THE SCIENCE OF LOGIC;
OR,
AN ANALYSIS OF THE LAWS OF THOUGHT.
BY REV. ASA MAHAN,
AUTHOR OF AN "INTELLECTUAL PHILOSOPHY,"
"A TREATISE ON THE WILL," ETC.
"Words are things;
A small drop of ink, falling like dew upon a thought,
Produces that which makes thousands, perhaps millions, think."
First published at
NEW YORK:
A. S. BARNES & CO., 51 & 53 JOHN-STREET.
1857.
[Copied with no changes by Rick Friedrich in September 1998.]
PREFACE.
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WHENEVER, in the development of any particular science there has been a misapprehension of its appropriate sphere, and especially when wrong principles have been introduced in development, a reconstruction of the whole science is of course demanded. The following treatise has been prepared in view of the assumption, that both these defects exist in important forms in the common treatises on this subject--treaties in which Dr. Whately's is one of the most prominent representatives. Every one is aware, that any given intellectual process having for its object the establishment of truth, may fail of its end for one or more of the three following reasons:
1. The process may be based throughout upon a misconception of the subject treated of.
2. Invalid premises may be introduced as the basis of conclusions deduced.
3. Or there may be a want of connection between the premises and the conclusions deduced from them.
All are equally aware, also, that every valid process is not only free from each of these defects, but possessed of the opposite excellences. In examining any such process, then, three questions are or should be always put, to wit: Has the author rightly apprehended his subject? Are the premises sound? Is there a valid connection between the premises and conclusions? In answering such questions, everyone feels the need of valid criteria by which he can determine whether the process is or is not valid in each of these particulars, and in one no less than in either of the others. The following treatise has been prepared upon the assumption, that the true and proper sphere of logic is to furnish all these different criteria, and thus to meet in full the real logical necessities of the human mind.
The common treatises are constructed upon the assumption that its true and proper sphere is to meet this want in the last particular only, that is, to furnish criteria by which we can distinguish valid from invalid deductions from given premises, and that irrespective of the character of the premises themselves. If we are right in our assumption--and the question whether we are not right, is fully discussed in the Introduction--then an enlargement of the sphere of the science beyond what is aimed at in ordinary treatises is demanded, and so far the science needs a reconstruction.
All such treatises that we have ever heard of--with one exception, "Thomson's Laws of Thought," which has never been reprinted in this country--have been constructed throughout upon the assumption, that "all negative propositions and no affirmative, distribute the predicate," and that in converting a universal affirmative proposition we must change its form from a universal to a particular; as, "All men are mortal,"--"Some mortal beings are men." Let us now suppose that as far as affirmative propositions are concerned, the above principles hold only in respect to a single class, while, in all other cases, such propositions as well as negative ones do, and from the nature of the relations between the subject and the predicate must, distribute the predicate as well as the subject. In that case undeniably, a reconstruction of the whole syllogism is demanded.
Now the truth of each of the above statements can be rendered demonstrably evident on a moment's reflection. Why is it, that in the proposition, for example, "All men are mortal," the subject only is distributed, and that its converse is, "Some mortal beings are men?" The reason is obvious. The term men represents a species of which the term mortal represents the genus. In other words, the former term represents what is called inferior, and the latter its superior, conception. The term mortal being applicable to a larger number of objects than the term men, must be understood, in the above proposition, as representing only a part of its significates. Such proposition, or course, can be converted, but by limitation, that is, changing its form from a universal to a particular. It is only in reference to this one class of propositions, however, that the principles under consideration do or can hold. When the sphere of the subject and predicate are, from the nature of the terms themselves, equal--as they are, in all cases but in reference to the single class referred to--then affirmative propositions distribute the predicate on the same principles that negative ones do.
We will mention here for illustration but a single class of propositions of this kind--the mathematical. In every universal affirmative proposition throughout the entire range of this science, the predicate as well as the subject is distributed; the converse as well as the exposita being universal also. This holds equally in regard to the principles and subsequent deductions of the this science. What is the converse, for example, of such propositions as the following? "Things equal to the same things are to one another,"--"The square of the hypothenuse of a right-angled triangle is equal to the sum of the squares of the two sides,"--"6 + 4 =10,"--"X = Z," &c.? The whole science of logic has been constructed upon principles of distribution and conversion, which would utterly mislead us, if applied to any of the universal affirmative propositions throughout the entire range of the science of the mathematics, or to any propositions but one of the single class above named.
In respect to the different figures of the syllogism, also, it has been laid down as holding universally, that the second yields only negative, and the third only particular, conclusions. This also holds true when, and only when, the propositions belong to the single class above named. In all other cases, we can obtain universal affimative or negative conclutions, in each and all The figures alike. Take the following for examples:
FIG. I. FIG. II. FIG. III.
M = X; X = M; M = X;
Z = M; Z = M; M = Z;
.: Z = X. .: Z = M. .: Z = X.
Every one will perceive at once that each of the above syllogisms is of equal validity, and that the converse of the conclusion is in each case universal, as well as the exposita.
The dictum, too, under which the syllogism has been constructed will be found to be applicable only to arguments constructed entirely from the single class of propositions named. These facts being undeniable, every one will perceive that science demands a reconstruction of the syllogism throughout. This we have attempted to do, and trust we have accomplished to the satisfaction of all who shall acquaint themselves with the following treatise. Before venturing to give our deductions in the important particulars now before us to the public, we submitted them to numbers of scientific men in whose judgment we have great confidence. From these we have received such expressions of approbation as to inspire us with the assurance, that these deductions will stand the test of the most rigid scientific scrutiny, which is most cordially invited.
The doctrine of fallacies, treated of in Part II., we have aimed to simplify by proper definitions, logical division, and arrangement of the whole subject, so as to render the doctrine luminous throughout and its principles of ready application in the reader's mind.
Almost no portion of the teatise does the author regard as of higher importance than the doctrine of method, as elucidated in Part III. We judge that the public will perceive that an important scientific want is there met.
In furnishing the examples presented in Part IV. we have had two special objects in view--to present fundamental suggestions in regard to important questions in science; and to furnish examples for criticism of corresponding importance. If, in any case or in all cases, it should turn out that we have erred in reasoning or in any other particular, and the error shall be discovered by the application of the principles previously elucidated, the great end of the work is answered, and the examples will still have their proper place in the work, just as they would if cited from another author as examples of fallacy in reasoning, or of error or defect in any other particular.
In the perusal of the following treatise the public will perceive that we are much indebted to three authors--Mr. Thomson, whose work we had never seen till we had progressed in our own to the very place where important citations from his first appear--Kant, whose treatise, in our judgment, excels by far in important respects any other that we have met with--and Sir William Hamilton, to whom the science of logic, and the author of this treatise especially, is more indebted than to any other author--the father of the science, of course, excepted. It is with the utmost gratification that we would record the fact, that in almost every particular in which we have departed from the beaten track in the development of the science, we are sustained throughout by such high authority as Sir William Hamilton. With these suggestions, the following treatise is commended to the careful examination and candid criticism of the public.
CONTENTS.
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INTRODUCTION.
Necessity of a correct definition of Logic.
All things occur according to rules.
Logic defined.
Relations of Logic to other sciences.
The idea of Logic developed in a form still more clear and distinct.
Divisions of Logic.
Correctness of the above definition verified.
Logic as distinguished from Esthetics.
Accordance of the above conception of Logic with that given by Kant.
Accordance of the above idea of Logic with that set forth by Sir William Hamilton.
Inadequate and false conceptions of this science.
1. The syllogistic idea.
2. Conceptions of Dr. Whately and others.
3. The idea that "the adequate object of Logic is language."
General division of topics.
PART I.--THE ANALYTIC.
CHAPTER I.--ANALYTIC OF CONCEPTIONS AND TERMS.
SECTION I.---Of Conceptions.
Conceptions defined.
Origin and constituent elements of Conceptions.
Error commences, not with Intuitions, but Conceptions.
Universal characteristics of all valid and invalid Conceptions.
Spontaneous and Reflective Conceptions.
First and second Conceptions.
Matter and sphere of Conceptions.
Individual, generic or generical, and specific or specificial Conceptions.
Highest genus and lowest species.
Empirical and rational Conceptions.
Presentative and representative Conceptions.
Abstract and concrete Conceptions.
Positive, privative, and negative Conceptions.
Conceptions classed under the principle of unity, plurality, and totality.
Inferior and superior Conceptions.
Concrete and characteristic Conceptions.
Laws of thought pertaining to the validity of Conceptions.
Particular, general, and abstract Conceptions.
Individual, specificial, and generical Conceptions.
Presentative and representative Conceptions.
Concrete and characteristic Conceptions.
Inferior and superior Conceptions.
Empirical and rational Conceptions.
SECTION II.---Of Terms.
Singular and common Terms.---Significates.
Relations of Logic to Terms.
CHAPTER II.--OF JUDGMENTS
SECTION I.---Of Judgments considered as Mental States.
Matter and form of Judgments.
Quantity of Judgments, universal, particular, individual or singular.
Quality of Judgments, affirmative, negative, indefinite.
Relation of Judgments, categorical, hypothetical, and disjunctive.
Remarks on these Judgments.
Categorical Judgments.
Disjunctive Judgments.
Modality of Judgments, problematical, assertative, contingent, necessary (appodictical)--Remarks.
Theoretical and practical Judgments.
Demonstrable, and indemonstrable or intuitive Judgments.
Analytical and synthetical Judgments.
Criteria of all first Truths.
Kant's definition of analytical and synthetical Judgments.
Tautological, identical, and implied Judgments.
Axioms, Postulates, Problems, and Theorems.
Corollarys, Lemmas, and scholia.
Criteria of Judgments, or characteristics of all valid Judgments.
General Criteria.
Particular and special Criteria.
Judgments relative to all valid Conceptions.
Individual (single), Particular, and Universal Judgments.
Individual Judgments (affirmative).
Individual Judgments (negative).
Particular (plurative) Judgments.
Universal Judgments (affirmative).
Universal Judgments (negative).
Judgments pertaining to the object of inferior and superior Conceptions.
Judgments pertaining to the objects of characteristic Conceptions (affrimative).
Judgments relative to objects of characteristic Conceptions (negative).
Hypothetical Judgments.
Hypothetical Judgments classed.
Criteria of such Judgments.
Disjunctive Judgments.
SECTION II.--Of Propositions.
Quality and Quantity of Propositions, &c.
Distribution of Terms.
Of Opposition.
Of the Conversion of the Predicate.
Quantification of the Predicate.
Parti-partial Negation.
Criteria by which Propositions properly falling under these different classes may be distinguished from each other.
CHAPTER III.--ANALYTIC OF ARGUMENTS OR SYLLOGISMS.
SECTION I.--Argument defined and elucidated.
Diverse Forms of the Syllogism.
SECTION II.--The Analytic and Synthetic Syllogism.
These distinct forms of the Syllogism elucidated.
SECTION III.--Figured and Unfigured Syllogisms.
Principles and Laws of the Unfigured Syllogism.
The Canon of this Syllogism.
General Remarks upon this form of the Syllogism.
SECTION IV.--The Firgured Syllogism.
This form defined.
Common assumption on the subject.
Influence of Assumptions.
Principles determining the distribution of the Predicate.
Fundamental mistake in developing the science of Logic.
Division of the present subject.
I. Those forms of the Syllogism which have been commonly treated of as including all forms of the categorical argument, to wit: those forms in which the terms employed are related to each other as Inferior and Superior Conceptions.
Preliminary Remarks upon this Form of the Figured Syllogism.
Only proximate conclusions obtained.
1. The principle of Extension and Intension, or of Breadth and Depth, as applied to the Syllogism.
2. Import of Judgments (Extension and Intension--Naming).
3. Direct and indirect conclusion.
4. Character of all the propositions employed in this form of the Syllogism.
Letters to be employed.
Canon and Laws of this Form of the Syllogism--Conditions on which we can obtain the different classes of Conclusions above named; that is, A, I, E, O.
Universal Affirmative Conclusions.
Universal Negative Conclusions.
Particular Affirmative Conclusions.
Particular Negative Conclusions.
All valid Conclusions deduced upon principles which accord with those above elucidated.
Analysis of the above relations.
The Canon of this Syllogsim.
Moods of the Syllogism.
Figure of the Syllogism--Form defined.
Number of the figures of the Syllogism.
Major and Minor Terms and Premises.
Order of the Premises.
Final abolishment of the Fourth Figure.
Opinions of Logicians upon the subject.
Our Reasons for the abolitions of this Figure.
Special Characteristics and Canon of each of the three Figures.
Figure I.
The Canon illustrated.
Figure II.
Canon of this figure.
Figure III.
Canon of this figure.
Absurdity of reducing the Syllogism of the other Figures to the first.
Nature of the Conclusions obtained in this form of the Syllogism.
Kind of arguments which appropriately belong to the different Figures.
A more brief view of this subject.
A scientific determination of the real number of Legitimate Moods in this form of the Syllogism.
Conditions of valid deductions of any kind in this form of the Syllogism.
Universal affirmative conclusions.
Particular affrimative conclusions.
PART II.--THE DIALECTIC, OR DOCTRINE OF FALLACIES.
CHAPTER I.--INVALID CONCEPTIONS.
CHAPTER II.--THE DIALECTIC--INVALID JUDGMENTS.
CHAPTER III.--THE DIALECTIC--FALLICIES OF REASONING.
PART III.--THE DOCTRINE OF METHOD.
PART IV.--APPLIED LOGIC.
INTRODUCTION.
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Necessity of a correct definition of Logic
Every science has a sphere peculiar to itself. Its end or aim also, in the occupancy of that sphere, is equally special and peculiar. The mathematics, for example, have an exclusive sphere, end, and aim, and metaphysics others equally special and exclusive. To enter intelligently and with the rational hope of the highest profit, upon the study of any particular science, its peculiar sphere, and special aim in the occupancy of the same, must be distinctly apprehended. Now while the sphere and aim of most of the sciences have been definitely determined, the opposite is most strikingly true in regard to logic. It would be difficult to name any two philosophers, with the exception perhaps of Kant and Sir William Hamilton, who fully agree in their ideas and definitions of this science.
By some it is defined as the art, by others as the science and by others still, as "the science and art of thought, and not the laws of reasoning, constitute the adequate object of the science." This definition, as the reader will readily perceive, is really identical with the following given by Kant: "This science of the necessary laws of the understanding and of reason in general, or of (what amounts to the same thing) the mere form (laws) of thinking in general, we name logic." These last two definitions, as we apprehend them, we regard as strictly correct, and as presenting the only true and adequate conception of the proper sphere and aim of the science. We will now proceed to elucidate the above definitions as we understand them, and to do so by giving our own independent definition of the science. As preparatory to this end, we would invite special attention to the following extract form our own work on Intellectual Philosophy.
"All things occur according to rules.
"'Every thing in nature,' says Kant, and this is one of his most important thoughts, 'as well in the inanimate as in the animate world, happens, or is done, according to rules, though we do not know them. Water falls according to the laws of gravitation, and the motion of walking is performed by animals according to rules. The fish in the water, the bird in the air, move according to rules.'
"Again: 'There is nowhere any want of rule. When we think we find that want, we can only say that, in this case, the rules are unknown to us.'
"The exercise of our intelligence is not an exception to the above remark. When we speak, our language is thrown into harmony with rules, to which we conform without, in most instances, a reflective consciousness of their existence. Grammar in nothing but a systematic development of these rules. So also, when we judge a proposition to be true or false, or to be proved or disproved, by a particular process of argumentation, or when we attempt to present to ourselves, for self-satisfaction, or to others for the purpose of convincing them, the grounds of our own convictions--that is, when we reason, our intelligence proceeds according to fixed rules. When we have judged or reasoned correctly, we find ourselves able, on reflection, to develop the rules in conformity to which we judged and reasoned, without a distinct consciousness of the fact. In the light of these rules we are then able to detect the reason and grounds of fallacious judgments and reasonings.
"Logic defined.
"The above remarks have prepared the way for a distinct statement of the true conception of logic. It is a systematic development of those rules in conformity to which the universal intelligence acts, in judging and reasoning. Logic, according to this conception, would naturally divide itself into tow parts--a development of those rules to which the intelligence conforms in all acts of correct judgment and reasoning, and a development of those principles by which false judgments may be distinguished from the true. A treatise on logic, in which the laws of judging and reasoning are evolved in strict conformity to the above conception, would realize the idea of science, as far as this subject is concerned. Logic, to judging and reasoning, is what grammar is to speaking and writing. Logic pertains not at all to the particular objects about which the intelligence is, from time to time, employed, but to the rules or laws in conformity to which it does act, whatever the objects may be.
"Relations of Logic to other sciences.
"In the chronological order of intellectual procedure, logic is preceded by judging and reasoning, just as speaking and writing precede grammar. In the logical order, however, it is the antecedent of all other sciences. In all sciences the intelligence, from given data, judges in regard to truths resulting from such data: we also reason from such data for the establishment of such truths. Logic develops the laws of thought which govern the action of the intelligence in all such procedures. As a science, it is distinct from all other sciences. Yet, it permeates them all, giving laws to the intelligence in all its judgments and reasonings, whatever the objects may be about which it is employed."
The idea of Logic developed in a form still more clear and distinct.
It will readily be perceived, we judge, that the above definitions and statements have made a somewhat near approach, to say the least, to the true idea of the science under consideration. To place the subject in a light still more clear and distinct, we would observe, that there are certain cognitions, certain processes of thought, which are universally regarded as valid for the truth of what is therein referred to. We examine, for example, the process of thought (statements and demonstrations) by which we are conducted to the conclusion, that the square of the hypothenuse of a right-angled triangle, is equal to the sum of the squares of its two sides. We affirm that, on account of what is contained in said process, that proposition is to be held as true; in other words, the process itself is valid for the truth of what is therein referred to. On the other hand, there are other processes which are not thus valid. What is true is sometimes professedly established by processes not at all valid for its reality, and through other processes what is not true is often affirmed to have been established as a reality. All processes of the first class are held as valid, and the two last named are regarded as invalid procedures of the intelligence. In each process alike, the valid, as well as the invalid, the intelligence has acted in accordance with certain fixed laws or principles, which we are able to determine. To develop, that is, determine, define, and elucidate these laws, and thus present universal criteria of valid and invalid procedures of the intelligence, when the object of such procedure is truth, is, as we understand the subject, the true and exclusive sphere and aim of logic as a science.
Divisions of Logic.
Logic, as a science, consequently divides itself into two parts:
1. A systematic development of those principles or laws to which the intelligence accords in all valid intellectual processes, processes whose object is truth.
2. A similar development of those principles to which the intelligence conforms, in all invalid processes of the class under consideration. Such is logic as a science, in the sense in which we understand the subject and in which we shall attempt to realize the idea. No one will dissent from the above conception, but upon a single assumption, to wit, that the sphere assigned to the science is too extensive, that sphere including all that has been commonly referred to the science and some things else supposed not to pertain to it. That this is the true and proper sphere of the science, we argue form the following considerations.
Correctness of the above definition verified.
1. The above definition gives a perfect unity and definiteness to our conceptions of the science, the very unity and definiteness which characterize all correct definitions of any other science. The truth of this statement is self-evident.
2. While the sphere here assigned to the science possesses not only perfect unity and definiteness, but also exclusiveness, occupying no department properly pertaining to any other science, it also has throughout a fixed and definite relation to all the other sciences, that is, it is what the science of logic should be, the true and proper antecedent to them all. It does not profess to teach what is true or what is false, in any sphere occupied by any one of the sciences; but it does aim to develop those laws and principles, by which we can determine whether any given procedure in the development of any of the sciences, is or is not valid for the truth of what is referred to in such process, and why such procedure is or is not thus valid. This is precisely what no one of the science professes, or aims, in any of its appropriate departments, to accomplish. Yet what this science aims to accomplish, is just what is needed, in all the sciences alike, in all intellectual processes having truth for their object and aim. We certainly need criteria by which valid processes may, in all cases, be determined and distinguished form those which are not valid. Hence we remark,
3. That this idea when realized meets a fundamental want of universal mind, a necessity which no other science does or can meet. The navigator, when abroad upon the ocean, no more needs tables and instruments by which he can determine his latitude and longitude, than does universal mind, educated mind especially, criteria by which it can judge correctly of the character of its own intellectual processes. Logic, as now defined, aims to meet this universal want, and when realized, does most fully and perfectly meet it. When its sphere is contracted within narrower limits than is here assigned to it, a fundamental want of universal mind is so far left unmet, and that when we gave no science, which, while moving in its proper sphere, does or can meet that want.
4. No adequate reason can be assigned, why any department of the sphere of this science, as above defined, should be assigned to logic, and any other department excluded from it. Nor can any other science be named to which the department excluded, can properly be assigned. We might, with the same propriety, include the latter department in our definition of the science and exclude the latter.
5. All treaties, or most, at least, attempt to realize the full idea of the science, as above defined, though not unfrequently in palpable contradiction to the fixed aim of the science, as previously defined in such treaties. The science is sometimes so defined, for example, that the only fallacies properly falling under its cognizance, are those belonging to one class exclusively, to wit, inferences deduced from premises whether true or false, with which they (the premises) have no logical connection. Yet, when such treatises come to treat of fallacies, they discuss not only this, but every other class of fallacies, and attempt to give us universal criteria by which valid intellectual processes may be distinguished from those which are not valid, the very sphere and aim of logic, as above defined. Hence in these illogical treatises, fallacies are discussed under three classes--the strictly logical, that is, those which fall within the proper sphere and cognizance of logic, as defined--the semi-logical, those which partly do, and partly do not, belong to the defined sphere of logic--and the non-logical, those that logic, as defined, has no business with whatever. It is just as wide a departure from all true principles of scientific procedure, to treat of non-logical fallacies, in a treatise on logic, as it would to include a treatise of arithmetic in a system of geometry. All fallacies are really and truly logical fallacies, or only a certain class of them should be discussed in a treatise on logic.
Logic as distinguished from Esthetics.
It may do something to render still more distinct and definite our conceptions of this science to compare its sphere and aim with those of another, the science of esthetics. This last has been commonly defined as the science of the beautiful in nature and art. As pertaining to mind, its appropriate sphere is the creations of the imagination, the object of which is to blend the elements of thought, not in harmony with things as they are, but with the ideas of beauty, grandeur, sublimity, perfection, &c. Esthetics, as a science, aims to develop those laws and principles in conformity to which this faculty must act, in order to realize the end referred to, to show what kind of elements must be blended into a given conception, and how they must be blended, so as to realize these ideas. Thus it presents criteria by which we can distinguish the truly beautiful from that which is not, in other words, the valid from the invalid procedures of the imagination.
The true and proper aim of the understanding and judgment, on the other hand, is to blend the elements of thought given by the primary faculties into conceptions and judgments in harmony with things, not as they might or should exist, but as they do exist. Logic aims to give those criteria by which we can distinguish those procedures of these faculties which are to be held as valid for realities, from those which are to be held as not thus valid. Esthetics might, with some approach to truth, be defined as the logic of the imagination, while logic proper has for its sphere the procedures of the understanding and judgment, in all processes the aim of which is to realize in processes of intuition, conception, judgment, and reasoning, the idea of truth.
Accordance of the above conception of Logic with that given by Kant.
The perfect accordance, in all essential particulars, of the conception of logic above developed, with that given by Kant, will appear manifest to all who are acquainted with his treatise on this science. To evince that accordance, we need only, cite the following passages from that treatise: "In logic we want to know," he says, "not how the understanding is and thinks, and how it has hitherto proceeded in thinking, but how it shall proceed. It is to teach the right use of the understanding," &c. Further on, after giving precisely similar distinctions between esthetics and logic that we have done, he presents the following division of the latter science, a division which must have its exclusive basis in a conception of the science strictly identical, in all essential particulars, if not in all others, with that which we have given: "We shall consequently have two parts of logic: the analytic, which propounds the formal criteria of truth; and the dialectic, which comprises the marks and the rules, by which we can know, that something does not agree with them. In this sense the dialectic would be of great use as a cathartic of the understanding." He then goes on to show that all other conceptions of the science not accordant with this are "improper" and "wrong."
Accordance of the above idea of Logic with that set forth by Sir William Hamilton.
In connection with the fact that Sir William Hamilton accords in general with conception of logic as given by Kant, the accordance of the idea of the former with that which we have presented will be made sufficiently manifest through the following paragraph selected from his article on Logic, found in his Discussion on Philosophy and Literature, p. 136, as published by the Harpers:
"We shall not dwell on what we conceive a very partial conception of the science, that Dr. Whately makes the process of reasoning not merely its principle, but even its adequate object, those of simple apprehension and judgment being considered not in themselves as constituent elements of thought, but simply as subordinate to argumentation. In this view logic is made controvertible with syllogistic. This view, which may be allowed in so far as it applies to the logic contained in the Aristotelic treatises now extant, was held by several of the Arabian schoolmen; borrowed from them by the Oxford Crackenthrope, it was adopted by Wallis; and from Wallis it passed to Dr. Whately. But, as applied to logic, in its own nature, this opinion has been long rejected, on grounds superfluously conclusive, by the immense majority even of the peripatetic dialecticians; and not a single reason has been alleged by Dr. Whately to induce us to waver in our belief, that the laws of thought, and the laws of reasoning, constitute the adequate object of the science. This error, which we cannot now refute, would, however, be of comparatively little consequence, did it not--as is notoriously the case, in Dr. Whately's Elements--induce a perfunctory consideration of the laws of those faculties of thought; these being viewed as only subsidiary to the process of reasoning."
The object of logic, we repeat, is not to reveal or affirm what is true or what is false in itself, that being the exclusive province of the various special departments of mental operation. Its exclusive object, on the other hand, is to develop and elucidate those laws of thought by which we can determine whether any given intellectual process, whatever its object may be, a process which professedly reveals and establishes the truth in respect to the object to which it pertains, is or is not valid for its truth, and why it is to be held as thus valid or not valid.
Inadequate and false conceptions of this science.
It will add somewhat to the distinctness and definiteness of our conceptions of this science, to compare the conceptions which we have set forth, with certain others which we regard as inadequate or wrong. Among these the following only demand special notice.
The syllogistic idea.
The first which we adduce is what may not inappropriately be denominated the syllogistic idea, that which affirms that the exclusive object of this science is to develop the laws of reasoning, that is, to state what, in a process of reasoning, are and must be the relations between the premises and conclusion, when the latter does or does not necessarily follow from the former. A very few considerations only are requisite to show how fundamentally inadequate this idea is to represent the true and appropriate sphere of this science. Take, as examples, the following syllogisms:
All men are mortal;
George is a man;
Therefore, he is mortal.
The conclusion, in this instance, is not only true, but it results as a necessary deduction from the premises. Take now another of a different character:
All mortal beings are men;
Every brute is a mortal being;
Therefore, ever brute is a man.
Here we have a false conclusion. It has the same necessary logical connection with the premises, however, that the conclusion of the former syllogism has. Again:
All bipeds are mortal;
All men are mortal;
Therefore, all men are bipeds.
In this case a true conclusion is deduced from premises with which it has no logical connection. Further:
All mortal beings are men;
All brutes are men;
Therefore, all brutes are mortal beings.
Here, also, we have a conclusion which is true in itself, but which is deduced from premises, both of which are false, and with which it has no logical connection. Again:
All animals are mortal;
All men are mortal;
Therefore, all men are animals.
In this syllogism, all the propositions are true; but the conclusion has no logical connection with the premises from which it is deduced. Once more:
All mortal beings are men;
George is a mortal being;
Therefore, he is a man.
The conclusion in this case is true, and is necessarily connected with the premises. Still there is a fallacy in the argument, one premise being false.
We have in the five last syllogisms, five different kinds of fallacies, and it would seem that the science of logic ought to give us principles by which we can determine, in each case alike, what is the nature and character of the fallacy, and why it is to be regarded as such. Yet with the first and last of the five, logic, according to the present definition, has nothing whatever to do. There being, in these cases, a necessary connection between the premises and conclusion, every condition required by the science has been fulfilled, and its mission is at an end in respect to them. At the same time, we have no other science to which it pertains to trace out the source of the fallacy in either case, and tell us where it may be found, and why it should be regarded as a fallacy. Numbers three, and four, and five, only, are logical fallacies, according to this definition, and would properly be designated as fallacies in reasoning by the science, as thus defined.
Of the six syllogisms, in three of them, numbers one, two, and six, the conclusions have a necessary connection with the premises, and the argument throughout, in each case, alike fulfills all the conditions of the science, as now defined: in the other three, though in the last two of them the intellectual procedure is fundamentally fallacious, and the propositions all true in the first, the whole of these syllogisms, we say must be classed together under the same category in a treatise upon this science, a treatise developed in strict consistency with such an idea of its exclusive sphere and design. Now we affirm that logic, when developed according to the true conception of its entire and proper domain and adequate aims as a science, will not thus confound things which so fundamentally differ.
In numbers one and two, each conclusion has the same necessary connection with its premises, yet the process of thought is in the first case valid for the truth of the conclusion, and not valid in the last. In the last four syllogisms, there is the same want of validity, whether the conclusion is true or false. Suppose we ask for the reason or grounds of the difference. To answer such an inquiry our investigations must, in every case take a wider range than the mere consideration of the logical connection between the premises and the conclusion, and must in all instances take into account the conceptions represented by the various terms of the syllogisms, the judgments represented in the propositions of which the syllogisms are constituted, and the connections between the premises and the conclusion in the same.
We will take the first syllogism in illustration. In this syllogism there are three conceptions represented by the terms men, mortal, and George. On examination they will be found to possess certain fundamental characteristics common to all others which appear in judgments really and truly valid for the reality and character of the objects to which they pertain, and which consequently distinguish all conceptions which must be held as true from those which must not, as elements of such judgments, be thus held. Relations equally fundamental and peculiar will be found to obtain between the subject and predicate in each of the premises of such a syllogism, and also between the premises themselves and the conclusion deduced from them. The characteristics of the conceptions, on the one hand, and those of the relations between the subject and predicate in each of the premises, and between said premises and the conclusion deduced from them, on the other, characteristics and relations which may be determined and defined, constitute the laws of thought by which all valid judgments and processes of reasoning may be distinguished from those which are not valid, inasmuch as all valid processes do and must possess throughout these identical characteristics, and all not valid must be thus regarded, for the reason that they violate these rules in some particular or other, some in the relations affirmed to exist between the subject and predicate in one or the other of the premises, or in both together, and others because they are constituted of invalid conceptions.
Now why should it be affirmed that one class of these laws of thought come within the appropriate sphere of logic, and that either of the others should be excluded from it? No reason whatever can be assigned for such an assumption. If any individual should accomplish what is needed in regard to any one of these departments, the relations between the premises and conclusion in processes of reasoning, for example, he would so far meet one important logical demand of universal mind. If, when he has done thus much, he should put forward the claim, that he has occupied the entire sphere of the science of logic, he would simply reveal the fact that he entertains too limited conceptions of that science.
Conceptions, judgments, and deductions from judgments presented as premises, these together, we repeat, constitute the proper sphere and object of this science. Its object is to develop and elucidate those laws of thought by which valid conceptions, valid judgments, and valid deductions, can be distinguished from those which are not valid, and by which it can be shown in what respects and for what reasons any given intellectual process is or is not thus valid.
Conceptions of Dr. Whately and others.
"Logic," says Dr. Whately, and we will give the definition in full, "in the most extensive sense which the name can with propriety be made to bear, may be considered as the science, and also as the art, of reasoning. It investigates the principles on which argumentation is conducted, and furnishes rules to secure the mind from error in its deductions. Its most appropriate office, however, is that of instituting an analysis of the process of the mind in reasoning; and in this point of view, it is, as has been state, strictly a science; while, considered in reference to the practical rules above-mentioned, it may be called the art of reasoning. This distinction, as will hereafter appear, has been overlooked, or not clearly pointed out by most writers on the subject; logic having been in general regarded as merely an art; and its claim to hold a place among the sciences having been expressly denied."
In the above paragraph there are, as shown most indubitably by Sir William Hamilton, at least three important errors.
The first that we notice is an historical one, the statement, that logicians have generally considered logic as an art, and not a science, whereas in the language of the author just named, "the great majority of logicians have regarded logic as a science, and expressly denied it to be an art. This is the oldest as well as the most general opinion."
The next error that we notice pertains to the nature of logic itself. It is in fact in no proper sense and art of reasoning, its fundamental aim, as far as reasoning is concerned, being not to teach us how to reason, but to enable us to judge, upon scientific principles, of processes of reasoning. We all know that an individual may be an excellent and scientific judge of processes of reasoning, and practically a very bad reasoner. Yet science tends to render practice more perfect. In this indirect and secondary sense logic is an art of reasoning.
The third and last error that we notice, is that of a too limited and inadequate conception of the true sphere and consequent full aim of the science. The error to which we now refer, consists in this supposition that the laws of reasoning, instead of the laws of thought, constitute the real sphere and object of the science. This error we have already exposed in another connection. Nothing in addition is therefore required on the subject.
The idea that "the adequate object of Logic is language."
As Dr. Whately proceeds in his elucidation of what he regards as the true and proper conception of this science, he has fallen into another important error, an error which has been so fully and so well exposed by Sir William Hamilton, that we will simply present his statement of it together with his exposition of the same, without any additional remarks of our own:
"But Dr. Whately is not only ambiguous; he is contradictory. We have seen that, in some places, he makes the process of reasoning the adequate object of logic; what shall we think, when we find, that, in others, he states that the total or adequate object of logic is language? But, as there cannot be two adequate objects, and as language and the operation of reasoning are not the same, there is, therefore, a contradiction. 'In introducing,' he says, 'the mention of language, previously to the definition of logic, I have departed from established practice, in order that it may be clearly understood, that logic is entirely conversant about language; a truth which most writers on the subject, if indeed they were fully aware of it themselves, have certainly not taken due care to impress on their readers' (p. 56). And again: 'Logic is wholly concerned in the use of language' (p. 74).
"The term logic (as also dialectic) is of ambiguous derivation. It may either be derived from logoV (endiaqetos), reason, or our intellectual faculties in general; or, from logoV (proforicos), speech or language, by which these are expressed. The science of logic may, in like manner, be viewed either--1. As adequately and essentially conversant about the former (the internal logoV, verbum mentale), and partially and accidentally, about the latter (the external logoV, verbum oris); or, 2. As adequately and essentially conversant about the latter, partially and accidentally about the former.
"The first opinion has been held by the great majority of logicians, ancient and modern. The second, of which some traces may be found in the Greek commentators of Aristotle, and in the more ancient Nominalists, during the middle ages (for the later scholastic Nominalists, to whom this doctrine is generally, but falsely attributed, held in reality the former opinion), was only fully developed in modern times by philosophers, of whom Hobbs may be regarded as principal. In making the analysis of the operation of reasoning the appropriate office of logic, Dr. Whately adopts the first of these opinions; in making logic entirely conversant about language, he adopts the second. We can hardly, however, believe that he seriously entertained this last. It is expressly contradicted by Aristotle (Analyt. Part i. 10, par. 7). It involves a psychological hypothesis in regard to the absolute dependence of the mental faculties on language, once and again refuted, which we are confident that Dr. Whately never could sanction; and, finally, it is at variance with sundry passages of the Elements, where a doctrine apparently very different is advanced. But, be his doctrine what it may, precision and perspicuity are not the qualities we should think of applying to it."
General division of topics.
We have now sufficiently indicated our own conception of the science under consideration. The way has thus been prepared to enter intelligently upon the elucidation of the different departments of our subject, which we shall treat of under the following general arrangement of topics:
I. The necessary laws of thought to which the intelligence does and must conform in all valid conceptions, judgments, and deductions, or processes of reasoning. This department of the science is denominated by Kant, the Analytic. For the sake of convenience we shall include what we have to say on this topic, under this same general title.
II. The doctrine of fallacies which the philosopher just named denominates the Dialectic, and which we shall attempt to elucidate under the same title.
III. The doctrine of Method, or the rules in conformity to which all scientific procedures should be conducted.
IV. Certain general and specific applications of the principles elucidated, applications adduced for the purpose of exemplifying the importance of the science, and the manner of applying its principles.
The first two topics embrace the entire field of logic considered as a science. The last two are presented for the purpose of elucidation.
LOGIC
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PART I.
THE ANALYTIC.
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CHAPTER I.
ANALYTIC OF CONCEPTIONS AND TERMS.
SECTION I.---OF CONCEPTIONS.
Conceptions defined.
A CONCEPTION, or notion, is a mental apprehension of some object or objects, and apprehension which we express by such terms as George, man, tree, plant, animal, &c. Such apprehensions we represent by the general term conception.
Origin and constituent elements of Conceptions.
Knowledge, with the human intelligence, begins not with conceptions but with intuitions, or a direct and immediate perception of the reality of objects. As shown in the Intellectual Philosophy (A System of Intellectual Philosophy, by Rev. Asa Mahan, pp. 476. New York, A. S. Barnes & Co.), and as now generally admitted by philosophers, the faculties of intuition, or original perception, are three,---SENSE, the faculty of external perception, the faculty which perceives the qualities of external material substances--CONSCIOUSNESS, the faculty of internal perception, the faculty which perceives and apprehends the operations or phenomena of the mind itself--and REASON, which apprehends the logical antecedents of phenomena perceived by Sense and consciousness, to wit, truths necessary and universal, such as space, time, substance, cause, personal identity, the infinite, &c.
In intuition each particular quality or phenomenon, together with its logical antecedent, is given singly and by itself. From the nature of the case, it cannot be otherwise, the quality being, in all instances, the object of direct and immediate perception or apprehension. By this we would not be understood as affirming that different qualities may not each be the object of simultaneous perception with others. This we believe. Yet, as each quality is itself individual and single, and is the object of direct and immediate perception, such quality must be originally given singly and by itself. The same holds true of the logical antecedent of such quality, as given by reason. Each quality has its special logical antecedent; and as the quality is originally given singly and by itself, the same must be held equally true of its logical antecedent. The logical antecedent of the reality of the quality of extension, for example, is that of an extended substance, quality necessarily supposing as the condition of its existence, the reality of substance, it being impossible to conceive of the reality of the former, without supposing that of the latter. The same holds true of all other qualities, or phenomena, of every kind.
The origin and constituent elements of conceptions of every kind now admit of a ready statement and explanation. When a quality is perceived, and its logical antecedent apprehended, we have a secondary operation of the intelligence, an operation in which the apprehension of the quality and that of its logical antecedent are united into a conception of a particular object. As other qualities of the same object together with their logical antecedents are perceived and apprehended, they are blended into the same conception, which thus becomes more or less complete, as it more or less fully represents its object. Thus if the object is material, for example, a conception of it is formed as a body existing in time and space, and having definite extension, form, color, &c. On the perception of subjective phenomena, we obtain, in a similar manner, the conception of mind, as a substance possessing the powers and susceptibility of thought, feeling, and voluntary determination. All the elements which do or can enter into conceptions must be given by the primary faculties referred to, as these are the only original sources of cognition. The function which thus blends the original elements of thought (intuitions) into conceptions, is denominated the understanding; and logic, so far as it pertains to conceptions, is the science of the laws of the understanding.
Error commences, not with Intuitions, but Conceptions.
As intuition, in all instances, pertains directly, immediately, and singly to its respective object, the fact of such intuitive perception must always be held as valid for the reality of its object. A denial of this principle is a formal impeachment of the validity of the intelligence, as a faculty of knowledge, and nullifies all attempts at knowledge of every kind. All forms of scientific procedure also have their basis in the assumed truth of this principle, the validity of intuition for the reality of its objects. Nor can any reasons be assigned for the assumption that any one class of intuitions should be regarded as thus valid, and others not. No principles, therefore, are required to enable us to distinguish valid from invalid intuitions.
One universal division of conceptions, however, is that of true and false. Here valid and invalid cognitions first appear in the process of thought, and hence the necessity of valid criteria by which the one class may be distinguished from the other.
Universal characteristics of all valid and invalid Conceptions.
The universal characteristics which distinguish all conceptions which should be held as valid for the reality and character of their respective objects, from conceptions which should not be thus held, may now be very readily and distinctly pointed out.
1. All conception which embrace those elements only, which have been really and truly given by intuition relatively to any object, must be held as valid throughout for the reality and character of such object.
2. All conceptions also must be held as thus valid which embrace such intuitions exclusively, together with their necessary logical antecedents. If the intuition is thus valid, so must all its necessary logical antecedents and consequents be. Of this there can be no doubt.
3. All conceptions, on the other hand, which embrace any elements not thus given in respect to the objects of said conceptions, must be held as not valid for such objects.
The truth of the above principles is self-evident. The only question to be determined is, how may we know when a given conception has one or the other of the above characteristics? To accomplish this end is the object of the following distinctions and elucidations.
Spontaneous and Reflective Conceptions.
There are two states in which each conception may be contemplated--to wit, as it first appears in the intelligence through the spontaneous action of the understanding; and as it appears when each element embraced in it has been the object of the distinct reflection, and the entire conception, with all its constituent elements, is presented in consciousness in a distinct and reflective form. All the elements embraced in the conception, in its reflective, is really found in it when in its spontaneous form. In the latter state, however, each element is given obscurely and indistinctly. In the former, in a form distinct and well defined, as a part of the whole conception.
First and second Conceptions.
Another important distinction between conceptions, a distinction for which we are indebted to Sir William Hamilton, and which was first developed, as he states, by Arabian philosophers, is that of first and second conceptions. "A first notion" (conception), says the writer above named, "is the concept of a thing as it exists in itself, and independent of any operation of thought, as John, man, animal, &c. A second notion is the concept, not of an object as it is in reality, but of the mode under which it is thought by the mind, as individual, species, genus, &c. The former is the concept of a thing--real--immediate--direct; the latter is the concept of a concept--formal--mediate--reflex." In other words, when a conception is contemplated as immediately pertaining to its object, as it is in itself, and that without reference to other conceptions, it is denominated a first conception. When it is contemplated in its relation to other conceptions, and as being capable of being classed with, or separated from them, then it is denominated a second conception. When, for example, we contemplate the conceptions represented by such terms as John, man, animal, &c., not as merely pertaining to some object, or class of objects, but in reference to the mode or form in which they pertain to them, that is, as individual, species, or genus, and consequently as capable of being classed with others which pertain, in a similar manner, to their object, these, we repeat, are denominated second conceptions. It is with conceptions of this class especially that logic, as a science, has to do. Phenomena must be classified, before their laws can be determined. So with conceptions. Before the laws of thought can be determined, thought itself must be classified by reflection.
Matter and sphere of Conceptions.
By the matter of the conception is meant, the intuitions actually included in it. By the sphere of a conception, we mean the number of individuals embraced under it. The conceptions represented by the term John, for example, as to its matter, represents all the elements given by intuition, in respect to this individual, and as to its sphere, is limited to this one person, it being applicable to none other. The conception represented by the term man, as to its matter, represents all intuitions, and those only which are common to all individuals of the race; and as to its sphere, it comprehends every such individual.
"The matter and sphere of a conception," as Kant observes, "bear to one another a converse relation." The more elements (intuitions) a conception embraces, that is, the more it contains so far as its matter is concerned, the less number of individuals does it represent, that is, the narrower is its sphere, and vise versa.
The greatness or narrowness of the sphere of a conception depends upon the number of individuals which take rank under it.
Individual, generic or generical, and specific or specificial Conceptions.
Conceptions which pertain to individuals are denominated individual conceptions. Those which pertain to kinds which embrace, not individuals as such, but sorts or classes of individuals (species) under them, are denominated generic or generical conceptions. Those, on the other hand, which pertain to the sorts (species) which are contained under the generic or generical conception, are denominated specific or specifical conceptions. The individual conception embraces all the elements given by intuition relatively to the one object to which it (the conception) pertains. The generic conception embraces only the intuitions which are common to all the specific conceptions which rank under it, and to all the individuals which rank under its various specific conceptions. The specific conception embraces all the elements of intuition belonging to the generic, and also all that belong to the particular class which it represents, and which are not found in the class from which the former is separated.
Highest genus and lowest species.
It is evident that a conception may be generic relatively to another and lower conception, and itself specificial, relatively to one pertaining to a higher genus. Thus the conception represented by the term man, is generic relatively to those which pertain to different orders of the race, and at the same time, specificial relatively to that of a higher genus represented by such terms as rational beings, including as a genus men, angels, &c.
A genus which is not a species is called the highest genus. A species which is not a genus, is called the lowest species. The following remarks of Kant upon this subject are worthy of special regard:
"If we conceive of a series of several conceptions subordinate to one another--for example, iron, metal, body, substance, thing--we may obtain higher and higher genera; for every species is always to be considered as a genus with regard to its inferior conception. For instance, the conception of a man being generical with regard to that of a philosopher, till we at last arrive at a genus that cannot be a species again. And one of that sort we must finally reach; because there must, at last, be a higher conception, from which, as such, nothing can be further abstracted without the whole conception vanishing. But in the whole series of species and of genera there is no such thing as a lowest conception of species, under which no other conception or species is contained; because one of that sort could not possibly be determined. For, if we have a conception, which we apply immediately to individuals, specific distinctions, which we do not notice, or to which we pay no attention, may exist in respect to it. There are no lowest conceptions, but comparatively, for use, which have obtained this signification, as it were, by convention, provided that we are agreed not to go deeper in a certain matter.
"Relatively to the determination of the specifical and of the generical conception, then, this universal law--There is a genus that cannot be any more a species; but there are no species but what may become genera again--holds good."
Empirical and rational Conceptions.
Intuitions are also classed as empirical and rational. All intuitions derived through perceptions external and internal, that is, through the intuitions of sense and consciousness, are called empirical, being derived through experience. Those, on the other hand, which sustain the relation of logical antecedents to empirical intuitions, such, for example, as the intuitions of space, time, cause, substance, &c., are denominated rational intuitions, being the intuitions of that faculty or function of the intelligence denominated the reason.
Now conceptions, the leading elements of which are intuitions of qualities of substances material and mental in the world within and around us, qualities which are the objects of perception, external and internal, are called empirical conceptions. All such conceptions are constituted of two classes of elements, the empirical and rational, that is, intuitions of sense and consciousness, on the one hand, and of reason on the other, all such objects, for example, being apprehended as substances or causes existing in time and space, &c., and as possessed of certain qualities and attributes. The latter class of elements are given by immediate perceptions, external or internal, and the former by the reason. Such conceptions are denominated empirical.
When the rational intuition becomes itself the object of reflection and abstraction, and the intelligence apprehends its object in a distinct and reflective form, as it is in itself, and in its relations to objects of empirical conceptions, we then have what is denominated rational conceptions: those of time, as the place of events; of space, as the place of bodies; of substances, as the subjects of qualities; and of causes, as the origin of events, &c. Rational conceptions sustain to the empirical the relations of logical antecedents, the reality of the objects of the latter being conceivable and possible, but upon the condition of that of the objects of the former class. Thus the reality of body is neither conceivable nor possible, but upon the supposition of the reality of space. So of time relatively to succession, of substance relatively to qualities, and of cause in respect to events. If there is no space, no time, no substance, or causes, there can be no bodies, succession, qualities, nor events. The conceptions of space, time, substance, cause, &c., are therefore denominated the logical antecedents of those of body, succession, qualities, and events. So in all other instances.
Presentative and representative Conceptions.
Sir William Hamilton has classed all our knowledge under two divisions--that which is derived by direct and immediate intuition of the qualities of objects--and that which pertains to such qualities mediately, through consciousness of sensations, for example. Of the first kind are our intuitions of the primary qualities of matter, those which belong to matter as such--for example, extension, form, &c. Our intuitions of the secondary qualities, such as taste, smell, and sound, are not direct and immediate, but indirect and mediate, that is, through the consciousness of sensations. Such intuitions are therefore called representative. Our intuitions of the secundo-primary qualities, on the other hand, those qualities which distinguish one class of material substances from another, such, for example, as gravity, cohesion, &c., are partly presentative and partly representative.
Conceptions constituted of presentative intuitions may be called presentative conceptions. Those constituted of the other class would then be denominated representative. The same conception may partake partly of one, and partly of the other character.
Abstract and concrete Conceptions.
Conceptions also are properly classed as abstract and concrete. The former pertain to some single quality given by intuition, irrespective of the particular object to which such quality belongs, or to which the intuition pertains--conceptions represented by such terms as redness, whiteness, roundness, rightness, &c.
Concrete conceptions pertain to their objects as they actually exist, and combine all the elements given by intuition relatively to such objects--conceptions expressed by such concrete terms as George, man, animal, &c.
Positive, privative, and negative Conceptions.
Conceptions which embrace those intuitions only which are actually given by intuition in respect to their objects, and refer to their objects as actually possessed of the qualities which such intuitions embrace, are called positive; such conceptions, for example, as are represented by such terms as sound, speech, a man speaking, &c. Conceptions which pertain to their objects as void of certain qualities which might be supposed to have been given by intuition as pertaining to the object, are denominated privative conceptions; conceptions, for example, expressed by such terms as deafness, dumbness, a man silent, &c. When, on the other hand, the conception pertains to its object, as merely void of certain characteristics, or as by no possibility possessed of them, then it is denominated a negative conception. Such conceptions are represented by such terms as a dumb statute, a lifeless corpse, &c.
Conceptions classed under the principle of unity, plurality, and totality.
Every conception pertains to its object as numerically one--and individual, John; or as many--a multitude; a number of individuals--as John, Thomas, Samuel, &c.; or as a totality, a whole of which each individual is an integral part--a troop of horses, &c. For this reason they are classed under the categories above named.
Inferior and superior Conceptions.
When one conception takes rank as species under another as its genus, as, for example, the conceptions of the various species of fruit-bearing and forest trees ranked under that of the genus tree, the former class of conceptions are denominated inferior, and the latter superior conceptions.
"The inferior conception," as Kant well observes, "is not contained in the superior, for it contains more in itself than the superior, but is contained under it, because the superior contains the ground of the cognition of the inferior." We know the apple-tree, as a tree, for example, through the superior conception represented by the term tree.
Concrete and characteristic Conceptions.
We commonly have two classes of conceptions relatively to the same object,--the one embracing in concrete all the elements given by intuition in respect to the object, and the other embracing those only which peculiarize and distinguish that object from all others. The former class of conceptions we have already designated. The latter may be denominated characteristic conceptions. It is through this conception that objects are distinguished one from another, recognized and classified.
Laws of thought pertaining to the validity of Conceptions.
We are now prepared to state the general laws of thought pertaining to the validity of conceptions. All conceptions, as we have seen, together with all their logical antecedents and consequents, are to be held as valid for their objects--conceptions which are constituted of real intuitions in respect to such objects. Just so far as any conception is constituted of intuitions not thus given, it is not thus valid. These are the general laws. A conception, we would further state, is valid when, and only when, all judgments legitimately deduced from it are themselves valid in respect to their object. How often, for example, when certain judgments are expressed in regard to persons or objects do we hear the reply, "You are totally mistaken in your conception of such person or object;" or, "You are right in your conception," &c. Wrong conceptions lead to wrong misjudgments. Let us now apply them to particular conceptions and to particular classes of conceptions.
Particular, general, and abstract Conceptions.
Particular conceptions are valid when, and only when, such conceptions embrace no elements but actual intuitions, empirical and rational in respect to such objects. Intuitions with all their necessary or logical antecedents and consequents, being thus valid, the same must be true of conceptions into which such intuitions only enter as constituent elements. This holds true, whether the conception relative to its object is complete or incomplete, that is, whether it represents the whole, or only a part of the qualities of its object; for whatever is necessarily implied in the existence of a quality, must be true of all objects to which the quality pertains, and that whether it exists in such objects in connection with other qualities or not.
For this reason, abstract conceptions, with all their necessary antecedents and consequents, must be valid for their objects. General conceptions are valid, when they embrace those elements only which are common to ever particular conception contained under it, and when each of the former embrace those elements only which are actually given by intuition relatively to its object. This for reasons above stated holds true, whether the general conception be complete or incomplete.
Individual, specificial, and generical Conceptions.
What has been said of particular, being applicable in all respects to all individual conceptions, nothing further need be added in respect to the latter.
When individual conceptions ranking under the specificial are valid, the latter are also valid for their objects, when they embrace all the elements contained in the generic, together with all those that are common to all the individual conceptions which rank under the specifical. Thus, for example, the specifical conception represented by the term apple-tree is valid, when said conception embraces all the elements contained in the conception represented by the term tree, together with all those common to all valid conceptions pertaining to all apple-trees of every kind and sort. So of all other specifical conceptions.
Generical conceptions are valid when they include those elements only strictly common to all valid specifical ones contained under the former.
Presentative and representative Conceptions.
Presentative conceptions, those, for example, which are constituted of intuitions pertaining to the primary and secundo-primary qualities of matter, must be valid absolutely for their objects. This is self-evident. All conceptions also, so far forth as they are constituted of such conceptions, are thus valid.
Representative conceptions, on the other hand, can, from the nature of the case, have only a relative validity. Our knowledge of the secondary qualities of matter, for example, is mediate, through the consciousness of sensations. The subjects of such qualities, therefore, are know to us only as the otherwise unperceived causes of such sensations. Our conceptions of them, therefore, are valid in the sense only, that constituted as our sensibility now is, there is in such objects a power thus to affect us. Our presentative conceptions are valid, not for ourselves merely, but for all intelligents. Our representative conceptions are valid only for beings constituted in respect to their sensitivity, as we are, and when in our circumstances, questions which can be resolved only by a reference to general experience.
The same conceptions are often constituted of presentative and representative intuitions, and are, consequently, in corresponding degrees absolutely and relatively valid.
Concrete and characteristic Conceptions.
Concrete conceptions are valid, when they are constituted exclusively of actual intuitions in respect to their object, and when they embrace all the intuitions thus given, and as given.
Characteristic conceptions are valid, when they are constituted of such intuitions of those qualities which belong exclusively to the object of said conceptions, and which are always connected with them. Let A, for example, represent some object or class of objects, and B a quality which belongs to no object but A, and is always present as a constituent element of A. The conception represented by the term B, is valid as a valid characteristic conception of A. When the quality represented by the term B appears, the presence of all that are represented by A may be affirmed.
A conception may often be assumed as valid for ordinary practical purposes, which should not be assumed as the basis of any truly scientific procedure.
Inferior and superior Conceptions.
The rules just stated in respect to individual, specifical, and generic conceptions, embrace all that need be said of inferior and superior ones, the latter being only different forms of representing the former.
Empirical and rational Conceptions.
All empirical conceptions fall directly under the laws and rules already defined and elucidated. We have occasion, therefore, to speak only of the latter class, those which sustain to the former the relation of logical antecedents. If any conception is to be held as valid for its object, all that is contained and implied in its logical antecedents must be regarded as equally valid for the same object. A fundamental element of our conception of body, for example, is that of a substance contained in space, and which occupies space. Whatever, therefore is necessarily implied in the conception of the latter, must be absolutely valid for the object of the former conception. The same holds true of all other rational intuitions. All the necessary logical antecedents of a valid intuition must be just as valid as the intuition itself in respect to the object of said intuition. The validity of the rational conception, therefore, can be denied but upon one assumption, the absolute objective invalidity of all empirical conceptions, together with that of the intuitions of which the former are constituted. This would be an utter and universal impeachment of the intelligence itself, as a faculty of knowledge, and would annihilate the validity of the impeachment itself.
All conceptions conforming to the principles above defined are to be held as valid. All violations, in whole or in part, of any of those principles are to be held as in a corresponding degree invalid. How conceptions became thus vitiated, it will be our object to show, when we come to the Dialectic, the investigation of the sources of fallacy.
SECTION II.---Of Terms.
Very little is requisite in regard to the subject of the present section, to wit, terms. In logic a conception, or notion, expressed in language is called a term. All that is employed for this purpose, that is, to represent the conception, is included in this definition.
It is evident from the above definition, that a term may consist of one, or many words; as, man, or a man on horseback, a horseman, or a troop of horses, &c.
Singular and common Terms.---Significates.
In the science of logic, terms are divided into two classes, singular and common. All terms which represent individuals, or single objects only, are called singular terms, as George, the Hundson, New York, &c. Those, on the other hand, which represent classes of individuals, as man, river, mountain, &c., are called common terms. The individuals which a common term represents are denominated its significates.
Relations of Logic to Terms.
Logic has to do with terms only indirectly, that is, as the representatives of conceptions. What is required in regard to the term is, that, according to its received import, it shall fully and distinctly represent its conception, and nothing more nor less. It must not, according to received usage, represent more no less elements than are included in the conception; for, in such cases wrong, and not the right conceptions are represented.
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CHAPTER II.
OF JUDGMENTS.
SECTION I.---OF JUDGMENTS CONSIDERED
AS MENTAL STATES.
A JUDGMENT is an intellectual apprehension in which a certain relation is mentally affirmed to exist between two or more conceptions. We have in our mind, for example, the conception of body and space. On reflection, we perceive a necessary relation between them, or rather between their objects, a relation of this character, to wit: the existence of the former can be conceived of as possible, but upon one condition, the admission of the reality of the latter. The mind then becomes distinctly conscious of the truth, that body supposes space. This mental affirmation is a judgment. We have in our minds also the conceptions represented by the terms man, on the one hand, and mortal, on the other; we perceive that, as a matter of fact, all that is included in the latter conception, holds true of every individual represented by the former. Mortality is, therefore, mentally affirmed of all men. This mental affirmation, also, is a judgment. So in all other instances. Whenever a certain relation is affirmed to exist between two or more conceptions, or between the objects of the same, this mental affirmation is a judgment.
Matter and form of Judgments.
Logic, as a science, as we have seen, pertains not at all directly to the particular objects about which the thoughts are employed in particular conceptions, judgments, and reasonings, but to the laws of thought itself relating to such objects. So it distinguishes between the matter and form of judgments, and takes cognizance directly only of the latter. The former consists of the special notions or judgments relating to their particular objects, one judgment pertaining to one object, or class of objects, and another to another. The latter, the form of the judgment, pertains to its character relative to other judgments, as affirmative or negative, universal or particular, &c.
Logic, as a science, considers specially the form of the judgment, and has to do with the matter thereof, only so far as to give the universal criteria, by which valid may be distinguished from invalid judgments.
Quantity of Judgments, universal, particular, individual or singular.
When judgments are contemplated relatively to the number of objets of the class to which they pertain, the number which is embraced in the judgment, we then refer to the quantity of judgments, as whether the relation affirmed is conceived of as holding true of all such objects, or of a part of them, or of some one individual. Relatively to quantity, judgments are accordingly classed as universal, particular, and individual, as in the case of those represented by the propositions, "All men are mortal; Some men are mortal; and, George is mortal." In the first case, as the relation is affirmed to hold true universally of all individuals represented by the term man, the judgment is called universal. In the second case, this relation is affirmed relatively to a part only of the individuals represented by this term. The judgment is accordingly called particular. In the last case, the relation is affirmed of one individual only. The judgment is therefore denominated individual. All judgments, as far as the relation of quantity is concerned, may be ranked as universal, particular, or individual.
According to Kant, particular judgments might more properly be called plurative, because they relate to more than one individual. In this he is no doubt, correct, and equally correct, while he expresses such preference, in adhering to common usage.
Individual judgments also are, in logic, treated practically as universal ones, because in the former, equally as in the latter the relation affirmed holds in regard to the whole subject without exception.
Quality of Judgments, affirmative, negative, indefinite.
As far as quality is concerned, their own intrinsic characteristics, judgments are classed, as affirmative, negative, and indefinite. When one conception (the subject) is thought of as coming under the sphere of another (the predicate), as in the judgment, "All men are mortal," all men being in the judgment place in the sphere, or class of mortal beings, the judgment is called affirmative. When one conception is thought of as excluded from the sphere of another conception, as in the judgment, "Mind is not matter," the former substance being thought of as excluded from the sphere or class of material substances, the judgments in that case is called negative. When, on the other hand, a conception is thought of not only as excluded from the sphere of another conception, but as included indefinitely in one excluded from the latter, we then have what is called an indefinite judgment. Thus in the judgment, "The human soul is not mortal," we separate the subject from the sphere or class of the mortal beings, and place it, but indefinitely, in a class excluded from the former, that is, among immortal beings. The distinction between negative and indefinite judgments is important to a correct understanding of the notion of judgments themselves. In logic, however, both are included under one, the negative, and all judgments are classed as affirmative or negative.
Relation of Judgments, categorical, hypothetical, and disjunctive.
When one conception is directly affirmed or denied of another, as in the judgments, "All men are mortal, and, the soul is not mortal," the judgment is denominated categorical. When conceptions are thought of in respect to one another in the relation of antecedent and consequent, as the judgment, "If Caesar was a usurper, he deserved death," the judgment is then denominated hypothetical.
When a conception is thought of as included in one member of a certain division, as in the judgment, "Caesar was a hero or a usurper," "A is in B, C, or D," the judgment is called disjunctive. From the nature of the relation of the subject and predicate in judgments, all judgments must be either categorical, hypothetical, or disjunctive.
REMARKS ON THESE JUDGMENTS
Categorical Judgments.
In categorical judgments, as Kant remarks, "the subject and the predicate make up the matter of the judgment; the form, by which the relation (of agreement or disagreement) between the subject and predicate is determined and expressed, is the Copula," which, when expressed in language, is always--is, or is not. Categorical judgments, as Kant further remarks, "make up the matter of other judgments." With the following remark of this great logician we cannot agree: "But from this we must not think, as several logicians do, that hypothetical and disjunctive judgments are nothing more than different dresses of categorical ones, and can therefore be all reduced to them. All the three judgments depend upon essentially distinct logical functions of the understanding, and consequently must be discussed according to their specific distinction." On a careful analysis of any hypothetical judgments, it will be found, that, in all cases, it is, as stated in the Intellectual Philosophy, a universal proposition expressed in the form of a particular. The proposition, for example, if Caesar was a usurper he deserved death, is nothing more than the universal proposition, "All usurpers deserve death," expressed in a concrete and particular form. A comparison of categorical and hypothetical syllogisms will also show that they are only different forms of the same thing. For example:
All usurpers deserve death;
Caesar was a usurper;
Therefore, he deserved death.
If Caesar was a usurper, he deserved death;
He was a usurper;
Therefore, he deserved death.
The same may be shown to hold true in all the forms which hypothetical judgments assume, and in regard to all the principles and laws pertaining to hypothetical syllogisms. Throughout they are nothing but categorical judgments, or syllogisms stated in particular form.
What has been said in regard to hypothetical judgments being so directly and manifestly applicable to the disjunctive, nothing in addition is required to show that this class also differs only in form from the categorical.
Disjunctive Judgments.
In the language of Kant, "the matter of these consists of two judgments, which are connected together as antecedent and consequent. The one of these judgments which contains the ground" (the subject of the universal categorical) "is the antecedent; the other; which stands in the relation of consequence to that" (that is, the predicate of the universal categorical judgment), "the consequent." The connection affirmed to exist between them is denominated the consequence. The antecedent and consequent in a hypothetical judgment, answer to the subject and predicate in the categorical, and the consequence in the former to the copula in the latter. A few passing remarks are deemed requisite on the following paragraph from Kant:
"Some think it easy to transform a hypothetical proposition to a categorical. But it is not practicable; because they are quite distinct in their very nature. In categorical judgments nothing is problematical, but every thing assertive; whereas in hypothetical ones, the consequence is only assertive or positive. In the latter we may therefore connect two false judgments together, for in this case the whole affair is the rightness in the connection--the form of the consequence; upon which the logical truth of these judgments depends. There is an essential distinction between these two propositions: 'All bodies are divisible, and, if all bodies are composed, they are divisible.' In the former, the thing is maintained directly: in the latter it is maintained on a problematically expressed condition only."
In reply, we remark:
1. That while it is true that "in categorical judgments nothing is problematical, but ever thing assertive, whereas in hypothetical ones, the consequence only assertive," it is equally true, that in both the same thing is asserted, only in different forms. This is manifest, from the fact, that in all hypothetical syllogisms, a categorical may be substituted for the hypothetical judgment (premise), and the argument will stand just as it did before. This we shall see hereafter.
2. Even in those hypothetical judgments which contain "two false judgments," with the connection of necessary consequence between them, a universally valid categorical judgment is always given--a judgment which alone renders valid the relation of consequence referred to. In the judgment, for example, "If Washington was a traitor to his country, he deserved death," we have the two false judgments, and the relation of necessary consequence, under consideration. In this very judgment, however, we have, in reality, the universal categorical one, "All traitors to their country deserve death," and upon the validity of this last judgment depends that of the consequence before us. The same holds true in all other instances.
3. The reason why there is "an essential distinction between these two propositions, all bodies are divisible, and if all bodies are composed they are divisible," is not, as Kant affirms, because a hypothetical proposition cannot be transformed into a categorical one, but because the two propositions before us do not in fact belong to the same class. The judgment, therefore, "If all bodies are composed they are divisible," cannot be transformed into this, "All bodies are divisible." The former judgment, however, may be transformed into this, "All substances which are composed (compounded) are divisible," because that, in these instances, what is affirmed in one case categorically, is affirmed in the other hypothetically. The examples adduced by our author lay no valid basis for the conclusion which he deduces from them.
Disjunctive Judgments.
A disjunctive judgment, is distinguished from all others by this peculiarity, to wit: it is constituted of a certain number of problematical judgments, all of which together sustain such a relation to a certain judgment known to be true, that the object of this judgment must be in one of the numbers referred to, to the exclusion of all the rest. For example, the judgment, which all will admit cannot but be true, that the final determining cause of the facts of the universe in creation and providence, is either an inhering law of nature, or some power out of and above nature, has its basis in the judgment which also must be true, that for the facts named some ultimate reason or cause must exist. A is know to exist. But it sustains such relations to B, C, and D, that it must be found in one of them, to the exclusion of all the rest. Hence the disjunctive judgment. A is in B, C, or D. The same principle obtains in all disjunctive judgments.
The several problematical judgments constitute the matter of the disjunctive judgments, and are called, as Kant observes, "members of the disjunction or opposition." Their mutual relations of disjunction or opposition, that is, the fact that each sustains such relations to all the other, that if it is true, they must be false, and if any of the others be true, each of the rest must be false, constitute the form of such judgments.
Modality of Judgments, problematical, assertative, contingent, necessary (appodictical)
When the connection between conceptions is conceived of as possible, that is, with the conviction that the relation may or may not exist, as in the proposition, "A may be in B," the judgment is called problematical. When the connection is conceived of as not only possible, but as actual, the judgment is called assertative. When the relation is conceived as actual, with the conviction that the facts might possibly have been otherwise, the judgment is denominated contingent; as in the proposition, "A died on yesterday," it being possible to conceive, while it is asserted, that he did die, at the time named, that he is yet alive, or that he died at some other time. When a relation between conceptions is conceived of as not only actual, but the conception is accompanied with the conviction that the facts can, by no possibility, be otherwise than they are, the judgment is said to be necessary or appodictical, as in the judgment, "Body supposes space, or an event, a cause." The contradictory of the problematical is the impossible, a relation which cannot be conceived of as exiting.
Remarks.
1. A judgment may be deemed necessary for either of two reasons--the nature of the relations between the conceptions, or the nature of the evidence in favor of the actual existence of such relations. Of the first class are the judgments, "Every event has a cause," "Two straight lines cannot inclose a space," &c. Of the second, is the judgment, "That the square of the hypothenuse of a right-angled triangle is equal to the sum of the square of its two sides." Judgments of the former class are called primitive, those of the latter, derivative.
2. An assertative judgment, while, from the nature of the relations between the conceptions themselves, it may be, and is contingent, may, relatively to the evidence of the existence of the relations referred to, be necessary. The judgments, "The world exists, and I exist," are of this character. Relatively to the nature of the relations between the subject and predicate in each of these judgments, the judgments themselves are merely assertative or contingent. Relatively to the nature of the affirmations of perception and consciousness, we say that these judgments must be true.
3. A judgment necessary, from the nature of the relations between the subject and predicate, is necessary in the absolute sense--the judgments, for example, "Body supposes space; and succession time," &c. A judgment necessary relatively to the perceptions of sense and consciousness, is said to be relatively necessary; as, for example, "Phenomena supposes substance." A necessary form of this judgment is this: "Substances are as their phenomena." The logical antecedent of the phenomenon of extension is the reality of an extended substance (body). The logical antecedent of the subjective phenomena of thought, feeling, and voluntary determination, is the reality of the self as possessed of the powers of intelligence, sensibility, and will. The above-named phenomena being given, the judgments, "Body is, and Self exists," are necessary, relatively so.
4. Assertative judgments, like the appodictical, are divided into two classes--primitve and derivative. The judgments, "Body is, and Self exists," are of the first class. The judgment "All bodies attract each other directly, as their matter, and inversely as the squares of their mean distances," is of the latter character.
5. All derivative judgments, as originally given, are problematical, and subsequently become assertative or appodictical, as the case may be; that is, they are originally given as possibly true or false, and consequently as capable of proof, and as wanting it.
Theoretical and practical Judgments.
Theoretical judgments affirm what does and what does not really belong to their objects. Practical judgments, on the other hand, express those forms or rules of action by which certain ends may be obtained, or those actions which ought or ought not to be performed.
Practical principles are treated as theoretical ones, when the question to be argued is, whether the former are, in reality, what they are judged to be. As thus contemplated only, would logic have to do with them.
Demonstrable, and indemonstrable or intuitive Judgments.
A demonstrable judgment is a problematical one, of the class which is capable of being proved. Indemonstrable (intuitive) judgments are those which are immediately certain, and for this reason, incapable of proof.
Judgments of the latter class, since every intellectual process properly denominated reasoning commences with them, are sometimes, and with unquestionable propriety, denominated primitive judgments. Those of the former, being in fact deduced from and evinced by the latter, are called derivative judgments.
Intuitive judgments by which the demonstrable may be evinced, but which cannot be subordinated to others, are called elemental judgments, and also principles, a principle in science being always a judgment which is itself immediately certain, and consequently not evincible through any other judgment.
A demonstrable judgment, when evinced, may become a principle relative to other demonstrable judgments; and a judgment which is derivative in one science, may be an elemental principle in another.
Analytical and synthetical Judgments.
Those judgments whose certainty is immediately evinced from an analysis of, or reflection on the conceptions constituting the subject and predicate of said judgments, are called analytical judgments; those judgments which are evincible only through other and more elementary ones, are called synthetical judgments.
On examination it will be found that all analytical judgments, that is, all judgments whose validity is immediately certain, divide themselves into two classes, and are and must be all comprehended in one or the other of them.
1. Those in which the predicate represents an essential quality of the subject, as in the judgment, "All bodies have extension." It is impossible for us to conceive of a body which has not extension. In the judgment before us, then, the predicate, extension, represents a fundamental element of our necessary conception of body. The judgment has, and must have, immediate certainty, or course. The same holds true in all similar judgments.
2. Those in which the conception represented by the predicate, sustains to that represented by the subject, the relation of logical antecedent, that is, when the reality of the object of the latter conception can be admitted but upon the supposition of that of the former. Of this kind is the judgment, "Body supposes space." The reality of the object represented by the term body, can be admitted but upon the condition of admitting that of the object of the conception represented by the term space. So of the judgments expressed by such propositions as "Succession supposes time; events a cause; phenomena substance," &c. All judgments of this character can but have, of themselves, immediate and intuitive certainty.
Now if we adduce any known indemonstrable judgment which has immediate certainty, we shall find, on examination, that it does, in fact, belong to one or the other of these classes, and that this is the exclusive ground of its certainty. Take, as an illustration, the axiom, "Things equal to the same things are equal to one another." On reflection, it wll be perceived, that the relation of equality among themselves, is the necessary condition of their being equal to the same things. In other words, the conception represented by the words, "equal to one another" (the predicate), is the logical antecedent of that represented by the words, "things equal to the same things" (the subject). Thus we might take up all similar judgments, and all other self-evident ones, and show that they do, in fact, belong to one or the other of the classes above elucidated.
Nor is it possible for us to conceive of any other grounds of the immediate certainty of judgments. In any other conceivable or definable case, the relation between the subject and predicate of the judgment would be such that the judgment would be, at the utmost, only problematical.
Criteria of all first Truths.
We have, then, in the relations before us, the fundamental and universal criteria by which first truths may be distinguished from all others. In all such judgments (first truths) the conception constituting the predicate either exclusively represents elements contained in that represented by the subject, or the former conception sustains to the latter the relation of logical antecedent. There are, and can be, no other first truths but these. The criteria of such truths commonly given, are rather external and circumstantial than intrinsically characteristic, as all scientific criteria should be. We refer to those criteria given by Dr. Reid, and concurred in by philosophers generally, such, for example, as the fact, that all men admit them as a matter of fact in all their reasoning; that even those who deny their validity act upon them; and if denied, the validity of all reasoning fails.
Kant's definition of analytical and synthetical Judgments.
According to Kant, we have but one class of analytical judgments, those in which the relation of identity referred to obtains between the predicate and subject. The other class he represents as synthetical judgments, which, according to him, embrace all judgments in which all the elements of the conception represented by the predicate are not embraced in the that represented by the subject. He accordingly divides synthetical judgments into two classes, the intuitive and problematical, though he gives us no explanations of the reasons why one is intuitive and the other not. In the Intellectual Philosophy, pp. 336-341, we have stated our objections to our author's definition of these two classes of judgments, the analytical and synthetical, and to the use which he has made of the latter. In this connection, we would simply add, that while our definition is just as plain, and of as ready application, as that of Kant, it presents a much more simple and easily understood classification of judgments. If any one, however, should prefer the definition of that philosopher, we would remind him, that in that case, he must divide synthetical judgments into two classes: those in which the conception represented by the predicate is, and those in which it is not, the logical antecedent of that represented by the subject, and that the former class, together, as first truths, and that no other judgments can be classed with them, as such truths. The logical and scientific bearings of each classification will then be, in all respects, the same, and nothing but a verbal difference remains.
Tautological, identical, and implied Judgments.
A tautological judgment is one in which the subject and predicate are identical, either in fact and in form; as, "John is John, Man is man," &c.; or, in all respects, in meaning, so that the predicate is, in no respect, even explicative of the subject; as, "Man is a human being," &c. Such judgments are of no use whatever.
Identical judgments, as distinguished from tautological, are those in which, while there is an identity in fact, there is such a diversity in form between the subject and predicate, that the latter is really and truly explicative of the former. Of this character are all correct definitions; as, for example, a triangle is a figure bounded by three straight lines. Of the same character is the class of analytical judgments, in which the predicate represents some element or quality of the subject; as, "All bodies have extension." Such judgments are, by no means, void of consequence, inasmuch as they render clear and distinct our conceptions of their objects.
An implied judgment is one which is really only another form of another judgment, but which presents some important element of the latter which was not distinctly expressed before. We often say: If this proposition is true, that is also true, because the latter is really implied in the former, that is, is only a different form of stating the same thing. Implied judgments have a very important use; indeed, a statement of them is often indispensable to the production of conviction.
Axioms, Postulates, Problems, and Theorems.
An axiom is an analytical judgment (analytical or intuitive synthetical judgment of Kant) which may be employed as a principle in the sciences in general, that is, a judgment by which other judgments may be evinced. As shown in the Intellectual Philosophy, pp. 257-8, the axioms which constitute the foundation-principles of each of the sciences are essentially identical with those of every other.
Postulates are analytical judgments which can be employed as principles only in particular sciences. Thus the axiom, "Things equal to the same things are equal to one another," is really, though often stated in a somewhat different form, identical with that which lies at the basis of every science that can be named; while the postulate, That a straight line may be drawn between any two points in space," pertains exclusively to geometry and kindred sciences.
A problem is a judgment which appears neither true nor false, and requires an answer to the question, Is it, or is it not true? or presents a number of judgments either of which apparently may be true, an but one can be, and requires an answer to the question, Which is true? or finally affirms that a certain thing may be done, and requires an answer to the question, How may it be done? In problems of the first kind most commonly classed above named, an answer of this kind is most commonly required, to wit, not what is, or what is not true, in the particular cases presented, but how may we determine, what is, and what is not true, in these cases? In the solution of particular problems, in this form, we obtain not only answers to the specific questions presented, but principles by which all other similar questions may be solved. Let us suppose, for example, that an event like the raising of Lazarus from the dead occurs in our presence. The question presents itself, Is this, or is it not a miracle? or, Is this event the result of the direct and immediate interposition of creative power, or of mere natural causes? In the first form, we have a problem of the first class named, and in the other of the second. Suppose, that we are required not merely to give a direct answer to these questions, but to give criteria by which we may know whether the event is, or is not, a miracle, or whether it was the result of a supernatural interposition of creative power, or of natural causes. In giving the solution in this form, we should not only obtain an answer to the specific questions above stated, but should also obtain criteria by which we can, in all other cases, distinguish events resulting from natural causes from real miracles. Suppose, on the other hand, we are required to give a rule, by which a given line may be divided into any specific number of equal parts. We then have a problem of the third class.
Theorems are theoretical judgments capable of proof, and requiring it; as, for example, the proposition, "All the angles of a triangle are equal to two right angles."
Corollarys, Lemmas, and scholia.
Corollaries are the immediate and intuitive consequences of preceding judgments.
A lemma is a judgment previously evinced, and now used as a principle in the demonstration of other judgments. In general it is not native in the particular science in which it is presupposed as evinced, but is taken from some other science, as when some ascertained truth in the science of geology, for example, is employed as a principle in the science of natural theology.
Scholia are explanatory notes or observations appended to evinced judgments, for the purpose of illustration.
CRITERIA OF JUDGMENTS, OR CHARACTERISTICS OF ALL VALID JUDGMENTS.
We are now prepared to give the universal criteria of judgments, or the universal and necessary characteristics of all valid judgments, as distinguished from those which are not valid.
General Criteria.
All universally valid judgments must have the following characteristics:
1. The conceptions constituting the subject and predicate of such judgments must be valid according to the criteria developed in the last chapter.
2. The judgment must be analytical according to the definition above given of such judgments.
Or, 3. It must be evinced as true, by means of judgments which are analytical.
All valid primitive judgments have the first two characteristics. All valid derivative ones have all the three together. Any judgment wanting these characteristics must be held as not valid.
Particular and special Criteria.
As necessarily involved in the above criteria, we present the following and special ones.
Judgments relative to all valid Conceptions.
1. All judgments must be held as valid in which any element of any valid conception is affirmed of the object or objects of such conception. Suppose, for example, that the conception represented by the term man, be assumed as valid, then any judgment in which any or every element of that conception is affirmed of all men or any one individual of the race, must be held as valid. So of all similar judgments relative to all valid conceptions.
2. All judgments must be held as valid, in which the necessary relations between a valid conception and its logical antecedent, or between any element of such conception and the logical antecedent of that element, are affirmed; as, for example, the judgments, "Body supposes space; succession time; events a cause; and phenomena substance," &c.
3. All judgments must be held as valid which affirm the immediate and necessary consequence of valid judgments. In other words, when one judgment must be held as valid, all others immediately implied in it must be held as valid also. If the judgment, "Every event must have a cause," is valid, then the judgment, "Every event must have a cause adequate and adapted to produce that event," must be held as valid also. If the judgment, "Phenomena or quality supposes substance," is valid, the judgment, "Substances are as their phenomena or qualities," must be held as valid also. If the judgment, "phenomenon or quality supposes substance," is valid, the judgment, "substances are as their phenomena or qualities," must be held as valid also. So in all other instances.
INDIVIDUAL (SINGLE), PARTICULAR, AND UNIVERSAL JUDGMENTS.
Individual Judgments (affirmative).
In regard to every individual (each particular object), the following judgments must be held as true:
1. All judgments which affirm of such object any element of any valid conception pertaining to it. Such judgments, being really analytical, must be valid.
2. All judgments which affirm of said object that it belongs to any class of objects with which it has common characteristics, the characteristics which peculiarize that class.
3. All judgments which affirm of such object any or all of the elements of the conception which represent that class.
4. All judgments which affirm of that individual any or all of the elements embraced in any superior conception of that just named.
The judgment, in the first instance, is really, as said above, analytical, and cannot but be valid. In the second case, we have the universal and immutable law of classification. Each object must take rank with all others with which it has common characteristics. The third case is necessarily involved in the second; for these are the necessary conditions of an object being entitled to take rank with a certain class. When, therefore, it is known to belong to a certain class it is, and must be, recognized as possessed of all the elements embraced in the conception which represents that class, and all judgments which affirm of it any or all of such elements must be valid. The elements embraced in the superior conceptions are embraced in the inferior. When all of the former may be affirmed of an object, or course any or all of the latter may be. All judgments of the fourth class, therefore, must be valid.
Individual Judgments (negative).
The following negative judgments in regard to such objects must be held as valid:
1. All judgments which deny of said object any and all elements and characteristics incompatible with any and all elements of valid conceptions and judgments in regard to it. When a given characteristic may be affirmed of any object, every thing incompatible with that characteristic may of course be denied of it. When, for example, it is admitted that matter has the quality of extension, and it is affirmed that the substance itself, in regard to its ultimate essence, is unknown to us, it may be denied absolutely that there is, or can be, in such substance, any thing incompatible with the idea of extension, and the judgment, that any theory in regard to the nature of that substance (any ontological conception of it) that affirms that it is not in reality an extended substance, is and must be false, must be held as valid. So in all other cases of the kind.
2. When it is undeniably true, that if an object does or did possess certain characteristics, those characteristics would appear, that is, would be given in intuition, and they do not appear, and have not appeared (are not given in intuition), then the judgments, which deny such characteristics of such objects, must be held as valid. It is undeniable, for example, that if Washington was under the controlling ambition of possessing monarchical or despotic power, he would, in the circumstances in which he was placed, have attempted to have gained that power over his countrymen, and the fact of such attempt would appear. The absence of the fact, renders valid the judgment, that he was not under the control of the principle before us. Again: if spontaneous production and the transmutation of species are the law of nature, and the order of creation, we should find somewhere in the present or past history of the earth, undeniable facts indicative of the truth of such theory. The total absence of any such facts within the knowledge of man, since his existence on earth, and the total absence of all abnormal specimens, of any intermediate creations, in the vast laboratory of geological science, render undeniably valid the judgment, "That the theory of spontaneous production and transmutation of species is not, and cannot be true." Very few of the laws of thought are of more importance than that under consideration, when legitimately employed.
3. All negative judgments are valid, which in matter, though not in form, are identical with valid affirmative ones. If the judgment, "A is mortal," is valid, the judgment, "A is not immortal," is also valid, inasmuch as the two propositions merely affirm one and the same thing. In argument, it is often expedient to state an affirmative judgment in it equivalent negative form.
A careful examination will show, we judge, that all valid individual judgments fall under one or the other of the classes above named, and that no judgment not belonging to one or the other of these classes should be held as valid.
Particular (plurative) Judgments.
All particular judgments of the following classes must be held as valid:
1. All judgments of this class which rank as subaltern judgments under universal ones which are valid. What is true of every member of a given class, may of course be affirmed to be true of some members of that class.
2. When a certain characteristic, or quality, belongs to a part, but not to all, of the members of a certain class, particular judgments which affirm that some of the members of that class have such characteristic or quality, must be held as valid.
3. In all such cases, the particular negative judgment which denies that characteristic or quality of some member of the class under examination, must be valid also. As wisdom, for example, pertains to a part, and not the whole, of the human race, the particular judgments, "Some men are wise, and some men are not wise," must be held as valid. So in all similar instances.
Universal Judgments (affirmative).
All affirmative universal judgments are valid which have either of the following characteristics, or all of them together:
1. Those in which any or all of the elements embraced in the conception which represents a class of objects, are affirmed of all the members of that class--any judgment, for example, which affirms of all men any or all the elements of the conception represented by the term man.
2. All which affirm universally of such a class any or all of the elements embraced in any conception, to which the conception representing that class sustains the relation of an inferior conception, that is, we may affirm of all the objects of a specifical conception, any or all of the elements of any of its superior or generical conceptions.
3. All judgments which affirm of all the members of a class any or all the elements embraced in the characteristic conception of such class.
Universal Judgments (negative).
All negative universal judgments must be admitted as valid which have the following characteristics:
1. All which deny of all the members of any one class or species any or all of the elements of any opposite specifical conception, those elements excepted which belong to superior conceptions under which each of the above take rank as inferior ones.
Thus, if we should deny of the conception represented by the term apple, any or all of the elements of the conception represented by the term peach, with the exception of those embraced in the superior conception represented by the term fruit, the affirmation would be valid, and that for the reason, that species under a genus are formed exclusively on the principle of contradiction. The same will hold equally true in all other similar cases.
2. All judgments in which any and all characteristics incompatible with any or all the elements of any valid conception, are denied of all objects represented by such conceptions. We may affirm absolutely, for example, that no untruth was ever given forth by inspiration of the Almighty. The reason is obvious. The thing denied is incompatible with all valid conceptions of Deity.
3. All universal negative judgments must be held as valid which are really equivalent to valid affirmative ones. Thus the judgment, "No man, physically considered, is immortal," must be held as valid, because it is in fact equivalent to the universally valid judgment expressed by the proposition, "All men are mortal." It is often of great importance, thus to substitute for a valid affirmative judgment, its equivalent negative one.
4. When it is undeniable, that a given characteristic, if it did attach to any member of a given class, would be given by intuition in connection with some members of the same, and is not given, then the judgment which denies such characteristic of all the members of that class, must be held as valid. Thus the judgment, "No plant is produced but through a seed, and no seed but through a plant," must be held as valid, because it is undeniable, that if the opposite judgments were true, facts corroborative of them would appear.
It is believed, that all valid universal negative judgments belong to one or the other of the classes above defined, and that we have fundamental criteria by which to determine the validity of such judgments.
Judgments pertaining to the object of inferior and superior Conceptions.
All that is required to be said relating to judgments pertaining to the objects of inferior and superior conceptions, has already been anticipated, and what is added, in this connection, is only for the sake of distinctness. On this subject we would simply add, that all judgments relative to such objects must be held as valid which have the following characteristics:
1. All judgments in which any object or class of objects having the elements represented in any conception is ranked or classed under that conception.
2. All judgments which affirm of any object of any inferior conception, not only any or all of the elements of that particular conception, but any or all of those of any superior one.
Judgments pertaining to the objects of characteristic Conceptions (affrimative).
When an object agrees with a characteristic conception, or possesses the elements embraced in such conception, the following judgments relative to it must be held as valid:
1. Any which rank of said object with the class to which the conception under consideration pertains.
2. All judgments which affirm of said object any or all the elements of the conception which represents that class, or all of any of the elements of any superior conception.
Suppose, for example, that an object is before us, that agrees with the characteristic conception of the class of substances represented by the term gold. For no other reason, we may affirm, that the object is gold, that it has any or all of the properties of gold. We may affirm, further, that it is a metal, a mineral; that it is matter, a substance; or affirm of it any or all of the elements, of any or of all the conceptions which these terms represent. So in all other instances.
Judgments relative to objects of characteristic Conceptions (negative).
Of all objects agreeing with characteristic conceptions, the following negative judgments must be held as valid:
1. All which deny of such objects any or all the elements represented in any opposite specifical conception, those excepted which are represented in the common superior conceptions. Thus, for example, if an object has the characteristic elements of gold, we may affirm, form such fact, that such object is not silver, copper, platinum, &c., and deny of it any of the peculiar and specificial qualities of such metals. So in all other instances.
2. All judgments which deny of such objects any or all of the elements represented by any incompatible conception. Thus, if we should affirm that any act having the undeniable characteristics of an act of perjury, did not proceed from an honest intention to speak the truth, the judgment would be valid.
3. All negative judgments which are equivalent to valid affirmative ones. In other connections, this principle has received a sufficient elucidation. Nothing, therefore, need be added in respect to it here.
HYPOTHETICAL JUDGMENTS.
It is a somewhat remarkable fact, that while all systems of logic treat of hypothetical and disjunctive judgments, in no such treatises do we find, so far as our knowledge extends, even an attempt to give us any criteria by which we may determine the validity of either class of these judgments. We will, therefore, attempt the accomplishment of this important result.
Hypothetical Judgments classed.
All hypothetical judgments may be divided into three classes:
1. Those in which the antecedent and consequent have different predicates, and each the same subject; as, "If A is in B, it is, or is not, in C."
2. Those in which both have the same predicate, and each a different subject: "If A is in B, C is, or is not, in B."
3. Those in which both have different subjects, and different predicates: "If A is B, C is, or is not, D."
Criteria of such Judgments.
Judgments of the first class are valid, when, and only when, the predicate of the consequent may be affirmed or denied universally, as the case may be, of the predicate of the antecedent. Thus, the judgment, "If A is in B, it also is in C," can be valid only when the judgment, "Every B is in C," is valid; and the former judgment must be valid when the latter is. So, also, we can affirm that, "If A is in B, it is not in C," when, and only when, the judgment, "B is never in C," is valid; and in that case, the former judgment must be true.
Judgments of the second class are valid, when, and only when, the subject of the antecedent may be affirmed or denied, as the case may be, universally of the subject of the consequent. Thus, the judgment, "If A is in B, C is in B," can be true but upon the supposition that C is always in A, and must be true in that case. The judgment, in its negative form, can be true, but upon the supposition, that C is never in A, and must, in that case, be always true.
Judgments of the third class can be true, but upon the condition that the relations between the subject and predicated of the antecedent, are the same as between the subject and predicate of the consequent. Equality or similarity of relations is the thing, and the only thing, really affirmed or denied in all such judgments. Unless, therefore, the judgment, "A sustains similar relations to B that C does to D," is valid, the judgment, "If A is B, C is D," cannot be valid. On the other hand, when the former judgment is valid, the latter, of course, must be. These remarks are so manifestly applicable to these judgments when given in the negative form, that nothing is called for on this point.
What may be affirmed, when the relations referred to are equal, may be affirmed when the relations are greater in degree. If, for example, we may say that A, possessing $100, is able to meet an indebtedness amounting to that sum, we may of course affirm, that B, possessing $10,000, is able to discharge an indebtedness amounting to $1,000.
Disjunctive Judgments.
Disjunctive judgments always partake of one or the other of these characteristics. A fact, or a class of facts (A), is known to exist, and their explanation is required. A certain given number of hypotheses, B, C, D, &c., two or more, present themselves, none others being, from the nature of the case, conceivable or possible, while one of them, to the exclusion of all the others, must be true. Hence we say, "A must be in B, C, or D." A judgment of this class is valid, when the facts A, are known to exist, and when all conceivable demonstrable judgments are specified in the judgment, "A is in B, C, or D," &c., and when, from the character of the facts, A must be found in one of these judgments, B, C, or D, to the exclusion of each of the others. Each judgment must be, in its nature, exclusive, and the whole together must, undeniably, exhaust the subject: for, if any one conceivable hypothesis is not included, the judgment is not valid.
Or it may be known that there is a cause, X, for a given class of facts, and the inquiry is, what is the nature of this cause? From the nature of the case, there can be but a certain number of answers to this question, and one of these, to the exclusion of each and all the others, must be true. In such a case, we say, "X is A, B, or C." Such a judgment is valid, when it undeniably embraces all conceivable or possible answers, and when each member of the judgment is in such disjunction with, or opposition to each and all of the others, that one of them, to the exclusion of each and all the others, must be true. If any possible answer to the question is omitted, or if each proposition is not, in its nature, exclusive of each and all the other, then the judgment is not valid. For example, All men believe, and must believe, that there is an ultimate reason why the facts of the universe are what they are, and not otherwise. Let X, for example, represent this ultimate or unconditioned cause. Now it is self-evident, that this cause X, must be an inherent law, or principle of nature, which we will call L, or a power out of and above nature, which we will denominate G, the god of theism. Hence, the judgment, "X is L or G," must be valid.
There is one form of the disjunctive judgment which, of course, must be valid, to wit: "Every X is A, or not A;" a form of the judgment which hardly differs form the tautological, and requires no elucidation.
We believe that all disjunctive judgments belong to one or the other of the above classes, and that we have, in the principles above given, universal criteria of their validity.
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SECTION II.
OF PROPOSITIONS.
Having treated sufficiently of judgments, it remains to make a few remarks in respect to propositions, which are judgments expressed in words. Logic treats only of assertative propositions, those which affirm or deny; as, "A is B, or A is not B."
Quality and Quantity of Propositions, &c.
Propositions, when contemplated with reference to their nature or substance, are divided into two classes, to wit: categorical, those which simply affirm or deny, as, "A is, or is not, B;" and hypothetical, those which affirm conditionally, as, "If A is B, C is D," &c.
When contemplated with reference to their quality, they are divided as affirmative: "A is B;" or negative, "A is not B."
In regard to the quantity, they are divided into universal, those in which the predicate is affirmed or denied of all the objects represented by the subject; as, "Every A is B, or no A is B;" and particular, those in which the predicate is affirmed or denied of a part only of the objects represented by the subject. As affirmative and negative propositions are each divided into two classes, universal and particular, we have four kinds of propositions: the universal affirmative, which is represented by the term, A; the universal negative, E; the particular affirmative, I; and the particular negative, O.
Distribution of Terms.
When a term stands for all its significates, that is, for every individual of the class which it represents, then it is said to be distributed. When it represents a part only of its significates, then it is said to be not distributed.
When the subject of a proposition is a common term, its distribution is commonly signified by such terms as "All, every, no," &c.; and when not distributed, by the term "Some," &c. When no sign is used, the question, whether the subject is to be understood as distributed or not, is always to be determined by the particular circumstances of the case, and not by a reference to the matter of the proposition. The quantity of a proposition, when no signs are used to indicate the distribution or non-distribution of terms, "is ascertained," says Dr. Whately, "by the matter, i. e. the nature of the connection between the extremes, which is either necessary, impossible, or contingent. In necessary and impossible matter, an indefinite is understood as a universal; e. g. 'Birds have wings,' i. e. all birds; 'Birds are not quadrupeds,' i. e. none. In contingent matter (i. e. where the terms partly--i. e. sometimes--agree, and partly not), an indefinite is understood as a particular; e. g. 'Food is necessary to life,' i. e. some food; 'Birds sing,' i. e. some do; 'Birds are not carnivorous,' i. e. some are not, or, all are not"
Here are two fundamental mistakes relatively to the science of logic,--the supposition that this science has any thing to do with the matter of the proposition--and the supposition that individuals always conform, in their use of terms, to the rules which our author has laid down; whereas the opposite is not unfrequently the case, and we should violate all the laws of language should we interpret their words according to any such rules.
Apply the principle we have laid down to the cases cited by Dr. Whately, and we shall at once see its validity. Suppose that the question is being argued, whether, as a matter of fact, all birds have wings. The individual maintaining the affirmative uses the phrase, "Birds have wings;" and on the opposite side it is affirmed, "Birds have not wings." The circumstances of the case require us to understand the first proposition as universal, and the second as particular, that is, the contradictory of the first. If, on the other hand, the question was this, "Are any birds quadrupeds?" and, on one side, it should be affirmed, "Birds are quadrupeds," and on the other, "Birds are not quadrupeds," we should be bound, by the circumstances of the case, to assume the first proposition as particular, and the second as universal. So in all other circumstances.
Singular propositions, those in which the subject is a proper name, or a common term, with a singular sign, are reckoned in logic as universals, because in such cases the predicate is affirmed of the whole subject. The following quotation from Dr. Whately presents the rules of distribution pertaining to the subject and predicate of propositions as commonly given, so distinctly, that we give it, without note or comment of our own:
It is evident, that the subject is distributed in every universal proposition, and never in a particular (that being the very difference between universal and particular propositions); but the distribution or non-distribution of the predicate depends (not on the quantity, but) on the quality of the propositions; for, if any part of the predicate agrees with the subject, it must be affirmed, and not denied of the subject; therefore, for an affirmative proposition to be true, it is sufficient that some part of the predicate agrees with the subject; and (for the same reason) for a negative to be true, it is necessary that the whole of the predicate should disagree with the subject; e. g. it is true that 'Learning is useful,' though the whole of the term 'useful' does not agree with the term 'learning,' (for many things are useful besides learning); but, 'No vice is useful,' would be false, if any part of the term 'useful' agreed with the term 'vice' (i. e. if you could find any one useful thing which was a vice). The two practical rules, then, to be observed respecting distribution, are:
"1st. All universal propositions (and no particular) distribute the subject.
"sd. All negative (and no affirmative)(* Here, as we shall see hereafter, is a fundamental mistake of logic.) [?--missing word distribute] the predicate. It may happen, indeed, that the whole of the predicate, in an affirmative, may agree with the subject; e. g. it is equally true that 'All men are rational animals;' and, 'All rational animals are men;' but this is merely accidental, and is not at all implied in the form of expression, which alone is regarded in logic."
Of Opposition.
Propositions are said to be opposed to each other, when the subject and predicate are the same, and they differ in quantity, quality, or both.
In respect to quantity, A and E are each opposed to I and O. From the nature of this opposition, the following rules, pertaining to the validity of proposition, arise:
1. If the universal is valid, so is the particular; that is, if A is true, I must be true also; and if E is true, O must be. If the proposition, "All men are mortal," is true, I, which affirms that "Some men are mortal," must be true also. If the proposition, "No birds are quadrupeds," is true, O, which affirms that "Some birds are not quadrupeds," must also be true.
2. If the particular, I or O, be false, its respective universal, A or E, must be false also; in other words, the denial of the particular involves a denial of the universal under which the former ranks. If the proposition, "Some men are mortal," is false, A, which affirms that "All men are mortal," cannot, of course, be true. So if the proposition, "Some men are not immortal," is false, E, which affirms that "No man is immortal," must be false also.
3. On the other hand, both the universals (A and E) may be false, and both the particulars (I and O) may be true; that is, the denial of the universal does not necessitate a denial of the particular. The proposition, "All men are liars," and "No men are liars," may both be false; and the propositions, "Some men are liars," and "Some men are not liars," may be true.
In respect to quality, A and I are each, respectively, opposed to E and O, and vice versa. The two universals are opposed throughout their whole extent; that is, what one affirms in regard to a whole class, the other denies in regard to every individual of that class. The universal of one is opposed to the particular of the opposite quality, A to O, E to I, simply and exclusively, in regard to one point, the question of universality. What the universal affirms as true of every individual of a certain class, the opposite particular denies in regard to some individuals of the same class. What I affirms as also true of some individuals of a given class, O denies, not of all, or of the same, but of some individuals of the same class. From the nature of this opposition, therefore, the following rules or axioms obtain:
1. If one universal is true, its opposite universal must be false. If "Every A is B," the proposition, "No A is B," must be false throughout.
2. The fact that one universal is false, does not imply that the opposite is true. The propositions, "Every A is B," and "No A is B," may both be false, and each of the particulars, to wit: "Some of A is B," and "Some of A is not B," may be true. The propositions, "All men are liars," and "No men are liars," are in fact, both false; and their respective particulars, "Some men are liars," and "Some men are not liars," are true.
3. If either particular is true, its opposite universal is false. If the proposition, "Some are liars," is true, the proposition, "No men are liars," must be false. So in all other instances.
4. The fact that one particular is true, does not imply that the opposite one is false. Both may be, and often are, true. The propositions, "Some men are virtuous," and "Some men are not virtuous," are both true.
5. If a universal is false, its opposite particular must be true; and if the particular is false, its opposite universal must be true. If the proposition, "No A is B," is false, the proposition, "Some A is B," must be true. So if the proposition, "Some A is B," is false, the proposition, "No A is B," must be true.
6. Both particulars can, in no case, be false, because both universals would then be true, which, as we have seen, is impossible.
The above principles will be found to be of very great importance, when understood and duly reflected on. * (See Tappan's Logic, pp. 318-320, where most of the above principles are stated and elucidated with great precision and clearness.)
Of the Conversion of the Predicate.
A proposition is said to be converted when, without a change of quality, its terms are transposed; that is, the subject is made the predicate, and the predicate the subject. When nothing more is done, we have what is called SIMPLE CONVERSION. The original proposition is called the exposita; when converted, it is denominated the converse.
Conversion is valid when, and only when, nothing is asserted in the converse which is not affirmed or implied in the exposita. Hence the universal rule of conversion, to wit: "no term must be distributed in the converse which was not distributed in the exposita." Whenever this is done, that is affirmed of the whole class which was before only asserted of a part of it; that is, more is affirmed in the converse than was implied in the exposita. The following are the necessary applications of this law:
1. E distributes both terms, and I neither. Each of these classes of propositions may always be converted simply, and the conversion will be illative; that is, the truth of the converse is implied in the truth of the exposita. If the proposition in E, "No virtuous man is a rebel," is true, its converse, "No rebel is a virtuous man," must be true also. If the proposition in I, "Some boasters are cowards," is true, its converse, "Some cowards are boasters," must also be true.
2. A, the universal affirmative, distributes only the subject.* (This proposition, as we shall see, holds when, and only when, the subject represents an inferior and the predicate a superior conception.) Its simple conversion, therefore, would not be illative. From the fact, that "All men are mortal," we cannot infer, or affirm, that all mortal beings are men. That fact being admitted, however, we can affirm, as necessarily implied in it, the truth of the proposition, that "Some mortal beings are men." Universal affirmatives, then, may always be converted by making the converse particular instead of universal. This has been denominated "conversion by limitation," or "per accident." As we are always permitted to affirm a particular, when a universal might be affirmed, the universal negative E can always be thus converted.
3. The particular negative distributes the predicate instead of the subject. Such propositions, therefore, cannot be converted simply; since, in that case, we should have the predicate distributed in the converse, when it was not distributed in the exposita. As Professor Tappan has observed: "According to a strict exposition of the form, a particular negative has no converse." From the fact, "That some men are not truthful," we cannot affirm, that "Some truthful persons are not men." The proposition is, in fact, incapable, as it stands, of conversion. It can be converted only by changing its form from a negative to a positive; that is, by attaching the term of negation to the predicate of the exposita. Take, for example, the proposition, "Some men are not truthful." From such a proposition, we may affirm, that "Some persons who are not truthful are men." This has been named conversion by negation. Since, as Dr. Whately remarks, "it is the same thing to affirm some attribute of the subject, as to deny the absence of that attribute," the universal affirmative may always be converted in the same manner. From the fact, for example, that "Every virtuous man is a true patriot," we may infer, that "Every one who is not a true patriot, is not a virtuous man," or, "None but true patriots can be virtuous."
Thus, as Dr. Whately states, "in one of these three ways, every proposition may be illatively converted, viz.: E and I simply; A and O by negation; A and E by limitation."
Hardly any department of logic needs to be more thoroughly studied and reflected upon than the department we have just passed over, when treating of the laws and principles of opposition and conversion of propositions. When a proposition is admitted as self-evident, or having been proved true, few persons seem to know what use to make of it, and that in consequence of not perceiving what is implied in it.
Quantification of the Predicate.
What we have said hitherto in regard to propositions, has been based on the assumption, that the quantity of propositions depends wholly upon the relations of the whole predicate to the subject. If the former is affirmed or denied of the whole subject, the proposition is universal. If it is affirmed or denied only of a part of the subject, the proposition is particular. We have said nothing (for the reason that logic, with the exception about to be named, has hitherto left the subject untouched) of the quantity of propositions so far as the predicate is concerned. To Sir William Hamilton the world is indebted for one of the most important attainments in this science which has been made for centuries, to wit: in the quantification of the predicate as well as of the subject. In all propositions alike, as he maintains, if we refer to the judgment itself, that is, to what is really thought in the mind, the predicate always has as real a quantity as the subject; and that, if we refer to the judgment, and not to the words of the proposition expressing it, conversion of propositions is always and exclusively simple, the subject and predicate being really, in all instances, definite in their meaning. Why, for example, is the converse of the proposition, "All men are animals," this: "Some animals are men?" The answer commonly given is: "This true, as far as the mere form of expression is concerned. If we refer to the thought in the mind, however, we shall find that the reason is, that, in the exposita, the subject is universal, and the predicate particular. What we really mean, when we say, "All men are animals," is not, that all men are any kind of animals, but some kind; rational, for example. The proposition before us, then, is really universally relative to the subject, and particular relative to the predicate. Hence, by simple conversion, we have the converse, "Some animals are men." The propositions, on the other hand, "Men are rational animals," and "All triangles are figures bounded by three straight lines," are universal in both particulars; and their converse would be, not "Some but all figures bounded by three straight lines are triangles." The proposition, "Men are wine-manufacturing and wine-drinking animals," however, is particular in respect to the subject, and universal in respect to the predicate; its real meaning being, "Some men are the only animals of this class that do exist," and its converse, "All wine-manufacturing and wine-drinking animals are men." The proposition, finally, "Some rational beings are animals," is particular, both in reference to subject and predicate, its real meaning being, "Some rational beings are some (some one class of) animals," and its converse, consequently, "Some animals are rational beings."
In negative propositions also, there is the same quantification of the predicate as in affirmative ones. In the proposition, for example, "No animal is immortal," the subject and predicate are both universal; the real meaning of the proposition being, "Any animal is not any one immortal being," and its converse, "Any immortal being is not an (any one) animal." In the proposition, on the other hand, "Money is not all that is valuable," the subject is universal, and the predicate, though universal in form, is particular in fact; that is, the thought which it represents is particular. The converse, "All that is valuable is not money," really means, "Some things that are valuable are not money." The real meaning of the exposita, then, is, "All of money that exists, is not some valuable things." In the proposition, "Some currency is not metal," the subject is particular, and the predicate universal, its real meaning being, that "Some one kind of currency is not any kind of metal." In the proposition, finally, "Some men are not like other men," both the subject and the predicate are particular, the real meaning being, "Some individuals of a class are not like others of a given class." So the proposition, "Some qualities of some individuals are not like other qualities of the same individual," is equivalent to the proposition, "Some of A (the quality B) is not some of A (the quality C)."
Rightly classified, then, we have eight instead of four classes (A, E, I, O) of propositions, as far as quantity is concerned, to wit: four classes of affirmative, and four of negative, propositions. Of the affirmative we have:
1st.) The "Toto-total=A f a," those in which both the subject and predicate are universal, as to quanlity="All A is all of B." "(All) triangles are (include all) figures bounded by three straight lines."
2d.) The "Toto-partial=A f i,"--the universal affirmative recognized by logicians,--those propositions in which the subject is universal, and the predicate particular, "All men are mortal (some mortal beings)"="All A is some B."
3d.) The "Parti-total=I f a"="Some A is all of B."
4th.) The "Parti-partial=I f i"="Some A is B," that is, some B--the particular affirmative of logicians.
Of negative propositions, we have:
5th.) The "Toto-total=A n a"="Any is not any"="Any man is not any irrational animal." This is E--the universal of logicians.
6th.) "Toto-partial= A n i"="Any is not some"=All of A is not B," that is, some of B. "All of money is not all of valuable things," that is, some valuable things.
7th.) "Parti-total=I n i"="Some is not any"="Some A is not B," that is, any part of B. "Some currency is not coin," that is, any coin. This is the particular negative of logicians.
8th.) "Parti-partial=I n i"="Some--is not some," that is, "Some of A (B) is not some of A (C)." "Some men are not like some other men."
This formula, though hitherto, as Sir William Hamilton affirms, "totally overlooked by logicians, is one of the most important and commonly used of all the others. It lies, indeed, at the basis of all the processes of specification and individualization, that is, the process by which a class (genus or species) is divided into its subject-parts, the counter-process, to wit: of quantification." We have before us, for example, a certain class of objects, we immediately begin to separate them into distinct sub-classes, and these last we individualize, separate, and distinguish as individuals. How is this done? It is wholly based upon the perception (judgment), that some portions of the same class; that is, upon the judgment, that "Some A is not some A." In the sub-classes, we may find, by means of the same formula, other specific differences, and thus continue the process till we have arrived at the lowest species. This last is individualized, as above stated. On the same principle, the qualities of the individual are separated form each other, till we come to elements incapable of division--the contradictory of the proposition--"Some is not some"--being the affirmation of absolute individuality, or indivisibility. For the sake of perspicuity and elucidation, as well as to bring out more fully the true aims of logic itself, we now give the following lengthy extract from Sir William Hamilton, an extract containing an objection to the formula under consideration, and the author's reply to the same.
Parti-partial Negation.
"To this Mr. de Morgan makes the following objection:
"'Thirdly, the proposition, "Some X's are not some Y's," has no fundamental proposition which denies it, and not even a compound of other propositions. It is then open to the above objection; and to others peculiar to itself. It is what I have called (F, L, p. 153) a spurious proposition, as long as either of its names applies to more than one instance. And the denial is as follows: "There is but one X, and but one Y, and X is Y." Unless we know beforehand, that there is but one soldier, and one animal, and that soldier the animal, we cannot deny "that some soldiers are not some animals." Whenever we know enough of X and Y to bring forward "some X's are not some Y's" as what could be conceived to have been false, we know more, namely, "no X is Y," which, when X and Y are singular, is true or false with "some X's are not some Y's.'"
"Here, also, Mr. de Morgan wholly misunderstands the nature and purport of the form which he professes to criticize. He calls it 'a spurious proposition.' Spurious, in law, means a bad kind of bastard. This is, however, not only a legitimate, for it expresses one of the eight necessary relations of prepositional terms, but, within its proper sphere, one of the most important of the forms which logic comprehends, and which logicians have neglected. It may, indeed, and that easily, be illogically perverted. It may be misemployed to perform the function which other forms are peculiarly adapted more effectually to discharge; it may be twisted to sever part of one notion from part of another, the two total notions being already, perhaps, thought as distinct;--and then, certainly, in this relation, it may be considered as useless;--but in no relation can it ever be denominated 'spurious.' For why? Whatever is operative in thought, must be taken into account, and, consequently, be overtly expressible in logic; for logic must be, as it professes to be, an unexclusive reflex of thought, and not merely an arbitrary selection--a series of elegant extracts, out of the forms of thinking. Whether the form that it exhibits as legitimate, be stronger or weaker, be more or less frequently applied;--that, as a material and contingent consideration, is beyond its purview. But, the form in question is, as said, not only legitimate--not 'spurious'--it is most important.
"What then is the function which this form is peculiarly--is indeed, alone, competent to perform? A parti-partial negative is the proposition in which, and in which exclusively, we declare a whole of any kind to be divisible. 'Some A is not some A,'--this is the judgment of divisibility and of division; the negation of this judgment (and of its corresponding intergrant) in the assertion, that "A has no some, no parts," is the judgment of indivisibility, of unity, of simplicity. This form is implicitly at work in all the sciences, and it has only failed in securing the attention of logicians, as an abstract form, because, in actual use, it is too familiar to be notorious, lying, in fact, unexpressed and latescent in every concrete application. Even in logic itself, it is indispensable. In that science it constitutes no less than the peculiar formula of the great principle of specification (and individualization), that is, the process by which a class (genus or species) is divided into its subject-parts--the counter-process, to wit, of generification. And this great logical formula is to be branded by logical writers as 'spurious!' No doubt, the particularity, as a quantity easily understood, is very generally elided in expression, though at work in thought; or it is denoted by a substitute. Meaning, we avoid saying--'Some men are not some men.' This we change, perhaps, into 'men are not men,' or 'how different are men from men,' or 'man from man,' or 'these from those,' or 'some from other,' &c. Still, 'some is not some,' lies at the root; and, when we oppose 'other,' 'some other,' &c., to 'some,' it is evident, that 'other' is itself only obtained as the result of the negation, which, in fact, it pleonastically embodies. For 'other than' is only a synonym for 'is not;' 'other (or some other) A,' is convertible with 'not some A;' while there is implied by 'this,' 'not that;' by 'that,' 'not this;' and by 'the other,' 'neither this nor that;' and so on. Here we must not confound the logical with the rhetorical, the necessary in thought with the agreeable in expression.
"Following Mr. de Morgan, in his selected example, and not ever transcending his more peculiar science, in the first place, as the instance of division, I borrow his logical illustration from the class 'solider.' Now in what manner is the genetic notion divided into species? We say to ourselves: 'Some soldier is not some soldier,' for 'some soldier is (all) infantry; some soldier is (all) cavalry,' &c., and '(any) infantry is not any cavalry.' A parti-partial negative is the only form of judgment for division, of what kind soever be the whole (and Mr. de Morgan can state for it no other). Again: in the second place, as the example of indivisibility: 'Some of this point is not some of this (same) point.' Such a proposition, Mr. de Morgan, as a mathematician, cannot admit; for a mathematical point is, ex hypothesi, 'without some--without some, and some'--without parts, same, and other; it is indivisible. He says, indeed, that a parti-partial negative cannot be denied. But if he be unable to admit, he must be able to deny; and it would be a curious--a singular anomaly, if logic offered no competent form for so ordinary a negation; if we could not logically deny that Socrates is a class-- that an individual is a universal--that the thought of an indivisible unit is the thought of a divisible plurality."
Criteria by which Propositions properly falling under these different classes may be distinguished from each other.
We will now attempt to give, what our author has not formally done, special criteria, by which we may distinguish propositions which fall under these different classes from one another. The following, we think, will be admitted as universally valid, as such criteria:
1. When the object of the proposition is to give a correct and full definition of a term or subject--or to assert the essential characteristics of an individual or class--or finally, to assert a real and perfect identity between the subject and predicate, then the proposition is to be classed as toto-total affirmative. Thus, in the definition, "A triangle is a figure bounded by three straight lines," we mean, all triangles include all such figures. So in all full definitions. When, on the other hand, we affirm that " All equilateral triangles are equiangular," the predicate represents a characteristic conception of the subject. Of course, it is found only in the subject, and always found in it. The subject and predicate, therefore, stand related; as, "All A is all of B." Of the same character is the proposition, "A good government is one that has the good of its subjects as its object." When we say, finally, "A Christian is a man who fears God," we mean that there is a real identity between the subject and predicate in this case. The proposition, therefore, like those before mentioned, is equivalent to "All A is all of B." The converse of all such propositions, consequently, is a universal affirmative.
2. When the judgment really affirmed in a proposition is, that individuals belong to a certain class, as, "John is a man," or that all the individuals represented by an inferior conception rank specifically under a superior conception, as, "All men are animals," "All men are mortal," &c., then the proposition is "toto-partial," the universal affirmative of logicians; that is, the subject is universal and the predicate particular; and the converse is a particular affirmative, "Some man is John," "Some mortal beings are men," &c.
3. When the judgment affirmed in a proposition is, that a quality assumed as attaching exclusivity to a certain class, but not to all the members thereof, belongs exclusively to that class--as, "Men posses wealth;" or, that a superior conception embraces under it all the individuals included under an inferior one--as, "Some animals are men," "A part of currency is gold coin," then the proposition is parti-total, the exposita being, "Some part of currency is all of gold coin," &c.; and the converse a toto-partial affirmative, to wit: "All of wealth is possessed by men (some men)," "All gold coin is currency (some part of currency)."
4. When the judgment affirmed in a proposition is, that some, not all, individuals of one class are like some, not all individuals of another, as, "Some men are long-lived animals," then the proposition is a parti-partial affirmative, and its converse of the same class, "Some long-lived animals are men."
5. When the judgment affirmed in a proposition is this, that no individual of one class is a member of another class, "No man is an angel;" or, that a certain individual is utterly void of given characteristics or class of characteristics, "John possesses no virtue;" or, that a certain individual does not belong to a certain class, "A is not an American," then the proposition is a toto-total negative, and its converse will be of the same character; as, "No angel is a man (any man)," "No virtue attaches to John," "No American is A," &c.
6. When one conception is admitted to represent all that another does, and other things besides, and when the object of the proposition is to deny that what is embraced in the former includes all that is embraced in the latter--as, "All of A is not all of B," that is, some of B--then the proposition is a toto-partial negative; and its converse a parti-total negative, "Some B is not A (any of A)". So when the object of a proposition is to deny of an individual the totality of characteristics represented by a given conception; as, "A has not all the vices," that is, some vices.
7. When the judgment affirmed in a given proposition denies that some individuals of a given class have any of the characteristics belonging to other individuals of the same class, or to any individual of another class--as, "Some members of the university are not studious," "Some Americans are not patriots," &c.; or, that all the individuals embraced under a superior conception are found among those embraced under an inferior one--as, "Some animals are not brutes;" the proposition is then parti-total, and its real converse would be a toto-partial negative, "All A is not some of B:" a certain class of studious persons does not include some members of the university, or any studious person is not some member of the university.
8. When the judgment affirmed in a given proposition denies the absolute indivisibility of any object, or the absolute likeness of all its qualities to one another--as, "Some A (the quality B) is not some A (the quality C);" or, that some members of a given class are not like other members of the same class--as, "Some men are not men," that is, do not belong to the class who properly represent humanity; then the proposition is a parti-partial negative, and its converse the same.
Such are the principles of classification of propositions, when respect is had to their sense, and not to the mere language in which the sense is expressed. The rules presented in the preceding section are applicable, when reference is had, not to the sense exclusively, but to the mere words of the propositions themselves.
Scholia 1. The most philosophical or scientific classification of propositions would be, as Sir William Hamilton observes, into two classes--the definite and indefinite. All universal and all individual propositions are definite, affirming or denying in regard to each and every individual referred to. The terms, "John, any man, no man," &c., are each alike and equally definite. The term, "Some (some men)," is indefinite. So the propositions, "John is an American," "Every man is mortal," "No man is a brute," &c., are each and all alike definite propositions; while the proposition, "Some men are learned," is indefinite. As all propositions are either individual, universal, or particular, and as the two classes first named are definite, and the latter class indefinite, all propositions, if strict scientific precision were observed, would be classed as definite or indefinite.
Scholia 2. Propositions whose subject and predicate are both definite, may properly be called definite-definite; those whose subject and predicate are both indefinite, might be called indefinite-indefinite propositions; those whose subject is universal and predicate particular, the definite-indefinite; and, finally, those whose subject is particular and predicate universal, the indefinite-definite. We thus have a complete and exhaustive system of classifying propositions.
Scholia 3. All conversion of propositions in accordance with the most perfect scientific procedure, is, as Sir William Hamilton has affirmed, exclusively simple. Example: "All men are mortal." Why is the converse of this proposition this, "Some mortal beings are men?" The reason is obvious, the subject of the exposita is, in fact universal, while the predicate is particular. The converse, on the other hand, as thought, is parti-total, to wit: "Some mortal beings are all of mankind." Hence, we have in reality, if we refer, not to the form, but to the matter of the judgment, that is, to what is given in the thought, but one form of conversion, that is, simple. Unless this principle is kept distinctly in mind, logic, as a science, will not be understood.
CHAPTER III.
ANALYTIC OF ARGUMENTS OR SYLLOGISMS.
SECTION I.--ARGUMENT DEFINED AND ELUCIDATED
AN argument is an intellectual process in which one judgment is deduced form another. All judgments are either intuitive or inferential, immediate or mediate. When the relation between two objects or conceptions is such, that the mind has, from the nature of said relation, a direct and immediate perception of the same, the judgment affirming such relation is called intuitive or immediate. When, on the other hand, this relation is dicerned through other judgments, the judgments affirming such relation is said to be inferential or mediate.
The characteristics of all valid immediate or intuitive judgments have already been given. When the relations between any two objects or conceptions, A or B. are not immediately discernible, it is self-evident that such relations can be discerned but upon one condition--that each of those objects sustain known or knowable relations to this known object, C. Through their discerned relations to this known object, we may infer (discern) their relations to each other. Thus, if A and B are both equal to C, we infer that they must be equal to each other. If, on the other hand, one agrees and the other disagrees with C, we infer that they must disagree with each other. On this principle, exclusively, all mediate judgments are deduced.
The term C, with which the others are compared, is called the middle term. Those compared with it (A and B), are called the extremes. Hence we remark:
1. That in no given argument can there be more than one middle term. If there was, then the extremes would not be compared with the same thing, and nothing pertaining to their relations to each other could be inferred from the comparison.
2. In such argument there must be two extremes, and there can be no more. If there were more than two, there would be a corresponding number of distinct arguments.
3. There must be, in such argument, when stated at length and in full, three, and no more, and no less, propositions: two called premises, in one of which, one, and in the other the remaining extreme, is compared with the middle term, and the conclusion or inference in which the relation of the two terms is affirmed. The truth of this statement is too evident to need any further elucidation.
NOTE.--The subject of the conclusion is, in logics generally, called the minor term, and the predicate of the conclusion the major term. The premise in which the minor term is compared with the middle, is called the minor premise, and that in which the major term is compared with the middle, is called the major premise.
4. When each premise, together with the conclusion, is stated in its proper form and order, the argument is then called a syllogism; and this is what is meant by the term syllogism. For example:
Every C is B;
Every A is C;
Therefore Every A is B.
5. From the nature of the syllogism, as above defined and elucidated, it is manifest that the following is, and must be, the universal canon or principle in conformity to which all valid conclusions must be deduced, namely: All conceptions or terms which agree with one and the same third conceptions or terms, the one agreeing and the other disagreeing with said common conception or term, disagree with the other. The validity of this principle is self-evident. All forms, also, which the syllogism can assume grow out of the diversified applications of this one principle; and the principle itself, always one and identical, assumes different forms according to the nature of the relations to which it is applied.
DIVERSE FORMS OF THE SYLLOGISM.
The syllogism assumes divers forms, each of which demands especial elucidation. Among these we notice in this connection the following:
SECTION II.--THE ANALYTIC AND SYNTHETIC SYLLOGISM.
When the conclusion (more properly the theorem or proposition to be proved) is stated first, and the propositions by which it is to be proven are subsequently stated, the syllogism is said to be analytic. For example:
Every A is B;
Because Every C is B;
And Every A is C; or,
"Caesar was a usurper," because, perforce, he seized the reins of government in Rome, and every one who does this is a usurper. On the other hand, when the premises are stated first in their proper order, and the conclusion last, the syllogism is then called synthetic. For example:
Every C is B;
Every A is C;
Therefore Every A is B.
Every one who forcibly seizes the reins of government is a usurper. Caesar did this. Therefore, "Caesar was a usurper." The following observations will sufficiently elucidate the nature and relations of these two distinct forms of the syllogism:
These distinct forms of the Syllogism elucidated.
1. They differ not at all in thought, but only in form. A mere inspection of the two forms of syllogisms, as given above, will render this statement self-evident. Each form consequently is equally valid.
2. The analytic is the most common and natural form of the syllogism, it being a far more common procedure in reasoning to state first the proposition to be proved (conclusion or thesis), and then to present the evidence of its truth, than it is to take the opposite course.
3. "In point of fact," to quote the language of Sir William Hamilton, to whom we would very gratefully acknowledge ourselves indebted for the above distinction, "the analytic syllogism is not only the more natural, it is even presupposed by the synthetic. To express in words, we must analyze in thought the organic whole--the mental simultaneity of a simple reasoning; and then we may reverse in thought the process, by a synthetic return. Further, we may now announce the reasoning in either order; but, certainly to express it in the essential, primary, or analytic order, is not only more natural, but more direct and simple, than to express it in the accidental, secondary, or synthetic."
4. The following citation from the same author will still further elucidate the importance of the distinction under consideration:
"This in the first place relieves the syllogism of two one-sided views. The Aristotelic syllogism is exclusively synthetic; the Epicurean (or Neocletian) syllogism was--for it has been long forgotten--exlusively analytic; while the hindoo syllogism is merely a clumsy agglutination of these counter-forms, being nothing but an operose repetition of the same reasoning, enounced, 1st. Analytically; 2d. Synthetically. In thought the syllogism is organically one; and it is only stated in an analytic or synthetic form from the necessity of adopting the one order or the other, in accommodation to the vehicle of its expression--language. For the conditions of language require that a reasoning be distinguished into parts, and these detailed before and after each other. The analytic and synthetic orders of enouncement are thus only accidents of the syllogistic process. This is, indeed, shown in practice; for our best reasonings proceed indifferently in either order.
"In the second place this central view vindicates the syllogism from the objection of petitio principii, which professing logically to annul logic, or at least to reduce it to an idle tautology, defines syllogistic--the art of avowing in the conclusion what has been already confessed in the premises. This objection (which has at least an antiquity of three centuries and a half) is only applicable to the synthetic or Aristotelic order of enouncement, which the objectors, indeed, contemplate as alone possible. It does not hold against the analytic syllogism; it does not hold against the syllogism considered aloof from the accident of its expression; and being proved irrelevant to these, it is easily shown in reference to the synthetic syllogism itself, that it applies only to an accident of its external form."*
*[The error involved in the above objection, even in its application to the synthetic syllogism, may be made by a single illustration. For example:
Gold is precious;
This substance is gold;
Therefore It is precious.
It is very true, that what is here announced in the conclusion, is, in a certain from, confessed in the premises. The object of the syllogism, however, is to announce in form, what has previously been ascertained by investigation. Suppose the conclusion to be denied; tests would then be applied to verify the minor premise. When its truth has been established, then, and not till then, it logically takes its place as a premise.]
5. As the analytic and synthetic syllogisms differ only in form and are identical in thought, they mutually elucidate each other. Suppose we have argued the truth of some proposition until we have, as we suppose, proved it. The argument has, as is almost universally the case, been conducted wholly in the analytic form. We now wish to test the validity of the argument. The best way to accomplish this will be, in most instances, to change the form from the analytic to the synthetic, and see whether the premises necessitate, as an inference, the truth of the proposition affirmed to have been proven.
6. For the reasons which have been already stated, the laws and principles which govern these two forms of the syllogism are one and identical. "Every especial variety in the one," to use the language of the author above referred to, "has its corresponding variety in the other."
7. The error, we remark in the last place, of modern and most of the ancient logicians, in treating the synthetic as the only and exclusive form of the syllogism , is now sufficiently manifest and no addition remarks upon the subject are necessary.
SECTION III.--FIGURED AND UNFIGURED SYLLOGISMS.
Science is indebted to Sir William Hamilton for another division of syllogisms of fundamental importance to a full and distinct understanding of the doctrine of the syllogism in general, or of the universal process of reasoning. We refer to his distinction between the figured and unfigured syllogism.
In the figured syllogism, as we shall see hereafter, the terms compared sustain to each other, in the several propositions, the relations of subject and predicate, the figure of the syllogism referring to the situation of the middle term in the premises relatively to the extremes.
In the unfigured syllogism, "the terms compared do not stand to each other in the reciprocal relation of subject and predicate, these being in the same proposition, on the other hand, both subject and predicate." For example:
All C and some B are equal;
All A and all B are equal;
Therefore All C and some A are equal; or,
C and A are unequal.
Again, a question arises whether C and A were together during the whole of a given journey taken by the latter. In reply, it is affirmed, that from sources perfectly reliable, it has been ascertained that in the journey referred to, C and B were in company only part of the distance traveled by the latter, and that from sources equally reliable, it has been ascertained that A and B were in company during the whole distance traveled by each. The inference is hence drawn that C traveled but a part of the distance referred to in company with A. This conclusion is perfectly valid, and the form of argumentation by which it is reached is as legitimate as any other, and withal quite as worthy to be elucidated in a treatise on logic; and that for the obvious reason that it is one of the most common forms of reasoning in almost all departments of thought. Indeed, logic, as a science, will be fundamentally incomplete and imperfect, while it overlooks this one form of the syllogism. Without further remarks, we shall now proceed to elucidate some of the laws and principles of the unfigured syllogism.
PRINCIPLES AND LAWS OF THE UNFIGURED SYLLOGISM.
The Canon of this Syllogism.
The canon of this syllogism we give in the language of the author above quoted from. "In as far as two notions (notions proper or individuals) either both agreeing, or one agreeing, and the other disagreeing, with a common third notion: just so far those notions do or do not agree with each other." Take the following examples in illustration:
All C and all or some B are equal;
All A and all B are equal;
Therefore All C and all or some A are equal;
And consequently, C and A are, or are not, equal to each other.
Again: All C and one-half of B are equal;
All A and all B are equal;
Therefore All C and one-half of A are equal; or,
C equals one-half of A
Again: A to B, and E to F, are in the same prepositional relations;
But, E is three times F;
Therefore A is three times B.
If the minor had been in this case, A is three times B, the conclusion would have been, that E is three times F; and the former couplet might as properly have been the minor, as the latter. Had the relation above named been that of analogy, the argument would be the same.
The following present other forms of the same Syllogism
All C and some B are equal to Y;
All A and all B are equal to Y;
Therefore Some C is equal to all A; or,
All A is equal to some C.
Suppose that it is known that the fortunes of C and B together are larger than that of Y (or all C and some B are equal to Y), while it has been ascertained that the united fortunes of A and B are just equal that of Y (all A and all B are equal to Y). We at once infer that the fortune of C is greater than that of A, for the obvious reason, that when each is added to the same thing the amounts differ as above stated.
Again: All C and half or all B are equal to Y;
All A and all B are equal to Y;
Therefore All C is equal to half or all A.
So if we should say that C minus, multiplied or divided by B, is equal to Y, and that A similarly related to B is equal to Y, the conclusion would be A=C. If C thus related to B is equal Y, and A thus related is greater or less than Y, we have the conclusion that C is greater or less than A, as the case may be.
The application of the above examples to negative conclusions is so obvious, that little need be said on this topic. In all instances in which the relation of equality between two conceptions has been proven, that of its absence and also that of greater or less may be denied. So when that of greater or less has been proved, the opposite of what is proven, together with the relation of equality, may be denied. For example:
All C and all B=Y;
All A and all B do not=Y;
Therefore C and A are not equal to each other.
So, also, when two conceptions pertain to their objects as always coexisting, and neither as existing separate form the other, or as sustaining to each other the relation of universal compatibility, &c., and when the object of a third conception is given as never coexisting, or as being incompatible with the object of either of the others, the same relation between this third and the remaining one may be denied. For example:
C and B always coexist,--or, are universally compatible;
A and B never coexist,--or, are wholly incompatible;
Therefore C and A never coexist,--or, are not compatible.
General Remarks upon this form of the Syllogism.
The following general remarks upon this form of the syllogism are deemed worthy of especial notice:
1. In it, the order of the propositions is, to use the language of Sir William Hamilton, "perfectly arbitrary." In other words, the unfigured syllogism has no proper major and minor terms or premises. A mere inspection of the able examples will render this statement self-evident.
2. In this syllogism, also, the terms of the conclusion are so manifestly and formally equivalent and definite, as far as distribution is concerned, that conversion is almost if not quite always simple, both in thought and form. Each term is given as universal or particular.
3. This syllogism may also, with perfect propriety, be given in the synthetic or analytic form. We may, for example, as properly say, "C and A are equal," because "A and B, on the one hand, and C and B on the other, are equal to Y," as to state the premises first, and then give the conclusion as an inference.
4. While this form of the syllogism had, until Sir William Hamilton presented it, been wholly overlooked by logicians, it presents one of the most common and necessary forms of valid reasoning among all classes of the community, and especially in the inductive sciences. Without this form of the syllogism, therefore, logic, as a science, would be wholly incomplete and limited in its applications.*
*(In justice to myself, and to truth, I would say, that before I had seen what Sir William Hamilton has written upon this subject, or had even heard that he had spoken or written any thing upon it, my own independent investigation had led me to a conception of this form of the syllogism, and to a careful inquiry into its principles and laws; and at the time when I read what he has written, my mind was employed in a vain attempt to find a place for it, in some department of the figured syllogism, and that under the apprehension, that what logicians had assumed as true, was so, to wit: that the latter is the only real form of the syllogism itself. I saw clearly, that in many forms of valid reasoning, the terms compared did not "stand to each other in the reciprocal relation of subject and predicate, being in the same proposition, either both subjects, or both predicates." I saw also, that the extremes in such cases, are not, as is true of the figured syllogism, each singly, and by itself, compared with the middle term; but, that both alike, first one and then the other, stand with the middle, in the common relation of subject and predicate; and that, in all such cases, it made no difference as to the order of the premises. Yet I was under the impression, that after all, they must have a place among the common forms of the syllogism, have no suspicion that there could be any other legitimate form. Form this perplexity I was relieved by the author referred to, and shall ever esteem it a high privilege to acknowledge the obligations which I thereby owe to him.)
SECTION IV.
THE FIGURED SYLLOGISM.
This form defined.
We now advance to a special consideration of the firgured syllogism. That which distinguishes this form of the syllogism from every other is this, the fact which we have already stated, that in all the propositions the terms are related to each other as subject and predicate.
Common assumption on the subject.
It has been commonly assumed that the terms employed in the various propositions, stand related to each other as inferior and superior conceptions, the subject being the inferior and the predicate the superior. On this assumption the universal rules of distribution are based, to wit: that while all universal propositions distribute the subject, all negative and no affirmative ones distribute the predicate. The latter principle can be true but upon the supposition, that the predicate is a superior and the subject an inferior conception. In the proposition, "All men are mortal," for example, the term mortal is not distributed, for the reason that it has a wider application than the term men. Suppose we say "X=Z;" then the predicate as well as the subject is distributed, and that for the obvious reason that Z, in this proposition, is conception in no form or sense inferior or superior to X. The converse of the former proposition is, "Some mortal beings are men," while that of the latter is "Z=X." In this last judgment neither conception is inferior or superior to the other, and, therefore, both terms are distributed.
Influence of Assumptions.
This fact presents another example of the influence of assumptions. When they once obtain a place in science as first truths or principles, the assumptions themselves are not examined, because their truth is always taken for granted. How true this is of the case before us! Since the days of Aristotle the principle has been assumed, that in all propositions, with accidental exceptions, the subject is the inferior and the predicate the superior conception; and from hence, the principle that no affrimative proposition distributes the predicate. "It may happen, indeed," says Dr. Whately, "that the whole of the predicate in an affrimative may agree with the subject; e. g. it is equally true, that 'All men are rational animals,' and 'All rational animals are men;' but this is merely accedental, and is not at all implied in the form of expression, which alone is regarded in logic."
It is true, as Dr. Whately observes, that in cases where the whole predicate in the affirmative proposition agrees with the whole subject, the fact does not appear from the mere form of expression; and it is equally true, on the other hand, that from the mere form of the expression it does not appear when the whole predicate does not agree with the whole subject. This fact is always to be determined by the nature of the conceptions compared, and the nature of the relations between them.
Principles determining the distribution of the Predicate.
We are now prepared for a distinct statement of the principles which determine the distribution and non-distribution, not only of the subject, but predicate in all judgments employed in reasoning. They are the following:
1. Whenever the subject and predicate stand related as inferior and superior conceptions, then they follow the rules of distribution commonly laid down in treatises on logic, to wit:
(1) All universal propositions (and no particular) distribute the subject;
(2) All negative (and no affirmative) the predicate.
2. Whenever the terms of a proposition belong to the same class, and are compared relatively to the principle of equality and difference, as equal, greater, or less, or when they fall under the relation of proximity or distance in time or place, &c., then in affirmative and negative propositions alike, the predicate follows the same principles of distribution as the subject. So, when the subject and predicate are correlative terms; as, "Father and son; cause and effect," &c., neither , as a conception, is superior to the other; and the predicate, when it as the correlative of the subject becomes by conversion the subject, its quantity is the same as that of the subject was. Finally, when the predicate is used to define the subject, the same principle obtains. The proposition, for example, "A is the cause of B," when converted becomes, "Be is the effect of A."
That the rules of distribution above stated are applicable universally to all propositions of the first class, is too evident to require much elucidation. In all cases where any class of facts are placed under a universal principle, as, for example, "Murder is criminal," "Such and such actions are right or wrong;" or, one or the other of these under a generical conception, as in such cases the predicate has a wider application than the subject, and is hence never distributed in affirmative propositions. Even in negative propositions, the term which has in itself the wider application is most commonly, though not always, the predicate. Thus, in the language of another, it is more natural to say, that "The apostles were no deceivers," than that "No deceivers are apostles."
Let us now look at propositions of the second class of judgments. When we say "X=Z," for example, the two terms are compared throughout their whole extent, and if one is distributed, the other of course must be, or the equality would not exist. Conversion, in all such cases, is simple, and never by limitation. If we say "X is greater than Z," the converse holds universally, "Z is less than X;" each term being alike and equally distributed in both cases. If we say, "X is the cause of Z," then the converse, Z is given equally universally, in the correlative form, as the effect of X. The distribution of the subject and the predicate in both cases is equal.
The same may be shown to hold true in all the cases which are given as falling under this class. From the nature of the case it cannot be otherwise. We are not here endeavoring to find under what superior conception a given inferior one ranks, or what inferior conception any given superior one includes. We are not inquiring under what general principle any given class of facts are to be classed. But we are inquiring in regard to objects of the same class, and that relatively to the question of their agreement or disagreement; as, whether they are equal or unequal, which is the greater and which the less, &c. In all such cases it makes no difference whatever which term is the subject and which the predicate; both, in all cases, being equally distributed.
Fundamental mistake in developing the science of Logic.
In all treatises on the science of logic, as far as we know, with the exception of Sir William Hamilton's works, and "Thomson's Laws of Thought," the figured syllogism has been considered as covering all forms of the categorical argument. In developing the syllogism it has also been assumed, as we have said, that the terms employed in the syllogism are related as inferior and superior conceptions. Now while the science of logic is developed upon such principles, it must remain as one of the most imperfect and unsatisfactory of all the science. Take the principle laid down as holding universally, that no affirmative propositions distribute the predicate, and apply it to any of the processes in the mathematics, and we shall find it wholly to fail; for these almost, if not quite universally, distribute the predicate equally with the subject. The entire science of the mathematics is based upon illogical principles, if this principle is correct. Every one of its principles is convertible, not by limitation, but simply. So of its subsequent deductions, not one of them accord with the principle, that no affirmative propositions distribute the predicate. Take, as an example, the proposition, "The square of the hypothenuse of a right-angled triangle is equal to the sum of the squares of it two sides." If no affirmative propositions distribute the predicate, and the universal affirmative ones can be converted but by limitation, then the converse of the above proposition would be this: "Some part of the sum of the square of the two sides of such triangle equals the square of the hypothenuse." But this is not the converse of the above proposition; that converse being universal and not particular, and that for the reason that all universal affirmative propositions of this class distribute the predicate as well as the subject. Nor are such propositions of unfrequent occurrence. We everywhere meet them in almost all departments of human thought. Indeed, it may be questioned which is most numerous, those universal affirmative propositions which do, and those which do not, distribute the predicate as well as the subject. Take another example form common life, to wit: "A resembles or is unlike B." The converse of all such propositions is not a particular but a universal affirmative, to wit: "B resembles or is unlike A." We need not add further illustrations.
DIVISION OF THE PRESENT SUBJECT.
In further elucidating the figured syllogism, we propose to pursue the following order of investigation:
1. Those forms of the syllogism which have been commonly treated of as including all forms of the categorical argument, to wit: those forms in which the terms employed are related to each other as inferior and superior conceptions.
2. Those forms in which affirmative propositions as well as negative distribute the predicate.
3. We shall then combine the two classes, and endeavor to develop the general laws of the figured syllogism as such.
I. THOSE FORMS OF THE SYLLOGISM WHICH HAVE BEEN COMMONLY TREATED OF AS INCLUDING ALL FORMS OF THE CATEGORICAL ARGUMENT, TO WIT: THOSE FORMS IN WHICH THE TERMS EMPLOYED ARE RELATED TO EACH OTHER AS INFERIOR AND SUPERIOR CONCEPTIONS.
In entering upon the investigations which follow, we would request the reader to keep distinctly in mind the kind of judgments to be treated of, to wit: those in which the subject and predicate represent respectively inferior and superior conceptions; conceptions related to each, as individual, specificial, and generical conceptions.
PRELIMINARY REMARKS UPON THIS FORM OF THE FIGURED SYLLOGISM.
Before we proceed further, we would invite special attention to the following preliminary remarks upon the department of the subject before us.
Only proximate conclusions obtained.
On a moment's reflection it will appear perfectly evident, that in this form of the syllogism we obtain only conclusions approximating the truth; that is, we determine not what individuals are in themselves, but with what class or classes they take rank. Take, for example, the following syllogism:
All men are mortal;
C is a man;
Therefore C is mortal, i. e. some mortal being.
We have here determined not the special characteristics of C, but the particular and special class to which he belongs. This is the character of all conclusions obtained through this form of the syllogism, and form the nature of the case it must be so.
The principle of Extension and Intension, or of Breadth and Depth, as applied to the Syllogism.
In our elucidation of superior and inferior conceptions we showed that, while the matter of the latter is much greater than that of the former, the sphere of the former is in a corresponding degree more extensive than that of the latter. In regard to matter, the individual conception embraces more elements than the specificial, and this last more than the generical. At the same time, this last conception is applicable to more objects than the second, and the second more than the first. The terms extension and intension, breadth and depth, are employed by Sir William Hamilton to represent these two opposite principles. In regard to depth (the matter of the conception), the individual is the lowest of all; that is, includes the greatest number of elements. In regard to the breadth, the number of objects which the conception represents, that is, relatively to its sphere, the generical conception is the most extensive of all others. The two quantities are in relations perfectly reverse to each other. The greater the depth, the less its breadth of a conception; and the greater it breadth, the less its depth. In regard to breadth, the inferior conception is contained under the superior. In regard to depth, the superior is contained in the inferior.
In this form of the figured syllogism the propositions always refer to one or the other of these principles. In affirmative propositions the subject is an inferior conception, and the predicate a superior. When of the two conceptions in a negative proposition one has the greater breadth than the other, this one, as we have before said, is commonly the predicate.
Now every proposition whose subject is an inferior and predicate a superior conception, may be understood relatively to the principle of intension (depth) or extension (breadth), and the meaning of the proposition will be as the principle to which it is referred. Thus the proposition, "All men are mortal," means, in regard to intension, that the quality represented by the term mortal, or mortality, belongs to every individual of the race; and in regard to extension, that all men belong to the class of mortal beings.
In further elucidation of this very important department of our subject, we here present the following extract form "Thomson's Laws of Thought." Of the last two examples cited at the close of the extract, we would remark that the term U designates toto-total affirmative propositions--those in which both subject and predicate are distributed; and Y parti-total affirmatives--those in which the subject is particular and the predicate universal; as, "Some X is all of Z."
"Import of Judgments (Extension and Intension--Naming).
"Upon the examination of any judgment which appears to express a simple relation between two terms, we shall find it really complex, and capable of more than one interpretation. 'All stones are hard,' means, in the first place, that the mark hardness is found among the marks or attributes of all stones; and in this sense of the judgment the predicate may be said to be contained in the subject, for a complete notion of stones contains the notion of hardness and something more. This is to read the judgment as to the intention (or comprehension) of its terms. Where it is a mere judgment of explanation, it will mean, 'the marks of the predicate are among what I have know to be among the marks of the subject;' but where it is the expression of a new step in our investigation of an accession of knowledge, it must mean, 'the marks of the predicate are among what I now find to be the marks of the subject.'
"Both subject and predicate, however, not only imply certain marks, but represent certain sets of objects. When we think of 'all stones,' we bring before us not only the set of marks--as hardness, solidity, inorganic structure, and certain general forms--by which we know a thing to be what we call a stone, but also the class of things which have the marks, the stones themselves. And we might interpret the judgment, 'All stones are hard,' to mean that, 'The class of stones is contained in the class of hard things.' This brings in only the extension of the two terms according to which, in the example before us, the subject is said to be contained in the predicate. Every judgment may be interpreted form either point of view; and a right understanding of this doctrine is of great importance. Let it be noticed, against a mistake which has been re-introduced into logic, that all conceptions, being general, represent a class; and that to speak of a 'general name' which is not the name of a class is a contradiction of terms. But this is very different from asserting that a class of things corresponding to the conceptions of 'giants,' 'centaur,' and 'siren,' are all of classes; but every one knows who realizes them, that the only region in which the classes really exist, is that of poetry and fiction. The mode of existence of the things which a conception denotes is a mark of the conception itself; and would be expressed in any adequate definition of it. It would be insufficient to define 'centaurs' as a set of monsters, half men and half horses, who fought with the Lapithae, so long as we left it doubtful whether they actually lived and fought, or only were feigned to have done so; and by some phrase, such as 'according to Ovid,' or, 'in the mythology,' we should probably express that their actual existence was not part of our conception of them.
"The judgment selected as our example contains yet a third statement. We observe marks; by them we set apart a class; and, lastly, we give a class or name a symbol to save the trouble of reviewing all the marks every time we would recall the conception. 'All stones are hard,' means that the name hard may be given to every thing to which we apply the name stones.
"All judgments, then, may be interpreted according to their intension, their extension, and their application of names or descriptions; as the following examples may help to show:
"A. 'All the metals are conductors of electricity,' means:
"Intension.--The attribute of conducting electricity belongs to all metals.
"Extension.--The metals are in the class of conductors of electricity.
"Nomenclature.--The name of conductors of electricity may be applied to the metals (among other things).
"E. 'None of the planets move in a circle.' means:
"Intension.--The attribute of moving in a circle does not belong to any planet.
"Extension.--None of the planets are in the class (be it real, or only conceivable) of things that move in a circle.
"Nomenclature.--The description of things that move in a circle cannot be applied to the planets.
"I. 'Some metals are highly ductile,' means:
"Intension.--The mark of great ductility is a mark of some metals.
"Extension.--Some metals are in the class of highly ductile things.
"Nomenclature.--The name of highly ductile things may be applied to some metals.
"O. 'Some lawful actions are not expedient,' means:
"Intension.--The attribute of expedience does not belong to some lawful actions.
"Extension.--Some lawful actions do not come into the class of expedient things.
"Nomenclature.--The name of expedient cannot be given to some lawful actions.
"U. 'Rhetoric is the art of persuasive speaking,' means:
"Intension.--The attributes of the art of persuasive speaking, and of rhetoric are the same.
"Exension.--Rhetoric is coextensive with the art of speaking persuasively.
"Nomenclature.--'The art of persuasive speaking' is an expression which may be substituted for rhetoric.
"Y. 'The class of animals includes the polyps,' means:
"Intesion.--The attributes of all the polyps belongs to some animals.
"Extension.--The polyps are in the class of animals.
"Nomenclature.--The name of polyps belongs to some animals."
Direct and indirect conclusion.
All are aware that in every valid syllogism there are two conclusions deducible from the premises laid down. One of these conclusions is direct and immediate, and the other often, though not always, as we shall see, indirect. In the premises, for example, "All M is X, and all Z is M," we have the direct conclusion, that "All Z is X." The converse of this is, "Some X is Z," and this last proposition may be called the indirect conclusion. It is optional, in view of the premises, to draw first the direct conclusion, and then by conversion to obtain the indirect conclusion, or to assume this last inference as implied in the premises.
Character of all the propositions employed in this form of the Syllogism.
The character of all the propositions of this form of the syllogism next claims our attention. Every premise and conclusion is either a universal affirmative proposition (A), a proposition with a distributed subject and an undistributed predicate; a particular affirmative (I) with both the subject and the predicate undistributed; a universal negative with both terms distributed (E); or, finally, a particular negative with the subject undistributed and the predicate distributed (O). All propositions constituted of inferior and superior conceptions must belong to one or the other of these classes.
Letters to be employed.
In further prosecuting our investigations we will, in elucidating the syllogism, make use of the letters X and Z to represent the extremes, and M to represent the middle term.
CANON AND LAWS OF THIS FORM OF THE SYLLOGISM--CONDITIONS ON WHICH WE CAN OBTAIN THE DIFFERENT CLASSES OF CONCLUSIONS ABOVE NAMED; THAT IS, A, I, E, O.
We now advance to a very important inquiry, to wit: the special relations of the extremes to the middle term, relations in which we can obtain these different classes of conclusions.
Universal Affirmative Conclusions.
There is but one conceivable relation of two such terms to a common third term, a relation from which a universal affirmative conclusion can be deduced, to wit: when all of the middle is contained in one extreme, and all of the other extreme is itself contained in said term. If all of M is in X and all of Z is M, then, of course, all of Z must be in X. Change the relations of the terms in any form or degree, and it will at once be perceived that no such conclusion can then be logically deduced. Stated in form this is the relation referred to:
All M is X;
All Z is M;
Therefore All Z is X.
Universal Negative Conclusions.
There are two relations of the extremes to the middle term from which universal negative conclusions arise, namely:
1. That in which all of the middle term is excluded from one extreme, and all of the other is included in said term. If none of M is in X, and all of Z is in M, then, of course, none of Z is in X. From this relation we have one form of argument:
No M is X;
All Z is M;
Therefore No Z is X.
2. When all of one extreme is included in the middle term, and all of the other is excluded from said term. If, for example, all of X is in M and none of Z is in M, of necessity, none of Z is in X. Here we have two forms, to wit:
No X is M; All X is M;
All Z is M; No Z is M;
Therefore No Z is X. Therefore No Z is X.
Particular Affirmative Conclusions.
There are three relations of two terms to a common third term, relations from which particular affirmative conclusions may be logically deduced. They are the following:
1. When all of the middle term is contained in one extreme, and part of the other extreme is contained in said term. So far as this part, which is common to the two extremes, is concerned, they must agree with each other, and a particular affirmative conclusion is logically valid. If all of M is in X and a part of Z is in M, then, of course, a part at least of Z must be in X; and the proposition, "Some Z is X," will be valid. Of this class we have one example, to wit:
All M is X;
Some Z is M;
Therefore Some Z is X.
2. When all of the middle term is contained in each extreme. If all of M is in both X and Z, then, so far as each contains M, they must agree, and the proposition, "Some Z is X," must be logically valid. Of this class, also, we have but one example:
All M is X;
All M is Z;
Therefore Some Z is X.
3. When all of the middle term is contained in one extreme and part of it in the other. So far as this part, which is common to the two extremes, is concerned, they must agree with each other, and the conclusion, "Some Z is X," must be held as logically valid. Under this division we have two forms of valid argument. For example:
All M is X; Some M is X;
Some M is Z; All M is Z;
Therefore Some Z is X. Therefore Some Z is X.
Particular Negative Conclusions.
In the following relations particular negative conclusions are valid.
1. When some of one extreme is contained in the middle term and the whole of the other is excluded from it. In this case the part of the one extreme contained in the middle must be excluded from the other extreme, all of which is excluded from the middle term, and the conclusion, "Some Z is not X," is valid. We would here remark that a part of one term is contained in another, when the former in the same proposition as the latter is the subject of a particular, or the predicate of a universal or particular affirmative proposition. A part of X, for example, is equally contained in M in the propositions, "Some X is M," "All M is X," and "Some M is X." In this relation we have the following forms:
(1) (2) (3)
No M is X; No X is M; No M is X;
Some Z is M; Some Z is M; All M is Z;
:. Some Z is not X. :. Some Z is not X. :. Some Z is not X.
(4) (5) (6)
No M is X; No X is M; No X is M;
Some M is Z; All M is Z; Some M is Z;
:. Some Z is not X. :. Some Z is not X. :. Some Z is not X.
2. When the whole of one extreme is contained in the middle and a part of the other excluded from it. In this case the part excluded from the middle must, of course, be excluded from the other extreme, all of which is included in the middle term. Of his form we have one example:
All X is M;
Some Z is not M;
Therefore Some Z is not X.
3. When a part of the middle term is excluded from one extreme and all of it contained in the other. In this relation, also, but one single form presents itself, to wit:
Some M is not X;
All M is Z;
Therefore Some Z is not X.
All valid Conclusions deduced upon principles which accord with those above elucidated.
From a careful examination of the above statements and examples, it will be seen not only that when the above relations do exist between the extremes and the middle term, the different forms of conclusions referred to do arise, but that to deduce any legitimate conclusions of any kind, relatively to inferior and superior conceptions related to each other as subject and predicate, these relations must exist. From no conceivable relations of X and Z to M, for example, can we affirm that "Every Z is X," but this, that "All of M is X and all of Z is M." Vary these relations in any form or degree whatever, and it will be seen at once that from such relations no such conclusion can be deduced. The same holds true in all the other cases named. Let us now analyze these relations for the purpose of deducing from them the general laws of the figured syllogism, especially in the form we are now considering it.*
*(With very few if any exceptions these principles apply to all forms of the syllogism, especially to the figured one. As thus applicable these principles should be studied, as they present the only relations between the extremes and the middle term which authorize inferences of any kind.)
Analysis of the above relations.
1. The fact which we first notice is this, that in all these forms of argument we have, at last, one affirmative premise. In all logically valid arguments, then, one premise at least must be affirmative; in other words, form exclusively negative premises no relations between the extremes can be affirmed or denied. From the fact that two terms disagree with a common third term, we cannot affirm that they agree or disagree with each other, for the reason the reason that while they both d[ ? ]s disagree with this term, they may either agree or disagree with each other. A and B may differ in size and hight from it, and one be equal or unequal in all particulars to the other.
2. We notice, also, the fact that when the conclusion is affirmative both premises are affirmative, and that when we have a negative conclusion one of the premises is negative. From the nature of the relations of the extremes to the middle term this must be the case. When the relation of the extremes to the middle term is positive, that is, when both agree with that term, their relations to each other must be positive also. When you affirm of the relation of one extreme to the middle term what you deny of the other, a corresponding disagreement must be of course, affirmed of the extremes themselves. Hence the general principle that when both premises are affirmative the conclusion must be affirmative, and when one premise is negative such must be the character of the conclusion.
3. We notice, further, that in all cases one of the premises is universal. From the fact that of the two extremes each partly agrees, or that one in part agrees, and the other similarly disagrees with the middle term, we can draw no legitimate inference in regard to their agreement or disagreement with each other; because the points of agreement or disagreement may not be the same at all, and the extremes, therefore, may not be compared with the same thing. Suppose, for illustration, that M has three, and only three, kinds of currency in his possession, to wit, gold, silver, and paper; while X has the first kind and Z the second. Each, in what he possesses, agrees in some respects with M, yet neither agrees with the other. From the fact, then, that two terms mutually agree or disagree in some respects with a third, we cannot legitimately affirm or deny any form of agreement or disagreement between those terms themselves. Suppose, further, that X has gold coin and Z copper; so far, then, the former agrees and the later disagrees with M. Form this fact, however, we cannot legitimately infer that Z has something (copper coin) which X has not; for the latter, from aught that appears in the premises, may have copper as well as gold coin, and thus agree with Z as well as M. In all legitimate forms of argument, therefore, one premise at least must be universal. In other words, from particular premises we can infer nothing.
4. From a careful examination of the above relations it will also be seen, that in every case the middle term is given as the subject of a universal, or the predicate of a negative proposition. In all legitimate forms of argument this condition is, and must be, fulfilled. From the fact that all X and all Z are in M, we cannot logically conclude that any part of Z is in X; for Z, from any thing presented in the premises, may be in one part of M and X in another, and neither have any form of agreement or disagreement with the same thing. So from the fact that all X is in M and some of M is not in Z, we cannot legitimately affirm that some part of Z is not in X; for all of Z may, notwithstanding what is affirmed in the premises, be in the part of M in which X is. In all forms of the argument, logically correct forms, which we are now speaking of, and which are included in the sphere of the figured syllogism, the middle term must be the subject of a universal or the predicate of a negative proposition; that is, must be distributed, at least once in the premises. Nor is it needful, as will appear from an analysis of the above cases, that it be distributed more than once. For if the whole of this term is compared, as it is in the relation supposed, with one extreme and a part only of it with the other, so far they must be compared with the same thing, and so far, therefore, their relations to each other may from hence be determined.
5. In all the cases before us, we remark again, that the terms of the conclusion are definite or indefinite; that is, distributed or not distributed just as they were in the premises. This is a universal law of the figured syllogism, and hence the rule: No term must be distributed in the conclusion which was not distributed in the premises. Where this rule is violated (the violation being called an illicit process of the terms as employed), something is affirmed universally in the conclusion which was only affirmed partially in the premises.
NOTE.--It is not necessary that every term which was distributed in the premises should be distributed in the conclusion, though such a use may always be made of it; but when a universal conclusion is valid, the particular which comes under it is valid also.
The Canon of this Syllogsim.
We are now prepared to state definitely the universal canon of this form of the figured syllogism, a canon which to be valid must embrace all of the principles above elucidated. As such a canon, we present the following, to wit: Whatever relations of subject and predicate exist between two terms and a common distributed third term, to which one at least of the former is positively related, exist between the terms themselves. This axiom will be found to include all cases which fall under this form of the figured syllogism, inasmuch as it implies all the relations above adduced.
Moods of the Syllogism.
Every proposition must, as we have seen, be universal or particular; affirmative or negative. When we have designated the propositions of a syllogism in order according to their respective quantity and quality, we have determined its mood. Thus, if all the propositions are universal affirmatives, we have the mood A, A, A, &c. The following extract from Dr. Whately expresses all that need be added on this subject with the exception subsequently stated:
"As there are four kinds of propositions and three propositions in each syllogism, all the possible ways of combining these four (A, E, I, O) by three are sixty-four. For any one of these four may be the major premise, each of these four majors may have four different minors, and of these sixteen pairs of premises each may have four different conclusions, 4 x 4 (=16) x 4 = 64. This is a mere arithmetical calculation of the moods without any regard to the logical rules; for many of these moods are inadmissible in practice from violating some of those rules; e. g. the mood E E E must be rejected as having negative premises; I O O for particular premises; and many others for the same faults; to which must be added I E O for an illicit process of the major in every figure. By examination then of all, it will be found that of the sixty-four there remain but eleven moods which can be used in a legitimate syllogism, viz.: A A A; A A I; A E E; A E O; A I I; A O O; E A E; E A O; E I O; I A I; O A O."
Dr. Whately states that the mood I E O involves "an illicit process of the major in every figure." This must be admitted if we grant that each figure alike has its proper major and minor terms and premises, which, as we shall hereafter see, is not the case. That, on the other hand, must be regarded as an allowable mood in which the conclusion necessarily results from the premises as presented. If we test the mood under consideration by this principle, we shall find that it has the same claim to be regarded as allowable as any of the others. That a legitimate and valid conclusion may be deduced form such an arrangement of the terms and premises, will be evident on a moment's reflection. For example:
Some X is M;
No Z is M;
Therefore No Z is some X.
Converse: Some X is not Z.
No one can deny that both of the above conclusions directly, immediately, and necessarily result from the premises. This, then, is an allowable mood, and we have twelve instead of "eleven moods which can be used in the legitimate syllogism."
FIGURE OF THE SYLLOGISM.
Form defined.
The figure of the syllogism is determined by the relation of the middle term to the extremes, and the number of the figures will be as the number of the relations which the terms admit.
Number of the figures of the Syllogism.
A moment's reflection will convince any one that there are three, and only three, such relations conceivable, to wit:
1. When the middle term is the subject of one extreme and the predicate of the other.
2. When it is the predicate of both extremes.
3. When it is the subject of both.
We conclude, then, that there are three, and only three, figures of the syllogism, and they are numbered according to the order above stated. We will give them in their order:
I. II. III.
M X; X M; M X;
Z M; Z M; M Z;
Z X; Z X; Z X.
Major and Minor Terms and Premises.
On a consideration of the relation of the extremes to the middle term in the first figure, it will be seen at once, that the extreme which is the predicate of the middle term, is, of all the terms employed, the widest extension, including first the middle term and then the other extreme, as included in the middle. The term, therefore, which thus includes both the others is properly called the major term; and that which is determined first by the middle term, and through it by the major, is called the minor term. The premise which contains the major, is called the major, and that which contains the minor term is called the minor, premise. On examining the other figures, it will be seen that in each alike the middle term sustain precisely the same relation to the extremes. In neither of these figures, therefore, is either extreme given as a conception superior or inferior to the other. In the second figure the middle term is given as alike superior, and in the third, as alike inferior to the each of the extremes. In these figures, therefore, we have no proper major or minor terms or premises. To place one as the major and the other as the minor term or premise is a mere arbitrary arrangement, and tends to obscure rather than throw light upon the subject.
Order of the Premises.
In the first figure it is more natural to place the major premise first, and then the minor; though this is by no means universally the case. The following extract form "Thomson's Laws of Thought" is worthy of very special attention on this subject: "Although an invariable order for the two premises and conclusion, namely, that the premise containing the predicate of the conclusion is first and the conclusion the last, is accepted by logicians, it must be regarded as quite arbitrary. The position of the conclusion may lead to the false notion that it never occurs to us till after the full statement of the premises; whereas in the shape of the problem or question it generally precedes them, and is the cause of their being drawn up. In this point the Hindoo syllogism is more philosophic than that which we commonly use. The premises themselves would assume a different order according to the occasion. It is a natural to begin with the fact and go on to the law, as it is to lay down the law and then mention the fact.
"I have an offer of a commission; now to bear a commission and serve in war, (or is not) against the divine law; therefore I am offered what it would (or would not) be against the divine law to accept.
"This is an order of reasoning employed every day, although it is the reverse of the technical; and we cannot call it forced or unnatural. The two kinds of sorties to be described below, are founded upon two different orders of the premises; the one going from the narrowest and the most intensive statement up to the widest, and the other from the widest and most extensive to the narrowest. The logical order cannot even plead the sanction of invariable practice. Neither the school of logicians who defend it, nor those who assail it, take a comprehensive view of the nature of inference. Both orders are right, because both are required at different times; the one is analytic, the other synthetic; the one most suitable to inquiry, and the other to teaching."
In the second and third figures, no order whatever of the premises is suggested by the relations of the extremes to the middle term; nor does the validity of the conclusion depend at all upon their order; either order is to be employed, as occasion requires.
FINAL ABOLISHMENT OF THE FOURTH FIGURE.
Opinions of Logicians upon the subject.
Logicians have commonly made four instead of three syllogistic figures, to wit: that in which the middle term is the subject of the major premise, and the predicate of the minor; that in which it is predicate of both extremes; that in which it is the subject of both; finally, that in which it is the predicate of the major premise and the subject of the minor.
When we met with the statement of Sir William Hamilton, that science requires the "final abolition of the fourth figure," a statement for which he gives no reasons in any of his writings that we have met with, we at first supposed that we had fallen upon the statement of an unnecessary attempt, if nothing more, at simplification in the science of logic. A careful examination of the figure, however, together with that of the possible relations of the extremes to the middle term, has convinced us of the truth and importance of this statement. We fully agree with this author that there can be, upon scientific principles, but "three syllogistic figures," and will proceed to give our reasons for that conviction, reasons for which we are alone responsible, as they are to us the exclusive result of our own investigations. Our reasons, among others, are the following:
Our Reasons for the abolitions of this Figure.
1. The relations which we have given embrace, as we have said, all conceivable relations which a single term can, as subject and predicate, sustain to two others in two given propositions, to wit: the subject of one extreme and the predicate of the other; the subject of both; and the predicate of both extremes. As but three relations are conceivable, science permits but three syllogistic figures.
2. The premises of the fourth figure are in fact nothing but those of the first transposed, such transposition being allowable and always understood as implying no change of the figure of the syllogism. For example:
All M is X; All X is M;
All Z is M. All M is Z.
In the first example we have the premises of Barbara in the first figure, and in the second of Brumantip of the fourth. Let X in the latter case take the place of Z and Z of X, and every one will perceive that we have nothing but the premises of Barbara changed. This is the case in all instances in the fourth figure. It is contrary to all the laws of science, therefore, to suppose a new figure to meet the case of a mere change of the order of the premises.
3. In the fourth figure, as given by logicians who retain it, the scientific major term is give as the minor and the minor as the major; so of the premises. Take Brumantip as an illustration:
All X is M;
All M is Z;
Therefore Some Z is X.
Who does not perceive that Z is here the superior, M the intermediate, and X the inferior conception? Z, in the first instance, as the superior conception contains M as its inferior conception, and then M as the superior contains X as its inferior conception. Z, then, according to all the laws of science, is the superior conception, and the consequent only proper major term. X is the proper minor; and Z the proper major term. The same holds true of all the moods of this figure.
4. In this figure, as given by logicians, the indirect is, in all instances, substituted for the direct conclusion. The direct conclusion from the premises of Brumantip, for example, is "All X is Z," and not "Some Z is X." If all X is in M and all of M in Z, then all of X must be in Z; and this is the direct, and only direct, conclusion. The proposition, "Some Z is X," is but the converse of the inference which the premises directly yield. The same holds true of every mood in this so called figure. No reasons whatever, then, exist for retaining it; all the laws and principles of true science, on the other hand, demand its "final abolition." It may be often convenient to change the order of the premises of the first figure, and to state its indirect conclusion as immediately evident from the premises, which is often done. For this reason, however, we should not confuse the principles of science by supposing a new figure.
SPECIAL CHARACTERISTICS AND CANON OF EACH OF THE THREE FIGURES.
On a careful examination of the three remaining figures, we shall perceive that in consequence of the peculiar relations of the middle term to the extremes in each, that each must have its peculiar and special characteristics, and be governed by laws equally special and peculiar. We will take them up in the order in which they are numbered:
FIGURE I.
In the first figure, the middle term, as the subject of the major term, is determined by said term, while it (the middle), as the predicate of the minor, itself determines the same, and in the immediate conclusion the determining extreme stands as the predicate, and the determined as the subject. In this figure consequently we have, from the relations of the terms to each other, our proper major and minor terms and equally proper major and minor premises. From these facts the proper order of the premises, as well as the relations of the extremes as subject and predicate in the conclusion, become perfectly manifest. In this figure, also, for the reasons just stated, we have one, and only one, direct, immediate, and proximately definite conclusion; and, mediately, the converse of the same. As an illustration of the above statement let us take, as an example, the mood Barbara:
All M is X;
All Z is M;
Therefore All Z is X.
Converse: Some X is Z.
Here it will be seen that we pass from one extreme (X) to the other (Z), through the middle term (M); X being given as containing all of M, that is, as determining it, and M in a similar manner as determining Z. In the conclusion, also, each term sustains to the other the identical relation which it did to the middle in the premises in which it appears. X contains Z, that is, determines it, as it did M in the major premise; and Z is contained in X, that is, is determined by it, as the former was by M in the minor premise. The relations of the extremes to each other in the conclusion, also, are necessarily determined by their relations to the middle term in the premises; no other order than that which gives X as the predicate and Z as the subject of the conclusion, being permitted by their relations in the premises to the middle term, through which their relations to each other, as expressed in the conclusion, are determined. It is by no arbitrary arrangement, therefore, that X is given as the major term, and the premise containing it as the major premise; and Z as the minor term, and premise containing it as the minor premise. From the nature of the relations of the terms in the premises, also, but one conclusion, Z is X, is directly and immediately given, and this conclusion is a proximately definite one.
Similar remarks are equally applicable, as a careful examination will show, to all the other moods of this syllogism. This figure, therefore, has a special canon which is the following, to wit:
Whatever relations of determining predicate and of determined subject exist between two terms and a common distributed third term, to which one at least is positively related, that relation said terms immediately, that is, directly, hold to each other; and mediately, that is, indirectly, its converse.
The Canon illustrated.
We will now, as a means of illustrating this canon, examine each of the moods in this figure. Barbara has already been sufficiently elucidated. We will, therefore, simply give an example of reasoning in this mood, without the use of letters. The case present is cited from Dr. Whately, and presents the celebrated argument of Aristotle (Eth., sixth book), to prove that the virtues are inseparable, viz.:
"He who possesses prudence possesses all virtue;
He who possesses one virtue must possess prudence;
Therefore, he who possesses one possesses all."
We will give Celarent in both forms, to wit, with and without letters:
No M is X;
Every Z is M;
Therefore No Z is X.
Converse: No X is Z.
Whatever is conformable to nature is not hurtful to society;
Whatever is expedient is confomable to nature;
Therefore: Whatever is expeident is not hurtful to society;
Converse: Whatever is hurtful to society is never expedient.
In both these examples alike there is a perfect conformity to the canon above given. The term included in or determined by the middle is the subject, and the one excluded from, and thus determines the character and relations of the extremes and of the premises also. We will now consider the two remaining moods, Darii and Ferio.
All M is X; No M is X;
Some Z is M; Some Z is M;
Therefore Some Z is X. Therefore: Some Z is not X.
Converse: Some X is Z. Converse: Some not X is Z.
Or better, perhaps: No X is some Z.
The remarks made above are so obviously applicable to these two moods, that we need add nothing in particular with respect to them. From an inspection of the four moods above given, it will appear that they present the only possible combinations of the premises according to the immutable laws of this figure. In this figure alone, also, can all of the four classes of propositions A, E, I, and O, be proven.
FIGURE II.
In elucidating the second figure, we will first present all its allowable moods, as given in the common treatises on logic. The letters prefixed will indicate the quantity of the propositions:
Cesare. Camestres. Festino. Baroko.
E. X is M; A. X is M; E. X is M; A. X is M;
A. Z is M; E. Z is M; I. Z is M; O. Z is M;
:. E. Z is X, or, :. E. Z is X, or, :. O. Z is not X, or, :. O. Z is not X, or
E. X is Z. E. X is Z. I. not X is Z, or, I. not X is Z, or
No X is some Z. No X is some Z.
In this figure, as will be readily perceived, we have in neither extreme a determining predicate as we have in the first. We have in each extreme alike, on the other hand, nothing but determined subjects. As a consequence we have no proper major or minor terms or premises, each extreme sustaining in these respects precisely similar relations to the middle term. The validity of the conclusion in no sense depends upon the order of the premises. In the first two moods, for example, we have by one order of the premises, Cesare, and by a simple change of the order we have Camestres. Nor can any reason be assigned why Z instead of X should be held as the minor term, or why the premise containing it should be considered as the minor premise. In the premises sometimes one and sometimes the other term is given as in part or wholly included in, and the other, in each case, as in whole or in part excluded from, one and the same term. By what law of intellectual procedure should one of the extremes be called the major term and its premise the major premise, and the other the minor term and its premise the minor premise? For the same reason we have no fixed law of subordination for the extremes in the conclusion. We have, on the other hand, in all instances two conclusions, each connected with the same distinctness and immediateness with the premises, to wit: "No Z is X, or, no X is Z;" "Some Z is not X, or, some not X is Z." A mere reference to the moods of this figure as above given, is all that is requisite to verify the above statement. In Camestres, for example, X sustains the precise relation to M that Z does in Cesare, and vice versa. The inference, then, "No X is Z," is just as directly and immediately deducible from the premises, as its converse "No Z is X." The same remarks are equally applicable to the conclusions, "Some Z is not X," and "Some not X is Z," obtained in Festino and Baroko. If, for example, "All X is in M," and "Some Z is not in M," the conclusion, "Some not X is Z," as immediately follows as its converse, "Some Z is not X." The difference here lies not in the connection of the conclusion with the premises, but in the fact that in one case we have an apparently affirmative conclusion when we have a negative premise. The conclusion, however, is, as far as mere conventional form is concerned affirmative, while in reality it is negative. So far, then, as this kind of affirmative propositions are concerned we may have in this, as we shall see in Figure III., an affirmative conclusion when we have one negative premise. What we desire to call especial attention to, is the fact, that this conclusion is as directly and immediately deducible from the premises, as its negative converse "Some Z is not X." In this figure, then, the premises always yield with the same distinctness and immediateness two conclusions. In consequence of the figure, we have, by a change of the order of the premises in the cases of Festino and Baroko, two additional allowable moods, making its real number six instead of four.
Canon of this Figure.
The following, then, is the special canon of this figure, to wit: Whatever relations of determined subject is held by two notions to a common distributed third, with which one is positively and one distributively, that is, negatively, related, that relation these conceptions hold indifferently to each other.
In illustrating this canon we will first take the case of Camestres. In this syllogism X is given as wholly agreeing, and Z as wholly disagreeing, with a common distributed third term, M, to which both stand related as determined subjects. In other words they, as determined subjects, wholly disagree in their relations to a common distributed third term. Similar relations of subject and predicate must they sustain to each other; and the propositions, "No X is Z," and "No Z is X," must be held as logically valid. In Cesare X is positively and Z negatively related to M. In all other respects, therefore, their relations to each other must be as in Camestres. In the other syllogisms of this figure X is given as wholly agreeing or wholly disagreeing with M, and Z as undistributed, and as such as sustaining in each case opposite relations to M. In other words, in these syllogisms these terms as determined subjects partially disagree in their relations to M. In their relations as subject and predicate to each other, therefore, they are given as partially disagreeing with each other. The canon includes every case that can fall under this figure.
FIGURE III.
The following are the syllogisms of this figure as commonly given, namely:
Darapti Disamis. Datisni.
A. M is X; I. M is X; A. M is X;
A. M is Z; A. M is Z; I. M is Z;
Therefore I. Z is X, or, Therefore I. Z is X, or, Therefore I. Z is X, or,
I. X is Z. I. X is Z. I. X is Z.
Felapton. Bokardo. Ferison.
E. M is X; O. M is not X; E. M is X;
A. M is Z; A. M is Z; I. M is Z;
Therefore O. Z is not X, or, :. O. Z is not X, or, :. O. Z is not X, or,
I. not X is Z. I. not X is Z. I. not X is Z.
In this figure the middle is in both premises alike the determined subject, and not the determining predicate, as in the second. As one extreme determines the middle in the precise form that the other does, we have here, also, no proper major and minor terms or premises. The order of the premises being indifferent, equally so is that of the terms in the conclusion. As each premise may stand indifferently as major or minor, so each extreme may be indifferently the subject or predicate of the conclusion. In other words, as in the second figure, so in this, the premises always yield with equal distinctness and immediateness two conclusions, one in which one extreme, and another in which the other extreme, is the subject. A careful examination of each of the above moods will perfectly evince the truth of all these statements, and will also show that, by a simple change of the order of the propositions in the case of three last-named moods, we have three more allowable ones in this figure.
Canon of this Figure.
The following, then, is the special canon of this figure, to wit: Whatever relations of determining predicate any two terms sustain to a common distributed third term, to which one, at least, of the former is positively related, those relations these terms sustain indifferently to each other. The application of this canon is too obvious to require any special elucidation.
NOTE.~In giving to each figure an especial canon, we have followed the example of Kant and of Sir William Hamilton. Our statement of these canons differs, not in thought but in form, from that found in the writings of these authors.
Absurdity of reducing the Syllogism of the other Figures to the first.
In the Intellectual Philosophy, page 320-1, we stated years ago our objections to a practice common to almost all treatises on logic, of reducing the syllogisms of the other figures to the first. We are quite happy to find our objections sustained by such authority as that of Sir William Hamilton. At the time we stated these objections we had never read or heard of his thoughts upon the subject, and he, of course, has never met with ours. Our objections to this practice, among others, are the following:
1. The laws of thought may be fully elucidated without any reference to figure. This we have already sufficiently shown in determining, wholly independent of any reference to the figure, the conditions on which all valid conclusions can be deduced.
2. Figure itself, as Sir William Hamilton observes, is "an unessential variation in syllogistic form." The middle term is just as really and truly compared with the extremes, and the conclusions thence deduced are just a valid, in one figure as in any other. Not a solitary ray of light is thrown upon the subject by the reduction. This we have already shown in the passage in the Philosophy above referred to.
3. The science of reasoning is, consequently, rather obscured than elucidated by the process. The pupil expects light and finds none; the disappointment obscures rather than illumines his vision of the principles of the science.
4. The pupil, we remark finally, is actually deceived by the process. He is made to think that the validity of one syllogism depends, not upon the relations of the extremes to the middle term, relations found in the syllogism itself, but upon that of other relations found in a syllogism of another and different figure, whereas the reverse of all this is in fact true. The validity of the process, in each syllogism alike, depends exclusively upon the relations to each other of the terms found in it. These considerations are abundantly sufficient to justify us in totally disregarding the custom under consideration.
Nature of the Conclusions obtained in this form of the Syllogism.
We have already stated that in this form of the syllogism, there is in reality but an approach towards the truth, that is, the whole truth pertaining to the objects of inquiry. It may be a matter of no little interest and importance to consider, for a few moments, the nature of the conclusions which we do obtain. What then is the nature of the agreement or disagreement between the subject and predicate really affirmed in said conclusions? Suppose that in the first figure we have obtained the conclusion, "All or some Z is X." That answer may be considered relatively to the principle of intension or extension. In reference to the former, the conclusion affirms that Z possesses the elements represented by the superior conception X. In reference to the latter, it affirms that all or some of the individuals represented by the individual or specificial conception Z, do belong to the class represented by the specificial or generical conception X. What pertains to Z in other respects is not affirmed or denied. So in the negative conclusion, "All or some Z is not X," we simply ascertain, that in so far as the qualities represented by the conception M are ever concerned, they differ, one having, and the other not having them. How far they may or may not agree in other respects, is not ascertained.
In the second figure, from the fact that one extreme does, and the other does not, rank in whole or in part under a given superior conception, we infer that they therefore so far disagree. This disagreement pertains simply and exclusively to the qualities or class represented by said superior conception. How far they agree or disagree in other particulars is not ascertained. Suppose, for example, that it has been ascertained that A is, and B is not, guilty of murder; in other words, that A is not B. In very many particulars, such as taking life and intentionally doing it, and doing it with the same weapons, they may agree. What has been ascertained is, that relatively to the peculiar elements represented by the term murder, the act of one does, and that of the other does not, involve said elements. This is the real character of the conclusions obtained in this figure.
In the third figure, in affirmative propositions, we ascertain, from the fact that certain elements represented by a certain conception M belong to a part of each of the classes represented by two conceptions Z and X, each superior to M, that some individuals ranking under each of these superior conceptions have either both the whole, or one all, and the other a part, of the qualities represented by M, and therefore, that they so far agree. The agreement ascertained pertains exclusively to the qualities referred to. In negative conclusions, from the fact that the elements referred to do belong to a part of one class and not to a part of another class, it is affirmed that so far portions of these classes do not agree with each other. The disagreement is always specific, and pertains exclusively to the elements represented by the inferior conception M.
Such is the character of all the conclusions obtained through this form of the syllogism. They are always in themselves specific and definite, but pertain only to a part of what really is true.
Kind of arguments which appropriately belong to the different Figures.
It may be important to occupy some time in considering the forms of argument which most properly belong to the different figures of the syllogism.
All cases in which the principle of extension on the one hand, and comprehension on the other, are in equilibrium, belong, as we have seen, exclusively to the first figure; and the question, whether in any given case these relations do obtain? may, in all instances, be very readily resolved. In this figure the minor as a determined subject ranks under another term, the middle; while said middle, as such a subject, ranks under, or is excluded from, the major term. This one peculiarity distinguishes all arguments in this figure from all which pertain to the others. Suppose, for example, the question to be argued is, Whether A in a certain act, taking the life of B, was guilty of murder, the fact of taking the life referred to being admitted. The advocate sustaining the charge first lays down the general principle, that, in the language of Coke and Blackstone, unlawfully killing a human being with premeditated malice, by a person of sound mind, is murder (All M is X), affirms and attempts to show, that A killed B in these very circumstances (All Z is M), and hence infers that A, in the act referred to, was guilty of murder (All Z is X). This is an argument in the mood Barbara. On the other hand, let us suppose that the advocate on the other side, after laying down the principle that taking life in self-defense is not murder (No M is X), affirms and attempts to prove that A took the life of B in self-defense (All Z is M), and hence concludes that the act referred to was not murder (No Z is X). We have in such a case an argument in the mood Celarent. The application of the above illustration to particular conclusions, affirmative and negative, belonging to this mood, are too obvious to require elucidation.
Let us suppose, now, that it is claimed or is likely to be, that two cases (X and Z) rank under one and the same principle or superior conception (M), and that we wish to disprove that assertion. In accomplishing this object, we first show that, on the principle of intention, X contains all of M, that is, as an inferior X is contained under M, as the superior conception (All X is M); we then show that Z has none of these elements, that is, as an inferior conception does not rank under M as its superior (No Z is M); we hence deduced the conclusion, "No Z is X," that is, X and Z do not rank under the same principle. In this case the argument is in the second figure, in the mood Camestres. If, on the other hand, it was argued that X is wholly void of certain fundamental characteristics which Z possesses, and that, therefore, X and Z do not belong to the same class, or that no Z is X, the argument would be in the same figure, but in the mood Cesare. On the same principle, in Festino and Barako a partial disagreement is disproved. Suppose it to be maintained, for example, that the miracles recorded in the Bible (X), and those claimed in behalf of other religions (Z), are in all essential characteristics alike, and, therefore, alike unworthy of credit; that is, the miracles recorded in the Bible are of the same essential characteristics as those claimed in behalf of other religions. The latter class are wholly unworthy of credit. Such, therefore, must be the character of the miracles chronicled in the Bible, an argument in the mood of Barbara. In opposition to this, we show, that the latter class of events have all of them certainly infallible marks of credibility (All X is M), that none of the former class, in fact, have any one of these characteristics (No Z is M), and hence deduce the conclusion, that these two classes of events do not belong to he same class at all (No Z is X). This, also, would be an argument in the second figure; the figure whose special province is such kind of refutations. Suppose once more that we wish to prove that certain individuals of each of two different classes have certain common characteristics, that is, that each class as the superior conception contains under it, in whole or in part, a common conception, and that there is consequently a partial resemblance between the classes themselves; or, that while part of one class has these characteristics, portions at least of the other class have them not, and that, consequently, there is this partial disagreement between these classes. Let us suppose, further, that it is asserted that all of these classes have the characteristics, or that all of one class and none of the other have them, and that we wish to disprove these propositions in their universal form. In all the above-named cases we naturally use some of the modes of the third figure. The argument will, in the first instance, stand thus: All of these characteristics do belong to one extreme, and all or a part of the same do or do not belong to the other, and, therefore, some of one class are or are not like some of the other; that is, "All of M is in X," and "All or some of it is or is not in Z," and, therefore, all or some of Z is or is not in X. When we desire to prove the contradictory of a universal proposition, whether affirmative or negative, we prove that some of the one, at least, are, and some of the other are not, in the state referred to, and that, therefore, the universal proposition we show, that no or some M is not in X, and that all or some M is in Z, and, therefore, some Z is not in X. In opposition to the universal negative proposition we show, that all M is in X, and that all or some M is in Z, and, therefore, some Z is in X. In all such positive arguments, and in all replies like those under consideration, the reasoning is commonly in the third figure; for example, "Prudence has for its object the benefit of individuals." This argument is in Darapti, and its object is to establish a fact or principle. Its form would be the same if its object was to refute the principle, that no form of real virtue has for its object the benefit of individuals. Suppose, for the sake of still further elucidation, that it is argue that a certain doctrine cannot be true, and that on account of a certain difficulty (M) which it involves. The argument in full stands thus: No doctrine involving this difficulty (M) can be true (X), or, "No M is X." This doctrine (Z) does involve this difficulty (M), or, "All Z is M," therefore this doctrine (Z) cannot be true, or, "No Z is X." To refute this argument we have only to show, that some one doctrine which cannot be denied involves this very difficulty. The argument in reply is in Darati, and stands, when stated in full, thus: This doctrine (M) involves this very difficulty (X), or "All M is X." This doctrine (M) is true (Z), or, "All M is Z." Therefore, some doctrine which is true involves this very difficulty, or "Some Z is X;" in other words, this objection is of no force against any doctrine. By carefully reflecting upon the above illustrations the pupil will be able to judge correctly in regard to the figure into which any particular argument is, or should be, thrown.
A more brief view of this subject.
To state the matter in still fewer words: when the middle term stands intermediate between the extremes, being inferior to one and superior to the other, then the argument is in the first figure. This we believe is generally the case when one premise is a general or universal principle. In this figure we always advance from the minor term through the middle to the major or superior conception. On the other hand, when the middle term is superior to each extreme, then the argument is in the second; and when it is in the relation of an inferior conception to each extreme, then the argument is in the third figure.
A SCIENTIFIC DETERMINATION OF THE REAL NUMBER OF LEGITIMATE MOODS IN THIS FORM OF THE SYLLOGISM.
Hitherto, in treatises on logic, the number of legitimate moods has been given as the result of mere experiment. Science demands that it shall be shown that, from the relations of the extremes to the middle term, there must be a certain number of legitimate moods, and that there can by no possibility be any more. This is what we now propose to accomplish.
Conditions of valid deductions of any kind in this form of the Syllogism.
The following, it must be born in mind, are the immutable conditions of any valid conclusions in the syllogism as thus far elucidated:
1. The middle term must be distributed at least once in the premises.
2. No term must be distributed in the conclusion which was not distributed in the premises.
3. One premise at least must be universal.
4. When the conclusion is universal both premises must be of the same character.
5. One premise, also, must be affirmative.
6. When the conclusion is affirmative both premises must be affirmative, and when one premise is negative the conclusion must be negative.
From these laws, which, as we have already seen, cannot but be valid, we must have a certain definite number of legitimate moods, and by no possibility can we have any more. This we will now proceed to show.
Universal affirmative conclusions.
Let us, in the first place, take a universal affirmative conclusion. To have such a conclusion, each premise must be both universal and affirmative. Unless X and Z are both given in the premises as agreeing universally with M, the former cannot, from their mutual relations to the latter, be affirmed to agree universally with each other. Such an agreement as legitimates such a conclusion does exist, as we have already seen, when the whole of one extreme is contained in the middle term, and the whole of said term is contained in the other extreme. A A A, then, is an allowable mood.
Particular affrimative conclusions.
To have a particular affirmative conclusion both premises must be affirmative, and one universal, of which the middle term is the subject, this being the condition of its being distributed in an affirmative proposition. Now there are but three possible forms in which these conditions can be fulfilled, to wit: when both premises are universal affirmatives~when the first premise is a universal, and the second a particular, affirmative~and, when the first is a particular, and the second a universal, affirmative. There can, then, be but three