connotation of Terms. If the question were aslced *' What is an animal ? " we can imagine two forms of answer being given : (a) an exact definition of the term; (b) an enumeration of the various classes of animals. The first answer might be expressed thus: — "An animal is a sentient, organised being." This definition tells us what must be the attributes of anything in the imiverse to which the name "animal" can be rightly applied. Such a definition is said to mark the connotation of the term. On the other hand the latter definition which proceeds to enumerate all the different classes of animals is said to mark the denotation of the term. A Term, therefore, in Logic is considered to discharge a double function: — 1. Connoting the attributes of things. 2. Denoting individual things. Notice that a term is a word which signifies a mental idea or concept. But in Logic we do not speak of the connotation or denotation of a concept. When speaking of concepts ^ve use the words intension and extension. '•' The intension of a concept corresponds to the connota- tion of the term signifying the concept. The extension of a concept corresponds to the denotation of its related term. The connotation of a term (or the intension of its corre- sponding concept) signifies the attributes implied in the meaning of the term. ♦ Some writers, however, speak of the intension and extension of termst and even the denotation and connotation of concepts. RELATION OP CONNOTATION AND DENOTATION. The denotation of a terra (or the extension of its corre- sponding concept) signifies the number of individual things to which the term is applicable in the same sense. The student is invitea to reflect upon these definitions. It will then bo seen that an important logical truth is involved. Every common term like man, bird, etc., stands for a number of individual things (different individual men or birds), and a quantity of attributes (rational being, feathered biped, etc.) Thought as expressed by "terms" is thus a kind of quantity, and aU our affirmations and assertions about terms are really a comparison of quantities. If I say " men are animals," I mean that " men " are a quantity of things contained in a greater quantity of things called *' animals." Now the two particular kinds of quantity we are consider- ing (connotation and denotation) have a mutual relation. For a moment's reflection will show that the wider or greater tlie denotation of a term becomes, the narrower or smaller must be its connotation. Thus compare the two terms "animal " and "man." The term animal embraces far more individual things under it than the term man, therefore its denotation is greater. But the term man implies a larger number of attributes than the term animal. For everything that you can say of animal you must say of man, but you also say of man certain things wliich you cannot say of all animals. Therefore the connotation of the term man is greater than that of the term animal. As a fairly correct general rule it may be said that as the denotation of a term is increased, the connotation is diminished, and vice versa. In other words the greater the number of individual things included under a common term, the fewer will be the number of attributes which can be predicated of the whole of them. This is expressed by the Logical rule that the connotation and denotation of a term (or the intensive and extensive quantity of a concept) are in inverse ratio. The greater THE PREDICABIiTCS. 29 ^4 the denotation, applicability or extent of a term— the less must be it 3 connotation or comprehensive quantity. The maximum of the one must in all cases be the minimum of the other, and vice versa. Now observe when two common terms are so related that the whole connotation of the one is included within the greater connotation of the other — the term which has the greater connotation is called the " Species," and the one which has the smaller, or included connotation, is called the ^^Genus.'* Thus taking the two related terms "man " and "animal," the^ term "man" implies all the attributes that the term "animal"* implies, as well as some further ones peculiar to itself. "Man" has the larger connotation, therefore "man" is a species of the genus *' animal." In the proposition "Man is an animal" we assert that "man" the species is included in "animal " the genus. Every affirmative proposition makes some such assertion respecting the subject of the proposition. The following problem, there- fore, arises : " Can the iJredicatcs of all propoditions be classified in relation to their subjects under certain definite heads? " Logic attempts this by the Doctrine of the Pre- dicables. The predicables, then, are a classification of all the possible relations of the predicate to the subject of a logical pro- position. The foUowing is the usual form of this classifica- tion ; — '1. Genus 2. Species 3. Differentia 4. Proprium ^^^ *^e subject, (property) 5. Accidens (accident) ^ These five heads of the predicates require consideration. Predicables are either- 30 GENUS AND SPECIES. 1. Genus is a common term, signifying a wider class which is made up of other narrower classes, e.f],^ animal, triangle. 2. Species is the name given to the narrower classes, included in a genus, e.^., Vertebrates, Invertebrates; equilateral triangle, etc. Genus and species, then, are relative terms, and must be considered together. A genus would be meajiingless apart from two or more species into which it is divided. A species would be equally meaningless apart from the genus in which it is contained. The student will notice that the same term may be at the s ime time a species of the next more general class, and a genus to the less general classes included undt-r it. Thus take the terra" triangle." Triangle is a species of the genus "recti- linear figures," whilst at the same time it is a genus of the different kinds of triangles : equilateral, isosocles, etc. From this it follows that every term may be both a genus and a species. But the technical language of Logic implies, however, that this is not universally the case. It implies that there is a genus Avhich is not a species of any higher genus ; and that there is a species w^hlch is not a genus to any lower species. For Logic speaks of : — 1. The highest genus ; 2. Intermediate genera or species ; o. The lowest species. The higlicst or most general genus, i.e?., which can have none above it, is such a one as " Being." This is called " the summum genus." The lowest relative species, which can have none' below it, is the name oi any indivlaual thing. This is called " the infima species." Any highest genus broken up into its component species, and these component species in turn regarded as genera again broken up into their component species, and the process TREE OF PORPHYRY. 81 tepeated iiniil you cannot proceed lurtber, (i.e., when an Infima species is reached) is called a " Predicamcntal Line." A process such as this is illustrated by the ancient " Tree of Porphyry " : Substance (a summum genus). Corporeal. I Incorporeal. Animate. Body. Living Being. Inanhnate. Sensible. I Animal. Insensible. Rational. Man. Irrational. I Socrates. Plato, and other individual men. Here Substance is the Summum Genus and Man is the Infima Species (i.e., man cannot be divided into any smaller species, but only into individual men). Each of the intermediate genera down the middle line (Body, Living Being, Animal), is called a subaltern genus or species, and the nearest genus to every term of which that term is itself a species, is called the proximum genus. • *|»-r SITWS- 82 DIFFERENTIA. 8. Differentia. — It has already being seen that a species has a larger connotation than its corresponding genus (i.e., the species implies more attributes). Now take any term used as a species and compare it with its next, or proximate, genus. The excess of the connotation of the species over the conno- tation of the genus is called the *^ Differentia " of the species. Thus:— Genus+Differentia= Species. Keferring to the Tree of Porphyry, "Living being" is a species of the genus " boJy." " Animate " is the attribute which forms the differentia of the species " living body," thus : — Body -f Animate = Living body (genus) (differentia) (species). 4. Froperty (Proprium).— By property is meant any attribute which is common to every individual in a given class, but which is not necessary for distinguishing that class. This will be clear from the following illustration. Take the term " triangle." A triangle is a figure bounded by three straight lines. *' Three-sided " is the differentia of a triangle. But triangles have many other properties, e.g., *' three -angled," " all their angles equal to two right angles," etc. 5. Accident (Accidens).— An accident is an attribute which has no necessary connection with the term to which it belongs. Thus the size of a triangle — i.e., big or little — is an accident. Size does not at all affect what !Euclid proves concerning triangles. Accidents are usually divided into Se_parahle accidents — e.g., how a man is dressed ; Inseparable accidents — e.g., the colour of his hair. IV EXERCISES. 88 EXERCISES ON CHAPTER V. 1. JJefiJKi dijferentta^ properly, and inscparahle accident, givittg examples. How far may iliese distinctions he interchanged. ^. To ivhich of the predicables luould you refer tlie predicates in tlie following propositions, and lohy : — (a) All men are animals." (b) Mr. Gladstone was a great statesman. (c) Tlie three angles of a triangle are togetJier equal to two right angles. (d) All dtccks are iveb-footfid, (e) John ruled badly. if) Alkalies hy their union tvUh acids form salts. 3. Ex2)lain clearly tlie connotation and the denotation of a term. JVJmt determines tlie connotation and denotation of terms ? Have all terms a denotation and connotation ? 4. Arrange tJie following terms in their order of extension : — Vertebrate, human, substance, child, organism, sclwolboy. 5. Explain tlie terms intension and extension as applied to terms in Logic, and distinguish genus and species, illustrating your explanation by the terms cart, eagle and man. 6. Distinguish between denotation and connotation, and slww the trnporiatice of tlie disiinciion in teaching. 7. Give tlie genus, tlie differentia, a proprium and an accident of silver, Dariuinian, square, house. 8. " A generic term denotes a larger riumber of objects tlmn a specific term; but it connotes a smaller niiuibcr of attributes" Explain this statement and illustrate it by examples. \ CHArTER VI. — o- Definition and Division of Terms. The definition of a terra is the explicit statement of the connotation of the term. Since every definition of a term must take the form of a proposition, it would be more convenient to have considered the logical doctrine of definition wKen we are dlscussmg propositions. But it is usual to consider the subject at this stage of our study. In a definition that which is defined is always the subject of a proposition. The predicate must declare with sufficient precision what the subject means. In other words, the predicate nmst show forth the attributes which separate the subject in question from all other subjects. All definitions are propositions, but all propositions are not de- finitions. Only thoSG proposliions are definitions in which the predicate so makes clear the attributes of the subject, as tO Separate it from all other subjects with which it might be confounded. The subject and predicate of a definition are, therefore, exactly co-extensive. The difference between them is this :— what was latent — wrapped up, as it were, in the subject — is fully unfolded or analysed in the predicate. Logic asserts that this result is achieved when the predicate of the defining proposition exposes^ the proximate genus and the differentia of RULES OP DEFINITION. 85 a term. For the genus hnplies all the attributes of the term considered as a species of the genus; whilst the differentia displays those attributes which distinguish the term as a species. In Logic, then. The definition of a term = proximate genus + differentia. Notice that there are some terms which are incapable of logical definition, e.g., a summnm genus, all proi)er names, etc- The former has no proximate gen s, the latter have such a multiplicity of attributes that we can only mention a number of them sufficient for the practical purpose of recognition. This enumeration, however, is "description" not definition. The student must not confound logical definition with " dictionary definition." In the latter all that is done is to substitute one word for another, assumed to have a similar connotation, on the ground that the new word is more familiar or intelligible than the one for which it is substituted. There are certain simple rules \vhich Logic lays down to which propositions must conform to entitle them to be regarded as good logical definitions. 1. The definition must bring into view the essential, dis- tinguishing attributes (differentia) of what is defined.- 2. The definition must be adequate, and applicable exclusively to what is defined. 3. We must not define by negations. 4. The definition must be expressed in unambiguous, intelligible language. Definition is a most important subject. Avoid confusing the definition of names witli the definition of things. The definition of a name is the settlement of what the name shall be, by which a thing or a concept sliall be designated. Any man is entitled to determine this as he pleases, so long as he adheres consistently to the name he has connected with the ♦ Obviously, to merely name properties or accidents can never be a logical definition. 36 LOGICAL DIVISION. concept or thing. Sounds or signs on paper, are in thenmelvcs indifferent to meaning. Each or any may be used to express any meaning that has been agreed upon by those who use the word. Definition of a tiling^ is not thus arbitrary. These definitions depend on what is involved in the essential nature of the thing defined. Men are apt to confound definitions of names with definitions of things, and to confuse both with that full analysis of the attributes implied in our concepts which it is the province of logical definition to bring into Hght. We are frequently asked to accept definitions of names as if they were the true definitions of things. Because we agree to employ a certain sound to express some meaning, it does not follow that the meaning so expressed corresponds to the essential attributes of the things signified by the sign. Logical Division. — Division is the analysis of the denotation of a term. It is always expressed in the form of a proposition, the term divided being the subject, and the exposition being the predicate. There are other familiar kinds of division with Avhich logical division must not be confounded, e.g., (a) Partition, which is the act of dividing some physical whole into its constituent parts, e.g., ship=hull, mast, sails, etc. ; man=head, trunk, limbs, etc. (h) Distinction of ambiguous or equivocal terms, e.g., Humanit3'=(l) human nature, or (2) the human race collectively; Vicc={l) a moral fault, or (2) a mechanical tool. (c) Eunmeration of individuals, e.g., naming all the books in a library. Logical division expounds the denotation of a term not by enumerating individuals. This would in most cases be im- possible. No one could enumerate all the different mcu RULES OF LOGICAIi DIVISION". 37 included under the term '' m£»,n." It proceeds by mentioning only the smaller groups denoted by the term. Collective and singular terms cannot be divided into smaller groups, and, therefore, cannot be logically divided. A collective term can be transformed into a common term, and 80 become capable of logical division. Thus "the fourteenth regiment" may be transformed into "soldiers of the fourteenth regiment," and in this form may be divided into ofHcers, privates, etc. When we proceed to divide a term into terms expressive of smaller groups, we seek some attribute which may be predicated of certain members of the group, but which cannot be predicated of the rest. This attribute is called the basis of division {fundamentum divisionis). Of course, the same genus may be variously divided by adopting different bases of division. Thus in dividing the genus "triangles" we may adopt the relative length of their sides as our basis, and so divide triangles into equilateral, isosceles, and scalene. Or we might adopt the size of their angles as the basis, and so divide triangles into right-angled, acute-angled, and obtuse- angled. But two or more bases of division must never be confused together m the same division, or we fall into the error called in Logic " Cross division." It would, e.g.^ be cross division to divide triangles into isosceles, right-angled, and scalene. There are certain rules to which a logical division must conform, viz. : — 1. Each act of division must have one and only one basis of division, or cross division will ensue. 2. The division must be exhaustive, i.e,^ the dividing members when taken together must be co-extensive with the divided whole. 3. If the division is a continued one {i,e., embraces more than one step), each step should, as far as possible, be a I ss CLASSIFICATION. proximate one — in other words " proceed stej) by step. e.g. :— t» Figure. Curvilinear. Tlect ill near. I Triangle. Equilateral. Quadrilateral. I Polygons. Isosceles. Scalene.* When we turn from the division of our concepts as expressed in terms, and proceed to consider material things the logical doctrine of division becomes a theory of logical scientific CLASSIFICATION. The object of classification is to so arrange the facts with \\ Inch we may be dealing that we can acquire the greatest command over them, and convey the greatest amount of information abont them in a few words. Classification is really a branch of Inductive Liogic. It is one of the important processes subsidiary to the application of the inductive canons. By its use we obtain a greater command over the knowledge we possess, and are put in the right avenue for obtaining additional information. It ri'ovides that our knowledge of thinirs shall be so arranged that the facts may be more easily remeniberecl, and that we may more readily perceive the laws by which they ai-e governed. ♦There is a further method of division in which each step is a division into con'esponding positive and negative terms, e.g. \— Figure. Rectilinear. I Non-rectilinear. ' Triangles, Non-triangles, etc. This is calletl division by Dichotomy. It is extremely cumbersome and of ismall iiuj^ortancc. K EXERCISES. 39 Logic considers all attempts at classification as either natural or artificial. By a natural classification is meant the grouping of facts in accordance with real natural distinctions. Thus an actual scientific knowledge of facts is a pre-supposed requisite for a natural classifica- tion. Different branches of science have different objects in view, and accordingly they often adopt a special basis for classification. The practical fairoer divides plants into those which are useful, and those which are weeds. Whilst the botanist adopts the division into monocotyledons and dicotyledons as his basis. The student who h is an elementary knowledge of Geology and Zoology will remember how differently fossils are classified iu the two Sciences. An artificial classification selects some point of resemblance amongst objects, and one which is easy to identify, and p.roceeds to classify related objects upon this basis. The Linnsean system of classification in Botany, which takes for its basis the number of stamens and pistils in a flowering plant, is a good illustration of an artificial system. In Zoology, wliere the primary basis of classification is into vertebrates and invertebrates, we have an example of a natural classification. EXERCISES ON CHAPTEU VI. J. Criticise the following definitions: — (a) Ignorance is a blind guide, (h) TJie cat i^ a domestic animal. ( c) Enjoymeni means pleasure. (d) Tranquillity is tlw absence of unrest. (e) Alcohol is a kind of medicine. 2. Define the terms gold, coal, legal nuisance^ civilization^ Cleopatra's Needle, bread, anger, Snowdon. 3. What do you understand by a perfect definition; and wJiat processes of tliought are employed in arriving at one ? Oive tiuo or three examples which err by being eitJier too wide or too narrow. 4. ]\^hat is tlie difference between (a) a description, (b) a defini- tion, (c) an explanation ? 5. Explain what is meant hy logical dwision, and hriefly siaU its rules. Oive instances which observe, and instatices which violate the rules. in EXERCISES. 7. tioji. 6. Comment on the folloicing as logical divisio7is : — (a) Pens into quill pens and steel pens. (b) Ireland into Ulster^ Minister ^ Lcinstcr and Connaught. ( c) Animals into vertebrate and invertebrate, (d) Colour into whiteness^ blackness and blueness. (e) Lights into artificial, bhce and red lights and moonlight. (f) Vice into an iimnoral act and a iiieclianical tool (cf) Englishmeiv into ricli and ^oor, consutnptive and bilious. Show the relatioit between Definition, Division and Classifica- 1 CHAPTER VII. Propositions. o- Having completed our investigation of the logical doctrine of ** terms," we now proceed to consider the teaching of Logic with regard to " propositions." Just as a " term " is the outward expression for the inward (psychological) fact, which is called a " concept," so a "proposition" is the translation into language of the inward mental act, which is called " judg- ment." Now, it has already been shown that a judgment is the simplest and most elementary manifestation of a complete thought. Every assertion or denial that we can frame in our minds is a judgment. When this mental act is expressed in language, we have what is called in Logic a proposition. A proposition, therefore, may be defined as The verbal expression of a truth or falsity, or A sentence making an affirmation or denial. Propositions which make simple assertions or denials, without any condition attached, are called Categorical. A Categorical Proposition is one vrhich simply asserts or denies some fact, e.g.. All men are mortal. No men are infaUible. Notice that in a categorical proposition we bring together two terms, and connect them by the copula. For logical purposes this copula is always the present tense of the verb *' to 6c," loith or ivithout the negative particle " not" 12 PROPOSITIONS : In ordinary language, of course, our categoHcal judgments axe expressed in various ways. But Logic considers that every simple assertion or denial can be expressed in one general form, and, for logical purposes, the assertion or denial must be reduced to this form. Hence the student must become accustomed to expressing the ordinary forms of simple assertions and denials in the precise form required by Logic There is no doubt that the logical form of an assertion will often appear awkward and -wordy," compared with ordinary conventional modes of expression, but the advantage gained by the precise exposition of our assertions is of the hjf'hest logical importance. Take as an illustration the assertion, "John was the brother of Richard. ' In order to get the present tense of the verb "to be" as the copula of this sentence, it must be expressed in some such form as :— " John is a person who was the brother of Richard." This trans- formation sometimes causes a little perplexity. Take, for example, the foHowing sentences : — (1) The bell will toll to-morrow. (2) None but the brave deserve the fair. (3) It does not rain. (4) Fire I These ordinary conventional sentences, when transformed into simple categorical propositions for logical purposes, become — Subject. Copula] Predicate. (1) The tolling of the bell (2) No not-brave persons (3) Rain (4) This property an event which will happen to-morrow. deserving of the fair. faUing. on fire. ^ jH' CATEGORICAL AND CONDITIONAL. 43 Observe, that when a sentence is being thus transforn.ed for logical purposes, and divided into its logical elements (subject, copula, predicate), if any one of the elements has been omitted in the conventional form, it must be supplied in the precise logical form. Thus the exclamation " Fire 1 " is sufficient, for practical purposes, to convey definite information, but until its subject and copula have been supplied, it is useless for logical purposes. A categorical proposition, then, is one which makes an unconditional assertion or denial. AVhen the assertion is expressed as a proposition display ing its logical elements, the copula is in all cases the peremptory " is" or " is not." But many of the assertions or denials that ^ve are making con- stantly are of such a nature as to forbid the employment of the unconditional "is " or " is not." To a large proportion of our judgments some condition or other is attached. Now, Logic draws a sharp distinction between judgments which are unconditional and those to which some condition is attached. The former are categorical, the latter conditional. We shall be chiefly concerned with categorical propositions, but it is needful to mention the two kinds or conditional propositions which are most common. Conditional propositions are usually distinguished as Hvro- THETICAL and Disjunctive. 1. Hiji)otheiical loropositions have a conjunctive condition The following are examples : — (a.) If A is B, then also C is D. (6) If Logic exercises the intellect, it ought to be studied. (c) Where ignorance is bliss, 'tis folly to be wise. Example (a) and similar examples, where symbols (A B, etc.) are used, are called abstract examples ; [h) and (c) are called concrete examples. 44 PROPOSITIONS : 2. Disjunctive propositions have an alternative condi- tion, e.f/. : — (a) A is either B or C. (b) He is either a knave or a fool. (c) All men are either good or bad. Sometimes we find propositions conditioned, at once conjunctively and tlisjunctively, e.g. : — If A is B, tlien C is either D or E. If a man becomes a soldier, then he must serve cither at home or abroad. Besides this obvious division of propositions into categorical (unconditional) and conditional, Logic further distinguishes them by their quality and their quantity. The quality of a proposition is determined by the copula. The copula may be either " is " or " is not." In the former case the proposition is affirmative, in the latter it is negative. A is B (affirmative) (1) A is not B (negative) (2) But we may also assert — All A is B, or only. Some A is B. The distinction of propositions, according as the affirmation or denial is made of the whole or only s^paH of the subject, is what is meant) by determining the quantity of a proposition.' rroposition?, in which the assertion or denial is made of the whole of the subject, are called universal propositions. Propositions, in which only part of the subject is affected are called PARTICULAR propositions. Notice carefully, that in universal propositions, the subject of the proposition is distributed, i.e., taken in its fuU denota- tion. In particular propositions the subject of the proposition is unaislrib,ilei7,i.e., the extent of its denotation is indefinite. i QUANTITY Al^D QUALITY. 45 Particular propositions are usually expressed in the form Some A's are B. Some A's are not B. The word " soixie " is absolutely indefinite; it may mean *' few " or " many," or indeed any proportion. In Logic it is the equivalent of " not-all." The student should also carefully note that in universal propositions the subject may be either : — (a) An undivided, whole class, of every member of which the predication is made, e.g., "Men are mortal"; i.e., All men and every individual man ; or (6) An indivisible individual, indicated b^ a proper na,nie ; e.r/., " John is n:iortal." Propositions, which have a proper name for their subject, arc sometimes called Singular Propositions. In most cases they may be considered only a sub-class of Universals. But instances arise which may cause perplexity. Thus : " John is sometimes eloquent," might be considered as universal with a somewhat complex predicate. (Tlie student will have found, ere this, that in expressing proposi- tions ill logical fonii, the predicate is often very complex). The pro- position in its full logical form would be: " John is a speaker who is sometimes eloquent." This is a true universal. On the other hand the proposition might be rendered : " Some of John's speeches are clociuent," in which case the subject is particular, not universal. These various ways of dividhig propositions may now be collected, thus : — Propositions are divided 1. On the basis of their quality into {a) affirmative, (6) negative. 2. On the basis of their quantity into (a) universal, {b) particular. The distinctions of quality and qinntily are considered as applying only to categorical propositions. To some extent the same distinctions can be applied to conditional propositions* 4G FOUU PHOi'OSlTIONAL FORMS But to atteu.pt this would be quite beyond the scope of this elementary treatise. From this we gather that all categorical assertions or denials may be grouped under four general forms. For, when our assertions axe expressed in logical form, we affirm that the subject is, cither (1) In its whole logical extent, or (2) In part of its lo-ical extent, ccM.taiued undex the logical extent of the predicate ; or, on the other hand, the proposition excludes either (3) The whole logical extent of its subject, or (4) Tart of the logical extent of its subject, from the logical extent of its predicate. This fourfold division answers to a combination of the divisions of propositions on the two bases of quality and quantity. Every categorical proposition, true or false, that Can be made on any subject whatever must find its place under one ot the following heads ; — 1. Universal affirmative, usually denoted by the symbol A. A Universal negative, „ ,, ^^ ^ 3. Particular affirmative, „ 4. Particular negative, „ " " " ' The symbols A, E. I. O, are taken from the Latin words affirmo ^ne^o. A ana I are the Hrst two vowels of the former WOfd. E the vowels of the latter word. ' The student should carefully consider the following silUDle examples of the four forms of which, in each case, an abstract exan.ple, a concrete example, and a diagrammatic illustration are given Notice the meaning of "is "in the propositionaJ forms. Is means '« is contained in " ; " is not " means " is not contained in." - AU X is Y " thus means ''All X is contained in Y." FOUR rROPOSITIONAL FORMS, Form A. — Universal Affirmative, All X is Y. All gold is yellow. 47 Form E. — Universal Negative. No X is Y. No man is infallible. Form T. — Particular Affirmativd Some X IS Y. Some men are wise. M iQ FOUR PROPOSITION AL FORMS. Form O. — Particular Negative.* Some X is not Y. Some men are not wise. T7ie folloiving ohscrvations on this fourfold form of Pro- positions are of the utmost importance : Foini A.— The subject is distributed, U., taken in its full extension : the predicate is not distributed. When we assert that ''all gold is yellow," we mean that gold, at all times and in all forms, is yellow ; therefore, the term " gold " is fully distributed. But the predicate is not distributed. For the proposition asserts only that amongst an indefinite number of yellow things, gold is always one. Form E.— Both the subject and the predicate are distributed. When we assert that " no man is infallible," we mean that the two terms "man " and ** infallibility '» are mutually exclusive. The attribute of infallibility cannot be predicated of any m^m in the whole universe. •In the diagrammatic illustrations tlic shaded parts always represent the subject Of the proposition. The student must note that the proposition o«;. contains tnformation about the part shaded. Thus in the diagram representing the proposition •« Some X is Y." our information is confined to the shaded part Of X entirely. We could not assume therefrom that some X is not Y The proposition only asserts that some portion of X is included within Y As a matter of fact more might be. but the proposition does not say so. These diagrammatic representations of propositions are called Euler's Circles and are open to mucli criticism. • * If f < , SIGNS OP QUANTITY IN LOGIC. 49 Form I.-Neither the subject nor the predicate is distributed. When we assert that " some men are wise," we mean that amongst men there is an indefinite number, forming an equally indefinite proportion of those beings of whom the attribute of wisdom may be predicated. Form O.— The predicate only is distributed. When we assert that ** some men are not wise," we mean that an indefinite number of men are excluded from the whole definite class of beings, of whom the attribute of wisdom may be predicated. These observations may be summarised ; — Form A distributes its subject only. E distributes both its subject and its predicate. I distributes neither its subject nor its predicate. O distributes its predicate only. }> it »» The student will notice that "this," "each,*' "every,*' "all," "no," and "some" are the only signs of quantity recognised by Logic. In ordinary speech many others are used, but they must be reduced to one of the signs given above before they can be considered in a logical reference. Note particularly that expressions like "few," "many," or such fractional terms as " three-fourths " are all considered equivalent to *' some." In short, " some " really stands for ** some at least " ; and beyond that, the word is altogether indefinite. '• Any " and similar expressions must be con- sidered as equivalent to " every." Cases will sometinios arise in which it is a matter of uncertainty whether a given expression is intended to be taken as a universal or a particular. This is especially so in current sayings and proverbs, e.(f., " Knowledge is power," " Haste makes waste." Such cases can only be determined by a careful survey of the facts the expressions are supposed to summarise. yriwF' i ww ^ I : * 1 'I ! CO EXERCISr.S. EXEECISES ON CHAPTER VII. 1. Dsjine a logical propositioyi ; and enumerate with examples, the various kinds of propositions, 2. What do you tinderstand as tJi2 exact tncaning of the logical copula ? 5. What are the signs of quantity recognised by Logic ? How do tliey compare with those used in grammar ? 4. Give the logical equivalent of each of the following expressions : "All are not" ; " Only tJiese are " ; " All except oiie " ; " Scarcely any " ; " Few are mot.''* 5. Reduce each of the following to strict logical form, and indicate wlietJier the vroposition is A, E, I, or : — (a) All birds have two wings. (h) All his sliots except tiuo hit the mark. (c) Tli2 more the merrier. (d) There^s not a joy tJie world can give like tliat it takes aioay. (e) All that glitters is not gold. (f) He jests at scars luho never felt a ivoiind. (g) None fail to remain poor tuJio are both ignorant and lazy. 6. The folUnuing sentences are soineiohat avibignous. Make at least two logical propositions of each: — (a) All are not clever ivho read much. (b) Some of the guests behaved disgracefully, (c) All the boohs cost a sovi're'inri. 7. What logical proposition is implied in each case, when tlie following are declared to be false : — (a) Honesty is tJie best policy, (b) All men- are liars. (c) Some horse dealers are honest. 8. Express in the simplest logical form you can the sense of the following passages : — (a) It never rains but it poztrs. (b) You cajinot have your cake and eat it. j'cj Unless help arrives xoc are beaten, (d) Many are called, hut few arc cltosen. 0. Say wlietlier tJie following is a categorical or hypotlictical proposition^ and tvhy : — Trespassers will be prosectited. CHAPTER Tin. o- Im mediate Inference, The whole of our study thus far has been a preparation for the investigation of inference or reasoning. Inference, in its Wider meaning, is the derivation of one proposition from one other proposition or from two other propositions. Those cases in which a conclusion is evolved from some one pro- position, witJwut the help of any other, are called Immediate Inferences. Thus, when we say " All animals are organised beings," we are able to infer directly from this that any particular animal is an organised being, and, again, that "no unorganised beings are animals." Every single assertion or denial that can be made will yield quite a number of otlier propositions, which differ from the original proposition in logical quantity or quality, or both. An Immediate Inference, then, is the inferential derivation of a new proposition from some one given proposition. The number and variety of conclusions which can be immediately derived from any single proposition, will be quite surprising to one who is not familiar with this kind of exercise. Take, for example, the following A (universal affirmative) proposition :— «• All X is Y." b 62 OPPOSITION. What inferences can be immediately derived from this ? Proceed thus : All X is Y ; No X is not-Y ; Some X is Y ; Some X is not not-Y; No not-Y is X; All not-Y is not-X; Some not-X is not-Y; Some not-X is not Y. This will be clearer if a concrete example is given : — '• All men are mortal." From this we may infer : " No men are not-mortal " ; •' Some men are mortal " ; " Some mortal beings are not not-men " ; " No not-mortal beings are men," etc. Now, without considering whether the examples just given are exhaustive, or whether all the conclusions are of practical importance, we will proceed to discuss the more important forms of Immediate Inference under the following heads : — I. Immediate Inferences of Opposition. n. »» . fi »» Conversion. III. „ „ „ Permutation. I. Inferences of Opposition. — Propositions are said to be opposed to each other \vhen they have the some subject and predicate respectively, but differ in quantity or quality, or both. Of the several kinds of opposition, that known as Contra- dictory Opposition is the most perfect and of the greatest logical value. This kind of opposition is an application of the ** law of the excluded middle," viz., that, of two contradictory propositions, one must be true and the other false. This occurs when an A proposition Is contradicted by an proposi- tion ; or an E proposition is contradicted by an I proposition. A.— All X is Y. Contradictory =0. — Some X is not Y. E.— No M is N. Contradictory = I. — Some M is N, OPPOSITION. 53 Taking either of these pairs of propositions, we see at once that both cannot be true and that they cannot both be false. Therefore, if either of the two propositions is affirmed to be true, we immediately infer the falsity of the other. Contrary Opposition is that which exists between an A and an E proposition, having the same subject and predicate. In this case, both propositions may be false, but both cannot be true, e.g, : — A. — ^All men are good. Contrary =E. — No men are good. This kind of opposition is of much less logical value. If we know that one proposition is true, we may immediately infer the falsity of the contrary. But if we know that one proposi- tion is false, we cannot infer the truth of its contrary. Sub-Contrary Opposition is that which exists between an I and an O proposition, which both have the same subject and predicate : — I. — Some men are wise. Sub-contrary 0. — Some men are not wise. In this case both of the propositions may be true, but both cannot be false. If we know that one of them is false, we can imiuediately infer the truth of its sub-contrary. Subaltern Opposition is that which exists between a universal and a particular proposition, i.e., propositions which both have the same subject and predicate, but differ in quantity : — A. — All men are mortal. Subaltern I. — Some men are mortal. From any universal proposition we can immediately infer the truth of any particular proposition of the same quality (an I from an A, or an from an E), but not vice versa. 54 OPPOSITION, An ancient square sets forth these various relations of opposition thus ; — A Contrcirtcs >VUI-4> jicx-O. • oub -contran^^ ^•^•— ^^opositions must always have the same subject'^''^"* and predicate before we can place them in opposition. There should now be found no difficulty in determining what inferences can be immediately drawn from the known truth or falsity of any one of the four ordinary proposibional forms. For convenience the student ia advised to commit the following to memory :— Contradictories cannot both be true, nor can they both be false. Contraries may both be false, but both cannot be true. Bub-contrarieg may both be true, but cannot both be false. Subalterns may both be true and both false. If the universal is true 80 is the particular; but the truth of the particular does not imply the truth of the universal. CONVERSION. 55 II. Inferences of Conversion. — By conversion is meant the immediate inferring of a new proposition from a given proposi- tion, in which the subject of the given proposition forms the predicate of the new proposition, and the predicate the subject. Tlius from " No stones are organised beings" — is obtained by conversion, " No organised beings are stones." The remarks in Chapter VII. on the distribution of the subject and predicate in the four prepositional forms, are of great consequence here. For, in converting a proposition, care must be taken that the two terms are used in precisely the same extent in the new (or mferred) proposition as they were in the original proposition. Now, in converting an E or an I j)roposition, no difficulty arises. ♦' No X is Y " distributes both its subject and its predicate. Hence, we may at once say " No Y is X." So, also, the I proposition " Some M is N " distributes neither its subject nor its predicate. Thus, we can immecliatoly say '* Some N is M." But in the A proposition '' All S is P," the subject S is distributed, but the predicate P is undistributed. If we converted this into " All P is S " we should distribute P in the new proposition, whereas it was not distributed in the original proposition. This we may not do. ITrom " All S is P" we can only infer " Some P is S." Hence we say that A propositions can only be converted "by limitation" (lier accidens). Summarizing these points wc learn that : — From an A proposition we can infer an I proposition by '* conversion by limitation." From an E proposition we can obtain another E proposi- tion by simple conversion. From an I proposition we can infer another I proposition by simple conversion. Lastly, we have to consider the case of O (particular, negative) propositions. Can these be converted ? Take, for instance, " Some X is not Y." Here X, the subject, is not i 1^ 55 PERMUTATION. distributed. If we convert the proposition and say " Some Y is not X," we distribute X in the new proposition. But, in conversion, we may never distribute a term in the new proposi- tion, which is undistributed in the original proposition. Hence. we conclude that O propositions cannot be converted. Practice in drawing immediate inferences by the conversion of given propositions is a most valuxble test of the student's progress in logical study. Both in ordinary discourse and in exannnations most ludicrous results follow from not observmg the rules of legitimate conversion. One examiner says that when he has asked for the converse of the proposition " None but the brave deserve the fair," students have said with perfect seriousness: "The fair deserve none but the brave," or *' No one ugly deserves the brave." The error in such cases arises from the fact that the student has omitted to put the given sentence into exact prepositional form, as logic requires. If this were done the sentence would become :— ''No one who is not-brave is deserving of the fair," and this is a simple E pro- position, and may therefore be converted simply into " No one deserving of the fair is not-brave," or, expressed more con- ventionally, '« No one deserving of the fair is a coward." III. Inferences of Permutation.-Of this kind of immediate inference there are several forms : (a) By OBVERSION.-Here we infer a new proposition, having for Its predicate the contradictory of the predicate, e.g. :— Original proposition.— All X is Y. Inference by Obversion.— No X is not-Y. We may ahvays obvert a proposition, if at the same time we change its quality. The rule of obversion is usually given thus: Substitute for the predicate term its contrapositive, and change the quality of the proposition. Thus'^^'Tr;? w/' ' '"'^^^''^ ^''^ ^°' ^^' opposite of a term. Thus "not-A- IS the contrapositive of "A." It is convenient to xise 'Z:^nr '''' :^-^-^^«t0^y" -y le used exclusively of i i> PERMUTATION. 57 Thus, MIX is Y yields No X is not-Y. No X is Y „ AH X is not-Y. Some X is Y „ Some X is not not-Y. Some X is not Y ,, Some X is not-Y. (h) By Contraposition. — In this case we infer a new pro- position which has the contrapositive of the original predicate for its subject, and the original subject for its predicate, e.g. :— Origmal proposition.— All X is Y. Contrapositive. — No not-Y is X. Immediate inference by contraposition is sometimes called the converse by contraposition. From A, E and O of the propositional forms we may infer -» contrapositive, but not from I. A E Original Proposition. All X is Y. No X is Y. Some X is not Y. Contrapositive. No not-Y is X. Some not-Y is X. Some not-Y is X. In drawing immediate inferences accuracy is all important. The exposition in this chapter has been iUustrated by symbols, but if the principles have been duly grasped it wiU not be difficult to apply them to concrete examples. In domg so the student must always reduce the sentences given as examples to strict logical form, if they are not already in that condition. The great importance of this subject makes it advisable that several worked examples should be presented for the reader's consideration. ■ ■ >'- .? ' t 1 h'- i 58 WORKED KXAMPLES. 1. What immediate inferences are derivable from the pro- position " All reallij happy men are virtuous " / (a) The Truth of the Subaltern : ♦' Some really happy men are virtuous." (6) The Falsity of the Contradictory : " Some really happy men are not virtuous." (c) The Falsity ol the Contrary: « No really happy men are virtuous." (d) By Conversion: "Some vktuous men are really happy." (e) By Obversion : " No really happy men are not virtuous." (/) By Contraposition: "No not-virtuous men are really happy." 2. Give the Converse, ihe Obverse and ihe ConiraposUlve of tlie following propositions :—{a) The longest road comes to an end; (6) Unasked advice is seldom acceptable. (Each of these propositions must first be reduced to logical form.) (a) This sentence=" The longest road Is limited." This is a universal affirmative. Its Converse is : " Some (one) limited thing is the longest road." Its Obverse is : " The longest road is not unlimited." Its Coiitrapositiveis: "No unlimited thing is the longest road." (6) This sentence=" Some unasked advice is unaccept- able." This is a particular affirmative proposition. Its Converse is: "Amongst (some) unacceptable things is unasked advice." Its Obverse is : "Some unasked advice is not acceptable." The sentence being an I proposition it has no contra- ' positive. I 1 ..; WORKED EXAMPLES. 59 3. Convert and contraposit the proposition, " For every wrong there is a legal remedy:' The proposition reduced to logical form is : " Every wrong is capable of a legal remedy." Its converse is : " Some things capable of legal remedy are wrongs. Its contrapositive is : "Nothing incapable of legal remedy is a wrong." 4. What eductions are possible from the proposition, " Amethysts are precious stones " ? (i^.^.-" Eduction " is a term frequently used for ''Immediate inference:') The given proposition is a universal affirmative, " All amethysts are precious stones," and may be treated as the* proposition in the first-worked example. 60 EXERCISES. I EXERCISES ON CHAPTER VIII. 5. fe^Zai„ ^ith illustrations, the difference between the contrary and the contradictory of apropositm. ^ 3 Explam why a universal ,^oMve proposition admits of the conversion of ^ts terms. " All eauilateral triangles are e^uialullT" 4 ei.e tn. converse, the contradictory mid contrary of " All A is JJ ; " Some vien are wise " a j ^i> ^ia <5. Give tU contradictory and the converse of:^ (a) Two blacks don't make a lohite. (h) James struck John. (C) Three-fourths of the candidates passed. 6. Assign the logical relation between each of th^ f^n^^ • mit.ns.iththe^oposition''AllcrystJrL!oms'°T"'''''' (a) Som crystals are solids. (6) No crystals are not solids. ( c) Some solids are crystals. 7. What is mediate inference ? Give where possible t!. converse the obverse and the contrapositive of:~ '^"■verse, (a) (said Hudibras) .• " I smell a rat." (b) ^\llere no oxen are, the crib is clean (c) Only Protestant princes can occupy tlu> English throne. 8. mat is opposition ? Which of l,^fo,;ns of opposition 1^ the greatest value and why i ^ I, 'fjf CHAPTEE IX. -o- Mediate Inference. — The Syllogism, Immediatk inference is the derivation of a new proposition from some given proposition. However useful this exercise may be, the new proposifcion is always recognised as only a different way of expressing the original proposition. Mediate inference professes to give a conclusion of a much more fruitful kind. In every example of a mediate inference, two propositions, and two only, are implied. In these two pro- positions the conclusion to be drawn is potentially contained, and out of these two propositions the conclusion is actually drawn by reasoning. The two propositions given are called the premisses of the conclusion. It will be seen afterwards that in ordinary discourse the two premisses and the conclusion are seldom fully expressed. One of the premisses is generally left to be understood, but, in spite of this, it is implied in the reasoning. When, however, the two premisses and the derived conclusion are fully and formally stated, the expression is called a Syllogism. Formal Logic assumes that, in every instance in which we draw a new and fruitful conclusion, the reasoning when fully expressed must take the form of a Syllogism, 62 THE SYLLOGISM. A Syllogism, then, is a conclusion expressly evolved from two propositions called its premisses. Each of the premisses of a syllogism must once have been a con- clasion from two other more remote premisses, unless one of the premisses is the statement of a truth which is axiomatic in its nature. All that we know, inferentially, about the universe, is known in the form of a vast number of conclusions drawn from other premisses. Knowledge is thus a net-work of conclusions, suspended nltimately upon a few axiomatic or self-evident truths. All arguing implies that there are certain remote premisses or assumptions, bearing logically on all questions, and about which the disputants must be agreed. It is worthy of observation that some persons, who are not acute reasoners, are yet able to see truth at a glance. Others are subtle and ready reasoners whose natural intuition (insight) is small. Argument and insight are often found in inverse ratio. It has beeu remarked that, generally, women are more strongly endowed with insight, and men with reasoning power. The following is a simple form of a Syllorrism :— All men may be educated.) Savages are men. j (^^emisses.) Therefore, Savages may be educated. (Conclusion.) Notice that in this example there are three propositions. Of these the first two are the premisses, and the last the conclusion. There are also three terms: '^ men," '' savac.es " and "educated," and the last two of these appear in" the conclusion. The term which forms the predicate of the COn- elusion (" educated ») is called the major term, and the term which fonns the suhject of the conclusion ("savages") is called the minor term. The term which appears in both the premisses, but which doos not appear in the conclusion ("men") is called the middle term. Further, the premiss which contains the major term (" All men may be educated ") IS caUed the major premiss; and the premiss which contains the minor term C*A11 savages are men") is caUed the minor premiss. These general definitions hold good for all kinds of syllogisms. ^i'^ \>. ill RULES OP THE SYLLOGISM. From these definitions it will be easy to see that a syllogism is the logical comparison of the two terms which appear in the conclusion, by means of a third, or middle, term. Liogic lays down three fundamental rules which apply to every varietj^ of syllogism. 1. Each syllogism must have three, and only three, terms ; it must have three, and only three, propositions. 2. Of the three terms thus involved in every syllogism, the middle term (i.e., the term common to both premisses) must be taken universally (i.e., it must be distributed), at least in one of the premisses ; and neither of the other terms, i.e., the major or the minor, can be taken univer- sally in the conclusion, unless it was taken universally in the premiss in which it occurred. 3. No conclusion can legitimately be drawn if both the premisses are negative ; or if both are particular ; and, if one of the premisses is particular, the conclusion must be particular ; or, if one of the premisses is negative, the conclusion must be negative. Notes on the Canons, or Rules of the Syllogism. Rule I. —We require to add that the terms must be used throughout in exactly the same sense. Owing to the ambiguity of words it sometimes happens that a syllogism will seem only to contain three terms when in reality there are four, i.e., one of the terms has been used in two distinct senses. In the fallacy quoted in the early part of this work we have an example of this : i He who is most hungry eats most, He who eats least is most hungry, Therefore he who eats least eats most. r 64 RULES OP THE SYLLOGISM. In this example a little reflection will show that terms are not being used throughout in the same sense, and that there are in reality more than three terms involved. ■ Rule 2. — The middle term must be once distributed, other- wise it cannot be a medium for comparing the other two terms. It must be either wholly in, or wholly out of one of the other terms before it can be the means of establishing a connection between them. If we use a diagrammatic illustration of the Syllogism, tha necessity of the distribution of the middle term is obvious. Thus let the Syllogism bo All M is P. All S is M. .-. All S is P. Here M is the middle term, and it is shown to be wholly in P. If, however, M were not wholly distributed we should have to represent it partially within P. Consequently we should not be able to say whether S was contained in the part of M within P or in that part of M which is without P and of which we are not supposed to know anything. A syllogism with an undistributed middle tenn is the most common form of erroneous reasoning. S\ I! BUL«8 OF THE SYLLOGISM. 65 A tenn must not be distributed in the conclusion that was not distributed in the premisses. Obviously, if an assertion is not made about the whole of a term in the premisses, we cannot make it of the whole of the term in the conclusion without going beyond what has been given. When this rule is broken m the case of the major term, it is called the IlUcit process of the Major ; and in the case of the minor term, Illicit jyrocess of the Minor. If we were to admit that a term might be taken universally in the conclusion, which was not so taken in the premisses, we should be admitting that the ** part is greater than the whole.'* Rules.— Two negatives cannot yield a conclusion. For two negative propositions are really a declaration that no connection exists between the major and minor term and the term by which they were to be compared — in other words there is no middle term, and no Syllogism can be formed with two negative premisses. That two particulars cannot give a valid conclusion, and that the conclusion follows the weakest premiss are corollaries from the previous rules. The general rules of the Syllogism depend upon one great canon, viz. : " Two terms that logically agree with the same third term, must logically agree with each other ; and two terms, one of which agrees while the other disagrees with the same third term must logically disagree with each other." The ultimate principle of reasoning thus defined, is expressed * its most general forms in the ''Dictum de ovmi et nullo " of Aristotli "Whatever is predicated affirmatively or negatively of any class, must, on pain of involving inconsistent (contradictory) thought be predicated of whatever is contained under that class." Aristotle regarded this as the axiom on which aU syllogistic inference is based. Every conclusion drawn in a syllogism, where the above general rules have been observed, is an aflfirmative or negative proposition deduced by means of a minor (or applying) premiss 'iN 66 THE SYLLOGISTIC MOODS. from a more general proposition that is assumed to be true, and in which the conclusion was virtually contained. It can also be shown that there must be four, and need not be more than four syllogistic forms. For, the general (major) proposition, which virtually contains the conclusion must be universal (either A or E), and the applying (minor) premiss must bring either the logical whole, or a part only, of its subject into comparison with the middle term. The minor premiss, therefore, will be either A or I. The general rules of the syllogism decide the conclusion. Hence, we may say that every reasoning may be exhibited by one or other of the following combinations of the four propositional forms : — AAA. AIL EAE. EIO. These letters, of course, tell us the quantity and quality of the two premisses, and the conclusion of the syllogism which each triplet forms. The arranging of the symbolic letters in different ways is called the Mood of the syllogism. Thus, AAA represents a syllogistic mood in which both the premisses and the conclusion are universal affirmatives. EIO represents a syllogistic mood in which the major premiss is a universal negative, the minor premiss a particular affirmative, and the cjonclusion a particular negative. The following are examples of four forms of syllogism : — Mood AAA, AH men may be educated. All savages are men. .•. All savages may be educated. All M is P. AU S is M. .-. AU S is P. [N.B. — This alone of all forms of syllogism gives a univerBsJ affirmative conclusion, and is, therefore, the one most convenient for expressing scientific reasonings with their universal afiorma* tive conclusions.] THE gYi:.r.OGI8TIC MOODS. Mood All. All educating influences are good, Some difficulties are educating influences, .*. Some difficulties are good. AH M is p. Some S is M. .-. Some S is P. Mood EAE. No Europeans are cannibals. All Englishmen are Europeans, .*. No Englishmen are cannibals. No M is P. All S is M. .-. No S is P. Mood EIO. Wliatever is followed by remorse is not desirable, Some pleasures are followed by remorse, .*. Some pleasures are not desirable. No M is P. Some S is M. .-. Some S is not P. 67 The student is advised at this point to transform some simple arguments from the form in which they arc ordinarily used, into precise syllogistic form. Consider ' for example the following : — 1. " There are no foreigners amongst the woimdtd, so no Frenchman received a woimdJ" Here we have given a major premiss and a conclusion. Tn order to express the statement in syllogistic form we must supply the minor prenjiss. The passage may then be written as a Syllogism in E A E : — No foreigners are wounded (All Frenchmen are foreigners) .'. No Frenchmen are wounded. 68 EXERCISES. 2. " No toar is long popular ; for every war increases taxation : and the popularity of anything that touches the pochet is sliort livecV This may be written as a Syllogism in E A E thus : — Nothing that increases taxation is long popular. Every war increases taxation. .'. No war is long popular. 8. " For some wars there has been no justijlcation ; for they have been harmfully aggressive^ and such aggression is ivithout excuse,'^ This may be expressed as a Syllogism in E I 0, thus :— No harmful aggression is justifiable, Some wars are harmfully aggressive, .*. Some wars are not justifiable. EXERCISES ON CHAPTER IX. 1. WJiat is understood by a proposition, a premiss, a conclusion^ and a syllogism ? Give an example of each. 2. *' From negative premisses you can infer nothing.'^ Explain and illustrate this statement. 3. SJioio how logical form as displayed in tlie syllogism tends to clearness of thought. 4. Give a clear explanation of tJie rule concerning the middle term of a syllogism, ,5. Enumerate the cases in which no valid conclusion can he draion from tioo premisses. 6, Supply a premiss that will make tJie following reasoning correct: ** liter e is no Englishman among the wounded, so no officer can have received a wound." 7. Pui the following argument into syllogistic form : — " Hoio can anyone maintain that pain is always an evil, who admits that remorse involves pain, and yet may sometimes be a real good ? " CHAPTER X. The Figures of the Syllogism. In one or other of these four sylloglalic forms all our reason- ings might be expressed, just as all our judgments could be expressed in one or other of the prepositional forms. But Logic takes cognisance of many other syllogistic forms besides these four. The question may suggest itself— why should we add to these four forms, if they are sufficient for the unabridged expression of aU sorts of reasonings ? The answer is that the addition is one of practical convenience. It will be found that many o£ the concrete reasonings of ordinary Ufe, though capable of being expressed in one of the four syllogistic forms, yet find a more convenient and natural expression in one or other of the additional forms. The way in which the additional syllogistic forms are obtained is by varying the position of the terms in the premisses. The four syllogistic forms already con- sidered have certain features in common. Thus, in each case the middle term is the subject of the major premiss and the predicate of the minor premiss. Also, the middle term is distributed in the major premiss but not in the minor premiss. On account of this similarity the four syUogistic forms already given are classed together, and constitute what is known as Figube I. of the syUogism. 'If I 70 FIGURES OF THE SYLLOGISM. But we can frame a series of syllogisms which violate none of the general rules of the syllogisms, in which the relations of Figure I. are varied. Thus we may have tJie middle term as the predicate of each proposition. In these cases the middle term will always be of greater logical extent than either of the other two. The syllogisms which exhibit these characteristics are classed together as Pigure II. Just as there were four valid moods under Figure I., so there are four vahd moods under Figure II. The student should construct concrete illustrations by reference to the following abstract examples of each of the moods of Figure II. : — AEE. All P is M. No S is M. No S is P. A 0. EAE. AUPisM. [ NoPisM. Some S is not M. j All S is M. Some S is not P. I No S is P. EIO. No P is M. Some S is M. Some S is not P. Notice that in Figure II. the conclusion in each mood is a negative one. Hence, this figure is the most convenient for expressing argumentative objections and refutations. AVhen tJie middle term is made the subject of each premiss, and is, therefore, of less logical extent than the other two terms, we get a series of syllogisms which are grouped together as forming Figure III. But for reasons that will affcerwards appear, we can form six valid moods of this figure. Tims ; — A A I. All M is P. All M is S. Some S is P. E 10. No M is P. Some M is S. Some S is not P. All. All M is P. Some M is S. Some S is P. I A I. Some M is P. All M is S. Some S is P. E A O No M is P. All M is S. Some S is not P. AO. Some M is not P. All M is S. Some S is not P. FIGURES OF THE SYLLOGISM 71 Notice tliat a particular conclusion onli/ is obtained in each mood of Figure III. Hence this mood is well fitted for pro- pounding examples argumentatively, or for establishing some particular or indefinite conclusion. There is yet a further group of syllogisms, known as Figure IV; in which the middle term is tbe predicate of the major premiss, and the subject of the iiiinor. This figure has five moods, viz : — A A I. AU P is M. AU l\f is S. Some S is P. AEE. All P is M. No M is S. No S is P. I A I. Some P is M. All M is s. Some S is P. E AO. No P is M. All M is S. Some S is not P. EIO. No P is M. Some M is S. Some S is not P. The fourth figure is clumsy and unnatural, and is omitted altogether by many logicians. It is worth while to notice the following results, obtained from a comparison of the conclusions in the various moods of the four figures : — A (universal affirmative) conclusions can only be obtained in one figure and in one mood of that figure. E (univepsal negative) conclusions can be obtained in three figures or four moods. I (particular affirmative) conclusions can be obtained in three figures or in six moods. (particular negative) conclusions can be obtained in each of the four figures or in eight moods. From this it follows that A conclusions are the most difficult to establish, and the easiest to overthrow. O conclusions, on the other hand, are the easiest to argue for, but the hardest to I i\ :] r^ I I I 72 FIGDBES OF THE SVLLOGISM. disprove. Or, more generally, universal and definite conclusions are most easil}^ overthrown, and particular and indefinite con- elusions are most easily maintained. Special Rules of the Figures of the Syllogisms. In addition to the general rules to which all syllogisms iiuist conform, logicians have deduced certain simple rules appUcable to the different Figures. In the First Figure, — {d) The major premiss must be universal. (6) The minor premiss must be affirmativ c. In the Second Figure, — (a) The major premiss must be universal.' (b) One premiss and the conclusion must be negative. In the Third Figure, — (a) The minor premiss must be afTirmative. (i) The conclusion must be particular. In the Fourth Figure, — (a) When the major premiss is afiirmative, the minor premiss must be universal. (b) When the minor premiss is affirmative, the con- clusion must be particular. (c) In negative moods, the major x^^emiss must be universal. These special rules of the figures do not introduce new material, tkey are only a concise statement deduced from results previously obtained. I 1 I EXERCISES. 78 EXERCISES ON CHAPTER X. 1. WJiat are tJie figures of the Syllogism? Examine whetJier IAI,EIO are valid or invalid in each of the figiores. 2. Which figure is most convenient (1) for overthrowing an adversary's conclusion ; (2) for establishing a negative conclusion; (3) for proving a universal truth. 3. Give tJie special rules of the Figures. 4. Express the following argument by a Syllogism of tJie third figure : — Some things which have a practical worth are also of theoretical value : for every science has a theoretical a^ well as a practical value. 6. What moods are good in tJie first figure and faulty in tJie Second, and vice versa 1 Why are they excluded in one figure and iwt in the other ? 6. From which syllogisms can you infer tmiversal, particular, negative inferences^ or none at all ? 7. Enumerate briefly the conditions of a valid deduction. 8. Construct a Syllogism in I A I to prove that some taxation is necessary. 1 ( 11 CHAPTER XI. o- The Reduction of Syllogisms. It lias already been observed that all reasonings may find Ibdr eipre«8iou in oiw of tbe foor moodit of Figun: L Tbd Olk « f Hcv^M «V0 olUn ««O«tMl0Qi for »>|*oei*] itULriMj«H>4, l>ui kuMcauefti as the flrat figuxo hi cotuilcMd Uio yiiooii dird^ft and perfect nkodo ol oxpar#M&ng: our rwi»ocnini;, I^o^jSo ii»ow^ hoirr uny KyUouuan of Fibres II., III.» and IV. (caUodl the tndiioel fiffurcs) uuiy be tnuiafoznieol into one of the moodfi ol the firsi fljgure. I'hU proeas i« caUed tAc lUdtteiion of Siflto^isms* Itkcreore ISftcoii ioooUh alicy|«(Uivr iii lliu Uioio itidirvci ftjjuiMt And ihirtcca ol them luj^ be reduced: (1) bj the ooa\'«nioo of oiM or XDoro ol tbe throe iiropositioius in the syllogism to be mluced» (2) by the tnioipoiition of the pnausses. or &) by bolti of these (iroceBies. [A 0» FijTure II. and A 0, Flguix« III. *ru cxceptioos and uill be con»dcred MparAtclj-J Hm ndnctiOQ ol tbc s^Uogisms of tbe indirect figures into direct or flc«l Agiixc tyllogisuis i:i oae of the iitost proDtable 0ae«rci90t in forti^il Ia^^k^ Tfoe proc<»« i» not oeaxl^' lus omj on u4ght appear. To enscre accuracy and rapfidllj la Ibe proetM on ii^ficniom rnncinoi^ hw b«en a»od by logicians tot mofftt than 5O0 ytojtA, ThU raneinoiun hnn \nyitn caIM ** (lie mag^ xtxtt of Lpgie.*' juid certainly the uxkrds of which it ia nEDUCnON OP SYIiT^OGTSMS. 76 composed are more full of uieaning than any similar com- bination ever made. The usual form of the mnemonic, which must be learnt by heart, is as follows : — Sarharay Oelarenl, Dariij i^crtoqiie, prloris, Cesare, Carnestres, Festino, Barolio, secundae, Tertia, Daraptiy Di^amis, Datisi, FelaptoHy Bokardot Ferison, habet, Quarta insuper addit, Bramantipy Camenes, Diniaris, FesapOy Fremon* [The words in italics are the significant words, the others being only connectives.] The following is the key to this famous mnemonic. Every mood in esbch of the lour figures is re))rc$»ented by a difierent woool. In ika qaat of ih>«' indtr«ct Ai^tiraM (It., TTF. and TV.), i)io nuMaftomb iella oa to whiU mooorduccd to the syllofpbno of the ilrsi ftgure, which iji rQp(r«»cntcd by a wonl having the some initial letter. Thus, the ayUogMua of the foarth ficn'v* ropc««c«itod by the vrovd CamonM, may bo redoccd to thri iiyllo^giiKn of Uie finiit fl({aro, ra^iN^ianitd by Ui« word CeliunenL 8. The letter ** »*' occurring in a word perfonnD adovbia f tmctioiL 11 it occurs in tlie miidU of a \^*ord a$ in Ceisre II lueaim llial« in Uie procet^ of zt7ductioin» the |)copo$atlon rvi>reaMiled by the provio. ;— " If your books art m ctrnfarmUg iri/Jb (Ms JCm^a, lA/y are superfluous ; if thfy «*t# al variance ttith ii ihi^ flW ptmUiofts* But, they miai ^ithit hs in w^formHif itiiii ikt Koran, ^at variance with U. Tiutsf^r^ Ihtff af€ either $^ftrfimm ^ pernicious," CHAPTER XTV. Fallacies of Deduction The study of logical forms, besides being a useful mental discipline, supplies a ready test for the detection of fallacies. Indeed, formal logic may be said to exist as a practical study for this purpose. The syllogistic forms may be regarded as a framework in which all our concrete reasonings may be unfolded or displayed, and one by which their weak points may be more readily discovered. A Fallacy is a reasoning apparently correct, which, never- theless, iuYolYes inconsistency In inferential thought The conclusion appears to follow from the premisses, but in reality it does not. We do not class palpable, downright blunders as fallacies. A fallacy is an error so wrapped up in words that the mistake is not at once perceived, and thus tends to produce conviction. Hence, the work of defining and exemplifying the different kinds of fallacies is in one respect the chief end of the science. But fallacious reasoning is so diverse that it is impossible to exemplify every variety of it. Nor is it possible, Boraetiines, to decide to what class a given fallacy ought to be referred. For fallacies, like consistent reasonings, are mostly expressed elliptically, and it is not always clear what the unabridged reasoning is supposed to be. Thus, when a person argues "that a country is iU-governed, because misery prevails there," the unabridged syllogism may take two different forms I (y INTERNAL FALLACIES. 89 neither of which is correct. (1) It may have for its omitted premiss, ** All miserable countries are ill -governed," which no reasonable opponent would admit ; or, (2) the omitted premiss maybe "Every ill-governed country is miserable," in which case the conclusion is invalid, for the middle term has not been distributed in either premiss. Again we do not consider wilful attempts to deceive as fallacies. To such attempts we apply a stronger term. When people who know the truth but suppress it by suggesting a wrong explanation [sujjpressio veri et suggcstio falsi), this is moral not logical error. Ir» short, a dishonest intention will evade all rules of Logic. Ordinary common sense is competent to expose most fallacious reasonings by its own sagacity. But it not infre- quently happens that common sense is aware of a fallacy in the course of argumentation, without being able to say exactly what is wrong. Arguments are felt to be wrong, but those unskilled in logical science are puzzled how to demonstrate the error or how to refute the fallacy. Logic supplies the needful help to enable students to localise and expose the error. It makes the student familiar with the common form of unsound inference. It keeps the attention fixed on the essential steps of all valid reasoning. It accustoms the student to mark accurately the exact meaning of terms used, and the relation of these terms to one another. And it shows the necessity of defining with precision the question in dispute. After a course of discipline like this, the mind forms a spontaneous habit of accurate judgment and self-consistent thought and reasoning. Fallacies are usually divided into two classes : — 1. Internal Fallacies, where the unsound element appears in the mode of expression. These are called fallacies "in iictione.'' All internal fallacies may be detected even by those who are ignorant of the viatter to wbich the reasoning relates. 90 INTERNAIi FALLACIES. Internal fallacies are subdivided into (i.) Purely formal fallacies, which are a breach of one or other of the rules of Logic. (ii.) Verbal fallacies, in which the error lies in some ambiguity in the words used. (i.) Purely formal fallacies are breaches of one or other of the rules governing mediate and immediate inference. All that is needed here is to remind the student of the most obvious pitfalls, viz. : — (a) Confusion of contradictory with contrary opposition of propositions. (6) Simple conversion of A propositions, (c) Syllogisms with an undistributed middle. {d) Illicit process of the major or mmor (see page 65). {e) Arguing from two negative or two particular premisses. (/) Neglect of the rules governing conditional syllogisms. (ii.) Verbal fallacies. These are often mere quibbles. The followmg are the chief varieties of verbal fallacies : — (a) Ambiguity of a word (equivocation). — A word is gometimes used in a different sense in the two proposi- tions of a syllogism in which it occurs, e.^., " Light is always cheering ; some afflictions are hght ; therefore some afflictions are cheering." Obviously, the middle term ** light " is used in a double sense, and there are four terms used instead of three. {b) Ambiguity in the gram,matical structure of a sentence (amphibology)^ e.g., *' Twice two and three." This is ambiguous, for the answer may be either seven or ten. " What he was beaten with was what I saw him beaten with. I saw him beaten with my eye. Therefore he was beaten with my eye." (c) Composition. This is the confusion of a universal with a collective term. When we assert something of M EXTJBKNAL, FALIiACIES. 91 ^ i < each and every member of a class, we may infer the same of the whole class. When we say that all the angles of a triangle are less than two right-angles, we use the word *'air' distributively ; but, when " all" is used collectively the sentence is incorrect. We could not say that " all the angles of a triangle talcen together are less than two right- angles. (d) Division. This fanacy is the converse of the faUacy of composition. What is said collectively may not be said of the various individuals included in the collective term. All the angles of a triangle taken together, are equal to two right-angles, but no indlviJual angle of a triangle is equal to two right-angles. [e) Fallacy of accent. This arises from the accent or emphasis being thrown on the wrong word in a sentence, e.g., ''And he said' saddle me the ass' ; and they saddled him.'' 2. External Fallacies. — The error here can only be recognised by those who are conversant with the matter about which the statement is made. These are said to be fallacies " extra dictionem:' It is not easy to give simple exaini)les of them. When the wrongful argument is stated in simple language, the error is easily seen. But, when the error is diluted over a speech of an hour's length, it is more difficult to detect it. The foUowing are the chief varieties of external fallacies : — i. Many Questions {plurium interrogationum). This fallacy is committed when several questions are so combined into one, that, if you answer "yes" or " no," you are committed to something more than your real meaning. A man asks : " Have you ceased ill using your mother 9 You would not care to answer " yes " or " no." Some- times in a court of law, questions of this kind are asked, and a plain answer "yes" or "no" demanded. Such 92 EXTERNAL FALLACIES. EXTERNAL PALLACIBB. 93 questions should be at once broken up into their several parts and each part answered singly. ii. Fallacy of the consequent, better known by the familiar phrase *^non sequitiir.^^ This is the general nan^ given to loose and pretended arguments, where there is no connection between the premisses and the conclusion. iii. The Fallacy of Accident is committed when we argue from a general rule to a particular case or vice versa. So, also, when a statement is made luith a qualification and we then use the statement as though it had been granted without qualification {a dido secundum quid ad dictum si7njpliciter). Thus, if it is granted that " I eat to-day what I bought yesterday," we must not follow this by arguing " What was bought yesterday was raw meat, there- fore raw meat was eaten to-day." In one case " meat " has the accidental quaHfication of ** rawness" added, where- as in the premiss an assertion is made without regard to any accidental qualification. This is a form of erroneous argument frequently used. iv. The Fallacy of False Cause (post hoc, propter hoc), where it is assumed that because one event follows another, the former event is the cause of the later. This is a purely inductive fallacy and will be considered later. V. Irrelevant Conclusion {ignoratio elenchi). This is a most important type of deductive fallacy. The name covers all those cases in \vhich a concUision is proved, which is really not the point in dispute, but which sufficiently resembles what was required to be i^roved, to be often mistaken for it. Scarcely any fallacy is so common or so dangerous as this. Arguing beside the point, distracting attention by irrelevant considerations, is as frequent as it is misleading. The incoherence of the ignoratio elenchi lies between the conclusion oflered ajid the proper answer ' to the question, but involves no breach of the rules of the syUogism. There are four varieties of this fallacy which should be noticed. {a) The argumentum ad hominem. This is confusion as to wliat the point at issue really is. Thus, if a new law is proposed, it is no proper argument to urge that the pro- poser is not the right person to bring the question forward. When we have advice given to us, it is not logic to retort that the preacher should practise what he preaches. If a man is accused of a crime, it is not relevant to assert that the accuser is just as bad. In all such cases, the argument proceeds not upon the merits of the case, but upon the character of the persons engaged in it. (6) Fallacies of ohjections.—\Nc commit this fallacy when we argue that a proposal should be rejected because it is open to objections. Such an argument is always a fallacy, if the alternative can be shown to be open to greater objections or difficulties. (c) Argumentum ad verecundiaM.—Hhi^ is an appeal to our respect for ancient or established authority. The fallacy lies in the assumption that whatever is old or well- established must ipso facto be good. (d) The argument in support of a change is the opposite of (c). The fallacy here Is the implication that all change is progress, whereas the contemplated change may occasion more or greater evils than would follow if no change were made. vi. The Surreptitious Assumption (Petitio pnwcipu).— Every example of deductive reasoning starts from some general principle (major premiss) about which the disputants are assumed to be agreed. If one disputant adopts as his premiss a statement which the other disputant does not accept, the question at issue remains unsettled and no conclusion can be drawn between them. " Begging the / .^ 94 THE DETECTIOIT OF FAt^riACIES. THE DETECTION OF PAt-TjACTES. 05 question '* and ** arguing in a circle " are familiar forms of the petitio principii. He who argues in a circle assumes the truth of his major premiss and by means of it reaches a conclui3ion, which he afterwards uses to establish the major premiss with which he started. Thus, an illogical divine might argue : ''We know that there is a God, because the Bible tells us so ; and we know that the Bible is true, because it is the Word of God." People are especially liable to fall into this fallacy when they use a mixture of English and classical words in the same reasoning. For they often seem to be proving one question by another which is identical with it, only expressed in words derived from another language ; e.g, : — " Consciousness is the immediate knowledge of an object ; for I cannot be said to know a thing unless my mind has been afEected by the thing itself." The detection of fallacies is such an important branch of 'logical study that a few typical fallacies are appended, with hints as to their solution. 1. Examine the following : — ^^ Every hint comes from an egg; every egg comes from a bird; therefore, every egg comes from an egg." The premisses written in logical form are : " Every bird is an egg-product; every egg is a bird-product," i.e.^ there are four terms, whereas a correct syllogism can have only three. For exercise, test the following in the manner above indicated: "Knowledge is power; consequently, smce power is desirable, knowledge is desirable." 2. Is the following a valid argument f — "To assault another is wrong ; conscquenflyj a soldier ivho aasaulta another does wrong.'' \ This is the fallacy of accidents. A soldier is a man with an accidental qualification, and we cannot argue from a general to a special in such a case. Examine in the same way: '' Intoxicants act as a poison to a drunkard, and everyone should avoid poison." 3. Examine the following :—'' He who is most hungry eats most; he who eats least is most hungry; therefore, he who eats least eats most.'' This is the fallacy of accidents ; *' eats most " is taken generally in the conclusion, but specifically in the premiss. 4. Examine :—^^ If Jack is a good hoy he will do as he is told; he is a good boy {for, if he will do as he is told, he is a good boy) ; therefore, he will do as he is told.'' A petitio principii — arguing in a circle. 5. Examine : — " The sea was the place where the incidents of my tale happened; there is the sea; therefore, my story is true.^^ An ignoratio elenchi. 6. Examine: — "-4 dog chases a tortoise : the tortoise has a hundred yards start, hut the dog runs ten yards to every one run by the tortoise. When the dog has run a hundred yards the tortoise will be ten yards ahead; when the dog has covered these ten yards, the torioise will he one yard ahead ; when the dog has covered this one yard, the tortoise will be ^th of a yard ahead, and so on. The tortoise will be always ahead and the dog ivill never overtal'e it." This is an ancient specimen of an ignoratio elenchi. The argument pretends to prove that the dog will never overtake the tortoise ; it really proves that the dog passes the tortoise between the Ulth and 112th yards. 7. Examine : — *' If I am to pass this examination I shall pass it, whether I answer correctly or not ; if I am not to pass it, I shall fail whether I answer correctly or not ; therefore, it does not matter how I answer the questions," \ 96 EXEBCISES. Here it is tacitly assumed that "whether I answer correctly or not " is not a link in the fated chain of events. It is assumed that fate does not work through correct answering of questions, and the conclusion is merely a repetition of this assumption. It is the veneration of "Fate" that draws away our attention from the error of this delightful petitio principii. EXERCISES ON CHAPTER XIV. 1. Describe any three fallacies in Logic, giving an example in each case. 2. Foint out some of the ordinctry forms of fallacy employed to mislead in argmneni or in oratory, and illustrate tlie forms named. 3. Explain and illustrate the terms " redtictio ad absurdum'' and ** begging the question." 4. Examine the following : (a) You are not what I ayn ; I am a man : therefore you are not a man. (6) A fish is a cold-blooded animal and breatlics by gills ; neitlier of tliese things is true of a whale ; tJierefore, it is not a fish. 5. Explain exactly the nature of the fallacies called ** accident," " argumentumad hominein,^* and " argtimentuni ad verecundiam.'*^ 6. Show how a logical training enables a student to detect fallacies. 7. Explain and illustrate by examples tlie following terms : — Ambiguity ; fallacy ; premiss ; suppressio veri et su( g stio falsi. CHAPTER XV. Inductive Logic. The Undamental ksSOn of De.5uctlve Logic has been that no conclusion may ever contain more than was contained in the premisses from which it was drawn. Particular premisses, wl saw, could not yield a general conclusion. The definition of Induchve Log. therefore, will seen, at the outset a paradox. For Inductive Lo^ic may be defined as the Inference from particu ars to the general, or from the known to the unknown. It IS the establishing of general laws or principles from observed particular facts or instances. John, Thomas, etc md.v.duai n.en, are mortal, from which the general inference IS drawn that "All men are mortal." Here we have a general conclusion about all men, derived from a^ ixadefinite number of particular instances. At first it .eems as if our conclusion was overdrawn. The conclusion contains more than is given in the premisses; it seems like a leap xn the dark. Modern science consists throughout of SUch general conclusions, based on particular facts. Because certain things resemble each other in certain observed ways, we assume that they wiU resemble each other in certain previously un- ObSerVed ways. But k ihk general assumption warranted . I>o any number of observed facts warrant a general universal conclusion. If the observed facts be two, or two hundred particular cases, are we warranted in making any assertion beyond the number observed ? What right have we to add tO 98 INDUCTION. what we actually observe, as we certainly do, whenever w© conclude, from seeing a number of particular events occur, that they will always occur ? We say " all animals die," but we have not seen all animals die. Similarly, " all bodies gravitate," but our experience does not extend beyond particular instances of gravitation. Yet we are certain that these inferences are legitimate. What, then, is the ground of this certainty? Why are we able to conclude that " All must be so-and-so," because we have observed that " Some are so- and-so " ; that " all " bodies gravitate because " some'' have been observed to gi-avitate ? This is the problem of induction. For, we must observe that this process of induction is attended with some perplexity. In some cases one single observation is enough to warrant a general conclusion, whilst in other cases we hesitate to draw a general conclusion from hundreds of observed instances. Euclid takes a single triangle and shows that its three angles are together equal to two right-angles. We therefore accept this as a general truth applying to triangles of every kind and everywhere. One single instance is Bufi&cient to establish a general rule. On the other hand, though every crow I have seen is a black one, I should have no hesitation in believing someone who told me that he had seen a grey one. Whence come the certainty in the one case and the uTiGGrtainty in the other. Induction is based upon one great axiom, viz, : " the COURSE OF NATURE IS UNIFORM." In other words nature is not a chaos, it is an orderly system. Any event does not follow any other event in a haphazard way. The relation of things to csLch other is governed by what we call " law." This truth is axiomatic in its nature. It is the assumption of all Induction and of all science. It is not a truth which we can prove, nor does it need proof. If any one cares to deny it, we can offer no demonstration of it, beyond showing the denier that he himself acts upon the assumption every hour of the day. CAUSE AND EFFECT. 99 This axiom is practically the assertion that things are related to each other by law, and the one general law, everywhere observable, is that of cause and effect. The truth, that every fact which has a beginning has its cause, is a truth coextensive with human experience. It is needful, then, to have a definite notion of w^hat is meant by cause. We may define it as follow^s : — A cause is that which immediately precedes any change, AND WHICH, EXISTING AT ANY TIME AND IN SIMILAR CIRCUMSTANCES HAS BEEN ALWAYS AND WILL BE ALWAYS FOLLOWED BY A SIMILAR CHANGE. In this sense of the word cause is synonymous with power, property, or quality. Thus : " Water has the power, property, or quality of melting salt " is the equivalent of *' Water is the cause of the melting of salt." Each statement means that when water is poured upon salt, the solid is transformed into liquid. Two parts of a sequence are thus before our minds (a) the addition of water to salt, fi) the transformation of a crystalline solid into a liquid. These are respectively cause and effect. The po^vers, properties, qualities, or causes of things are not to be regarded as anything superadded to the thing. These are not the things plus their powers^ but things alone. Things are the invariable antecedents of changes in similar circumstances. The changes occur in an order or with a uniformity, which we believe to be regular. It is this general fact which enables us to reason about nature and to draw general inferences. If the changes which we see continually happening were chaotic, without uniformity, there could be no reasoning about them, either inductive or deductive. Now, in our actual experience, causal connections are mixed up with casual or merely accidental coincidences. Even the un- scientific man remarks that some sequences repeat themselves, whilst others do not. He watches the secjuences and the coincidences happeuing within his range of observation. Me OTHER CANONS. 107 absence of air in the receiver is the only particular in which the circumstances differ. The phenomenon of retardation, occurr- ing in one case but not in the other, is at once described as the effect of which " ak " is in some way the cause. ^ These two canons will sufliciently illustrate the logical conditions which regulate inferences about the laws of nature, and in particular the law of causation. They are calculated contrivances for finding out when causal connection really exists. All scientific work exemplifies these rules. There are three other canons somewhat more complex which are, how- ever, deductions or combinations of the two already discussed. For the sake of completeness they are printed here, but the elementary studeiit will scarcely be required to discuss them. 3. The Joint Method of Agreement and Difference. If two or more instances in which the phenomenon occurs, have only one circumstance in common, while two or more instances in which it does not occur have nothing in common save tke absence of that circumstance; the circumstance in which alone the two sets of instances differ, is the effect or the cause or an indispensable part of the cause of the phenomenon." 4. Methodof Concomitant Yariations.—*' Whatever phe- nomenon varies in any manner whenever another phenome- non varies in some particular manner, is either a cause or an effect of that phenomenon, or is connected with it through some fact of causation." 5. Method of Residues. — " Subtract from any phenome- non such part as previous induction has shown to be the effect of certain antecedents, and the residue of the phe- nomenon is the effect of the remaining antecedents." \\ INDUCTIVE FALLACIES. 109 CHAPTER XVIII. -o- Arguments Similar to Induction. I. Analogy. — In ordinary life we are often as mucli obliged to act upon what is probably true as upon what we know is certainly true. Analogy is a form of reasoning which aims only at giving more or less probable certainty. If we find two things closely resembling each other in certain observed ways, we argue that they ivill jirohahly resemble each other in ways ivhich we have not observed. This is the formula of analogy. Induction argues : " These sequences have been found in some instances, therefore they will be found in all instances." Analogy argues: "These two things resemble each other in certain qualities, therefore they probably resemble each other in other qualities. Some of the planets are known to resemble the earth in certaia respects, therefore they probably resemble the earth in being inhabited." Butler's great work on the " Analogy of Religion " argues that, because nature and revealed religion have many resemblances, there- fore it is j)robable that they have a common Author, Logic lays down the following rules for good analogical reasoning : — 1. The ratio or proportion in mmaber of resemblances must be contrasted with the number of known differ- euces. If the former are many and the latter few the < y analogical conclusion is increased in probability, and vice VeTsd. N.B. — If one of the things about which we are arguing is only little known, the unknown points must be added to the points of difiference in contrast with the resemblances. Thus, the argument about the planets being inhabited is weakened by the fact that we know very little about them. 2. The kind of resembling and differentiating circum- stances must be carefully considered, and the general result compared with what we know of the laws of the universe. Thus, in the case of the planets, we know that life as it exists on the earth can only exist within certain definite limits of temperature and in connection with atmospheric air. Mercury is too hot, Saturn is too cold, whilst the moon has no atmosphere. All the resemblances, therefore, count for nothing when we consider the Iciiid of differences that exist. II. Inductio per enumepationem simplicem. — This argument is a kind of inductive fallacy. It argues that, because a case happens to be true in every histance in our experiGnce, there- fore it is a general law or truth. Before we can assume that a thing is universally true, because we have never known an instance to the contrary, we must have reason to suppose that, if there had been instances to the contrary, we should have heard of them. III. Post hoc, ergo propter hoc— This is the fallacy of Induction. It is the confusion of casual with causal connec- tion, against which the Inductive Canons are designed to guard US. Thus, we have a National Debt and we have national prosperity. We are arguing post hoc ergo propter hoc, if we ascribe the prosperity to the Debt. " After, therefore because of " is the generic name for imperfect proof of causation from observed facts of succession. 110 DEDUCTION AND INDUCTION, EXERCISES. Ill IV. Perfect Induction is the name given to tlie conclusion, when all possible cases have been duly examined, and we have summarised the result in a general proposition. If, however, Induction is defined as an inference from the known to the unknown, it is obvious that "perfect induction" is reaUy no induction at all. Relation of Deduction to Induction. Having thus briefly considered the aim and scope of Deductive and Inductive Logic, it only remains to get into clear perspective the relition between the two. Some modern logicians, seeing the vast practical importance of Inductive Logic as the logic of the physical sciences, have been led to doubt ibe value of Deductive Logic altogether. They argue that the syllogism is only a petitio principii. When we argue that, because aU men are mortal, Socrates, being a man, is mortal, the conclusion was " begged " in the general proposition. But the number and variety of fallacies of deduction which abound in ordinary life, are a sufficient warrant to ensure the study of Deductive Logic a permanent and important place in a liberal education. The simplest way of expressing the relation between the two branches is to consider Inductive Logic as the orderly state- ment of those laws by which we arrive at general conclusions. The general conclusions have then validity, based securely on the principle of the uniformity of nature and the all-pervading law of universal causation. The general conclusions become a sort of memoranda in which our conclusions are expressed. But these memoranda require to be correctly interpreted and reasonably applied to particular cases. This is the proper work of Deductive Logic. Iti short, the one is the counter* vart oj the other. EXERCISES ON INDUCTIVE LOGIC. 1. Explain and illmtrate the difference between the Inductive and Deductive inethods of arriving at truth. Wliat are tJie chief dangers in reasoning from analogy ? Explain and illmtrate tlve terms " reductio ad absurdum" and " begging the qucstiott.** 2. Illustrate tlie statement that in all discoveries of natural science, the processes of induction and deduction folloio each oilier before a complete verification of a law can be obtained. d. Tmt is the exact difference between inductm and deductm reasoning ? Give a simple example of each process in connection with some subject of instruction in an elementary scliool course. 4. Oive some familiar examples of false induction, and say what school exercises are best calculated to encourage a habit of making a true me of the inductive process. 5. Distinguish between analogy and induction, hypothesis and theory. What is needed besides induction for ascertaining scientific truth ? 6. By what i^rocesses of reasoning would you prove that tlie earth IS round, or ilial the room in which you are is not empty, but filled with something I or, by examining a bird that it was an anirnal made to live in the air ? WJmt name ivould you give to tlie process in the last case ? 7. Distingtiish between generalisation and reasoning from analogy, and give an htstance of each. 8. Distinguish between observation and experim,ent, and show how we may learn by experiment what we could not lear^i merely from observation. 9, ''Induction is really tlie inverse process of Deduction' Explain this. 112 EXERCISES. 10. J]/ha,i: is meant hy Induciio per enur.ierallon sinipVicetn ? 11. Why is so called ''perfect induction " not cojisidered a really inductive process ? 12. State exactly tulmt you understand by tlie terms " Cause and Effect;' an I " tJie Plurality of Causes." 13. Whit is the meaning and significance of the principle known as " the Uniformity of Nature " ? 14. Explain the principle of tJie MetJwd of Agreejnent and the Mdhd of Differom rpspcctirnhj, and say to wliat uses ih two metJiods are apjjropiate. INDEX Absolute terms Abstract terras Accent, fallacy of Accident, fal acy of „ Separable & Inseparable Accidents A dicto secundum, quid Affirmative propositions ... Agreement, method of Ambiguity, fallacies of Ambiguity of " all," etc. ... „ „ terras Amphibology, fallacy of ... Analogy Antecedent and consequent Argumentum ad hominem ... Argumsntiim ad verecundum Aristotle's dictum Art and Science Page. ,.. 46 23 24 91 92 82 82 . 92 . 46 . 104 . 90 . 49 26,90 , 90 . 108 83 , 93 98 65 14 93 63 104 29 41 0g 94 88 24 23 91 88 15 62 Begging the question Canons of syllogisma Canons, inductive ... ... Categoreraatlc words Categorical propositions ... Cause and effect ... Circle, arguing in a Classifleation Collective terms Common tenns Compogftion, fallacy of ... Conditional ayllogisms Concept Conclusion of syllogism ... Conoomitant varIatloD8,method of 107 ... 24 ... 43 ... 82 ,.. 102 ... 89 ... 27 ... 24 ... 92 ... 67 ... 63 ... 62 ... 55 ... 44 ... 37 15, 110 Concrete terms Conditional propositions... „ syllogiEms ... Conjecture Conjunctive syllogism Connotation ... ConnotatlvG tenni Consequent ... „ fallacy of tbe Contraposition Contrary opposition Contradictory opposition... Conversion, inferences of... Copula Cross division Deduction Paob. Definition 84 „ ■ of liogio 12 Denotation 28 Description 35 Detection of fallacies 94 Dichotomy 88 Dictum (le omni et nullo 65 Difference, method of 106 Differentia 32 Dilemma 86 Disjunctive propositions 43 „ syllogisms 84 Distributed, meaning of 44 Distribution 48 Division 86 „ fallacy of 91 Dogmatic hypotiesia 102 •-• ••• ■•■ ••• ••• ••• 40 Enthyraene 79 Epicheireraa 80 Episyllogism 80 Equivocal termi 26 Equivocation ... ... ... 90 Euler'8 diagrams 47, 48, 64 Evidence ... ... ... ... 17 Excluded middle, law of 20 Exercises 11, 18, 21, 26, 93, 99, 40, 50, 60, 68, 73, 78, 81, 87, 96, 111, 112 Experiment 101 Extension and Intension ... 27 Externalfallacies 91 Fallacies, detection of 94 Fallacy 88 False cause, fallacy of 92 Figures of the syllogism 69 Formal fallacies 90 Fundamentum diviiionii 87 Genus ^ ... 29 Grammar 17 Hypothesis 102 Hypothetical propositions ... 43 „ syllogism 83 I 4Q Identity, law of ,"„ '.',', ','.'. 20 Ignoratio elenchl 92 Illicit process ... ... ... 65 Inimefliate inference 52 u. INDEX. Page. Induction 16,98,110 Jnductio per enumerationem nm- plicem 109 Inductive Logio, exercises on ... Ill Inferences 13 Inflraa species ... ... ... 80 Inseparable accident 83 Intension 27 Internal fallacies ... ... ... 89 Irrelevant ooncl sion 92 Judgments •Jwn III .11 III III III Laws of thought ... , ... Legitimate bypotbesis, conditions of liOgio, derivation of the word ... Major premiss „ term 13 12 19 108 14 62 62 91 61 89 104 106 oonoonutant variations 107 Many questions, fallacy of Meaiate inference Metaphysics ... Method of agreement difference • •• tu „ „ residues 107 Middle term 62 Minor premiss 62 „ terms 62 Mnemonio lines 76 Modus ponens... ... ... ... 88 „ tollena 88 Moods of the Syllogism 66 Negative and positive terms ... 24 Negative propositions 45 Non-oontradiotion, Law of ... 20 Non aequltur 92 O 46 Observation and experiment . . . 101 Obversion 56 Opposition, Inferences of ... ... 62 Particular propositions 45 Peracddena 55 Perfect figure 74 „ induction 110 Permutation, inferences of ... 56 PlOK. Petitio prineipii 93 Philosophy and its branches ... 7 Polysyllogism 80 Porphyry, tree of 31 Positive terms ... • 24 Post hoc, propter hoe ... ... 109 Predicables 29 Premiss 61 Proper terms 23 Property 82 Proposition 13, 42 Prosyllogism 80 Proximate genus 81 Psychology 9 Qualitjr of propositions 44 Quantity of propositions 44 Mtductio ad ahiurdum 77 Reduction 74 Relation of deduction and induo* tion 110 Relative terms 25 Residues, method of ... ... 107 Rules Of the figures of tbe syUo* gism ... ... ... ... .. 72 Rules of the syllogism ... ... 63 Boienoe 12 Simple apprehension 15 Singular terms 28 Sorites 80 Species 29 Subaltern opposition 53 Sub-contrary opposition 69 Sufficient reason, law of ... ... 21 Sunmumgenui 80 Syllogism 63 Synoategorematic words 23 Terms ; 17,22 Thought la Undistributed, meaning of Uniformity of nature Universal propositions Univooal terms Variations, concomitant, method "' ••< ••• •!• ,,, II, Verbal fallacies 44 98 45 26 107 90 For Certifleate Students, JTulii, 1900. 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