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DEDUCTIVE LOGIC
BY
ST. GEORGE STOCK, M.A.
PEMBROKE COLLEGE, OXFORD
PREFACE.
One critic, who was kind enough to look at this book in manuscript,
recommended me to abandon the design of Publishing it, on the ground
that my logic was too like all other logics; another suggested to me
to cut out a considerable amount of new matter. The latter advice I
have followed; the former has encouraged me to hope that I shall not
be considered guilty of wanton innovation. The few novelties which I
have ventured to retain will, I trust, be regarded as legitimate
extensions of received lines of teaching.
My object has been to produce a work which should be as thoroughly
representative of the present state of the logic of the Oxford Schools
as any of the text-books of the past. The qualities which I have aimed
at before all others have been clearness and consistency. For the task
which I have taken upon myself I may claim one qualification--that of
experience; since more than seventeen years have now elapsed since I
took my first pupil in logic for the Honour School of Moderations, and
during that time I have been pretty continuously engaged in studying
and teaching the subject.
In acknowledging my obligations to previous writers I must begin with
Archbishop Whately, whose writings first gave me an interest in the
subject. The works of Mill and Hamilton have of course been freely
drawn upon. I have not followed either of those two great writers
exclusively, but have endeavoured to assimilate what seemed best in
both. To Professor Fowler I am under a special debt. I had not the
privilege of personal teaching from him in logic,--as I had in some
other subjects; but his book fell into my hands at an early period in
my mental training, and was so thoroughly studied as to have become a
permanent part of the furniture of my mind. Much the same may be said
of my relation to the late Professor Jevons's Elementary Lessons in
Logic. Two other books, which I feel bound to mention with special
emphasis, are Hansel's edition of Aldrich and McCosh's Laws of
Discursive Thought. If there be added to the foregoing Watts's Logic,
Thomson's Outlines of the Laws of Thought, Bain's Deductive Logic,
Jevons's Studies in Deductive Logic and Principles of Science,
Bradley's Principles of Logic, Abbott's Elements of Logic, Walker's
edition of Murray, Ray's Text-book of Deductive Logic, and
Weatherley's Rudiments of Logic, I think the list will be exhausted of
modern works from which I am conscious of having borrowed. But, not to
forget the sun, while thanking the manufacturers of lamps and candles,
I should add that I have studied the works of Aristotle according to
the measure of my time and ability.
This work has had the great advantage of having been revised, while
still in manuscript, by Mr. Alfred Robinson, Fellow of New College, to
whom I cannot sufficiently express my obligation. I have availed
myself to the full of the series of criticisms which he was kind
enough to send me. As some additions have been made since then, he
cannot be held in anyway responsible for the faults which less kindly
critics may detect.
For the examples at the end I am mainly indebted to others, and to a
large extent to my ingenious friend, the Rev. W. J. Priest of Merton
College.
My thanks are due also to my friend and former pupil, Mr. Gilbert
Grindle, Scholar of Corpus, who has been at the pains to compose an
index, and to revise the proofs as they passed through the press.
And last, but not least, I must set on record my gratitude to
Commander R. A. Stock, R.N., one of Her Majesty's Knights of Windsor,
without whose brotherly aid this work might never have been written,
and would certainly not have assumed exactly its present shape.
OXFORD,
_October_ 22, 1888.
CONTENTS.
PREFACE.
INTRODUCTION, §§ 1-56.
PART I. Of Terms, §§ 57-171.
CHAP. I. Of the Term as distinguished from other words, §§ 57-76.
II. Of the Division of Things, §§ 77-85.
III. Of the Divisions of Terms, §§ 86-165.
IV. Of the Law of Inverse Variation of Extension and Intension, §§
166-171.
PART II. Of Propositions, §§ 172-185.
CHAP. I. Of the Proposition as distinguished from other Sentences,
§§ 172-185.
II. Of the Copula, §§ 186-201.
III. Of the Divisions of Propositions, §§ 202-273.
IV. Of the Distribution of Terms, §§ 274-294.
V. Of the Quantification of the Predicate, §§ 295-312.
VI. Of the Heads of Predicables, §§ 313-346.
VII. Of Definition, §§ 347-384.
VIII. Of Division, §§ 385-425.
PART III. Of Inferences, §§ 426-884.
CHAP. I. Of Inferences in general, §§ 426-441.
II. Of Deductive Inferences, §§ 442-448.
III. Of Opposition, §§ 449-478.
IV. Of Conversion, §§ 479-495.
V. Of Permutation, §§ 496-502.
VI. Of Compound Forms of Immediate Inference, §§ 503-532.
VII. Of Other Forms of Immediate Inference, §§ 533-539.
VIII. Of Mediate Inferences or Syllogisms, §§ 540-557.
IX. Of Mood and Figure, §§ 558-568.
X. Of the Canon of Reasoning, §§ 569-581.
XI. Of the General Rules of Syllogism, §§ 582-598.
XII. Of the Determination of the Legitimate Moods of Syllogism, §§
599-605.
XIII. Of the Special Rules of the Four Figures, §§ 606-620.
XIV. Of the Determination of the Moods that are valid in the Four
Figures, §§ 621-632.
XV. Of the Special Canons of the Four Figures, §§ 633-647.
XVI. Of the Special Uses of the Four Figures, §§ 648-655.
XVII. Of the Syllogism with Three Figures, §§ 656-666.
XVIII. Of Reduction, §§ 667-700.
XIX. Of Immediate Inference as applied to Complex Propositions, §§
701-730.
XX. Of Complex Syllogisms, §§ 731-743.
XXI. Of the Reduction of the Partly Conjunctive Syllogism, §§
744-752.
XXII. Of the Partly Conjunctive Syllogism regarded as all Immediate
Inference, §§ 753-759.
XXIII. Of the Disjunctive Syllogism, §§ 760-765.
XXIV. Of the Reduction of the Disjunctive Syllogism, §§ 766-769.
XXV. Of the Disjunctive Syllogism regarded as an Immediate
Inference, §§ 770-777.
XXVI. Of the Mixed Form of Complex Syllogism, §§ 778-795.
XXVII. Of the Reduction of the Dilemma, §§ 796-797.
XXVIII. Of the Dilemma regarded as an Immediate Inference, §§
798,799.
XXIX. Of Trains of Reasoning, §§ 800-826.
XXX. Of Fallacies, §§ 827-884.
EXERCISES.
INDEX.
INTRODUCTION.
§ 1. LOGIC is divided into two branches, namely--
(1) Inductive,
(2) Deductive.
§ 2. The problem of inductive logic is to determine the actual truth
or falsity of propositions: the problem of deductive logic is to
determine their relative truth or falsity, that is to say, given such
and such propositions as true, what others will follow from them.
§ 3. Hence in the natural order of treatment inductive logic precedes
deductive, since it is induction which supplies us with the general
truths, from which we reason down in our deductive inferences.
§ 4. It is not, however, with logic as a whole that we are here
concerned, but only with deductive logic, which may be defined as The
Science of the Formal Laws of Thought.
§ 5. In order fully to understand this definition we must know exactly
what is meant by 'thought,' by a 'law of thought,' by the term
'formal,' and by 'science.'
§ 6. Thought, as here used, is confined to the faculty of
comparison. All thought involves comparison, that is to say, a
recognition of likeness or unlikeness.
§ 7. The laws of thought are the conditions of correct thinking. The
term 'law,' however, is so ambiguous that it will be well to determine
more precisely in what sense it is here used.
§ 8. We talk of the 'laws of the land' and of the 'laws of nature,'
and it is evident that we mean very different things by these
expressions. By a law in the political sense is meant a command
imposed by a superior upon an inferior and sanctioned by a penalty for
disobedience. But by the 'laws of nature' are meant merely certain
uniformities among natural phenomena; for instance, the 'law of
gravitation' means that every particle of matter does invariably
attract every other particle of matter in the universe.
§ 9. The word 'law' is transferred by a metaphor from one of these
senses to the other. The effect of such a command as that described
above is to produce a certain amount of uniformity in the conduct of
men, and so, where we observe uniformity in nature, we assume that it
is the result of such a command, whereas the only thing really known
to us is the fact of uniformity itself.
§ 10. Now in which of these two senses are we using the term 'laws of
thought'? The laws of the land, it is plain, are often violated,
whereas the laws of nature never can be so [Footnote: There is a sense
in which people frequently speak of the laws of nature being violated,
as when one says that intemperance or celibacy is a violation of the
laws of nature, but here by 'nature' is meant an ideal perfection in
the conditions of existence.]. Can the laws of thought be violated in
like manner with the laws of the land? Or are they inviolable like the
laws of nature?
§ 11. In appearance they can be, and manifestly often are violated-for
how else could error be possible? But in reality they can not. No man
ever accepts a contradiction when it presents itself to the mind as
such: but when reasoning is at all complicated what does really
involve a contradiction is not seen to do so; and this sort of error
is further assisted by the infinite perplexities of language.
§ 12. The laws of thought then in their ultimate expression are
certain uniformities which invariably hold among mental phenomena, and
so far they resemble the laws of nature: but in their complex
applications they may be violated owing to error, as the laws of the
land may be violated by crime.
§ 13. We have now to determine the meaning of the expression 'formal
laws of thought.'
§ 14. The distinction between form and matter is one which pervades
all nature. We are familiar with it in the case of concrete things. A
cup, for instance, with precisely the same form, may be composed of
very different matter-gold, silver, pewter, horn or what not?
§ 15. Similarly in every act of thought we may distinguish two
things--
(1) the object thought about,
(2) the way in which the mind thinks of it.
The first is called the Matter; the second the Form of Thought.
§ 16. Now Formal, which is another name for Deductive Logic, is
concerned only with the way in which the mind thinks, and has nothing
to do with the particular objects thought about.
§ 17. Since the form may be the same, whilst the matter is different,
we may say that formal logic is concerned with the essential and
necessary elements of thought as opposed to such as are accidental and
contingent. By 'contingent' is meant what holds true in some cases,
but not in others. For instance, in the particular case of equilateral
triangles it is true to say, not only that 'all equilateral triangles
are equiangular,' but also that 'all equiangular triangles are
equilateral.' But the evidence for these two propositions is
independent. The one is not a formal consequence of the other. If it
were, we should be able to apply the same inference to all matter, and
assert generally that if all A is B, all B is A, which it is notorious
that we cannot do.
§ 18. It remains now for the full elucidation of our definition to
determine what is meant by 'science.'
§ 19. The question has often been discussed whether logic is a science
or an art. The answer to it must depend upon the meaning we assign to
these terms.
§ 20. Broadly speaking, there is the same difference between Science
and Art as there is between knowing and doing.
Science is systematized knowledge;
Art is systematized action.
Science is acquired by study;
Art is acquired by practice.
§ 21. Now logic is manifestly a branch of knowledge, and does not
necessarily confer any practical skill. It is only the right use of
its rules in thinking which can make men think better. It is
therefore, in the broad sense of the terms, wholly a science and not
at all an art.
§ 22. But this word 'art,' like most others, is ambiguous, and is
often used, not for skill displayed in practice, but for the knowledge
necessary thereto. This meaning is better conveyed by the term
'practical science.'
§ 23. Science is either speculative or practical. In the first case we
study merely that we may know; in the latter that we may do.
Anatomy is a speculative science;
Surgery is a practical science.
In the first case we study the human frame in order that we may
understand its structure; in the second that we may assist its
needs. Whether logic is a speculative or a practical science depends
entirely upon the way in which it is treated. If we study the laws of
thought merely that we may know what they are, we are making it a
speculative science; if we study the same laws with a view to deducing
rules for the guidance of thought, we are making it a practical
science.
§ 24. Hence logic may be declared to be both the science and the art
of thinking. It is the art of thinking in the same sense in which
grammar is the art of speaking. Grammar is not in itself the right
use of words, but a knowledge of it enables men to use words
correctly. In the same way a knowledge of logic enables men to think
correctly, or at least to avoid incorrect thoughts. As an art logic
may be called the navigation of the sea of thought.
§ 25. The laws of thought are all reducible to the three following
axioms, which are known as The Three Fundamental Laws of Thought.
(1) The Law of Identity--
Whatever is, is;
or, in a more precise form,
Every A is A.
(2) The Law of Contradiction--
Nothing can both be and not be;
Nothing can be A and not A.
(3) The Law of Excluded Middle--
Everything must either be or not be;
Everything is either A or not A.
§ 26. Each of these principles is independent and self-evident.
§ 27. If it were possible for the law of identity to be violated, no
violation of the law of contradiction would necessarily ensue: for a
thing might then be something else, without being itself at the same
time, which latter is what the law of contradiction militates
against. Neither would the law of excluded middle be infringed. For,
on the supposition, a thing would be something else, whereas all that
the law of excluded middle demands is that it should either be itself
or not. A would in this case adopt the alternative of being not A.
§ 28. Again, the violation of the law of contradiction does not
involve any violation of the law of identity: for a thing might in
that case be still itself, so that the law of identity would be
observed, even though, owing to the law of contradiction not holding,
it were not itself at the same time. Neither would the law of excluded
middle be infringed. For a thing would, on the supposition, be both
itself and not itself, which is the very reverse of being neither.
§ 29. Lastly, the law of excluded middle might be violated without a
violation of the law of contradiction: for we should then have a thing
which was neither A nor not A, but not a thing which was both at the
same time. Neither would the law of identity be infringed. For we
should in this case have a thing which neither was nor was not, so
that the conditions of the law of identity could not exist to be
broken. That law postulates that whatever is, is: here we have a thing
which never was to begin with.
§ 30. These principles are of so simple a character that the
discussion of them is apt to be regarded as puerile. Especially is
this the case with regard to the law of identity. This principle in
fact is one of those things which are more honoured in the breach than
in the observance. Suppose for a moment that this law did not
hold--then what would become of all our reasoning? Where would be the
use of establishing conclusions about things, if they were liable to
evade us by a Protean change of identity?
§ 31. The remaining two laws supplement each other in the following
way. The law of contradiction enables us to affirm of two exhaustive
and mutually exclusive alternatives, that it is impossible for both to
be true; the law of excluded middle entitles us to add, that it is
equally impossible for both to be false. Or, to put the same thing in
a different form, the law of contradiction lays down that one of two
such alternatives must be false; the law of excluded middle adds that
one must be true.
§32. There are three processes of thought
(1) Conception.
(2) Judgement.
(3) Inference or Reasoning.
§ 33. Conception, which is otherwise known as Simple Apprehension, is
the act of forming in the mind the idea of anything, e.g. when we form
in the mind the idea of a cup, we are performing the process of
conception.
§ 34. Judgement, in the sense in which it is here used [Footnote:
Sometimes the term 'judgement' is extended to the comparison of
nameless sense-impressions, which underlies the formation of
concepts. But this amounts to identifying judgement with thought in
general.] may be resolved into putting two ideas together in the
mind, and pronouncing as to their agreement or disagreement, e.g. we
have in our minds the idea of a cup and the idea of a thing made of
porcelain, and we combine them in the judgement--'This cup is made of
porcelain.'
§ 35. Inference, or Reasoning, is the passage of the mind from one or
more judgements to another, e.g. from the two judgements 'Whatever is
made of porcelain is brittle,' and 'This cup is made of porcelain,' we
elicit a third judgement, 'This cup is brittle.'
§ 36. Corresponding to these three processes there are three products
of thought, viz.
(1) The Concept.
(2) The Judgement.
(3) The Inference.
§ 37. Since our language has a tendency to confuse the distinction
between processes and products, [Footnote: E.g. We have to speak quite
indiscriminately of Sensation, Imagination, Reflexion, Sight, Thought,
Division, Definition, and so on, whether we mean in any case a process
or a product.] it is the more necessary to keep them distinct in
thought. Strictly we ought to speak of conceiving, judging and
inferring on the one hand, and, on the other, of the concept, the
judgement and the inference.
The direct object of logic is the study of the products rather than of
the processes of thought. But, at the same time, in studying the
products we are studying the processes in the only way in which it is
possible to do so. For the human mind cannot be both actor and
spectator at once; we must wait until a thought is formed in our minds
before we can examine it. Thought must be already dead in order to be
dissected: there is no vivisection of consciousness. Thus we can never
know more of the processes of thought than what is revealed to us in
their products.
§ 38. When the three products of thought are expressed in language,
they are called respectively
(1) The Term.
(2) The Proposition.
(3) The Inference.
§ 39. Such is the ambiguity of language that we have already used the
term 'inference' in three different senses--first, for the act or
process of inferring; secondly, for the result of that act as it
exists in the mind; and, thirdly, for the same thing as expressed in
language. Later on we shall have to notice a further ambiguity in its
use.
§ 40. It has been declared that thought in general is the faculty of
comparison, and we have now seen that there are three products of
thought. It follows that each of these products of thought must be the
result of a comparison of some kind or other.
The concept is the result of comparing attributes.
The judgement is the result of comparing concepts.
The inference is the result of comparing judgements.
§ 41. In what follows we shall, for convenience, adopt the phraseology
which regards the products of thought as clothed in language in
preference to that which regards the same products as they exist in
the mind of the individual. For although the object of logic is to
examine thought pure and simple, it is obviously impossible to discuss
it except as clothed in language. Accordingly the three statements
above made may be expressed as follows--
The term is the result of comparing attributes.
The proposition is the result of comparing terms.
The inference is the result of comparing propositions.
§ 42. There is an advantage attending the change of language in the
fact that the word 'concept' is not an adequate expression for the
first of the three products of thought, whereas the word 'term' is. By
a concept is meant a general notion, or the idea of a class, which
corresponds only to a common term. Now not only are common terms the
results of comparison, but singular terms, or the names of
individuals, are so too.
§ 43. The earliest result of thought is the recognition of an
individual object as such, that is to say as distinguished and marked
off from the mass of its surroundings. No doubt the first impression
produced Upon the nascent intelligence of an infant is that of a
confused whole. It requires much exercise of thought to distinguish
this whole into its parts. The completeness of the recognition of an
individual object is announced by attaching a name to it. Hence even
an individual name, or singular term, implies thought or
comparison. Before the _child_ can attach a meaning to the word
'_mother_,' which to it is a singular term, it must have
distinguished between the set of impressions produced in it by one
object from those which are produced in it by others. Thus, when
Vergil says
Incipe, parve puer, risu cognoscere matrem,
he is exhorting the beatific infant to the exercise of the faculty of
comparison.
§ 44. That a common term implies comparison does not need to be
insisted upon. It is because things resemble each other in certain of
their attributes that we call them by a common name, and this
resemblance could not be ascertained except by comparison, at some
time and by some one. Thus a common term, or concept, is the
compressed result of an indefinite number of comparisons, which lie
wrapped up in it like so many fossils, witnessing to prior ages of
thought.
§ 45. In the next product of thought, namely, the proposition, we have
the result of a single act of comparison between two terms; and this
is why the proposition is called the unit of thought, as being the
simplest and most direct result of comparison.
§ 46. In the third product of thought, namely, the inference, we have
a comparison of propositions either directly or by means of a
third. This will be explained later on. For the present we return to
the first product of thought.
§ 47. The nature of singular terms has not given rise to much dispute;
but the nature of common terms has been the great battle-ground of
logicians. What corresponds to a singular term is easy to determine,
for the thing of which it is a name is there to point to: but the
meaning of a common term, like 'man' or 'horse,' is not so obvious as
people are apt to think on first hearing of the question.
§ 48. A common term or class-name was known to mediæval logicians
under the title of a Universal; and it was on the question 'What is a
Universal 7' that they split into the three schools of Realists,
Nominalists, and Conceptualists. Here are the answers of the three
schools to this question in their most exaggerated form--
§ 49. Universals, said the Realists, are substances having an
independent existence in nature.
§ 50. Universals, said the Nominalists, are a mere matter of words,
the members of what we call a class having nothing in common but the
name.
§ 51. Universals, said the Conceptualists, exist in the mind alone,
They are the conceptions under which the mind regards external
objects.
§ 52. The origin of pure Realism is due to Plato and his doctrine of
'ideas'; for Idealism, in this sense, is not opposed to Realism, but
identical with it. Plato seems to have imagined that, as there was a
really existing thing corresponding to a singular term, such as
Socrates, so there must be a really existing thing corresponding to
the common term 'man.' But when once the existence of these general
objects is admitted, they swamp all other existences. For individual
men are fleeting and transitory--subject to growth, decay and
death--whereas the idea of man is imperishable and eternal. It is only
by partaking in the nature of these ideas that individual objects
exist at all.
§ 53. Pure Nominalism was the swing of the pendulum of thought to the
very opposite extreme; while Conceptualism was an attempt to hit the
happy mean between the two.
§ 54. Roughly it may be said that the Realists sought for the answer
to the question 'What is a Universal?' in the matter of thought, the
Conceptualists in the form, and the Nominalists in the expression.
§ 55. A full answer to the question 'What is a Universal?' will bring
in something of the three views above given, while avoiding the
exaggeration of each. A Universal is a number of things that are
called by the same name; but they would not be called by the same name
unless they fell under the same conception in the mind; nor would they
fall under the same conception in the mind unless there actually
existed similar attributes in the several members of a class, causing
us to regard them under the same conception and to give them the same
name. Universals therefore do exist in nature, and not merely in the
mind of man: but their existence is dependent upon individual objects,
instead of individual objects depending for their existence upon
them. Aristotle saw this very clearly, and marked the distinction
between the objects corresponding to the singular and to the common
term by calling the former Primary and the latter Secondary
Existences. Rosinante and Excalibur are primary, but 'horse' and
'sword' secondary existences.
§ 56. We have seen that the three products of thought are each one
stage in advance of the other, the inference being built upon the
proposition, as the proposition is built upon the term. Logic
therefore naturally divides itself into three parts.
The First Part of Logic deals with the Term;
The Second Part deals with the Proposition;
The Third Part deals with the Inference.
PART I.--OF TERMS.
CHAPTER 1.
_Of the Term as distinguished from other words._
§ 57. The word 'term' means a boundary.
§ 58. The subject and predicate are the two terms, or boundaries, of a
proposition. In a proposition we start from a subject and end in a
predicate (§§ 182-4), there being nothing intermediate between the two
except the act of pronouncing as to their agreement or disagreement,
which is registered externally under the sign of the copula. Thus the
subject is the 'terminus a quo,' and the predicate is the 'terminus ad
quem.'
§ 59. Hence it appears that the term by its very name indicates that
it is arrived at by an analysis of the proposition. It is the
judgement or proposition that is the true unit of thought and
speech. The proposition as a whole is prior in conception to the terms
which are its parts: but the parts must come before the whole in the
synthetic order of treatment.
§ 60. A term is the same thing as a name or noun.
§ 61. A name is a word, or collection of words, which serves as a mark
to recall or transmit the idea of a thing, either in itself or through
some of its attributes.
§ 62. Nouns, or names, are either Substantive or Adjective.
A Noun Substantive is the name of a thing in itself, that is to say,
without reference to any special attribute.
§ 63. A Noun Adjective is a name which we are entitled to add to a
thing, when we know it to possess a given attribute.
§ 64. The Verb, as such, is not recognised by logic, but is resolved
into predicate and copula, that is to say, into a noun which is
affirmed or denied of another, plus the sign of that affirmation or
denial. 'The kettle boils' is logically equivalent to 'The kettle is
boiling,' though it is by no means necessary to express the
proposition in the latter shape. Here we see that 'boils' is
equivalent to the noun 'boiling' together with the copula 'is,' which
declares its agreement with the noun 'kettle.' 'Boiling' here is a
noun adjective, which we are entitled to add to 'kettle,' in virtue of
certain knowledge which we have about the latter. Being a verbal noun,
it is called in grammar a participle, rather than a mere
adjective. The word 'attributive' in logic embraces both the adjective
and participle of grammar.
§ 65. In grammar every noun is a separate word: but to logic, which is
concerned with the thought rather than with the expression, it is
indifferent whether a noun, or term, consists of one word or many. The
latter are known as 'many-worded names.' In the following passage,
taken at random from Butler's Analogy--'These several observations,
concerning the active principle of virtue and obedience to God's
commands, are applicable to passive submission or resignation to his
will'--we find the subject consisting of fourteen words, and the
predicate of nine. It is the exception rather than the rule to find a
predicate which consists of a single word. Many-worded names in
English often consist of clauses introduced by the conjunction 'that,'
as 'That letters should be written in strict conformity with nature is
true': often also of a grammatical subject with one or more dependent
clauses attached to it, as
'He who fights and runs away,
Will live to fight another day.'
§ 66. Every term then is not a word, since a term may consist of a
collection of words. Neither is every word a term. 'Over,' for
instance, and 'swiftly,' and, generally, what are called particles in
grammar, do not by themselves constitute terms, though they may be
employed along with other words to make up a term.
§ 67. The notions with which thought deals involve many subtle
relations and require many nice modifications. Language has
instruments, more or less perfect, whereby such relations and
modifications may be expressed. But these subsidiary aids to
expression do not form a notion which can either have something
asserted of it or be asserted itself of something else.
§ 68. Hence words are divided into three classes--
(1) Categorematic;
(2) Syncategorematic;
(3) Acategorematic.
§ 69. A Categorematic word is one which can be used by itself as a
term.
§ 70. A Syncategorematic word is one which can help to form a term.
§ 71. An Acategorematic word is one which can neither form, nor help
to form, a term [Footnote: Comparatively few of the parts of speech
are categorematic. Nouns, whether substantive or adjective, including
of course pronouns and participles, are so, but only in their
nominative cases, except when an oblique case is so used as to be
equivalent to an attributive. Verbs also are categorematic, but only
in three of their moods, the Indicative, the Infinitive, and the
Potential. The Imperative and Optative moods clearly do not convey
assertions at all, while the Subjunctive can only figure as a
subordinate member of some assertion. We may notice, too, that the
relative pronoun, unlike the rest, is necessarily syncategorematic,
for the same reason as the subjunctive mood. Of the remaining parts of
speech the article, adverb, preposition, and conjunction can never be
anything but syncategorematic, while the interjection is
acategorematic, like the vocative case of nouns and the imperative and
optative moods of verbs, which do not enter at all into the form of
sentence known as the proposition.].
§ 72. Categorematic literally means 'predicable.' 'Horse,' 'swift,'
'galloping' are categorematic. Thus we can say, 'The horse is swift,'
or 'The horse is galloping.' Each of these words forms a term by
itself, but 'over' and 'swiftly' can only help to form a term, as in
the proposition, 'The horse is galloping swiftly over the plain.'
§ 73. A term then may be said to be a categorematic word or collection
of words, that is to say, one which can be used by itself as a
predicate.
§ 74. To entitle a word or collection of words to be called a term, it
is not necessary that it should be capable of standing by itself as a
subject. Many terms which can be used as predicates are incapable of
being used as subjects: but every term which can be used as a subject
(with the doubtful exception of proper names) can be used also as a
predicate. The attributives 'swift' and 'galloping' are terms, quite
as much as the subject 'horse,' but they cannot themselves be used as
subjects.
§ 75. When an attributive appears to be used as a subject, it is owing
to a grammatical ellipse. Thus in Latin we say 'Boni sapientes sunt,'
and in English 'The good are wise,' because it is sufficiently
declared by the inflexional form in the one case, and by the usage of
the language in the other, that men are signified. It is an accident
of language how far adjectives can be used as subjects. They cease to
be logical attributives the moment they are so used.
§ 76. There is a sense in which every word may become categorematic,
namely, when it is used simply as a word, to the neglect of its proper
meaning. Thus we can say--'"Swiftly" is an adverb.' 'Swiftly' in this
sense is really no more than the proper name for a particular
word. This sense is technically known as the 'suppositio materialis'
of a word.
CHAPTER II.
_Of the Division of Things._
§ 77. Before entering on the divisions of terms it is necessary to
advert for a moment to a division of the things whereof they are
names.
§ 78. By a 'thing' is meant simply an object of thought--whatever one
can think about.
§ 79. Things are either Substances or Attributes. Attributes may be
sub-divided into Qualities and Relations.
Thing
_______________|_______________
| |
Substance Attribute
_____________|____________
| |
Quality Relation
§ 80. A Substance is a thing which can be conceived to exist by
itself. All bodies are material substances. The soul, as a thinking
subject, is an immaterial substance.
§ 81. An Attribute is a thing which depends for its existence upon a
substance, e.g. greenness, hardness, weight, which cannot be
conceived to exist apart from green, hard, and heavy substances.
§ 82. A Quality is an attribute which does not require more than one
substance for its existence. The attributes just mentioned are
qualities. There might be greenness, hardness, and weight, if there
were only one green, hard and heavy substance in the universe.
§ 83. A Relation is an attribute which requires two or more substances
for its existence, e.g. nearness, fatherhood, introduction.
§ 84. When we say that a substance can be conceived to exist by
itself, what is meant is that it can be conceived to exist
independently of other substances. We do not mean that substances can
be conceived to exist independently of attributes, nor yet out of
relation to a mind perceiving them. Substances, so far as we can know
them, are only collections of attributes. When therefore we say that
substances can be conceived to exist by themselves, whereas attributes
are dependent for their existence upon substances, the real meaning of
the assertion reduces itself to this, that it is only certain
collections of attributes which can be conceived to exist
independently; whereas single attributes depend for their existence
upon others. The colour, smoothness or solidity of a table cannot be
conceived apart from the extension, whereas the whole cluster of
attributes which constitutes the table can be conceived to exist
altogether independently of other 'such clusters. We can imagine a
table to exist, if the whole material universe were annihilated, and
but one mind left to perceive it. Apart from mind, however, we cannot
imagine it: since what we call the attributes of a material substance
are no more than the various modes in which we find our minds
affected.
§ 85. The above division of things belongs rather to the domain of
metaphysics than of logic: but it is the indispensable basis of the
division of terms, to which we now proceed.
CHAPTER III.
_Of the Division of Terms._
§ 86. The following scheme presents to the eye the chief divisions of
terms.
Term
Division of terms according to their place in thought.
Subject-Term
Attributive
according to the kind of thing signified.
Abstract
Concrete
according to Quantity in Extension.
Singular
Common
according to Quality.
Positive
Privative
Negative
according to number of meanings.
Univocal
Equivocal
according to number of things involved in the name.
Absolute
Relative
according to number of quantities.
Connotative
Non-connotative
_Subject-term and Attributive._
§ 87. By a Subject-term is meant any term which is capable of standing
by itself as a subject, e.g. 'ribbon,' 'horse.'
§ 88. Attributives can only be used as predicates, not as subjects,
e.g. 'cherry-coloured,' 'galloping.' These can only be used in
conjunction with other words (syncategorematically) to make up a
subject. Thus we can say 'A cherry-coloured ribbon is becoming,' or 'A
galloping horse is dangerous.'
§ 89. Attributives are contrivances of language whereby we indicate
that a subject has a certain attribute. Thus, when we say 'This paper
is white,' we indicate that the subject 'paper' possesses the
attribute whiteness. Logic, however, also recognises as attributives
terms which signify the non-possession of attributes. 'Not-white' is
an attributive equally with 'white.'
§ 90. An Attributive then may be defined as a term which signifies the
possession, or non-possession, of an attribute by a subject.
§ 91. It must be carefully noticed that attributives are not names of
attributes, but names of the things which possess the attributes, in
virtue of our knowledge that they possess them. Thus 'white' is the
name of all the things which possess the attribute whiteness, and
'virtuous' is a name; not of the abstract quality, virtue, itself, but
of the men and actions which possess it. It is clear that a term can
only properly be said to be a name of those things whereof it can be
predicated. Now, we cannot intelligibly predicate an attributive of
the abstract quality, or qualities, the possession of which it
implies. We cannot, for instance, predicate the term 'learned' of the
abstract quality of learning: but we may predicate it of the
individuals, Varro and Vergil. Attributives, then, are to be regarded
as names, not of the attributes which they imply, but of the things in
which those attributes are found.
§ 92. Attributives, however, are names of things in a less direct way
than that in which subject-terms may be the names of the same
things. Attributives are names of things only in predication, whereas
subject-terms are names of things in or out of predication. The terms
'horse' and 'Bucephalus' are names of certain things, in this case
animals, whether we make any statement about them or not: but the
terms 'swift' and 'fiery' only become names of the same things in
virtue of being predicable of them. When we say 'Horses are swift' or
'Bucephalus was fiery,' the terms 'swift' and 'fiery' become names
respectively of the same things as 'horse' and 'Bucephalus.' This
function of attributives as names in a secondary sense is exactly
expressed by the grammatical term 'noun adjective.' An attributive is
not directly the name of anything. It is a name added on in virtue of
the possession by a given thing of a certain attribute, or, in some
cases, the non-possession.
§ 93. Although attributives cannot be used as subjects, there is
nothing to prevent a subject-term from being used as a predicate, and
so assuming for the time being the functions of an attributive. When
we say 'Socrates was a man,' we convey to the mind the idea of the
same attributes which are implied by the attributive 'human.' But
those terms only are called attributives which can never be used
except as predicates.
§ 94. This division into Subject-terms and Attributives may be
regarded as a division of terms according to their place in
thought. Attributives, as we have seen, are essentially predicates,
and can only be thought of in relation to the subject, whereas the
subject is thought of for its own sake.
_Abstract and Concrete Terms_.
§ 95. An Abstract Term is the name of an attribute, e.g. whiteness
[Footnote: Since things cannot be spoken of except by their names,
there is a constantly recurring source of confusion between the thing
itself and the name of it. Take for instance 'whiteness.' The
attribute whiteness is a thing, the word 'whiteness' is a term.],
multiplication, act, purpose, explosion.
§ 96. A Concrete Term is the name of a substance, e.g. a man, this
chair, the soul, God.
§ 97. Abstract terms are so called as being arrived at by a process of
Abstraction. What is meant by Abstraction will be clear from a single
instance. The mind, in contemplating a number of substances, may draw
off, or abstract, its attention from all their other characteristics,
and fix it only on some point, or points, which they have in
common. Thus, in contemplating a number of three-cornered objects, we
may draw away our attention from all their other qualities, and fix it
exclusively upon their three-corneredness, thus constituting the
abstract notion of 'triangle.' Abstraction may be performed equally
well in the case of a single object: but the mind would not originally
have known on what points to fix its attention except by a comparison
of individuals.
§ 98. Abstraction too may be performed upon attributes as well as
substances. Thus, having by abstraction already arrived at the notion
of triangle, square, and so on, we may fix our attention upon what
these have in common, and so rise to the higher abstraction of
'figure.' As thought becomes more complex, we may have abstraction on
abstraction and attributes of attributes. But, however many steps may
intervene, attributes may always be traced back to substances at
last. For attributes of attributes can mean at bottom nothing but the
co-existence of attributes in, or in connection with, the same
substances.
§ 99. We have said that abstract terms are so called, as being arrived
at by abstraction: but it must not be inferred from this statement
that all terms which are arrived at by abstraction are abstract. If
this were so, all names would be abstract except proper names of
individual substances. All common terms, including attributives, are
arrived at by abstraction, but they are not therefore abstract terms.
Those terms only are called abstract, which cannot be applied to
substances at all. The terms 'man' and 'human' are names of the same
substance of which Socrates is a name. Humanity is a name only of
certain attributes of that substance, namely those which are shared by
others. All names of concrete things then are concrete, whether they
denote them individually or according to classes, and whether directly
and in themselves, or indirectly, as possessing some given attribute.
§ 100. By a 'concrete thing' is meant an individual Substance
conceived of with all its attributes about it. The term is not
confined to material substances. A spirit conceived of under personal
attributes is as concrete as plum-pudding.
§ 101. Since things are divided exhaustively into substances and
attributes, it follows that any term which is not the name of a thing
capable of being conceived to exist by itself, must be an abstract
term. Individual substances can alone be conceived to exist by
themselves: all their qualities, actions, passions, and
inter-relations, all their states, and all events with regard to them,
presuppose the existence of these individual substances. All names
therefore of such things as those just enumerated are abstract
terms. The term 'action,' for instance, is an abstract term. For how
could there be action without an agent? The term 'act' also is equally
abstract for the same reason. The difference between 'action' and
'act' is not the difference between abstract and concrete, but the
difference between the name of a process and the name of the
corresponding product. Unless acts can be conceived to exist without
agents they are as abstract as the action from which they result.
§ 102. Since every term must be either abstract or concrete, it may be
asked--Are attributives abstract or concrete? The answer of course
depends upon whether they are names of substances or names of
attributes. But attributives, it must be remembered, are never
directly names of anything, in the way that subject-terms are; they
are only names of things in virtue of being predicated of
them. Whether an attributive is abstract or concrete, depends on the
nature of the subject of which it is asserted or denied. When we say
'This man is noble,' the term 'noble' is concrete, as being the name
of a substance: but when we say 'This act is noble,' the term 'noble'
is abstract, as being the name of an attribute.
§ 103. The division of terms into Abstract and Concrete is based upon
the kind of thing signified. It involves no reference to actual
existence. There are imaginary as well as real substances. Logically a
centaur is as much a substance as a horse.
_Terms._
§ 104. A Singular Term is a name which can be applied, in the same
sense, to one thing only, e.g. 'John,' 'Paris,' 'the capital of
France,' 'this pen.'
§ 105. A Common Term is a name which can be applied, in the same
sense, to a class of things, e.g. 'man,' 'metropolis,' 'pen.'
In order that a term may be applied in the same sense to a number of
things, it is evident that it must indicate attributes which are
common to all of them. The term 'John' is applicable to a number of
things, but not in the same sense, as it does not indicate attributes.
§ 106. Common terms are formed, as we have seen already (§ 99), by
abstraction, i. e. by withdrawing the attention from the attributes in
which individuals differ, and concentrating it upon those which they
have in common.
§ 107. A class need not necessarily consist of more than two
things. If the sun and moon were the only heavenly bodies in the
universe, the word 'heavenly body' would still be a common term, as
indicating the attributes which are possessed alike by each.
§ 108. This being so, it follows that the division of terms into
singular and common is as exhaustive as the preceding ones, since a
singular term is the name of one thing and a common term of more than
one. It is indifferent whether the thing in question be a substance or
an attribute; nor does it matter how complex it may be, so long as it
is regarded by the mind as one.
§ 109. Since every term must thus be either singular or common, the
members of the preceding divisions must find their place under one or
both heads of this one. Subject-terms may plainly fall under either
head of singular or common: but attributives are essentially common
terms. Such names as 'green,' 'gentle,' 'incongruous' are applicable,
strictly in the same sense, to all the things which possess the
attributes which they imply.
§ 110. Are abstract terms then, it may be asked, singular or common?
To this question we reply--That depends upon how they are used. The
term 'virtue,' for instance, in one sense, namely, as signifying moral
excellence in general, without distinction of kind, is strictly a
singular term, as being the name of one attribute: but as applied to
different varieties of moral excellence--justice, generosity,
gentleness and so on--it is a common term, as being a name which is
applicable, in the same sense, to a class of attributes. Similarly the
term 'colour,' in a certain sense, signifies one unvarying attribute
possessed by bodies, namely, the power of affecting the eye, and in
this sense it is a singular term: but as applied to the various ways
in which the eye may be affected, it is evidently a common term, being
equally applicable to red, blue, green, and every other colour. As
soon as we begin to abstract from attributes, the higher notion
becomes a common term in reference to the lower. By a 'higher notion'
is meant one which is formed by a further process of abstraction. The
terms 'red,' 'blue,' 'green,' etc., are arrived at by abstraction from
physical objects; 'colour' is arrived at by abstraction from them, and
contains nothing, but what is common to all. It therefore applies in
the same sense to each, and is a common term in relation to them.
§ 111. A practical test as to whether an abstract term, in any given
case, is being used as a singular or common term, is to try whether
the indefinite article or the sign of the plural can be attached to
it. The term 'number,' as the name of a single attribute of things,
admits of neither of these adjuncts: but to talk of 'a number' or 'the
numbers, two, three, four,' etc., at once marks it as a common
term. Similarly the term 'unity' denotes a single attribute, admitting
of no shades of distinction: but when a writer begins to speak of 'the
unities' he is evidently using the word for a class of things of some
kind or other, namely, certain dramatical proprieties of composition.
Proper _Names_ and _Designations_.
§ 112. Singular terms may be subdivided into Proper Names and
Designations.
§ 113. A Proper Name is a permanent singular term applicable to a
thing in itself; a Designation is a singular term devised for the
occasion, or applicable to a thing only in so far as it possesses some
attribute.
§ 114. 'Homer' is a proper name; 'this man,' 'the author of the Iliad'
are designations.
§ 115. The number of things, it is clear, is infinite. For, granting
that the physical universe consists of a definite number of
atoms--neither one more nor one less--still we are far from having
exhausted the possible number of things. All the manifold material
objects, which are made up by the various combinations of these atoms,
constitute separate objects of thought, or things, and the mind has
further an indefinite power of conjoining and dividing these objects,
so as to furnish itself with materials of thought, and also of fixing
its attention by abstraction upon attributes, so as to regard them as
things, apart from the substances to which they belong.
§ 116. This being so, it is only a very small number of things, which
are constantly obtruding themselves upon the mind, that have singular
terms permanently set apart to denote them. Human beings, some
domestic animals, and divisions of time and place, have proper names
assigned to them in most languages, e.g. 'John,' 'Mary,' 'Grip,'
'January,' 'Easter,' 'Belgium,' 'Brussels,' 'the Thames,' 'Ben-Nevis.'
Besides these, all abstract terms, when used without reference to
lower notions, are of the nature of proper names, being permanently
set apart to denote certain special attributes, e.g. 'benevolence,'
'veracity,' 'imagination,' 'indigestibility, 'retrenchment.'
§ 117. But the needs of language often require a singular term to
denote some thing which has not had a proper name assigned to it. This
is effected by taking a common term, and so limiting it as to make it
applicable, under the given circumstances, to one thing only. Such a
limitation may be effected in English by prefixing a demonstrative or
the definite article, or by appending a description, e.g. 'this pen,'
'the sofa,' 'the last rose of summer.' When a proper name is unknown,
or for some reason, unavailable, recourse may be had to a designation,
e.g. 'the honourable member who spoke last but one.'
_Collective Terms_.
§ 118. The division of terms into singular and common being, like
those which have preceded it, fundamental and exhaustive, there is
evidently no room in it for a third class of Collective Terms. Nor is
there any distinct class of terms to which that name can be given. The
same term may be used collectively or distributively in different
relations. Thus the term 'library,' when used of the books which
compose a library, is collective; when used of various collections of
books, as the Bodleian, Queen's library, and so on, it is
distributive, which, in this case, is the same thing as being a common
term.
§ 119, The distinction between the collective and distributive use of
a term is of importance, because the confusion of the two is a
favourite source of fallacy. When it is said 'The plays of Shakspeare
cannot be read in a day,' the proposition meets with a very different
measure of acceptance according as its subject is understood
collectively or distributively. The word 'all' is perfectly ambiguous
in this respect. It may mean all together or each separately--two
senses which are distinguished in Latin by 'totus' or 'cunctus,' for
the collective, and 'omnis' for the distributive use.
§ 120. What is usually meant however when people speak of a collective
term is a particular kind of singular term.
§ 121. From this point of view singular terms may be subdivided into
Individual and Collective, by an Individual Term being meant the name
of one object, by a Collective Term the name of several considered as
one. 'This key' is an individual term; 'my bunch of keys' is a
collective term.
§ 122. A collective term is quite as much the name of one thing as an
individual term is, though the thing in question happens to be a
group. A group is one thing, if we choose to think of it as one. For
the mind, as we have already seen, has an unlimited power of forming
its own things, or objects of thought. Thus a particular peak in a
mountain chain is as much one thing as the chain itself, though,
physically speaking, it is inseparable from it, just as the chain
itself is inseparable from the earth's surface. In the same way a
necklace is as much one thing as the individual beads which compose
it.
§ 123. We have just seen that a collective term is the name of a group
regarded as one thing: but every term which is the name of such a
group is not necessarily a collective term. 'London,' for instance, is
the name of a group of objects considered as one thing. But 'London'
is not a collective term, whereas 'flock,' 'regiment,' and 'senate'
are. Wherein then lies the difference? It lies in this--that flock,
regiment and senate are groups composed of objects which are, to a
certain extent, similar, whereas London is a group made up of the most
dissimilar objects--streets and squares and squalid slums, fine
carriages and dirty faces, and so on. In the case of a true collective
term all the members of the group will come under some one common
name. Thus all the members of the group, flock of sheep, come under
the common name 'sheep,' all the members of the group 'regiment' under
the common name, 'soldier,' and so on.
§ 124. The subdivision of singular terms into individual and
collective need not be confined to the names of concrete things. An
abstract term like 'scarlet,' which is the name of one definite
attribute, may be reckoned 'individual,' while a term like 'human
nature,' which is the name of a whole group of attributes, would more
fitly be regarded as collective.
§ 126. The main division of terms, which we have been discussing, into
singular and collective, is based upon their Quantity in
Extension. This phrase will be explained presently.
§ 126. We come now to a threefold division of terms into Positive,
Privative and Negative. It is based upon an implied two-fold division
into positive and non-positive, the latter member being subdivided
into Privative and Negative.
Term
_______________|_______________
| |
Positive Non-Positive
_____________|____________
| |
Privative Negative
If this division be extended, as it sometimes is, to terms in general,
a positive term must be taken to mean only the definite, or
comparatively definite, member of an exhaustive division in accordance
with the law of excluded middle (§ 25). Thus 'Socrates' and 'man' are
positive, as opposed to 'not-Socrates' and 'not-man.'
§ 127. The chief value of the division, however, and especially of the
distinction drawn between privative and negative terms, is in relation
to attributives.
From this point of view we may define the three classes of terms as
follows:
A Positive Term signifies the presence of an attribute, e.g.: 'wise,'
'full.'
A Negative Term signifies merely the absence of an attribute,
e.g. 'not-wise,' 'not-full.'
A Privative Term signifies the absence of an attribute in a subject
capable of possessing it, e.g. 'unwise,' 'empty'. [Footnote: A
privative term is usually defined to mean one which signifies the
absence of an attribute where it was once possessed, or might have
been expected to be present, e.g. 'blind.' The utility of the slight
extension of meaning here assigned to the expression will, it is
hoped, prove its justification.]
§ 128. Thus a privative term stands midway in meaning between the
other two, being partly positive and partly negative--negative in so
far as it indicates the absence of a certain attribute, positive in so
far as it implies that the thing which is declared to lack that
attribute is of such a nature as to be capable of possessing it. A
purely negative term conveys to the mind no positive information at
all about the nature of the thing of which it is predicated, but
leaves us to seek for it among the universe of things which fail to
exhibit a given attribute.
A privative term, on the other hand, restricts us within a definite
sphere. The term 'empty' restricts us within the sphere of things
which are capable of fulness, that is, if the term be taken in its
literal sense, things which possess extension in three dimensions.
§ 129. A positive and a negative term, which have the same matter,
must exhaust the universe between them, e.g. 'white' and 'not-white,'
since, according to the law of excluded middle, everything must be
either one or the other. To say, however, that a thing is 'not-white'
is merely to say that the term 'white' is inapplicable to it.
'Not-white' may be predicated of things which do not possess extension
as well as of those which do. Such a pair of terms as 'white' and
'not-white,' in their relation to one another, are called
Contradictories.
§ 130. Contrary terms must be distinguished from
contradictory. Contrary terms are those which are most opposed under
the same head. Thus 'white' and 'black' are contrary terms, being the
most opposed under the same head of colour. 'Virtuous' and 'vicious'
again are contraries, being the most opposed under the same head of
moral quality.
§ 131. A positive and a privative term in the same matter will always
be contraries, e.g. 'wise' and 'unwise,' 'safe' and 'unsafe': but
contraries do not always assume the shape of positive and privative
terms, but may both be positive in form, e.g. 'wise' and 'foolish,'
'safe' and 'dangerous.'
§ 132. Words which are positive in form are often privative in
meaning, and vice versâ. This is the case, for instance, with the word
'safe,' which connotes nothing more than the absence of danger. We
talk of a thing involving 'positive danger' and of its being
'positively unsafe' to do so and so. 'Unhappy,' on the other hand,
signifies the presence of actual misery. Similarly in Latin 'inutilis'
signifies not merely that there is no benefit to be derived from a
thing, but that it is _positively injurious_. All such questions,
however, are for the grammarian or lexicographer, and not for the
logician. For the latter it is sufficient to know that corresponding
to every term which signifies the presence of some attribute there may
be imagined another which indicates the absence of the same attribute,
where it might be possessed, and a third which indicates its absence,
whether it might be possessed or not.
§ 133. Negative terms proper are formed by the prefix 'not-' or
'non-,' and are mere figments of logic. We do not in practice require
to speak of the whole universe of objects minus those which possess a
given attribute or collection of attributes. We have often occasion to
speak of things which might be wise and are not, but seldom, if ever,
of all things other than wise.
§ 134. Every privative attributive has, or may have, a corresponding
abstract term, and the same is the case with negatives: for the
absence of an attribute, is itself an attribute. Corresponding to
'empty,' there is 'emptiness'; corresponding to 'not-full' there may
be imagined the term 'not-fulness.'
§ 135. The contrary of a given term always involves the contradictory,
but it involves positive elements as well. Thus 'black' is
'not-white,' but it is something more besides. Terms which, without
being directly contrary, involve a latent contradiction, are called
Repugnant, e.g. 'red' and 'blue.' All terms whatever which signify
attributes that exclude one another may be called Incompatible.
§ 136. The preceding division is based on what is known as the Quality
of terms, a positive term being said to differ in quality from a
non-positive one.
_Univocal and Equivocal Terms_.
§ 137. A term is said to be Univocal, when it has one and the same
meaning wherever it occurs. A term which has more than one meaning is
called Equivocal. 'Jam-pot,' 'hydrogen' are examples of univocal
terms; 'pipe' and 'suit' of equivocal.
§ 138. This division does not properly come within the scope of logic,
since it is a question of language, not of thought. From the
logician's point of view an equivocal term is two or more different
terms, for the definition in each sense would be different.
§ 139. Sometimes a third member is added to the same division under
the head of Analogous Terms. The word 'sweet,' for instance, is
applied by analogy to things so different in their own nature as a
lump of sugar, a young lady, a tune, a poem, and so on. Again, because
the head is the highest part of man, the highest part of a stream is
called by analogy 'the head.' It is plainly inappropriate to make a
separate class of analogous terms. Rather, terms become equivocal by
being extended by analogy from one thing to another.
_Absolute and Relative Terms_.
§ 140. An Absolute term is a name given to a thing without reference
to anything else.
§ 141. A Relative term is a name given to a thing with direct
reference to some other thing.
§ 142. 'Hodge' and 'man' are absolute terms. 'Husband' 'father,'
'shepherd' are relative terms. 'Husband' conveys a direct reference to
'wife,' 'father' to 'Child,' 'shepherd' to 'sheep.' Given one term of
a relation, the other is called the correlative, e.g. 'subject' is
the correlative of 'ruler,' and conversely 'ruler' of 'subject.' The
two terms are also spoken of as a pair of correlatives.
§ 143. The distinction between relative and absolute applies to
attributives as well as subject-terms. 'Greater,' 'near, 'like,' are
instances of attributives which everyone would recognise as relative.
§ 144. A relation, it will be remembered, is a kind of attribute,
differing from a quality in that it necessarily involves more
substances than one. Every relation is at bottom a fact, or series of
facts, in which two or more substances play a part. A relative term
connotes this fact or facts from the point of view of one of the
substances, its correlative from that of the other. Thus 'ruler' and
'subject' imply the same set of facts, looked at from opposite points
of view. The series of facts itself, regarded from either side, is
denoted by the corresponding abstract terms, 'rule 'and 'subjection.'
§ 145. It is a nice question whether the abstract names of relations
should themselves be considered relative terms. Difficulties will
perhaps be avoided by confining the expression 'relative _term_'
to names of concrete things. 'Absolute,' it must be remembered, is a
mere negative of 'relative,' and covers everything to which the
definition of the latter does not strictly apply. Now it can hardly be
said that 'rule' is a name given to a certain abstract thing with
direct reference to some other thing, namely, subjection. Rather
'rule' and 'subjection' are two names for identically the same series
of facts, according to the side from which we look at them. 'Ruler'
and 'subject,' on the other hand, are names of two distinct
substances, but each involving a reference to the other.
§ 146. This division then may be said to be based on the number of
things involved in the name.
_Connotative and Non-Connotative Terms._
§ 147. Before explaining this division, it is necessary to treat of
what is called the Quantity of Terms.
_Quantity of Terms._
§ 148. A term is possessed of quantity in two ways--
(1) In Extension;
(2) In Intension.
§ 149. The Extension of a term is the number of things to which it
applies.
§ 150. The Intension of a term is the number of attributes which it
implies.
§ 151. It will simplify matters to bear in mind that the intension of
a term is the same thing as its meaning. To take an example, the term
'man' applies to certain things, namely, all the members of the human
race that have been, are, or ever will be: this is its quantity in
extension. But the term 'man' has also a certain meaning, and implies
certain attributes--rationality, animality, and a definite bodily
shape: the sum of these attributes constitutes its quantity in
intension.
§ 152. The distinction between the two kinds of quantity possessed by
a term is also conveyed by a variety of expressions which are here
appended.
Extension = breadth = compass = application = denotation.
Intension = depth = comprehension = implication = connotation.
Of these various expressions, 'application' and 'implication' have the
advantage of most clearly conveying their own meaning. 'Extension' and
'intension,' however, are more usual; and neither 'implication' nor
'connotation' is quite exact as a synonym for 'intension.' (§ 164.)
§ 153. We now return to the division of terms into connotative and
non-connotative.
§ 154. A term is said to connote attributes, when it implies certain
attributes at the same time that it applies to certain things distinct
therefrom. [Footnote: Originally 'connotative' was used in the same
sense in which we have used 'attributive,' for a word which directly
signifies the presence of an attribute and indirectly applies to a
subject. In this, its original sense, it was the subject which was
said to be connoted, and not the attribute.]
§ 155. A term which possesses both extension and intension, distinct
from one another, is connotative.
§ 156. A term which possesses no intension (if that be possible) or in
which extension and intension coincide is non-connotative.
§ 157. The subject-term, 'man,' and its corresponding attributive,
'human,' have both extension and intension, distinct from one
another. They are therefore connotative. But the abstract term,
'humanity,' denotes the very collection of attributes, which was
before connoted by the concrete terms, 'man' and 'human.' In this
case, therefore, extension and intension coincide, and the term is
non-connotative.
§ 158. The above remark must be understood to be limited to abstract
terms in their singular sense. When employed as common terms, abstract
terms possess both extension and intension distinct from one
another. Thus the term 'colour' applies to red, blue, and yellow, and
at the same time implies (i.e. connotes), the power of affecting the
eye.
§ 159. Since all terms are names of things, whether substances or
attributes, it is clear that all terms must possess extension, though
the extension of singular terms is the narrowest possible, as being
confined to one thing.
§ 160. Are there then any terms which possess no intension? To ask
this, is to ask--Are there any terms which have absolutely no meaning?
It is often said that proper names are devoid of meaning, and the
remark is, in a certain sense, true. When we call a being by the name
'man,' we do so because that being possesses human attributes, but
when we call the same being by the name, 'John,' we do not mean to
indicate the presence of any Johannine attributes. We simply wish to
distinguish that being, in thought and language, from other beings of
the same kind. Roughly speaking, therefore, proper names are devoid of
meaning or intension. But no name can be entirely devoid of
meaning. For, even setting aside the fact, which is not universally
true, that proper names indicate the sex of the owner, the mere act of
giving a name to a thing implies at least that the thing exists,
whether in fact or thought; it implies what we may call 'thinghood':
so that every term must carry with it some small amount of intension.
§ 161. From another point of view, however, proper names possess more
intension than any other terms. For when we know a person, his name
calls up to our minds all the individual attributes with which we are
familiar, and these must be far more numerous than the attributes
which are conveyed by any common term which can be applied to
him. Thus the name 'John' means more to a person who knows him than
'attorney,' 'conservative,' 'scamp,' of 'vestry-man,' or any other
term which may happen to apply to him. This, however, is the acquired
intension of a term, and must be distinguished from the original
intension. The name 'John' was never meant to indicate the attributes
which its owner has, as a matter of fact, developed. He would be John
all the same, if he were none of these.
§ 162. Hitherto we have been speaking only of christening-names, but
it is evident that family names have a certain amount of connotation
from the first. For when we dub John with the additional appellation
of Smith, we do not give this second name as a mere individual mark,
but intend thereby to indicate a relationship to other persons. The
amount of connotation that can be conveyed by proper names is very
noticeable in the Latin language. Let us take for an example the full
name of a distinguished Roman--Publius Cornelius Scipio Æmilianus
Africanus minor. Here it is only the prænomen, Publius, that can be
said to be a mere individual mark, and even this distinctly indicates
the sex of the owner. The nomen proper, Cornelius, declares the wearer
of it to belong to the illustrious gens Cornelia. The cognomen,
Scipio, further specifies him as a member of a distinguished family in
that gens. The agnomen adoptivum indicates his transference by
adoption from one gens to another. The second agnomen recalls the
fact of his victory over the Carthaginians, while the addition of the
word 'minor' distinguishes him from the former wearer of the same
title. The name, instead of being devoid of meaning, is a chapter of
history in itself. Homeric epithets, such as 'The Cloud-compeller,'
'The Earth-shaker' are instances of intensive proper names. Many of
our own family names are obviously connotative in their origin,
implying either some personal peculiarity, e.g. Armstrong, Cruikshank,
Courteney; or the employment, trade or calling of the original bearer
of the name, Smith, Carpenter, Baker, Clark, Leach, Archer, and so on;
or else his abode, domain or nationality, as De Caen, De Montmorency,
French, Langley; or simply the fact of descent from some presumably
more noteworthy parent, as Jackson, Thomson, Fitzgerald, O'Connor,
Macdonald, Apjohn, Price, Davids, etc. The question, however, whether
a term is connotative or not, has to be decided, not by its origin,
but by its use. We have seen that there are some proper names which,
in a rough sense, may be said to possess no intension.
§ 163. The other kind of singular terms, namely, designations (§ 113)
are obviously connotative. We cannot employ even the simplest of them
without conveying more or less information about the qualities of the
thing which they are used to denote. When, for instance, we say 'this
table,' 'this book,' we indicate the proximity to the speaker of the
object in question. Other designations have a higher degree of
intension, as when we say 'the present prime minister of England,'
'the honourable member who brought forward this motion to-night.'
Such terms have a good deal of significance in themselves, apart from
any knowledge we may happen to possess of the individuals they denote.
§ 164. We have seen that, speaking quite strictly, there are no terms
which are non-connotative: but, for practical purposes, we may apply
the expression to proper names, on the ground that they possess no
intension, and to singular abstract terms on the ground that their
extension and intension coincide. In the latter case it is indifferent
whether we call the quantity extension or intension. Only we cannot
call it 'connotation,' because that implies two quantities distinct
from one another. A term must already denote a subject before it can
be said to connote its attributes.
§ 165. The division of terms into connotative and non-connotative is
based on their possession of one quantity or two.
CHAPTER IV.
_Of the Law of Inverse Variation of Extension and Intension._
§ 166. In a series of terms which fall under one another, as the
extension decreases, the intension increases, and vice versâ. Take for
instance the following series--
Thing
|
Substance
|
Matter
|
Organism
|
Animal
|
Vertebrate
|
Mammal
|
Ruminant
|
Sheep
|
This sheep.
Here the term at the top possesses the widest possible extension,
since it applies to everything. But at the same time it possesses the
least possible amount of intension, implying nothing more than mere
existence, whether in fact or thought. On the other hand, the term at
the bottom possesses the greatest amount of intension, since it
implies all the attributes of, an individual superadded to those of
the class to which it belongs: but its extension is the narrowest
possible, being limited to one thing.
§ 167. At each step in the descent from the term at the top, which is
called the 'Summum genus,' to the individual, we decrease the
extension by increasing the intension. Thus by adding on to the bare
notion of a thing the idea of independent existence, we descend to the
term 'substance,' This process is known as Determination, or
Specialisation.
§ 168. Again, by withdrawing our attention from the individual
characteristics of a particular sheep, and fixing it upon those which
are common to it with other animals of the same kind, we arrive at the
common term, 'sheep.' Here we have increased the extension by
decreasing the intension. This process is known as Generalisation.
§ 169. Generalisation implies abstraction, but we may have abstraction
without generalisation.
§ 170. The following example is useful, as illustrating to the eye how
a decrease of extension is accompanied by an increase of intension. At
each step of the descent here we visibly tack on a fresh
attribute. [Footnote: This example is borrowed from Professor Jevons.]
Ship
|
Steam-ship
|
Screw steam-ship
|
Iron screw steam-ship
|
British iron screw steam-ship.
Could we see the classes denoted by the names the pyramid would be
exactly inverted.
§ 171. The law of inverse variation of extension and intension must of
course be confined to the inter-relations of a series of terms of
which each can be predicated of the other until we arrive at the
bottom of the scale. It is not meant to apply to the extension and
intension of the same term. The increase of population does not add to
the meaning of 'baby.'
PART II.--OF PROPOSITIONS.
CHAPTER I.
_Of the Proposition as distinguished from Other Sentences_.
§ 172. As in considering the term, we found occasion to distinguish it
from words generally, so now, in considering the proposition, it will
be well to begin by distinguishing it from other sentences.
§ 173. Every proposition is a sentence, but every sentence is not a
proposition.
§ 174. The field of logic is far from being conterminous with that of
language. Language is the mirror of man's whole nature, whereas logic
deals with language only so far as it gives clothing to the products
of thought in the narrow sense which we have assigned to that term.
Language has materials of every sort lying strewn about, among which
the logician has to seek for his proper implements.
§ 175. Sentences may be employed for a variety of purposes--
(1) To ask a question;
(2) To give an order;
(3) To express a feeling;
(4) To make a statement.
These various uses give rise respectively to
(1) The Interrogative Sentence;
(2) The Imperative Sentence;
(3) The Exclamatory Sentence;
(4) The Enunciative Sentence; Indicative Potential.
It is with the last of these only that logic is concerned.
§ 176. The proposition, therefore, corresponds to the Indicative and
Potential, or Conditional, sentences of grammar. For it must be borne
in mind that logic recognises no difference between a statement of
fact and a supposition. 'It may rain to-morrow' is as much a
proposition as 'It is raining now.'
§ 177. Leaving the grammatical aspect of the proposition, we must now
consider it from the purely logical point of view.
§ 178. A proposition is a judgement expressed in words; and a
judgement is a direct comparison between two concepts.
§ 179. The same thing may be expressed more briefly by saying that a
proposition is a direct comparison between two terms.
§ 180. We say 'direct comparison,' because the syllogism also may be
described as a comparison between two terms: but in the syllogism the
two terms are compared indirectly, or by means of a third term.
§ 181. A proposition may be analysed into two terms and a Copula,
which is nothing more than the sign of agreement or disagreement
between them.
§ 182. The two terms are called the Subject and the Predicate (§ 58).
§ 183. The Subject is that of which something is stated.
§ 184. The Predicate is that which is stated of the subject.
§ 185. Hence the subject is thought of for its own sake, and the
predicate for the sake of the subject.
CHAPTER II.
Of _the Copula_.
§ 186. There are two kinds of copula, one for affirmative and one for
negative statements.
§ 187. Materially the copula is expressed by some part of the verb 'to
be,' with or without the negative, or else is wrapped up in some
inflexional form of a verb.
§ 188. The material form of the copula is an accident of language, and
a matter of indifference to logic. 'The kettle boils' is as logical a
form of expression as 'The kettle is boiling.' For it must be
remembered that the word 'is' here is a mere sign of agreement between
the two terms, and conveys no notion of actual existence. We may use
it indeed with equal propriety to express non-existence, as when we
say 'An idol is nothing.'
§ 189. When the verb 'to be' expresses existence in fact it is known
in grammar as 'the substantive verb.' In this use it is predicate as
well as copula, as when we say 'God is,' which may be analysed, if we
please, into 'God is existent.'
§ 190. We have laid down above that there are two kinds of copula,
affirmative and negative: but some logicians have maintained that the
copula is always affirmative.
§ 191. What then, it may be asked, on this view, is the meaning of
negative propositions! To which the answer is, that a negative
proposition asserts an agreement between the subject and a negative
term. When, for instance, we say 'The whale is not a fish,' this would
be interpreted to mean 'The whale is a not-fish.'
§ 192. Undoubtedly any negative proposition may be exhibited in an
affirmative form, since, by the law of excluded middle, given a pair
of contradictory terms, wherever the one can be asserted, the other
can be denied, and vice versâ. We shall find later on that this
principle gives rise to one of the forms of immediate inference. The
only question then can be, which is the more natural and legitimate
form of expression. It seems simpler to suppose that we assert the
agreement of 'whale' with 'not-fish' by implication only, and that
what we directly do is to predicate a disagreement between 'whale' and
the positive attributes connoted by 'fish.' For since 'not-fish' must
apply to every conceivable object of thought except those which fall
under the positive term 'fish,' to say that a whale is a 'not-fish,'
is to say that we have still to search for 'whale' throughout the
whole universe of being, minus a limited portion; which is only a more
clumsy way of saying that it is not to be found in that portion.
§ 193. Again, the term 'not-fish' must be understood either in its
intension or in its extension. If it be understood in its intension,
what it connotes is simply the absence of the positive qualities which
constitute a fish, a meaning which is equally conveyed by the negative
form of proposition. We gain nothing in simplicity by thus confounding
assertion with denial. If, on the other hand, it is to be taken in
extension, this involves the awkwardness of supposing that the
predicative power of a term resides in its extensive capacity.
§ 194. We therefore recognise predication as being of two
kinds--affirmation and negation--corresponding to which there are two
forms of copula.
§ 195. On the other hand, other logicians have maintained that there
are many kinds of copula, since the copula must vary according to the
various degrees of probability with which we can assert or deny a
predicate of a subject. This view is technically known as the doctrine
of
_The Modality of the Copula._
§ 196. It may plausibly be maintained that the division of
propositions into affirmative and negative is not an exhaustive one,
since the result of an act of judgement is not always to lead the mind
to a clear assertion or a clear denial, but to leave it in more or
less doubt as to whether the predicate applies to the subject or
not. Instead of saying simply A is B, or A is not B, we may be led to
one of the following forms of proposition--
A is possibly B.
A is probably B.
A is certainly B.
The adverbial expression which thus appears to qualify the copula is
known as 'the mode.'
§ 197. When we say 'The accused may be guilty' we have a proposition
of very different force from 'The accused is guilty,' and yet the
terms appear to be the same. Wherein then does the difference lie? 'In
the copula' would seem to be the obvious reply. We seem therefore
driven to admit that there are as many different kinds of copula as
there are different degrees of assurance with which a statement may be
made.
§ 198. But there is another way in which modal propositions may be
regarded. Instead of the mode being attached to the copula, it may be
considered as itself constituting the predicate, so that the above
propositions would be analysed thus--
That A is B, is possible.
That A is B, is probable.
That A is B, is certain.
§ 199. The subject here is itself a proposition of which we predicate
various degrees of probability. In this way the division of
propositions into affirmative and negative is rendered exhaustive. For
wherever before we had a doubtful assertion, we have now an assertion
of doubtfulness.
§ 200. If degrees of probability can thus be eliminated from the
copula, much more so can expressions of time, which may always be
regarded as forming part of the predicate. 'The sun will rise
to-morrow' may be analysed into 'The sun is going to rise to-morrow.'
In either case the tense belongs equally to the predicate. It is often
an awkward task so to analyse propositions relative to past or future
time as to bring out the copula under the form 'is' or 'is not': but
fortunately there is no necessity for so doing, since, as has been
said before (§ 188), the material form of the copula is a matter of
indifference to logic. Indeed in affirmative propositions the mere
juxtaposition of the subject and predicate is often sufficient to
indicate their agreement, e.g. 'Most haste, worst speed,' chalepha
tha kala. It is because all propositions are not affirmative that we
require a copula at all. Moreover the awkwardness of expression just
alluded to is a mere accident of language. In Latin we may say with
equal propriety 'Sol orietur cras' or 'Sol est oriturus cras'; while
past time may also be expressed in the analytic form in the case of
deponent verbs, as 'Caesar est in Galliam profectus'--'Caesar is gone
into Gaul.'
§ 201. The copula then may always be regarded as pure, that is, as
indicating mere agreement or disagreement between the two terms of the
proposition.
CHAPTER III.
_Of the Divisions of Propositions_.
§ 202. The most obvious and the most important division of
propositions is into true and false, but with this we are not
concerned. Formal logic can recognise no difference between true and
false propositions. The one is represented by the same symbols as the
other.
§ 203. We may notice, however, in passing, that truth and falsehood
are attributes of propositions and of propositions only. For something
must be predicated, i.e. asserted or denied, before we can have
either truth or falsehood. Neither concepts or terms, on the one hand,
nor reasonings, on the other, can properly be said to be true or
false. In the mere notion of a Centaur or of a black swan there is
neither truth nor falsehood; it is not until we make some statement
about these things, such as that 'black swans are found in Australia,'
or 'I met a Centaur in the High Street yesterday,' that the question
of truth or falsehood comes in. In such expressions as a 'true friend'
or 'a false patriot' there is a tacit reference to propositions. We
mean persons of whom the terms 'friend' and 'patriot' are truly or
falsely predicated. Neither can we with any propriety talk of true or
false reasoning. Reasoning is either valid or invalid: it is only the
premisses of our reasonings, which are propositions, that can be true
or false. We may have a perfectly valid process of reasoning which
starts from a false assumption and lands us in a false conclusion.
§ 204. All truth and falsehood then are contained in propositions; and
propositions are divided according to the Quality of the Matter into
true and false. But the consideration of the matter is outside the
sphere of formal or deductive Logic. It is the problem of inductive
logic to establish, if possible, a criterion of evidence whereby the
truth or falsehood of propositions may be judged (§ 2).
§ 205. Another usual division of propositions is into Pure and Modal,
the latter being those in which the copula is modified by some degree
of probability. This division is excluded by the view which has just
been taken of the copula, as being always simply affirmative or simply
negative.
§ 206. We are left then with the following divisions of
propositions--
Proposition
according to Form
Simple
Complex
Conjunctive
Disjunctive
Universal
Singular
General
according to Matter
Verbal
Real
according to Quantity
Universal
Singular
General
Particular
Indefinite
(strictly) Particular
according to Quality
Affirmative
Negative
_Simple and Complex Propositions_.
§ 207. A Simple Proposition is one in which a predicate is directly
affirmed or denied of a subject, e.g. 'Rain is falling.'
§ 208. A simple proposition is otherwise known as Categorical.
§ 209. A Complex Proposition is one in which a statement is made
subject to some condition, e.g. 'If the wind drops, rain will fall.'
§ 210. Hence the complex proposition is also known as Conditional.
§ 211. Every complex proposition consists of two parts--
(1) Antecedent;
(2) Consequent.
§ 212. The Antecedent is the condition on which another statement is
made to depend. It precedes the other in the order of thought, but may
either precede or follow it in the order of language. Thus we may say
indifferently--'If the wind drops, we shall have rain' or 'We shall
have rain, if the wind drops.'
§ 213. The Consequent is the statement which is made subject to some
condition.
§ 214. The complex proposition assumes two forms,
(1) If A is B, C is D.
This is known as the Conjunctive or Hypothetical proposition.
(2) Either A is B or C is D.
This is known as the Disjunctive proposition.
§ 215. The disjunctive proposition may also appear in
the form
A is either B or C,
which is equivalent to saying
Either A is B or A is C;
or again in the form
Either A or B is C,
which is equivalent to saying
Either A is C or B is C.
§ 216. As the double nomenclature may cause some confusion, a scheme
is appended.
Proposition
____________|_____________
| |
Simple Complex
(Categorical) (Conditional)
___________|__________
| |
Conjunctive Disjunctive.
(Hypothetical)
§ 217. The first set of names is preferable. 'Categorical' properly
means 'predicable' and 'hypothetical' is a mere synonym for
'conditional.'
§ 218. Let us examine now what is the real nature of the statement
which is made in the complex form of proposition. When, for instance,
we say 'If the sky falls, we shall catch larks,' what is it that we
really mean to assert? Not that the sky will fall, and not that we
shall catch larks, but a certain connection between the two, namely,
that the truth of the antecedent involves the truth of the
consequent. This is why this form of proposition is called
'conjunctive,' because in it the truth of the consequent is conjoined
to the truth of the antecedent.
§ 219. Again, when we say 'Jones is either a knave or a fool,' what is
really meant to be asserted is--'If you do not find Jones to be a
knave, you may be sure that he is a fool.' Here it is the falsity of
the antecedent which involves the truth of the consequent; and the
proposition is known as 'disjunctive,' because the truth of the
consequent is disjoined from the truth of the antecedent.
§ 220. Complex propositions then turn out to be propositions about
propositions, that is, of which the subject and predicate are
themselves propositions. But the nature of a proposition never varies
in thought. Ultimately every proposition must assume the form 'A is,
or is not, B.' 'If the sky falls, we shall catch larks' may be
compressed into 'Sky-falling is lark-catching.'
§ 221. Hence this division turns upon the form of expression, and may
be said to be founded on the simplicity or complexity of the terms
employed in a proposition.
§ 222. In the complex proposition there appears to be more than one
subject or predicate or both, but in reality there is only a single
statement; and this statement refers, as we have Seen, to a certain
connection between two propositions.
§ 223. If there were logically, and not merely grammatically, more
than one subject or predicate, there would be more than one
proposition. Thus when we say 'The Jews and Carthaginians were Semitic
peoples and spoke a Semitic language,' we have four propositions
compressed into a single sentence for the sake of brevity.
§ 224. On the other hand when we say 'Either the Carthaginians were of
Semitic origin or argument from language is of no value in ethnology,'
we have two propositions only in appearance.
§ 225. The complex proposition then must be distinguished from those
contrivances of language for abbreviating expression in which several
distinct statements are combined into a single sentence.
_Verbal and Real Propositions_.
§ 226. A Verbal Proposition is one which states nothing more about the
subject than is contained in its definition, e.g. 'Man is an animal';
'Men are rational beings.'
§ 227. A Real Proposition states some fact not contained in the
definition of the subject, e.g. 'Some animals have four feet.'
§ 228. It will be seen that the distinction between verbal and real
propositions assumes a knowledge of the precise meaning of terms, that
is to say, a knowledge of definitions.
§ 229. To a person who does not know the meaning of terms a verbal
proposition will convey as much information as a real one. To say 'The
sun is in mid-heaven at noon,' though a merely verbal proposition,
will convey information to a person who is being taught to attach a
meaning to the word 'noon.' We use so many terms without knowing their
meaning, that a merely verbal proposition appears a revelation to many
minds. Thus there are people who are surprised to hear that the lion
is a cat, though in its definition 'lion' is referred to the class
'cat.' The reason of this is that we know material objects far better
in their extension than in their intension, that is to say, we know
what things a name applies to without knowing the attributes which
those things possess in common.
§ 230. There is nothing in the mere look of a proposition to inform us
whether it is verbal or real; the difference is wholly relative to,
and constituted by, the definition of the subject. When we have
accepted as the definition of a triangle that it is 'a figure
contained by three sides,' the statement of the further fact that it
has three angles becomes a real proposition. Again the proposition
'Man is progressive' is a real proposition. For though his
progressiveness is a consequence of his rationality, still there is no
actual reference to progressiveness contained in the usually accepted
definition, 'Man is a rational animal.'
§ 231. If we were to admit, under the term 'verbal proposition,' all
statements which, though not actually contained in the definition of
the subject, are implied by it, the whole body of necessary truth
would have to be pronounced merely verbal, and the most penetrating
conclusions of mathematicians set down as only another way of stating
the simplest axioms from which they started. For the propositions of
which necessary truth is composed are so linked together that, given
one, the rest can always follow. But necessary truth, which is arrived
at 'a priori,' that is, by the mind's own working, is quite as real as
contingent truth, which is arrived at 'a posteriori,' or by the
teachings of experience, in other words, through our own senses or
those of others.
§ 232. The process by which real truth, which is other than deductive,
is arrived at 'a priori' is known as Intuition. E.g. The mind sees
that what has three sides cannot but have three angles.
§ 233. Only such propositions then must be considered verbal as state
facts expressly mentioned in the definition.
§ 234. Strictly speaking, the division of propositions into verbal and
real is extraneous to our subject: since it is not the province of
logic to acquaint us with the content of definitions.
§ 235, The same distinction as between verbal and real proposition, is
conveyed by the expressions 'Analytical' and 'Synthetical,' or
'Explicative' and 'Ampliative' judgements.
§ 236. A verbal proposition is called analytical, as breaking up the
subject into its component notions.
§ 237. A real proposition is called synthetical, as attaching some new
notion to the subject.
§ 238. Among the scholastic logicians verbal propositions were known
as 'Essential,' because what was stated in the definition was
considered to be of the essence of the subject, while real
propositions were known as 'Accidental.'
_Universal AND PARTICULAR Propositions_.
§ 239. A Universal proposition is one in which it is evident from the
form that the predicate applies to the subject in its whole extent.
§ 240. When the predicate does not apply to the subject in its whole
extent, or when it is not clear that it does so, the proposition is
called Particular.
§ 241. To say that a predicate applies to a subject in its whole
extent, is to say that it is asserted or denied of all the things of
which the subject is a name.
§ 242. 'All men are mortal' is a universal proposition.
§ 243. 'Some men are black' is a particular proposition. So also is
'Men are fallible;' for here it is not clear from the form whether
'all' or only 'some' is meant.
§ 244. The latter kind of proposition is known as Indefinite, and must
be distinguished from the particular proposition strictly so called,
in which the predicate applies to part only of the subject.
§ 245. The division into universal and particular is founded on the
Quantity of propositions.
§ 246. The quantity of a proposition is determined by the quantity in
extension of its subject.
§ 247. Very often the matter of an indefinite proposition is such as
clearly to indicate to us its quantity. When, for instance, we say
'Metals are elements,' we are understood to be referring to all
metals; and the same thing holds true of scientific statements in
general. Formal logic, however, cannot take account of the matter of
propositions; and is therefore obliged to set down all indefinite
propositions as particular, since it is not evident from the form that
they are universal.
§ 248. Particular propositions, therefore, are sub-divided into such
as are Indefinite and such as are Particular, in the strict sense of
the term.
§ 249. We must now examine the sub-division of universal propositions
into Singular and General.
§ 250. A Singular proposition is one which has a singular term for its
subject, e.g. 'Virtue is beautiful.'
§ 251. A General proposition is one which has for its subject a common
term taken in its whole extent.
§ 252. Now when we say 'John is a man' or 'This table is oblong,' the
proposition is quite as universal, in the sense of the predicate
applying to the whole of the subject, as when we say 'All men are
mortal.' For since a singular term applies only to one thing, we
cannot avoid using it in its whole extent, if we use it at all.
§ 253. The most usual signs of generality in a proposition are the
words 'all,' 'every,' 'each,' in affirmative, and the words 'no,'
'none,' 'not one,' &c. in negative propositions.
§ 254. The terminology of the division of propositions according to
quantity is unsatisfactory. Not only has the indefinite proposition to
be set down as particular, even when the sense manifestly declares it
to be universal; but the proposition which is expressed in a
particular form has also to be construed as indefinite, _so_ that
an unnatural meaning is imparted to the word 'some,' as used in
logic. If in common conversation we were to say 'Some cows chew the
cud,' the person whom we were addressing would doubtless imagine us to
suppose that there were some cows which did not possess this
attribute. But in logic the word 'some' is not held to express more
than 'some at least, if not all.' Hence we find not only that an
indefinite proposition may, as a matter of fact, be strictly
particular, but that a proposition which appears to be strictly
particular may be indefinite. So a proposition expressed in precisely
the same form 'Some A is B' may be either strictly particular, if some
be taken to exclude all, or indefinite, if the word 'some' does not
exclude the possibility of the statement being true of all. It is
evident that the term 'particular' has become distorted from its
original meaning. It would naturally lead us to infer that a statement
is limited to part of the subject, whereas, by its being opposed to
universal, in the sense in which that term has been defined, it can
only mean that we have nothing to show us whether part or the whole is
spoken of.
§ 255. This awkwardness of expression is due to the indefinite
proposition having been displaced from its proper position. Formerly
propositions were divided under three heads--
(1) Universal,
(2) Particular,
(3) Indefinite.
But logicians anxious for simplification asked, whether a predicate in
any given case must not either apply to the whole of the subject or
not? And whether, therefore, the third head of indefinite propositions
were not as superfluous as the so-called 'common gender' of nouns in
grammar?
§ 256. It is quite true that, as a matter of fact, any given predicate
must either apply to the whole of the subject or not, so that in the
nature of things there is no middle course between universal and
particular. But the important point is that we may not know whether
the predicate applies to the whole of the subject or not. The primary
division then should be into propositions whose quantity is known and
propositions whose quantity is unknown. Those propositions whose
quantity is known may be sub-divided into 'definitely universal' and
'definitely particular,' while all those whose quantity is unknown are
classed together under the term 'indefinite.' Hence the proper
division is as follows--
Proposition
__________|____________
| |
Definite Indefinite
_____|_______
| |
Universal Particular.
§ 257. Another very obvious defeat of terminology is that the word
'universal' is naturally opposed to 'singular,' whereas it is here so
used as to include it; while, on the other hand, there is no obvious
difference between universal and general, though in the division the
latter is distinguished from the former as species from genus.
_Affirmative and Negative Propositions._
§ 258. This division rests upon the Quality of propositions.
§ 259. It is the quality of the form to be affirmative or negative:
the quality of the matter, as we saw before (§ 204), is to be true or
false. But since formal logic takes no account of the matter of
thought, when we speak of 'quality' we are understood to mean the
quality of the form.
§ 260. By combining the division of propositions
according to quantity with the division according to quality,
we obtain four kinds of proposition, namely--
(1) Universal Affirmative (A).
(2) Universal Negative (E).
(3) Particular Affirmative (I).
(4) Particular Negative (O).
§ 261. This is an exhaustive classification of propositions, and any
proposition, no matter what its form may be, must fall under one or
other of these four heads. For every proposition must be either
universal or particular, in the sense that the subject must either be
known to be used in its whole extent or not; and any proposition,
whether universal or particular, must be either affirmative or
negative, for by denying modality to the copula we have excluded
everything intermediate between downright assertion and denial. This
classification therefore may be regarded as a Procrustes' bed, into
which every proposition is bound to fit at its proper peril.
§ 262. These four kinds of propositions are represented respectively
by the symbols A, E, I, O.
§ 263. The vowels A and I, which denote the two affirmatives, occur in
the Latin words 'affirmo' and 'aio;' E and O, which denote the two
negatives, occur in the Latin word 'nego.'
_Extensive and Intensive Propositions._
§ 264. It is important to notice the difference between Extensive and
Intensive propositions; but this is not a division of propositions,
but a distinction as to our way of regarding them. Propositions may be
read either in extension or intension. Thus when we say 'All cows are
ruminants,' we may mean that the class, cow, is contained in the
larger class, ruminant. This is reading the proposition in
extension. Or we may mean that the attribute of chewing the cud is
contained in, or accompanies, the attributes which make up our idea of
'cow.' This is reading the proposition in intension. What, as a matter
of fact, we do mean, is a mixture of the two, namely, that the class,
cow, has the attribute of chewing the cud. For in the ordinary and
natural form of proposition the subject is used in extension, and the
predicate in intension, that is to say, when we use a subject, we are
thinking of certain objects, whereas when we use a predicate, we
indicate the possession of certain attributes. The predicate, however,
need not always be used in intension, e.g. in the proposition 'His
name is John' the predicate is not intended to convey the idea of any
attributes at all. What is meant to be asserted is that the name of
the person in question is that particular name, John, and not
Zacharias or Abinadab or any other name that might be given him.
§ 265. Let it be noticed that when a proposition is read in extension,
the predicate contains the subject, whereas, when it is read in
intension, the subject contains the predicate.
_Exclusive Propositions._
§ 266. An Exclusive Proposition is so called because in it all but a
given subject is excluded from participation in a given predicate,
e.g. 'The good alone are happy,' 'None but the brave deserve the
fair,' 'No one except yourself would have done this.'
§ 267. By the above forms of expression the predicate is declared to
apply to a given subject and to that subject only. Hence an exclusive
proposition is really equivalent to two propositions, one affirmative
and one negative. The first of the above propositions, for instance,
means that some of the good are happy, and that no one else is so. It
does not necessarily mean that all the good are happy, but asserts
that among the good will be found all the happy. It is therefore
equivalent to saying that all the happy are good, only that it puts
prominently forward in addition what is otherwise a latent consequence
of that assertion, namely, that some at least of the good are happy.
§ 268. Logically expressed the exclusive proposition when universal
assumes the form of an E proposition, with a negative term for its
subject
No not-A is B.
§ 269. Under the head of exclusive comes the strictly particular
proposition, 'Some A is B,' which implies at the same time that 'Some
A is not B.' Here 'some' is understood to mean 'some only,' which is
the meaning that it usually bears in common language. When, for
instance, we say 'Some of the gates into the park are closed at
nightfall,' we are understood to mean 'Some are left open.'
_Exceptive Propositions._
§ 270. An Exceptive Proposition is so called as affirming the
predicate of the whole of the subject, with the exception of a certain
part, e.g. 'All the jury, except two, condemned the prisoner.'
§ 271. This form of proposition again involves two distinct
statements, one negative and one affirmative, being equivalent to 'Two
of the jury did not condemn the prisoner; and all the rest did.'
§ 272. The exceptive proposition is merely an affirmative way of
stating the exclusive--
No not-A is B = All not-A is not-B.
No one but the sage is sane = All except the sage are mad.
_Tautologous or Identical Propositions_
§ 273. A Tautologous or Identical proposition affirms the subject of
itself, e.g. 'A man's a man,' 'What I have written, I have written,'
'Whatever is, is.' The second of these instances amounts formally to
saying 'The thing that I have written is the thing that I have
written,' though of course the implication is that the writing will
not be altered.
CHAPTER IV.
_Of the Distribution of Terms._
§ 274. The treatment of this subject falls under the second part of
logic, since distribution is not an attribute of terms in themselves,
but one which they acquire in predication.
§ 275. A term is said to be distributed when it is known to be used in
its whole extent, that is, with reference to all the things of which
it is a name. When it is not so used, or is not known to be so used,
it is called undistributed.
§ 276. When we say 'All men are mortal,' the subject is distributed,
since it is apparent from the form of the expression that it is used
in its whole extent. But when we say 'Men are miserable' or 'Some men
are black,' the subject is undistributed.
§ 277. There is the same ambiguity attaching to the term
'undistributed' which we found to underlie the use of the term
'particular.' 'Undistributed' is applied both to a term whose quantity
is undefined, and to one whose quantity is definitely limited to a
part of its possible extent.
§ 278. This awkwardness arises from not inquiring first whether the
quantity of a term is determined or undetermined, and afterwards
proceeding to inquire, whether it is determined as a whole or part of
its possible extent. As it is, to say that a term is distributed,
involves two distinct statements--
(1) That its quantity is known;
(2) That its quantity is the greatest possible.
The term 'undistributed' serves sometimes to contradict one of these
statements and sometimes to contradict the other.
§ 279. With regard to the quantity of the subject of a proposition no
difficulty can arise. The use of the words 'all' or 'some,' or of a
variety of equivalent expressions, mark the subject as being
distributed or undistributed respectively, while, if there be nothing
to mark the quantity, the subject is for that reason reckoned
undistributed.
§ 280. With regard to the predicate more difficulty may arise.
§ 281. It has been laid down already that, in the ordinary form of
proposition, the subject is used in extension and the predicate in
intension. Let us illustrate the meaning of this by an example. If
someone were to say 'Cows are ruminants,' you would have a right to
ask him whether he meant 'all cows' or only 'some.' You would not by
so doing be asking for fresh information, but merely for a more
distinct explanation of the statement already made. The subject being
used in extension naturally assumes the form of the whole or part of a
class. But, if you were to ask the same person 'Do you mean that cows
are all the ruminants that there are, or only some of them?' he would
have a right to complain of the question, and might fairly reply, 'I
did not mean either one or the other; I was not thinking of ruminants
as a class. I wished merely to assert an attribute of cows; in fact, I
meant no more than that cows chew the cud.'
§ 282. Since therefore a predicate is not used in extension at all, it
cannot possibly be known whether it is used in its whole extent or
not.
§ 283. It would appear then that every predicate is necessarily
undistributed; and this consequence does follow in the case of
affirmative propositions.
§ 284. In a negative proposition, however, the predicate, though still
used in intension, must be regarded as distributed. This arises from
the nature of a negative proposition. For we must remember that in any
proposition, although the predicate be not meant in extension, it
always admits of being so read. Now we cannot exclude one class from
another without at the same time wholly excluding that other from the
former. To take an example, when we say 'No horses are ruminants,' the
meaning we really wish to convey is that no member of the class,
horse, has a particular attribute, namely, that of chewing the
cud. But the proposition admits of being read in another form, namely,
'That no member of the class, horse, is a member of the class,
ruminant.' For by excluding a class from the possession of a given
attribute, we inevitably exclude at the same time any class of things
which possess that attribute from the former class.
§ 285. The difference between the use of a predicate in an affirmative
and in a negative proposition may be illustrated to the eye as
follows. To say 'All A is B' may mean either that A is included in B
or that A and B are exactly co-extensive.
[Illustration]
§ 286. As we cannot be sure which of these two relations of A to B is
meant, the predicate B has to be reckoned undistributed, since a term
is held to be distributed only when we know that it is used in its
whole extent.
§ 287. To say 'No A is B,' however, is to say that A falls wholly
outside of B, which involves the consequence that B falls wholly
outside of A.
[Illustration]
§ 288. Let us now apply the same mode of illustration to the
particular forms of proposition.
§ 289. If I be taken in the strictly particular sense, there are, from
the point of view of extension, two things which may be meant when we
say 'Some A is B'--
(1) That A and B are two classes which overlap one another, that is
to say, have some members in common, e.g. 'Some cats are black.'
[Illustration]
(2) That B is wholly contained in A, which is an inverted way of
saying that all B is A, e.g. 'Some animals are men.'
[Illustration]
§ 290. Since we cannot be sure which of these two is meant, the
predicate is again reckoned undistributed.
§ 291. If on the other hand 1 be taken in an indefinite sense, so as
to admit the possibility of the universal being true, then the two
diagrams which have already been used for A must be extended to 1, in
addition to its own, together with the remarks which we made in
connection with them (§§ 285-6).
§ 292. Again, when we say 'Some A is not B,' we mean that some, if not
the whole of A, is excluded from the possession of the attribute B. In
either case the things which possess the attribute B are wholly
excluded either from a particular part or from the whole of A. The
predicate therefore is distributed.
[Illustration]
From the above considerations we elicit the following--
§ 293. Four Rules for the Distribution of Terms.
(1) All universal propositions distribute their subject.
(2) No particular propositions distribute their subject,
(3) All negative propositions distribute their predicate.
(4) No affirmative propositions distribute their predicate.
§ 294. The question of the distribution or non-distribution of the
subject turns upon the quantity of the proposition, whether universal
or particular; the question of the distribution or non-distribution of
the predicate turns upon the quality of the proposition, whether
affirmative or negative.
CHAPTER V.
_Of the Quantification of the Predicate._
§ 295. The rules that have been given for the distribution of terms,
together with the fourfold division of propositions into A, E, 1, 0,
are based on the assumption that it is the distribution or
non-distribution of the subject only that needs to be taken into
account in estimating the quantity of a proposition.
§ 296. But some logicians have maintained that the predicate, though
seldom quantified in expression, must always be quantified in
thought--in other words, that when we say, for instance, 'All A is B,'
we must mean either that 'All A is all B' or only that 'All A is some
B.'
§ 297. If this were so, it is plain that the number of possible
propositions would be exactly doubled, and that, instead of four
forms, we should now have to recognise eight, which may be expressed
as follows--
1. All A is all B. ([upsilon]).
2. All A is some B. ([Lambda]).
3. No A is any B. ([Epsilon]).
4. No A is some B. ([eta]).
5. Some A is all B. ([Upsilon]).
6. Some A is some B. ([Iota]).
7. Some A is not any B. ([Omega]).
8. Some A is not some B. ([omega]).
§ 298. It is evident that it is the second of the above propositions
which represents the original A, in accordance with the rule that 'No
affirmative propositions distribute their predicate' (§ 293).
§ 299. The third represents the original E, in accordance with the
rule that 'All negative propositions distribute their predicate.'
§ 300. The sixth represents the original I, in accordance with the
rule that 'No affirmative propositions distribute their predicate.'
§ 301. The seventh represents the original O, in accordance with the
rule that 'All negative propositions distribute their predicate.'
§ 302. Four new symbols are required, if the quantity of the predicate
as well as that of the subject be taken into account in the
classification of propositions. These have been supplied, somewhat
fancifully, as follows--
§ 303. The first, 'All A is all B,' which distributes both subject and
predicate, has been called [upsilon], to mark its extreme
universality.
§ 304. The fourth, 'No A is some B,' is contained in E, and has
therefore been denoted by the symbol [eta], to show its connection
with E.
§ 305. The fifth, 'Some A is all B,' is the exact converse of the
second, 'All A is some B,' and has therefore been denoted by the
symbol [Upsilon], which resembles an inverted A.
§ 306. The eighth is contained in O, as part in whole, and has
therefore had assigned to it the symbol [omega],
§ 307. The attempt to take the predicate in extension, instead of, as
it should naturally be taken, in intension, leads to some curious
results. Let us take, for instance, the u proposition. Either the sign
of quantity 'all' must be understood as forming part of the predicate
or not. If it is not, then the u proposition 'All A is all B' seems
to contain within itself, not one proposition, but two, namely, 'All A
is B' and 'All B is A.' But if on the other hand 'all' is understood
to form part of the predicate, then u is not really a general but a
singular proposition. When we say, 'All men are rational animals,' we
have a true general proposition, because the predicate applies to the
subject distributively, and not collectively. What we mean is that
'rational animal' may be affirmed of every individual in the class,
man. But when we say 'All men are all rational animals,' the predicate
no longer applies to the subject distributively, but only
collectively. For it is obvious that 'all rational animals' cannot be
affirmed of every individual in the class, man. What the proposition
means is that the class, man, is co-extensive with the class, rational
animal. The same meaning may be expressed intensively by saying that
the one class has the attribute of co-extension with the other.
§ 308. Under the head o u come all propositions in which both subject
and predicate are singular terms, e.g. 'Homer was the author of the
Iliad,' 'Virtue is the way to happiness.'
§ 309. The proposition [eta] conveys very little information to the
mind. 'No A is some B' is compatible with the A proposition in the
same matter. 'No men are some animals' may be true, while at the same
time it is true that 'All men are animals.' No men, for instance, are
the particular animals known as kangaroos.
§ 310. The [omega] proposition conveys still less information than the
[eta]. For [omega] is compatible, not only with A, but with
[upsilon]. Even though 'All men are all rational animals,' it is still
true that 'Some men are not some rational animals': for no given human
being is the same rational animal as any other.
§ 311. Nay, even when the [upsilon] is an identical proposition,
[omega] will still hold in the same matter. 'All rational animals are
all rational animals': but, for all that, 'Some rational animals are
not some others.' This last form of proposition therefore is almost
wholly devoid of meaning.
§ 312. The chief advantage claimed for the quantification of the
predicate is that it reduces every affirmative proposition to an exact
equation between its subject and predicate. As a consequence every
proposition would admit of simple conversion, that is to say, of
having the subject and predicate transposed without any further change
in the proposition. The forms also of Reduction (a term which will be
explained later on) would be simplified; and generally the
introduction of the quantified predicate into logic might be attended
with certain mechanical advantages. The object of the logician,
however, is not to invent an ingenious system, but to arrive at a true
analysis of thought. Now, if it be admitted that in the ordinary form
of proposition the subject is used in extension and the predicate in
intension, the ground for the doctrine is at once cut away. For, if
the predicate be not used in its extensive capacity at all, we plainly
cannot be called upon to determine whether it is used in its whole
extent or not.
CHAPTER VI.
_Of the Heads of Predicables_.
§ 313. A predicate is something which is stated of a subject.
§ 314. A predicable is something which can be stated of a subject.
§ 315. The Heads of Predicables are a classification of the various
things which can be stated of a subject, viewed in their relation to
it.
§ 316. The treatment of this topic, therefore, as it involves the
relation of a predicate to a subject, manifestly falls under the
second part of logic, which deals with the proposition. It is
sometimes treated under the first part of logic, as though the heads
of predicables were a classification of universal notions, i.e. common
terms, in relation to one another, without reference to their place in
the proposition.
§ 317. The heads of predicables are commonly reckoned
as five, namely,
(1) Genus.
(2) Species.
(3) Difference.
(4) Property.
(5) Accident.
§ 318. We will first define these terms in the sense in which they are
now used, and afterwards examine the principle on which the
classification is founded and the sense in which they were originally
intended.
(1) A Genus is a larger class containing under it smaller
classes. Animal is a genus in relation to man and brute.
(2) A Species is a smaller class contained under a larger one. Man
is a species in relation to animal.
(3) Difference is the attribute, or attributes, which distinguish
one species from others contained under the same genus. Rationality
is the attribute which distinguishes the species, man, from the
species, brute.
N.B. The genus and the difference together make up the Definition of
a class-name, or common term.
(4) A Property is an attribute which is not contained in the
definition of a term, but which flows from it.
A Generic Property is one which flows from the genus.
A Specific Property is one which flows from the difference.
It is a generic property of man that he is mortal, which is a
consequence of his animality. It is a specific property of man that
he is progressive, which is a consequence of his rationality.
(5) An Accident is an attribute, which is neither contained in the
definition, nor flows from it.
§ 319. Accidents are either Separable or Inseparable.
A Separable Accident is one which belongs only to some members of a
class.
An Inseparable Accident is one which belongs to all the members of a
class.
Blackness is a separable accident of man, an inseparable accident of
coals.
§ 320. The attributes which belong to anything may be distinguished
broadly under the two heads of essential and non-essential, or
accidental. By the essential attributes of anything are meant those
which are contained in, or which flow from, the definition. Now it may
be questioned whether there can, in the nature of things, be such a
thing as an inseparable accident. For if an attribute were found to
belong invariably to all the members of a class, we should suspect
that there was some causal connection between it and the attributes
which constitute the definition, that is, we should suspect the
attribute in question to be essential and not accidental. Nevertheless
the term 'inseparable accident' may be retained as a cloak for our
ignorance, whenever it is found that an attribute does, as a matter of
fact, belong to all the members of a class, without there being any
apparent reason why it should do so. It has been observed that animals
which have horns chew the cud. As no one can adduce any reason why
animals that have horns should chew the cud any more than animals
which have not, we may call the fact of chewing the cud an inseparable
accident of horned animals.
§ 321. The distinction between separable and inseparable accidents is
sometimes extended from classes to individuals.
An inseparable accident of an individual is one which belongs to him
at all times. A separable accident of an individual is one which
belongs to him at one time and not at another.
§ 322. It is an inseparable accident of an individual that he was born
at a certain place and on a certain date. It is a separable accident
of an individual that he resides at a certain place and is of a
certain age.
§ 323. There are some remarks which it may be well to make about the
above five terms before we pass on to investigate the principle upon
which the division is based.
§ 324. In the first place, it must of course be borne in mind that
genus and species are relative terms. No class in itself can be either
a genus or a species; it only becomes so in reference to some other
class, as standing to it in the relation of containing or contained.
§ 325. Again, the distinction between genus and difference on the one
hand and property on the other is wholly relative to an assumed
definition. When we say 'Man is an animal,' 'Man is rational,' 'Man is
progressive,' there is nothing in the nature of these statements
themselves to tell us that the predicate is genus, difference, or
property respectively. It is only by a tacit reference to the accepted
definition of man that this becomes evident to us, Similarly, we
cannot know beforehand that the fact of a triangle having three sides
is its difference, and the fact of its having three angles a
property. It is only when we assume the definition of a triangle as a
three-sided figure that the fact of its having three angles sinks into
a property. Had we chosen to define it, in accordance with its
etymological meaning, as a figure with three angles, its
three-sidedness would then have been a mere property, instead of being
the difference; for these two attributes are so connected together
that, whichever is postulated, the other will necessarily follow.
§ 326. Lastly, it must be noticed that we have not really defined the
term 'accident,' not having stated what it is, but only what it is
not. It has in fact been reserved as a residual head to cover any
attribute which is neither a difference nor a property.
§ 327. If the five heads of predicables above given were offered to us
as an exhaustive classification of the possible relations in which the
predicate can stand to the subject in a proposition, the first thing
that would strike us is that they do not cover the case in which the
predicate is a singular term. In such a proposition as 'This man is
John,' we have neither a predication of genus or species nor of
attribute: but merely the identification of one term with another, as
applying to the same object. Such criticism as this, however, would be
entirely erroneous, since no singular term was regarded as a
predicate. A predicable was another name for a universal, the common
term being called a predicable in one relation and a universal in
another-a predicable, extensively, in so far as it was applicable to
several different things, a universal, intensively, in so far as the
attributes indicated were implied in several other notions, as the
attributes indicated by 'animal' are implied in 'horse,' 'sheep,'
'goat,' &c.
§ 328. It would be less irrelevant to point out how the classification
breaks down in relation to the singular term as subject. When, for
instance, we say 'Socrates is an animal,' 'Socrates is a man,' there
is nothing in the proposition to show us whether the predicate is a
genus or a species: for we have not here the relation of class to
class, which gives us genus or species according to their relative
extension, but the relation of a class to an individual.
§ 329. Again, when we say
(1) Some animals are men,
(2) Some men are black,
what is there to tell us that the predicate is to be regarded in the
one case as a species and in the other as an accident of the subject?
Nothing plainly but the assumption of a definition already known.
§ 330. But if this assumption be granted, the classification seems to
admit of a more or less complete defense by logic.
For, given any subject, we can predicate of it either a class or an
attribute.
When the predicate is a class, the term predicated is called a Genus,
if the subject itself be a class, or a Species, if it be an
individual.
When, on the other hand, the predicate is an attribute, the attribute
predicated may be either the very attribute which distinguishes the
subject from other members of the same class, in which case it is
called the Difference, or it may be some attribute connected with the
definition, i.e. Property, or not connected with it, i.e. Accident.
§ 331. These results may be exhibited in the following scheme--
Predicate
________________|_________________
| |
Class Attribute
______|_______ __________|________
| | | |
(Subject a (Subject a (The (Not the
common singular distinguishing distinguishing
term) term) Attribute) attribute)
Genus Species Difference
|___________________
| |
(Connected (Not connected
with the with the
definition) definition)
Property Accident
§ 332. The distinction which underlies this division between
predicating a class and predicating an attribute (in quid or in quale)
is a perfectly intelligible one, corresponding as it does to the
grammatical distinction between the predicate being a noun substantive
or a noun adjective. Nevertheless it is a somewhat arbitrary one,
since, even when the predicate is a class-name, what we are concerned
to convey to the mind, is the fact that the subject possesses the
attributes which are connoted by that class-name. We have not here the
difference between extensive and intensive predication, since, as we
have already seen (§ 264), that is not a difference between one
proposition and another, but a distinction in our mode of interpreting
any and every proposition. Whatever proposition we like to take may be
read either in extension or in intension, according as we fix our
minds on the fact of inclusion in a class or the fact of the
possession of attributes.
§ 333. It will be seen that the term 'species,' as it appears in the
scheme, has a wholly different meaning from the current acceptation in
which it was defined above. Species, in its now accepted meaning,
signifies the relation of a smaller class to a larger one: as it was
originally intended in the heads of predicables it signifies a class
in reference to individuals.
§ 334. Another point which requires to be noticed with regard to this
five-fold list of heads of predicables, if its object be to classify
the relations of a predicate to a subject, is that it takes no account
of those forms of predication in which class and attribute are
combined. Under which of the five heads would the predicates in the
following propositions fall?
(1) Man is a rational animal.
(2) Man is a featherless biped.
In the one case we have a combination of genus and difference; in the
other we have a genus combined with an accident.
§ 335. The list of heads of predicables which we have been discussing
is not derived from Aristotle, but from the 'Introduction' of
Porphyry, a Greek commentator who lived more than six centuries later.
_Aristotle's Heads of Predicables_.
§ 336. Aristotle himself, by adopting a different basis of division,
has allowed room in his classification for the mixed forms of
predication above alluded to. His list contains only four heads,
namely,
Genus ([Greek: génos])
Definition ([Greek: òrismós])
Proprium ([Greek: îdion])
Accident ([Greek: sumbebekós])
§ 337. Genus here is not distinguished from difference. Whether we
say 'Man is an animal' or 'Man is rational,' we are equally understood
to be predicating a genus.
§ 338. There is no account taken of species, which, when predicated,
resolves itself either into genus or accident. When predicated of an
individual, it is regarded as a genus, e.g. 'Socrates is a man'; when
predicated of a class, it is regarded as an accident, e.g. 'Some
animals are men.'
§ 339. Aristotle's classification may easily be seen to be
exhaustive. For every predicate must either be coextensive with its
subject or not, i.e. predicable of the same things. And if the two
terms coincide in extension, the predicate must either coincide also
in intension with the subject or not.
A predicate which coincides both in extension and intension with its
subject is exactly what is meant by a definition. One which coincides
in extension without coinciding in intension, that is, which applies
to the same things without expressing the whole meaning, of the
subject, is what is known as a Proprium or Peculiar Property.
If, on the other hand, the two terms are not co-extensive, the
predicate must either partially coincide in intension with the subject
or not. [Footnote: The case could not arise of a predicate which was
entirely coincided in intension with a subject with which it was not
co-extensive. For, if the extension of the predicate were greater than
that of the subject, its intension would be less, and if less,
greater, in accordance with the law of inverse variation of the two
quantities (§ 166).] This is equivalent to saying that it must either
state part of the definition of the subject or not. Now the definition
is made up of genus and difference, either of which may form the
predicate: but as the two are indistinguishable in relation to a
single subject, they are lumped together for the present purpose under
the one head, genus. When the predicate, not being co-extensive, is
not even partially co-intensive with its subject, it is called an
Accident.
§ 340. Proprium, it will be seen, differs from property. A proprium
is an attribute which is possessed by all the members of a class, and
by them alone, e.g. 'Men are the only religious animals.'
§ 341. Under the head of definition must be included all propositions
in which the predicate is a mere synonym of the subject, e.g. 'Naso is
Ovid,' 'A Hebrew is a Jew,' 'The skipper is the captain.' In such
propositions the predicate coincides in extension with the subject,
and may be considered to coincide in intension where the intension of
both subject and predicate is at zero, as in the case of two proper
names.
§ 342. Designations and descriptions will fall under the head of
'proprium' or peculiar property, e.g. 'Lord Salisbury is the present
prime minister of England,' 'Man is a mammal with hands and without
hair.' For here, while the terms are coincident in extension, they are
far from being so in intension.
§ 343. The term 'genus' must be understood to include not only genus
in the accepted sense, but difference and generic property as well.
§ 344. These results may be exhibited in the following
scheme--
Predicate
___________________|______________
| |
Coextensive with not
the subject coextensive
________|_________ _____|________
| | | |
Co-intensive not partially not at all
with the subject cointensive cointensive [Greek: sumbubekós]
[Greek: òrismós] [Greek: îdion] [Greek: génos] Accident
______|_____ ______|_____________ |________________
| | | | | | | |
Defini- Synonym Designa- Descrip- Peculiar Genus Differ- Generic
tion tion tion Property ence Property
§ 345. Thus Aristotle's four heads of predicables may be split up, if
we please, into nine--
1. Definition \
> [Greek: òrismós].
2. Synonym /
3. Designation \
|
4. Description > [Greek: îdion].
|
5. Peculiar Property/
6. Genus \
|
7. Difference > [Greek: génos].
|
8. Generic Property/
9. Accident--[Greek: sumbebekós].
§ 346. We now pass on to the two subjects of Definition and Division,
the discussion of which will complete our treatment of the second part
of logic. Definition and division correspond respectively to the two
kinds of quantity possessed by terms.
Definition is unfolding the quantity of a term in intension.
Division is unfolding the quantity of a term in extension.
CHAPTER VII.
_Of Definition._
§ 347. To define a term is to unfold its intension, i.e. to explain
its meaning.
§ 348. From this it follows that any term which possesses no intension
cannot be defined.
§ 349. Hence proper names do not admit of definition, except just in
so far as they do possess some slight degree of intension: Thus we can
define the term 'John' only so far as to say that 'John' is the name
of a male person. This is said with regard to the original intension
of proper names; their acquired intension will be considered later.
§ 350. Again, since definition is unfolding the intension of a term,
it follows that those terms will not admit of being defined whose
intension is already so simple that it cannot be unfolded further. Of
this nature are names of simple attributes, such as greenness,
sweetness, pleasure, existence. We know what these things are, but we
cannot define them. To a man who has never enjoyed sight, no language
can convey an idea of the greenness of the grass or the blueness of
the sky; and if a person were unaware of the meaning of the term
'sweetness,' no form of words could convey to him an idea of it. We
might put a lump of sugar into his mouth, but that would not be a
logical definition.
§ 351. Thus we see that, for a thing to admit of definition, the idea
of it must be complex. Simple ideas baffle definition, but at the same
time do not require it. In defining we lay out the simpler ideas
which are combined in our notion of something, and so explain that
complex notion. We have defined 'triangle,' when we analyse it into
'figure' and 'contained by three lines.' Similarly we have defined
'substance' when we analyse it into 'thing' and 'which can be
conceived to exist by itself.'
§ 352. But when we get to 'thing' we have reached a limit. The Summum
Genus, or highest class under which all things fall, cannot be defined
any more than a simple attribute; and for the very good reason that it
connotes nothing but pure being, which is the simplest of all
attributes. To say that a thing is an 'object of thought' is not
really to define it, but to explain its etymology, and to reclaim a
philosophical term from its abuse by popular language, in which it is
limited to the concrete and the lifeless. Again, to define it
negatively and to say that a thing is 'that which is not nothing' does
not carry us any further than we were before. The law of contradiction
warrants us in saying as much as that.
§ 353. Definition is confined to subject-terms, and does not properly
extend to attributives. For definition is of things through names, and
an attributive out of predication is not the name of anything. The
attributive is defined, so far as it can be, through the corresponding
abstract term.
§ 354. Common terms, other than attributives, ought always to admit of
definition. For things are distributed by the mind into classes owing
to their possessing certain attributes in common, and the definition
of the class-name can be effected by detailing these attributes, or at
least a sufficient number of them.
§ 355. It is different with singular terms. Singular terms, when
abstract, admit of definition, in so far as they are not names of
attributes so simple as to evade analysis. When singular terms are
concrete, we have to distinguish between the two cases of proper names
and designations. Designations are connotative singular terms. They
are formed by limiting a common term to the 'case in hand.' Whatever
definition therefore fits the common term will fit also the
designation which is formed from it, if we add the attributes implied
by the limitations. Thus whatever definition fits the common term
'prime minister' will fit also the singular term 'the present prime
minister of England' by the addition to it of the attributes of place
and time which are indicated by the expression. Such terms as this
have a definite amount of intension, which can therefore be seized
upon and expounded by a definition.
§ 356. But proper names, having no original intension of their own,
cannot be defined at all; whereas, if we look upon them from the point
of view of their acquired intension, they defy definition by reason of
the very complexity of their meaning. We cannot say exactly what
'John' and 'Mary' mean, because those names, to us who know the
particular persons denoted by them, suggest all the most trifling
accidents of the individual as well as the essential attributes of the
genus.
§ 357. Definition serves the practical purpose of enabling us mentally
to distinguish, or, as the name implies, 'mark off' the thing defined
from all other things whatsoever. This may seem at first an endless
task, but there is a short cut by which the goal may be reached. For,
if we distinguish the thing in hand from the things which it is most
like, we shall, 'a fortiori,' have distinguished it from things to
which it bears a less resemblance.
§ 358. Hence the first thing to do in seeking for a definition is to
fix upon the class into which the thing to be defined most naturally
falls, and then to distinguish the thing in question from the other
members of that class. If we were asked to define a triangle, we
would not begin by distinguishing it from a hawser, but from a square
and other figures with which it is more possible to confound it. The
class into which a thing falls is called its Genus, and the attribute
or attributes which distinguish it from other members of that class
are called its Difference.
§ 359. If definition thus consists in referring a thing to a class, we
see a further reason why the summum genus of all things cannot be
defined.
§ 360. We have said that definition is useful in enabling us to
distinguish things from one another in our minds: but this must not be
regarded as the direct object of the process. For this object may be
accomplished without giving a definition at all, by means of what is
called a Description. By a description is meant an enumeration of
accidents with or without the mention of some class-name. It is as
applicable to proper names as to common terms. When we say 'John Smith
lives next door on the right-hand side and passes by to his office
every morning at nine o'clock,' we have, in all probability,
effectually distinguished John Smith from other people: but living
next, &c., cannot be part of the intension of John Smith, since John
Smith may change his residence or abandon his occupation without
ceasing to be called by his name. Indirectly then definition serves
the purpose of distinguishing things in the mind, but its direct
object is to unfold the intension of terms, and so impart precision to
our thoughts by setting plainly before us the meaning of the words we
are using.
§ 361. But when we say that definition is unfolding the intension of
terms, it must not be imagined that we are bound in defining to unfold
completely the intension of terms. This would be a tedious, and often
an endless, task. A term may mean, or convey to the mind, a good many
more attributes than those which are stated in its definition. There
is no limit indeed to the meaning which a term may legitimately
convey, except the common attributes of the things denoted by it. Who
shall say, for instance, that a triangle means a figure with three
sides, and does not mean a figure with three angles, or the surface of
the perpendicular bisection of a cone? Or again, that man means a
rational, and does not mean a speaking, a religious, or an aesthetic
animal, or a biped with two eyes, a nose, and a mouth? The only
attributes of which it can safely be asserted that they can form no
part of the intension of a term are those which are not common to all
the things to which the name applies. Thus a particular complexion,
colour, height, creed, nationality cannot form any part of the
intension of the term 'man.' But among the attributes common to a
class we cannot distinguish between essential and unessential, except
by the aid of definition itself. Formal logic cannot recognise any
order of priority between the attributes common to all the members of
a class, such as to necessitate our recognising some as genera and
differentiae and relegating others to the place of properties or
inseparable accidents.
§ 362. The art of giving a good definition is to seize upon the
salient characteristics of the thing defined and those wherefrom the
largest number of other attributes can be deduced as consequences. To
do this well requires a special knowledge of the thing in question,
and is not the province of the formal logician.
§ 363. We have seen already, in treating of the Heads of Predicables
(§ 325), that the difference between genus and difference on the one
hand and property on the other is wholly relative to some assumed
definition. Now definitions are always to a certain extent arbitrary,
and will vary with the point of view from which we consider the thing
required to be defined. Thus 'man' is usually contrasted with 'brute,'
and from this point of view it is held a sufficient definition of him
to say that he is 'a rational animal,' But a theologian might be more
anxious to contrast man with supposed incorporeal intelligences, and
from this point of view man would be defined as an 'embodied spirit.'
§ 364. In the two definitions just given it will be noticed that we
have really employed exactly the same attributes, only their place as
genus and difference has been reversed. It is man's rational, or
spiritual, nature which distinguishes him from the brutes: but this is
just what he is supposed to have in common with incorporeal
intelligences, from whom he is differentiated by his animal nature.
[Illustration]
This illustration is sufficient to show us that, while there is no
absolute definition of anything, in the sense of a fixed genus and
difference, there may at the same time be certain attributes which
permanently distinguish the members of a given class from those of all
other classes.
§ 365. The above remarks will have made it clear that the intension of
a term is often much too wide to be conveyed by any definition; and
that what a definition generally does is to select certain attributes
from the whole intension, which are regarded as being more typical of
the thing than the remainder. No definition can be expected to exhaust
the whole intension of a term, and there will always be room for
varying definitions of the same thing, according to the different
points of view from which it is approached.
§ 366. Names of attributes lend themselves to definition far more
easily than names of substances. The reason of this is that names of
attributes are primarily intensive in force, whereas substances are
known to us in extension before they become known to us in
intension. There is no difficulty in defining a term like 'triangle'
or 'monarchy,' because these terms were expressly invented to cover
certain attributes; but the case is different with such terms as
'dog,' 'tree,' 'plant,' 'metal,' and other names of concrete
things. We none of us have any difficulty in recognising a dog or
tree, when we see them, or in distinguishing them from other animals
or plants respectively. We are therefore led to imagine that we know
the meaning of these terms. It is not until we are called upon for a
definition that we discover how superficial our knowledge really is of
the common attributes possessed by the things which these names
denote.
§ 367. It might be imagined that a common name would never be given to
things except in virtue of our knowledge of their common
attributes. But as a matter of fact, the common name was first given
from a confused notion of resemblance, and we had afterwards to detect
the common attributes, when sometimes the name had been so extended
from one thing to another like it, that there were hardly any definite
attributes possessed in common by the earlier and later members of the
class.
§ 368. This is especially the case where the meaning of terms has been
extended by analogy, e.g. head, foot, arm, post, pole, pipe, &c.
§ 369. But in the progress of thought we come to form terms in which
the intensive capacity is everything. Of this kind notably are
mathematical conceptions. Terms of this kind, as we said before, lend
themselves readily to definition.
§ 370. We may lay down then roughly that words are easy or difficult
of definition according as their intensive or extensive capacity
predominates.
§ 371. There is a marked distinction to be observed between the
classes made by the mind of man and the classes made by nature, which
are known as 'real kinds.' In the former there is generally little or
nothing in common except the particular attribute which is selected as
the ground of classification, as in the case of red and white things,
which are alike only in their redness or whiteness; or else their
attributes are all necessarily connected, as in the case of circle,
square and triangle. But the members of nature's classes agree in
innumerable attributes which have no discoverable connection with one
another, and which must therefore, provisionally at least, be regarded
as standing in the relation of inseparable accidents to any particular
attributes which we may select for the purposes of definition. There
is no assignable reason why a rational animal should have hair on its
head or a nose on its face, and yet man, as a matter of fact, has
both; and generally the particular bodily configuration of man can
only be regarded as an inseparable accident of his nature as a
rational animal.
§ 372. 'Real kinds' belong to the class of words mentioned above in
which the extension predominates over the intension. We know well
enough the things denoted by them, while most of us have only a dim
idea of the points of resemblance between these things. Nature's
classes moreover shade off into one another by such imperceptible
degrees that it is often impossible to fix the boundary line between
one class and another. A still greater source of perplexity in dealing
with real kinds is that it is sometimes almost impossible to fix upon
any attribute which is common to every individual member of the class
without exception. All that we can do in such cases is to lay down a
type of the class in its perfect form, and judge of individual
instances by the degree of their approximation to it. Again, real
kinds being known to us primarily in extension, the intension which we
attach to the names is hable to be affected by the advance of
knowledge. In dealing therefore with such terms we must be content
with provisional definitions, which adequately express our knowledge
of the things denoted by them, at the time, though a further study of
their attributes may induce us subsequently to alter the
definition. Thus the old definition of animal as a sentient organism
has been rendered inadequate by the discovery that so many of the
phenomena of sensation can be exhibited by plants,
§ 373. But terms in which intension is the predominant idea are more
capable of being defined once for all. Aristotle's definitions of
'wealth' and 'monarchy' are as applicable now as in his own day, and
no subsequent discoveries of the properties of figures will render
Euclid's definitions unavailable.
§ 374. We may distinguish therefore between two kinds of definition,
namely,
(1) Final.
(2) Provisional.
§ 375. A distinction is also observed between Real and Nominal
Definitions. Both of these explain the meaning of a term: but a real
definition further assumes the actual existence of the thing
defined. Thus the explanation of the term 'Centaur' would be a
nominal, that of 'horse' a real definition.
It is useless to assert, as is often done, that a nominal definition
explains the meaning of a term and a real definition the nature of a
thing; for, as we have seen already, the meaning of a term is whatever
we know of the nature of a thing.
§ 376. It now remains to lay down certain rules for correct
definition.
§ 377. The first rule that is commonly given is that a definition
should state the essential attributes of the thing defined. But this
amounts merely to saying that a definition should be a definition;
since it is only by the aid of definition that we can distinguish
between essential and non-essential among the common attributes
exhibited by a class of things. The rule however may be retained as a
material test of the soundness of a definition, in the sense that he
who seeks to define anything should fix upon its most important
attributes. To define man as a mammiferous animal having two hands, or
as a featherless biped, we feel to be absurd and incongruous, since
there is no reference to the most salient characteristic of man,
namely, his rationality. Nevertheless we cannot quarrel with these
definitions on formal, but only on material grounds. Again, if anyone
chose to define logic as the art of thinking, all we could say is that
we differ from him in opinion, as we think logic is more properly to
be regarded as the science of the laws of thought. But here also it is
on material grounds that we dissent from the definition.
§ 378. Confining ourselves therefore to the sphere with which we are
properly concerned, we lay down the following
_Rules for Definition._
(1) A definition must be co-extensive with the term defined.
(2) A definition must not state attributes which imply one another.
(3) A definition must not contain the name defined, either directly
or by implication.
(4) A definition must be clearer than the term defined.
(5) A definition must not be negative, if it can be affirmative.
Briefly, a definition must be adequate (1), terse (2), clear (4); and
must not be tautologous (3), or, if it can be avoided, negative (5).
§ 379. It is worth while to notice a slight ambiguity in the term
'definition' itself. Sometimes it is applied to the whole proposition
which expounds the meaning of the term; at other times it is confined
to the predicate of this proposition. Thus in stating the first four
rules we have used the term in the latter sense, and in stating the
fifth in the former.
§ 380. We will now illustrate the force of the above rules by giving
examples of their violation.
Rule 1. Violations. A triangle is a figure with three equal sides.
A square is a four-sided figure having all its sides equal.
In the first instance the definition is less extensive than the term
defined, since it applies only to equilateral triangles. This fault
may be amended by decreasing the intension, which we do by eliminating
the reference to the equality of the sides.
In the second instance the definition is more extensive than the term
defined. We must accordingly increase the intension by adding a new
attribute 'and all its angles right angles.'
Rule 2. Violation. A triangle is a figure with three sides and three
angles.
One of the chief merits of a definition is to be terse, and this
definition is redundant, since what has three sides cannot but have
three angles.
Rule 3. Violations. A citizen is a person both of whose parents were
citizens.
Man is a human being.
Rule 4. Violations. A net is a reticulated fabric, decussated at
regular intervals.
Life is the definite combination of heterogeneous changes, both
simultaneous and successive, in correspondence with external
co-existences and sequences.
Rule 5. Violations. A mineral is that which is neither animal nor
vegetable.
Virtue is the absence of vice.
§ 381. The object of definition being to explain what a thing is, this
object is evidently defeated, if we confine ourselves to saying what
it is not. But sometimes this is impossible to be avoided. For there
are many terms which, though positive in form, are privative in force.
These terms serve as a kind of residual heads under which to throw
everything within a given sphere, which does not exhibit certain
positive attributes. Of this unavoidably negative nature was the
definition which we give of 'accident,' which amounted merely to
saying that it was any attribute which was neither a difference nor a
property.
§ 382. The violation of Rule 3, which guards against defining a thing
by itself, is technically known as 'circulus in definiendo,' or
defining in a circle. This rule is often apparently violated, without
being really so. Thus Euclid defines an acute-angled triangle as one
which has three acute angles. This seems a glaring violation of the
rule, but is perfectly correct in its context; for it has already been
explained what is meant by the terms 'triangle' and 'acute angle,' and
all that is now required is to distinguish the acute-angled triangle
from its cognate species. He might have said that an acute-angled
triangle is one which has neither a right angle nor an obtuse angle:
but rightly preferred to throw the same statement into a positive
form.
§ 383. The violation of Rule 4 is known as 'ignotum per ignotius' or
'per aeque ignotum.' This rule also may seemingly be violated when it
is not really so. For a definition may be correct enough from a
special point of view, which, apart from that particular context,
would appear ridiculous. From the point of view of conic sections, it
is correct enough to define a triangle as that section of a cone which
is formed by a plane passing through the vertex perpendicularly to the
base, but this could not be expected to make things clearer to a
person who was inquiring for the first time into the meaning of the
word triangle. But a real violation of the fourth rule may arise, not
only from obscurity, but from the employment of ambiguous language or
metaphor. To say that 'temperance is a harmony of the soul' or that
'bread is the staff of life,' throws no real light upon the nature of
the definiend.
§ 384. The material correctness of a definition is, as we have already
seen, a matter extraneous to formal logic. An acquaintance with the
attributes which terms imply involves material knowledge quite as much
as an acquaintance with the things they apply to; knowledge of the
intension and of the extension of terms is alike acquired by
experience. No names are such that their meaning is rendered evident
by the very constitution of our mental faculties; yet nothing short of
this would suffice to bring the material content of definition within
the province of formal logic.
CHAPTER VIII.
_Of Division._
§ 385. To divide a term is to unfold its extension, that is, to set
forth the things of which it is a name.
§ 386. But as in definition we need not completely unfold the
intension of a term, so in division we must not completely unfold its
extension.
§ 387. Completely to unfold the extension of a term would involve
stating all the individual objects to which the name applies, a thing
which would be impossible in the case of most common terms. When it is
done, it is called Enumeration. To reckon up all the months of the
year from January to December would be an enumeration, and not a
division, of the term 'month.'
§ 388. Logical division always stops short at classes. It may be
defined as the statement of the various classes of things that can be
called by a common name. Technically we may say that it consists in
breaking up a genus into its component species.
§ 389. Since division thus starts with a class and ends with classes,
it is clear that it is only common terms which admit of division, and
also that the members of the division must themselves be common terms.
§ 390. An 'individual' is so called as not admitting of logical
division. We may divide the term 'cow' into classes, as Jersey,
Devonshire, &c., to which the name 'cow' will still be applicable, but
the parts of an individual cow are no longer called by the name of the
whole, but are known as beefsteaks, briskets, &c.
§ 391. In dividing a term the first requisite is to fix upon some
point wherein certain members of the class differ from others. The
point thus selected is called the Fundamentum Divisionis or Basis of
the Division.
§ 392. The basis of the division will of course differ according to
the purpose in hand, and the same term will admit of being divided on
a number of different principles. Thus we may divide the term 'man,'
on the basis of colour, into white, black, brown, red, and yellow; or,
on the basis of locality, into Europeans, Asiatics, Africans,
Americans, Australians, New Zealanders, and Polynesians; or again, on
a very different principle, into men of nervous, sanguine, bilious,
lymphatic and mixed temperaments.
§ 393. The term required to be divided is known as the Totum Divisum
or Divided Whole. It might also be called the Dividend.
§ 394. The classes into which the dividend is split up are called the
Membra Dividentia, or Dividing Members.
§ 395. Only two rules need be given for division--
(1) The division must be conducted on a single basis.
(2) The dividing members must be coextensive with the divided whole.
§ 396. More briefly, we may put the same thing thus--There must be no
cross-division (1) and the division must be exhaustive (2).
§ 397. The rule, which is commonly given, that each dividing member
must be a common term, is already provided for under our definition of
the process.
§ 398. The rule that the dividend must be predicable of each of the
dividing members is contained in our second rule; since, if there were
any term of which the dividend were not predicable, it would be
impossible for the dividing members to be exactly coextensive with it.
It would not do, for instance, to introduce mules and donkeys into a
division of the term horse.
§ 399. Another rule, which is sometimes given, namely, that the
constituent species must exclude one another, is a consequence of our
first; for, if the division be conducted on a single principle, the
constituent species must exclude one another. The converse, however,
does not hold true. We may have a division consisting of mutually
exclusive members, which yet involves a mixture of different bases,
e.g. if we were to divide triangle into scalene, isosceles and
equiangular. This happens because two distinct attributes may be found
in invariable conjunction.
§ 400. There is no better test, however, of the soundness of a
division than to try whether the species overlap, that is to say,
whether there are any individuals that would fall into two or more of
the classes. When this is found to be the case, we may be sure that we
have mixed two or more different fundamenta divisionis. If man, for
instance, were to be divided into European, American, Aryan, and
Semitic, the species would overlap; for both Europe and America
contain inhabitants of Aryan and Semitic origin. We have here members
of a division based on locality mixed up with members of another
division, which is based on race as indicated by language.
§ 401. The classes which are arrived at by an act of division may
themselves be divided into smaller classes. This further process is
called Subdivision.
§ 402. Let it be noticed that Rule 1 applies only to a single act of
division. The moment that we begin to subdivide we not only may, but
must, adopt a new basis of division; since the old one has, 'ex
hypothesi,' been exhausted. Thus, having divided men according to the
colour of their skins, if we wish to subdivide any of the classes, we
must look out for some fresh attribute wherein some men of the same
complexion differ from others, e.g. we might divide black men into
woolly-haired blacks, such as the Negroes, and straight-haired blacks,
like the natives of Australia.
§ 403. We will now take an instance of division and
subdivision, with a view to illustrating some of the
technical terms which are used in connection with the
process. Keeping closely to our proper subject, we will
select as an instance a division of the products of thought,
which it is the province of logic to investigate.
Product of thought
_______________|____________________________
| | |
Term Proposition Inference
____|___ ______|_____ _____|______
| | | | | |
Singular Common Universal Particular Immediate Mediate
___|___ ___|___
| | | |
A E I O
Here we have first a threefold division of the products of thought
based on their comparative complexity. The first two of these, namely,
the term and the proposition, are then subdivided on the basis of
their respective quantities. In the case of inference the basis of the
division is again the degree of complexity. The subdivision of the
proposition is carried a step further than that of the others. Having
exhausted our old basis of quantity, we take a new attribute, namely,
quality, on which to found the next step of subdivision.
§ 404. Now in such a scheme of division and subdivision as the
foregoing, the highest class taken is known as the Summum Genus. Thus
the summum genus is the same thing as the divided whole, viewed in a
different relation. The term which is called the divided whole with
reference to a single act of division, is called the summum genus
whenever subdivision has taken place.
§ 405. The classes at which the division stops, that is, any which are
not subdivided, are known as the Infimae Species.
§ 406. All classes intermediate between the summum genus and the
infimae species are called Subaltern Genera or Subaltern Species,
according to the way they are looked at, being genera in relation to
the classes below them and species in relation to the classes above
them.
§ 407. Any classes which fall immediately under the same genus are
called Cognate Species, e.g. singular and common terms are cognate
species of term.
§ 408. The classes under which any lower class successively falls are
called Cognate Genera. The relation of cognate species to one another
is like that of children of the same parents, whereas cognate genera
resemble a line of ancestry.
§ 409. The Specific Difference of anything is the attribute or
attributes which distinguish it from its cognate species. Thus the
specific difference of a universal proposition is that the predicate
is known to apply to the whole of the subject. A specific difference
is said to constitute the species.
§ 410. The specific difference of a higher class becomes a Generic
Difference with respect to the class below it. A generic difference
then may be said to be the distinguishing attribute of the whole class
to which a given species belongs. The generic difference is common to
species that are cognate to one another, whereas the specific
difference is peculiar to each. It is the generic difference of an A
proposition that it is universal, the specific difference that it is
affirmative.
§ 411. The same distinction is observed between the specific and
generic properties of a thing. A Specific Property is an attribute
which flows from the difference of a thing itself; a Generic Property
is an attribute which flows from the difference of the genus to which
the thing belongs. It is a specific property of an E proposition that
its predicate is distributed, a generic property that its contrary
cannot be true along with it (§ 465); for this last characteristic
flows from the nature of the universal proposition generally.
§ 412. It now remains to say a few words as to the place in logic of
the process of division. Since the attributes in which members of the
same class differ from one another cannot possibly be indicated by
their common name, they must be sought for by the aid of experience;
or, to put the same thing in other words, since all the infimae
species are alike contained under the summum genus, their distinctive
attributes can be no more than separable accidents when viewed in
relation to the summum genus. Hence division, being always founded on
the possession or non-possession of accidental attributes, seems to
lie wholly outside the sphere of formal logic. This however is not
quite the case, for, in virtue of the Law of Excluded Middle, there is
always open to us, independently of experience, a hypothetical
division by dichotomy. By dichotomy is meant a division into two
classes by a pair of contradictory terms, e.g. a division of the
class, man, into white and not-white. Now we cannot know,
independently of experience, that any members of the class, man,
possess whiteness; but we may be quite sure, independently of all
experience, that men are either white or not. Hence division by
dichotomy comes strictly within the province of formal logic. Only it
must be noticed that both sides of the division must be hypothetical.
For experience alone can tell us, on the one hand, that there are any
men that are white, and on the other, that there are any but white
men.
§ 413. What we call a division on a single basis is in reality the
compressed result of a scheme of division and subdivision by
dichotomy, in which a fresh principle has been introduced at every
step. Thus when we divide men, on the basis of colour, into white,
black, brown, red and yellow, we may be held to have first divided men
into white and not-white, and then to have subdivided the men that are
not-white into black and not-black, and so on. From the strictly
formal point of view this division can only be represented as
follows--
Men
___________________|_____
| |
White (if any) Not-white (if any)
_________________|_____
| |
Black (if any) Not-black (if any)
__________________|____
| |
Brown (if any) Not-brown (if any)
____________________|____
| |
Red (if any) Not-red (if any).
§ 414. Formal correctness requires that the last term in such a series
should be negative. We have here to keep the term 'not-red' open, to
cover any blue or green men that might turn up. It is only experience
that enables us to substitute the positive term 'yellow' for
'not-red,' since we know as a matter of fact that there are no men but
those of the five colours given in the original division.
§ 415. Any correct logical division always admits of being arrived at
by the longer process of division and subdivision by dichotomy. For
instance, the term quadrilateral, or four-sided rectilinear figure, is
correctly divided into square, oblong, rhombus, rhomboid and
trapezium. The steps of which this division consists are as follows--
Quadrilateral
__________|_________
| |
Parallelogram Trapezium
_____|_____________________
| |
Rectangle Non-rectangle
___|___ _____|_____
| | | |
Square Oblong Rhombus Rhomboid.
§ 416. In reckoning up the infimae species in such a scheme, we must
of course be careful not to include any class which has been already
subdivided; but no harm is done by mixing an undivided class, like
trapezium, with the subdivisions of its cognate species.
§ 417. The two processes of definition and division are intimately
connected with one another. Every definition suggests a division by
dichotomy, and every division supplies us at once with a complete
definition of all its members.
§ 418. Definition itself, so far as concerns its content, is, as we
have already seen, extraneous to formal logic: but when once we have
elicited a genus and difference out of the material elements of
thought, we are enabled, without any further appeal to experience, to
base thereon a division by dichotomy. Thus when man has been defined
as a rational animal, we have at once suggested to us a division of
animal into rational and irrational.
§ 419. Again, the addition of the attributes, rational and irrational
respectively, to the common genus, animal, ipso facto supplies us with
definitions of the species, man and brute. Similarly, when we
subdivided rectangle into square and oblong on the basis of the
equality or inequality of the adjacent sides, we were by so doing
supplied with a definition both of square and oblong--'A square is a
rectangle having all its sides equal,' and 'An oblong is a rectangle
which has only its opposite sides equal.'
§ 420. The definition of a square just given amounts to the same thing
as Euclid's definition, but it complies with a rule which has value as
a matter of method, namely, that the definition should state the
Proximate Genus of the thing defined.
§ 421. Since definition and division are concerned with the intension
and extension of terms, they are commonly treated of under the first
part of logic: but as the treatment of the subject implies a
familiarity with the Heads of Predicables, which in their turn imply
the proposition, it seems more desirable to deal with them under the
second.
§ 422. We have already had occasion to distinguish division from
Enumeration. The latter is the statement of the individual things to
which a name applies. In enumeration, as in division, the wider term
is predicable of each of the narrower ones.
§ 423. Partition is the mapping out of a physical whole into its
component parts, as when we say that a tree consists of roots, stem,
and branches. In a partition the name of the whole is not predicable
of each of the parts.
§ 424. Distinction is the separation from one another of the various
meanings of an equivocal term. The term distinguished is predicable
indeed of each of the members, but of each in a different sense. An
equivocal term is in fact not one but several terms, as would quickly
appear, if we were to use definitions in place of names.
§ 425. We have seen that a logical whole is a genus viewed in relation
to its underlying species. From this must be distinguished a
metaphysical whole, which is a substance viewed in relation to its
attributes, or a class regarded in the same way. Logically, man is a
part of the class, animal; metaphysically, animal is contained in
man. Thus a logical whole is a whole in extension, while a
metaphysical whole is a whole in intension. From the former point of
view species is contained in genus; from the latter genus is contained
in species.
PART III.--OF INFERENCES.
CHAPTER I.
_Of Inferences in General_.
§ 426. To infer is to arrive at some truth, not by direct experience,
but as a consequence of some truth or truths already known. If we see
a charred circle on the grass, we infer that somebody has been
lighting a fire there, though we have not seen anyone do it. This
conclusion is arrived at in consequence of our previous experience of
the effects of fire.
§ 427. The term Inference is used both for a process and for a product
of thought.
As a process inference may be defined as the passage of the mind from
one or more propositions to another.
As a product of thought inference may be loosely declared to be the
result of comparing propositions.
§ 428. Every inference consists of two parts--
(1) the truth or truths already known;
(2) the truth which we arrive at therefrom.
The former is called the Antecedent, the latter the Consequent. But
this use of the terms 'antecedent' and 'consequent' must be carefully
distinguished from the use to which they were put previously, to
denote the two parts of a complex proposition.
§ 429. Strictly speaking, the term inference, as applied to a product
of thought, includes both the antecedent and consequent: but it is
often used for the consequent to the exclusion of the
antecedent. Thus, when we have stated our premisses, we say quite
naturally, 'And the inference I draw is so and so.'
§ 430. Inferences are either Inductive or Deductive. In induction we
proceed from the less to the more general; in deduction from the more
to the less general, or, at all events, to a truth of not greater
generality than the one from which we started. In the former we work
up to general principles; in the latter we work down from them, and
elicit the particulars which they contain.
§ 431. Hence induction is a real process from the known to the
unknown, whereas deduction is no more than the application of
previously existing knowledge; or, to put the same thing more
technically, in an inductive inference the consequent is not contained
in the antecedent, in a deductive inference it is.
§ 432. When, after observing that gold, silver, lead, and other
metals, are capable of being reduced to a liquid state by the
application of heat, the mind leaps to the conclusion that the same
will hold true of some other metal, as platinum, or of all metals, we
have then an inductive inference, in which the conclusion, or
consequent, is a new proposition, which was not contained in those
that went before. We are led to this conclusion, not by reason, but by
an instinct which teaches us to expect like results, under like
circumstances. Experience can tell us only of the past: but we allow
it to affect our notions of the future through a blind belief that
'the thing that hath been, it is that which shall be; and that which
is done is that which shall be done; and there is no new thing under
the sun.' Take away this conviction, and the bridge is cut which
connects the known with the unknown, the past with the future. The
commonest acts of daily life would fail to be performed, were it not
for this assumption, which is itself no product of the reason. Thus
man's intellect, like his faculties generally, rests upon a basis of
instinct. He walks by faith, not by sight.
§ 433. It is a mistake to talk of inductive reasoning, as though it
were a distinct species from deductive. The fact is that inductive
inferences are either wholly instinctive, and so unsusceptible of
logical vindication, or else they may be exhibited under the form of
deductive inferences. We cannot be justified in inferring that
platinum will be melted by heat, except where we have equal reason for
asserting the same thing of copper or any other metal. In fact we are
justified in drawing an individual inference only when we can lay down
the universal proposition, 'Every metal can be melted by heat.' But
the moment this universal proposition is stated, the truth of the
proposition in the individual instance flows from it by way of
deductive inference. Take away the universal, and we have no logical
warrant for arguing from one individual case to another. We do so, as
was said before, only in virtue of that vague instinct which leads us
to anticipate like results from like appearances.
§ 434. Inductive inferences are wholly extraneous to the science of
formal logic, which deals only with formal, or necessary, inferences,
that is to say with deductive inferences, whether immediate or
mediate. These are called formal, because the truth of the consequent
is apparent from the mere form of the antecedent, whatever be the
nature of the matter, that is, whatever be the terms employed in the
proposition or pair of propositions which constitutes the
antecedent. In deductive inference we never do more than vary the form
of the truth from which we started. When from the proposition 'Brutus
was the founder of the Roman Republic,' we elicit the consequence 'The
founder of the Roman Republic was Brutus,' we certainly have nothing
more in the consequent than was already contained in the antecedent;
yet all deductive inferences may be reduced to identities as palpable
as this, the only difference being that in more complicated cases the
consequent is contained in the antecedent along with a number of other
things, whereas in this case the consequent is absolutely all that the
antecedent contains.
§ 435. On the other hand, it is of the very essence of induction that
there should be a process from the known to the unknown. Widely
different as these two operations of the mind are, they are yet both
included under the definition which we have given of inference, as the
passage of the mind from one or more propositions to another. It is
necessary to point this out, because some logicians maintain that all
inference must be from the known to the unknown, whereas others
confine it to 'the carrying out into the last proposition of what was
virtually contained in the antecedent judgements.'
§ 436. Another point of difference that has to be noticed between
induction and deduction is that no inductive inference can ever attain
more than a high degree of probability, whereas a deductive inference
is certain, but its certainty is purely hypothetical.
§ 437. Without touching now on the metaphysical difficulty as to how
we pass at all from the known to the unknown, let us grant that there
is no fact better attested by experience than this--'That where the
circumstances are precisely alike, like results follow.' But then we
never can be absolutely sure that the circumstances in any two cases
are precisely alike. All the experience of all past ages in favour of
the daily rising of the sun is not enough to render us theoretically
certain that the sun will rise tomorrow We shall act indeed with a
perfect reliance upon the assumption of the coming day-break; but, for
all that, the time may arrive when the conditions of the universe
shall have changed, and the sun will rise no more.
§ 438. On the other hand a deductive inference has all the certainty
that can be imparted to it by the laws of thought, or, in other words,
by the structure of our mental faculties; but this certainty is purely
hypothetical. We may feel assured that if the premisses are true, the
conclusion is true also. But for the truth of our premisses we have to
fall back upon induction or upon intuition. It is not the province of
deductive logic to discuss the material truth or falsity of the
propositions upon which our reasonings are based. This task is left to
inductive logic, the aim of which is to establish, if possible, a test
of material truth and falsity.
§ 439. Thus while deduction is concerned only with the relative truth
or falsity of propositions, induction is concerned with their actual
truth or falsity. For this reason deductive logic has been termed the
logic of consistency, not of truth.
§ 440. It is not quite accurate to say that in deduction we proceed
from the more to the less general, still less to say, as is often
said, that we proceed from the universal to the particular. For it may
happen that the consequent is of precisely the same amount of
generality as the antecedent. This is so, not only in most forms of
immediate inference, but also in a syllogism which consists of
singular propositions only, e.g.
The tallest man in Oxford is under eight feet.
So and so is the tallest man in Oxford.
.'. So and so is under eight feet.
This form of inference has been named Traduction; but there is no
essential difference between its laws and those of deduction.
§ 441. Subjoined is a classification of inferences, which will serve
as a map of the country we are now about to explore.
Inference
________________________|__________
| |
Inductive Deductive
_________________|_______________
| |
Immediate Mediate
___________|__________ ______|______
| | | |
Simple Compound Simple Complex
______|________________ | ______|_____________|_
| | | | | | |
Opposition Conversion Permutation | Conjunctive Disjunctive Dilemma
|
_________|________
| |
Conversion Conversion
by by
Negation position