Deductive Logic
by
St. George Stock
Part 1 out of 6
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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
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