THE
^XEMENTS OF LOGIC.
BY
T^ K. ABBOTT, B.D,
FELLOW AND TUTOR OF TRINITY COLLEGE, DUBLIN.
SECOND EDITION.
DUBLIN : HODGES, FIGGIS, & CO., GRAFTON-STREET.
LONDON : LONGMANS, GREEN, & CO.
1885.
BCic
f\3
DUBLIN :
PRINTED AT THE UNIVERSITY PRESS,
BY PONSONBY AND WELDRICK.
3 S^^^3
rn
TO THE READER.
In the following treatise I have endeavoured to
unite conciseness and scientific accuracy, while
adhering, as far as possible, to the traditional
lines of the Aristotelian Logic. Critical discus-
sions have been avoided, as out of place in a
purely Elementary Treatise, but the student who
desires to pursue the subject further will have
nothing to unlearn. A few things have been in-
cluded which could not with propriety have been
omitted, even in an Elementary Treatise, but
which may be passed over by the student who
only seeks a minimum of knowledge of Logic.
The paragraphs containing these are printed with
close lines, and are (with insignificant exceptions)
further distinguished in the Table of Contents by
a prefixed [f].
Digitized by the Internet Archive
in 2007 with funding from
IVIicrosoft Corporation
http://www.archive.org/details/elementsoflogicOOabborich
CONTENTS.
N.B. Paragraphs printed closely^ and marked [f] in this Table,
may be passed over by the beginner.
PART FIRST.
OF TERMS.
PAGE
Preliminary, . . 3
Of Terms in General, 3
Abstract of Concrete Terms, . . . . . . 4
Common, Singular, and Collective Terms, .... 5
Of Denotation and Connotation, 5
Of Abstraction and of Genus and Species, . . . . 7 ^
[t] Remarks on Concepts, 8
Non-connotative Terms, 8
Contradictory and Contrary Terms, 10
Clearness and Distinctness, 10
Of Definition, 11
Of Division, , , , , , , , . .12
viii CONTENTS.
PART SECOND.
OF SIMPLE PROPOSITIONS.
PAGE
Preliminary Definitions, 14
The Copula, 15
Quality of Propositions, 16
Quantity of Propositions, 16
Distribution of Terms, 18
[t] Quantification of the Predicate, 20
Of the Import of Propositions, 21
Verbal and Real, or Analytical and Synthetical Proposi-
tions, 23
Copulative Propositions, 24
Modal Propositions, 25
APPENDIX TO PART II.
[tj Of the Predicables, 26
[t] Of the Ten Categories, 27
PART THIRD.
OF INFERENCES.
CHAPTER I.
0/ Immediate Inferences,
Subalternation, 29
Opposition, 30
Conversion, ......... 34
Contraposition, 36
Remarks on Conversion, 38
Of the Principle of Substitution, . . . . -39
CONTENTS.
IX
CHAPTER 11.
0/ Mediate Reasoning, o?' Syllogism.
PAGE
Of Mediate Reasoning Generally, 40
General Rules of Syllogism, 42
[t] Inferences from premisses with Undistributed Middle, . 44
K: Of the Four Figures, 44
Special Rules of the Four Figures, 47
[t] Remark on the Special use of each Figure, ... 50
OfMoods, 51
[t] Remark on the Validity of the Moods enumerated, . . 53
Of Aristotle' s Dictum, 55
OfReduction, 57
[t] Of Reductio ad Impossibile, 61
Of the Unfigured Syllogism, 62
Of the Enthymeme, 63
Of Sorites, . 63
CHAPTER HI.
Of Complex Propositions and Syllogisms.
Of Complex Propositions, . . . . .
. 65
Of Complex Syllogisms,
. 66
Conditional Syllogisms,
. 67
Disjunctive Syllogisms,
. 68
Of the Dilemma,
. 69
Of the Reduction of Complex Syllogisms, .
• 70
X CONTENTS.
CHAPTER IV.
0/ Probable Reasoning.
Of Chains of Probability, ....
Of Cumulative Probabilities,
[ t] Of Inductive Reasoning, ....
[t] Of the Logical Basis of Induction,
[t] Mr. Mill's View of the Type of Reasoning,
Of Analogy,
PART FOURTH.
CHAPTER I.
Of Fallacies,
Of Fallacies in General,
Of Logical Fallacies, .
Illicit Process,
Undistributed Middle, .
Two Middle Terms,
Fallacies in Complex Syllogisms,
Of Semi-logical Fallacies, .
Ambiguous Terms,
Composition and Division.
Fallacy of Accident,
Of Material Fallacies, .
Ignoratio Elenchi,
Petitio Principii, .
Arguing in a Circle,
A non causa pro causa,
Fallacy of Many Questions,
Fallacy of False Analogy,
Argumentum ad Hominem,
CONTENTS. XI
CHAPTER II.
Of Methods of Proof and of Exposition.
PAGE
Analytic and Synthetic Exposition, . . • • • '90
A priori and A posteriori Proof, 9^
Of Explanations and Deduction, ^^ ^
APPENDIX.
Exercises, ^^
.101
Index,
CORRIGENDA.
Page 5., line 3 -^y^- Premisses ^ I E O, however, is illegitimate, for the , ^
term IS particular m its premiss and universal in the co
Page 52, line 3. --/^?r twelve r<ra^ eleven.
Page 52, line 4.— i^<?r eleven read ten.
{(UNIVERSITY
THE ELEMENTS OF LOGIC.
INTRODUCTION.
WHAT IS LOGIC ?
I. Everything in nature takes place according to laws.
This, which is true of material things, is also true of the
operations of the mind. These follow laws of which we
are not always aware. Language, for instance, follows
laws of grammar, and these laws in any particular lan-
guage are the same, whatever the subject may be of which
we are speaking. The exercise of the understanding in
particular is itself governed by laws. Some of these only
apply to special kinds of subject-matter, as, for example,
the laws of number in mathematical reasoning. But there
are others which hold good in all cases, whatever the
subject-matter of our thoughts or reasonings may be. It j
is of these laws that Logic treats. Just as the rules of
grammar are the same whatever the subject spoken of
may be, so the rules of Logic are the same, whatever the
subject thought about or reasoned about*
B
2 THE ELEMENTS OF LOGIC.
2. The distinction involved^in this statement is the im-
portant one between Form and Matter, Matter is the
variable part ; rorm the invariable. In language, for
example, the lexicon deals with the Matter ; grammar
deals with the Form.
3. Logic, then, is the Science of the Form of Thought.
4. Some writers assign to Logic a much wider sphere
than this ; they regard it as including the rules of estima-
tion of evidence, and of the operations of the understand-
ing subsidiary thereto. The narrower science here treated
of may, for the sake of precision, be called Formal
Logic, and the exposition of the rules just referred to
may be called Applied Logic.
( 3 }
FART FIRST.
OF TERMS.
Preliminary,
5. The simplest act of _t hou £ht is a Judgmentj_ which
expressed in words is a Proposition. Propositions so con-
nected that one follows from one or more others constitute
Reasoning. Propositions, however, are compound in ex-
pression, and consist of words or terms. Hence we begin
with Terms ; we treat in the next place of Propositions,
and. in the third place of Reasoning.
Of Terms in General,
6. In a proposition we either affirm or deny something
of something else. ' Horses are quadrupeds ; ' \' horses
are useful animals.' Here we affirm * quadrupeds,' * useful
animals,' of ' horses,' Words which can be used in this
way, i, e. as naming something which we affirm or deny, or
as naming that of which something is affirmed or denied,
are called Terms. That which is affirmed or denied
is called the Predicate ; that of which, it is affirmed
or denied is the Subject.
7. A Term, therefore, may be defined as a word or
combination of words, which can be used by itself as
subject or predicate of a proposition It must be ob-
served that a Term may consist of several words. In
B 2
4 THE ELEMENTS OF LOGIC. [part i.
the proposition given above, * useful animal,' is a single
term ; * Queen of Great Britain and Ireland ' is also a single
term.
8. On the other hand, there are many words which can-
not stand by themselves as terms. Such are prepositions,
adverbs (except of time or place), conjunctions, oblique
cases of nouns, &c. These can be used as terms only
with something understood. We might say: ***of" is a
preposition,' i, e. the word or sound ' of.'
' Term ' is derived from * terminus,' used by logicians
to represent the Greek opos ; the subject and predicate
being, as it were, the boundaries of the proposition.
Of Abstract and Concrete Terms,
g.^erms may be names of Things, or of Attributes
of Things. For example, * man ' is a name of Things ;
* humanity' is the name of an Attribute. Names of
Things are concrete ; Names of Attributes are Ab-
stract. Thus * whiteness ' is Abstract ; * white thing '
is Concrete.
10. Concrete Terms may be grammatically either sub-
stantives or adjectives (including participles). *This horse
is white ' means the same as * this horse is a white horse '
or * a white thing.' An adjective cannot always be used
alone as the subject of a proposition ; this is because it is,
grammatically, an incomplete name, and requires a substan--
tive either expressed or understod to complete it. When
used as a predicate, the subject is understooa ; as in the
example just given^ — * This horse is white ' (/. e, * a white
horse'). Greek and Latin are more free in their use of
adjectives.
The same word may be sometimes used as an abstract,
PART I.] OF TERMS, 5
and sometimes as a concrete term, ex, gr. * green is an
agreeable colour/ Here * green ' = ' green colour.' ^'
Of Commofiy Singular ^ and Collective Terms,
11. Names of things are divided into Singular, Common,
and Collective.
12. A Singular Term is one which names an individual
thing, as : Socrates, the present Prime Minister, this
person, the oldest inhabitant. A Common or Univer-
sal Term is applicable to any one of a class, which
may include many things, as : man, tree, phoenix, prime
minister.
13. A Collective Term is applicable to a group of
things as a whole, but not to each, member of it, as :
the British Parliament, the fifty-third regiment.
14. A singular term cannot be a predicate unless the
subject is also a singular term. There are some abstract
terms which are names of classes of attributes, as : colour,
sound.
Of Denotation (or Extension) and Connotation,
15. Universal terms have a double signification. By
the word * boat,' for example, we mean a small open
vessel constructed to float on the water, and to carry one
or more persons, etc. But if asked of what is the word
* boat ' a name, we might answer either by giving this defi-
nition, or by enumerating different kinds of boats, as
barges, pinnaces, gigs, etc., or all individual boats. The
* ' Abstract ' is sometimes used, especially by the older writers, in
' the sense of ' general.'
)
6 TJIF ELEMENTS OF LOGIC. [part i.
term 'boat' is said to 'denote' all these, that is, every-
thing which can be called a boat, and to 'connote' or
note therewith the attributes implied in the name.
1 6. The things to which the name is applicable con-
stitute its Denotation, or, as it is sometimes called, its
Extension. The attributes implied by the name con-
stitute its Connotation or comprehension. )The answer
to the question, What is a boat, i, e, what is the mean-
ing of the word, is its Connotation. The answer to
the question, Of what things can we say that they are
boats, is its Denotation or Extension. The form of the
question may be such as to admit of either answer. ' What
are boats ? ' ' What are the metallic elements } ' ' Who
are the greatest men in history.?' maybe understood in
either way.
17. It is obvious that the larger the number of different
things included in the Extension (= Denotation) the fewer
will be the common attributes ; and, on the other hand,
the greater the number of attributes taken into the Con-
notation (= Comprehension) the fewer, in general, will be
the individual things to which the term is applicable. In
other words : By increasing the Connotation we diminish
the Extension, and by enlarging the Extension we diminish
the Connotation, j Thus, if we take the term ' house,'
and add the attribute 'intended for dwelling in,' we get
' dwelling-house,' which includes (or denotes) fewer ob-
jects, that is, has a less Extension. On the other hand, if
we wish to use a term which will apply to (or denote)
a greater number of things, such as ' structure,' we must
necessarily leave out of the Connotation those attributes
which distinguish houses from other structures. y*
18. It may happen in particular cases that we may add
PART I.] OF TERMS, 7
to the Connotation of a term without increasing the exten-
sion. For instance, we may add to ' rational animal ' the
attribute ' biped ; ' and since all rational animals are bipeds,
the extension remains unaltered.
Of AhstractioUy and of Genus and Species,
19. If we take any term whose connotation is known,
and leave out some of the attributes connoted, we thereby
form a new notion which denotes, or may denote, a larger
number of individuals. It is called a more general notion.
This leaving out of attributes is called * abstracting ' from
them, and the process itself is * Abstraction.'
20. The notion thus arrived at is called the ' Genus ' of
the less general ; the less general is the * Species.' When
two common terms are so related that the connotation
of one includes the connotation of the other, that
which has the larger connotation is called the Species
of the other, which is called the Genus.
21. Thus, if we take the term * riding horse,' and leave
out the notion implied in the first part, we arrive at the
notion ' horse,' which is the Genus, of which * riding
horse' is Species. On the other hand, if we add to the
latter the notion * small,' we arrive at ' riding pony,' which
is a Species, of which * riding horse ' is Genus.*
22. The more general notion is said to be higher than
the less general, which is lower.
23. It is clear that whatever attribute belongs to the
*\Nr. bT — The terms 'Genus ' and ' Species ' have a more limited and
technical sense in treatises on Natural History.
8 THE ELEMENTS OF LOGIC, [part i.
higher notion belongs also to the lower, and whatever is
inconsistent with the higher is inconsistent with the lower.
For the notion of the lower includes or contains in it the
notion of the higher. On the other hand, the extension
of the lower is part of the extension of the higher, and,
therefore, whatever individual belongs to the lower class
belongs also to the higher.
24. A Genus which has none above it is a Genus sum-
mum ; a Species which has none below it is a Species
infima. A Summum Genus with its subordinate genera,
and these again subdivided into lower genera, and so on to
the lowest species, is a Predicamental line.
Remarks.
25. These universal terms maybe regarded as names of
notions or concepts, the concept being the group of attri-
butes which constitutes the meaning of the term.
26. In arriving at concepts or universal terms, we gene-
rally begin by grouping together things which have a certain
general resemblance not yet definitely stated, /. e, we begin
by using the term to designate a certain denotation. We
may afterwards proceed to study the common attributes
implied in the term, and we may find that in our applica-
tion of the term we have gradually dropped out of sight
one attribute after another until there is little in common
between the things first and those last named. Thus * boat'
is applied to a steam-packet, as well as to a row-boat.
Of Non-Connotative Terms,
27. Some terms have no connotation, e. g. proper names.
The fact that the name * Walter Scott ' calls up to our
mind the idea of the 'Author of Waverley' does not prove
that the name connotes or implies this. It was not be-
cause he wrote *Waverley' that he was called Walter
Scott ; and if it were proved that he bad not written it
PART I.] OF TERMS, 9
he would not cease to be so called, nor would the actual
author acquire a right to the name. The name simply
denotes a person living in a certain time and place (unless
w^e say that it connotes the male sex). It may be asked :
* If the name John Smith does not suggest a certain appear-
ance, &c., how am I to know him when I see him ? ' The
answer is, it may suggest, but that is not the same as con-
noting, any^more than the cart is part of the horse that
draws it. I The name ' earth-worm ' may suggest * Charles
Darwin' (because he wrote a treatise about earth-worms),
but it does not connote it. What is decisive is, that if
another man were found very like my mental picture of
John Smith, I should not call him by that name. So the
name * London ' calls up to my mind the notion * the
largest and richest city in the world.' But this is not con-
noted by the name. The city was called London when it
was not the largest and richest ; it would continue to be so
called although we should discover that Pekin was larger
and richer, and we should not in that case call the latter
city Londoi^J
28. A term which, though not a proper name, applies
only to a single individual, may be connotative. Thus,
* the Author of Waverley ' denotes an individual and con-
notes his attribute. It is not, however, a proper name,
only we happen to know that it can apply only to one
individual.
29. The name of a single attribute is also non-connota-
tive ; a name like * colour/ which applies to several attri-
butes, connotes that in which they agree ; in this case,
causing a certain kind of visual sensation.
10 THE ELEMENTS OF LOGIC, [part i.
Of Contradictory and Contrary Terms,
30. Terms or attributes, of which one is simply the
negative of the other, are called Contradictory, as : wise,
^.not wise.
^^ 31. Contradictory Attributes cannot at the same time
belong to the same object. This is called the Law of
Contradiction.
32. Of two Contradictory Attributes one or other muit
be predicable of every object. This is called the Law of
Excluded Middle ; middle having here the sense of mean.
Between * white ' and * not white ' there is no mean.
33. Terms or attributes which are the most opposed of
those coming under the sagi^ class are called Contrary,
as : wise, foolish ; white, black.
34. Contrary terms cannot at the same time be predi-
cated of the same object.
Of Clearness and Distinctness,
35. A notion, or concept, or meaning of a term, is said
to be clear when we are able to distinguish it from other
notions or concepts. It is distinct when we are able to
distinguish an d enumerate the parts (attributes) of which
v\ it is composed.
36. A notion may be tolerably clear, and yet far from
being distinct. Thus, we sometimes say we know very
well what the thing is, but cannot define it, i, e. we know
of what things it may be predicated. This is the case
with natural objects generally, and with many other objects
of experience. We may have a sufficiently clear notion of
horses and of trees, but might find it impossible to analyze
PARTI.] OF lERMS, '- II
either notion with any correctness. We may know what
is poetry and what is not, but should find it hard to define
what constitutes poetry.
^*^^ 37. The opposite of clear is obscure. The opposite
of distinct is confused. ^^-o-^
^-^/''^^''^y Of Definition. ^.
38. Definition is a statement of the connotation of a
term.
39. The rules of Definition are : —
{a) It must be adequate {adcequata), that is, its jgxten:;
sion must be exactly equal to that of the term defined. If
this rule is violated the definition is either too wide or too
narrow.
40. (b) It must be precise , that is, it must contain
nothing superfluou s. Thus it would be incorrect to define
a parallelogram as a rectilinear quadrilateral whose oppo-
site sides are. equal and parallel ; for the equality follows
from the parallelism. Such a definition would imply that
we could have a rectilinear quadrilateral whose opposite
sides were parallel and not equal, or equal and not pa-
rallel.
This rule might be otherwise expressed by saying that
the attributes enumerated must be distinct.
41. (c) It must not contain any term equivalen t-^to the
term defined ; for instance, we must not define Life as the
Operation of Vital Forces, for ' vital ' means ' of life.'
The violation of this rule constitutes a * circle in defini-
tion.'
42. {d) If possible, it should not be by negative attri-
butes.
12 THE ELEMENTS OF LOGIC, [part i.
43. It is to be observed that definitions of natural ob- ._
jects or kinds gf^thiags, such as we know by experience,
can seldom be complet e. We can only define the name so
as to mark out clearly the thing intended. Natural kinds
of things possess an indefinite number of attributes in.
common, and it is to a certain extent arbitrary which of
these we select as the connotation of the name. Even ^
after we have made this selection, the discovery of a thing 1
diflfering in one or two of these attributes might induce us
to alter the connotation.
44. The definitions actually adopted by naturalists,
botanists, &c., are not based on the ordinary connotation
^of the names, but qn^a^dection.of attributes supposed to
/ I be more fixed or connected with a greater number of other
^ "attributes. ^
45. Descriptio n is an enumeration of attributes of the [
thing described sufiicient to distinguish it from other
^ tljmgs.
^ ^6. Individual things can only be described, not de-
fined. V ',
Of Division.
47. Division is an enumeration of the parts of the Ex-
tension or Denotation of a Term. Thus Animal is divided \
into Vertebrate and Invertebrate.
48. The chief rules of Division are : —
• [a) It must be adequ ate, that is, all the members \
taken together must be equal to the whoje. \
49. (^) It must be distinct ; that is, the members must
be mutually exclusive ; in other words, each must be capa-
ble of being denied universally of the rest. For instance,
PART I.] OF TERMS. 13
we must not divide European into French, German, Prus- «
sian, &c.
50. {c) If possible y there should be only one Principle
of Division {fundamentum divisionis)^ that is, all the
dividing attributes should come under one motion. For
instance, we might divide Book according to contents into
histories, novels, &c., or according to size into folios,
quartos, &c., or according to binding into bound and un-
bound; but it would be incorrect to divide Book into
folios, quartos, novels, &c. If the third rule is observed
the second will be also fulfilled, but it is not always possi-
ble to observe this third rule.
51. Individuals are so called as not being capable of
logical division.
( 14 )
PART SECOND.
OF SIMPLE PROPOSITIONS,
Preliminary Definitions,
52. A Simple Proposition asserts or denies one term
of another term.
Gold is heavy ;
Some planets are bright ;
No planets are self-luminous.
53. We must repeat some definitions :
The term of which something is affirmed or denied is
the Subje ct.
That which is affirmed or denied is the Predicate .
The Predicate is said to be predicated aflarmatively or
negatively of the subject.
The connecting word (is ; is not) is the Copula.
54. The type of the simple proposition is —
A is B ; or, A is not B.
A is the Subject ; B, the Predicate ; ' is ' or ' is not,' the
Copula.
PART II.] OF SIMPLE PROPOSITIONS, 15
55. If we use the symbol AB to signify A that is B, we
may write the proposition Every A is B, thus :
Every A is AB.
Thus if A = crow and B = black, the proposition :
All crows are black
becomes
All crows are black crows.
Of the Copula, "^
56. The only Copula admitted by logicians is the pre-
sent tense of the verb ' to be.' The following remarks
must be attended to : — •
57. {a) All other verbal forms combine predicate and
copula, and for logical purposes must be resolved into
these two elements. Thus :
All planets revolve round the sun
is equivalent to
All planets are revolving (or bodies that revolve)
round the sun.
58. ib) The Copula does not involve the notion of
time, although it has the grammatical form of the present
tense. If the assertion refers specially to past, present, or
future time, the note of time must be put in the predi-
cate. Thtis : Every man was a boy = Every man is what
was formerly a boy. Babylon was once the capital of a
great empire = Babylon is a city once the capital, &c.
59. {c) The Copula does not involve the notion of
existence. Thus: Centaurs are imaginary beings, half
riiah, half horse, is a proposition logically correct.
1 6 THE ELEMENTS OF LOGIC, [part ii.
Of the Quality of Propositions,
60. By the Quality of a proposition {qualitas) is meant
the answer to the question : Of what sort {qualis) is the
predication ? i. e. is it Affirmative or Negative ?
As to duality, then, propositions are divided into
Affirmative and Negative.
61. Every negative proposition may be treated as an
affirmative by affixing the particle ' not ' to the predicate
instead of to the copula-^
A is not B = A is - not-B ;
Ex, gr. Ostriches do not. fly
= Ostriches are creatures that do not fly.
62. Similarly, Affirmatives may be changed into Nega-
tives. Thus:
Worms are invertebrate
= Worms are not vertebrate.
These are called Equipollent Propositions.
Of the Quantity of Propositions.
63. The Quantity of a Proposition is the answer to
the question : What is the extent of the assertion }
In respect to Quantity, then. Propositions are divided
into Universal, Particular, and Singular.
64. A Universal Proposition is one in which the
Subject is a common term taken in its whole exten-
sion :
Every crow is black, or All crows are black ;
No reptiles fly.
In symbols,
Every A is B ;
No A is B,
PART II.] OF SIMPLE PROPOSITIONS. 17
65. A Particular Proposition is one in which the
Subject is a common term taken in part of its exten-
sion :
Some swans are black ;
Some swans are not black.
In symbols,
Some As are B ;
Some As are not B.
66. It must be observed that ' some ' in logic means
* one at least/ and does not mean * some only.' ' Some
As are IIS'' is true if one A is B and if every A is B.
67. A Singular Proposition is one in which the
subject is a Singular, Collective, or Abstract term,
as —
John is tall ;
The English nation is of mixed origin ;
Truthfulness is a virtue.
Cases where a term originally abstract is used as a common .
term are of course not included.
68.. For logical purposes Singular Propositions may
generally be regarded as universal, the subject being
taken in its whole extension (which is only one thing).
69. It is possible, however, to make a Proposition which
has a singular term as its subject, particular, by distin-
guishing different times of the individual's existence.
Ex. gr, * John is not always busy.' Here the real subject
may be considered to be ' the times of John's existence, '
and the assertion is, that some of these times are not busy
times. So with abstract terms : *Law is not always justice'
is equivalent to ' Some things legal are not just.' * Justice
is ever equal ' is equivalent to 'All justice is equal.*
1 8 THE ELEMENTS OF LOGIC, [part ii.
70. Propositions are sometimes expressed indefinitely,
as : Men are cruel to men. But before we can deal with
such a proposition the subject must be ' quantified/ that
is, it must be specified whether the assertion is to be
understood of all men or of some. This can be done only
by reference to the 'matter;' and not by any rules of
logic. In the absence of further information we must
assume that such propositions are particular. ^
71. Since propositions are twofold as to quality and
twofold as to quantity, we have four kinds of propositions,
and these are symbolically represented as follows ; —
A universal affirmative is called A.
A universal negative „ E.
A particular affirmative ,, I.
A particular negative „ O.
(The vowels A and I were chosen as the first two of
* affirmo ; ' E and O as the vowels of ' nego.')
72. It must be observed that by the idiom of the English
language * Every A is not B,' or ' All As are not B,' means
* Not every A is B,' i, e, ' Some As are not B.*
Of the Distribtction of the Terms of a Proposition.
73. A common term is said to be distributed when it
is tak^p^in^its _w]iol^_jjxtension ; otherwise it is undis-
tributed.
74. The quantity of a proposition has been already de-
fined to be the quantity of its subject, i, e. a universal
proposition is one whose subject is distributed, and a
particular proposition is one whose subject is undistri-
buted. The quantity of the subject and the quantity of
the proposition are therefore one and the same.
PART II.] OF SIMPLE PROPOSITIONS, 19
75. The quantity of the predicate depends on the
quality of the proposition.
The predicate of an affirmative proposition is undis-
tributed (=is particular).
The predicate of a negative proposition is distributed
(=is universal).
76. First, when we say ^ Crows (all or some) are black,'
we do not imply that other things may not be black also.
Hence the predicate * black ' is not taken in its whole
extension. We merely assert that crows are reckoned
amongst black things ; i. e. the extension of * crow ' is
part (at least) of the extension of * black.' The same
may be said of every affirmative proposition. * A is B '
does not mean that A is the only thing that is B.
77. Hence the predicate of every affirmative proposition
is particular (= is undistributed). It may happen that the
extension of the predicate does not exceed 4hat of the
subject, as when we say ' All equilateral triangles are equi-
angular,' and it is also true that all equiangular triangles
are equilateral. But this is accidental, and was not con-
tained in the former proposition.
78. Secondly, when I say * No crows are white ' I assert
that no crow is any white thing ; i. e. no white thing is
identified with a crow. Therefore * white ' is taken in its
whole extension.
79. So if the proposition is a particular negative : * Some
swans are not white,' i. e. are not any white thing, I assert
that some swans are excluded from the whole extension of
white things.
80. Hence the predicate of every negative proposition
is distributed (= is universal).
20 THE ELEMENTS OF LOGIC, [part ii.
8 1. Using the symbols introduced in the last section: —
A has its subject universal and predicate p artic ular.
E has both, subject and predicate universal.
I has both subject and predicate particular.
has its subject particular and predicate universal.
Of the Qua7itification of the Predicate,
82. Some logicians maintain that in every proposition
we in thought assign a definite quantity to the 'predicate,
and therefore ought to do so in words when we wish to
make the proposition logically complete. They hold that
when we say * All crows are black/ we mean * All crows
are some black things;* and when we say 'Equilateral
triangles are equiangular/ we mean * All equilateral tri-
angles are all equiangular triangles.'
83. On this view, then, affirmative propositions may
have a universal predicate, and negative propositions may
have a particular predicate, as : (All or some) A is not
some ;(certain) B. This is called * quantifying ' the pre-
dicate.
84. Every affirmative proposition is on this doctrine
viewed as an equation in extension ; every negative as an
inequality ; and there are, of course, eight forms of pro-
positions, instead of four, A E I O being each subdivided
into two, according to the quantity of the predicate.
85. It deserves to be noticed that in these forms the
words * all,' * some,' are used in a modified sense. *A11'
does not mean 'every,' but 'all together,' i,e. it is not
taken distributively, but collectively. ' Some ' is also taken
collectively and definitely, as * some certain ' (= ' quidam').
86. In common language we sometimes quantify the
predicate; ex.gr, by using such words as 'alone,' 'only,'
' constitute,' ' consist of,' &c. Thus :
PART II.] OF SIMPLE PROPOSITIONS 1 1
The virtuous alone are truly happy = A alone is B = A is
allB.
Dead languages are not the only ones worth studying
= A is not some B.
A caustic is a kind of curve = A is some B.
There are patriots and patriots = There are different
kinds of patrTots = Some A is not some A.
87. Some of these propositions would be regarded on
the common theory as compound. For instance, 'All A
is all B ^ would be regarded as including two assertions :
* All A is B ' and ' All B is A/ ' The virtuous alone are
truly happy ' means * The virtuous are happy, and the not
virtuous are not happy.'
Of the Import of Propositions,
88. Although affirmative propositions have all been re-
duced to the type (Every or Some) A is B, it does not
follow that the relation |between A and B is in all cases
the same.
The distinction most important to notice is, that some
propositions express complete coincidence or equality ;
others partial coincidence.
89. Coincidence is expressed in the following cases : —
{a) When the subject and predicate are both singular,
both collective, or both abstract terms (used in their
proper sense, i, e. not figuratively). Both singular, as :
Victoria is Queen of Great Britain and Ireland. Both
collective, as : The House of Lords is the Hereditary
Chamber. Both abstract, in which case they must be
synonymous : Freedom is Liberty.
(b) When the predicate is a definition ; Water is H2O.
{c) In mathematical propositions of quantity : The square
of four is sixteen : The line A is equal to the line B ; i» e.
The magnitude of A is the magnitude of B.
22 TRE ELEMENTS OF LOGIC, [part ii.
90. In other cases what is implied in affirmative propo-
sitions is partial coincidence : the attributes connoted by
the predicate are asserted to co-exist with those connoted
by the subject, and hence the extension of the subject is
contained under the extension of the predicate ; in other
words, coincides with part of it, ex. gr. Whales are mam-
malia ; Whales are of prodigfou^ size.
9 1 . Where the predicate is an adjective, the proposition
usually asserts merely the possession of an attribute :
where it is a substantive, the proposition asserts that the
subject belongs to a certain class. But this is only an
accident of language, and does not necessitate any logical
distinction.
92. In general, therefore, in affirmative propositions,
the extension of the subject coincides with part, or the
whole, of the extension of the predicate.
93. N. B. — If A and B each connote onjy one attribute,
or definite group of attributes, then what is expressed is
usually not mere co-existence, but dependence ; i. e. the
attribute connoted by B depends on that connoted by A ;
ex, gr, : \
The diligent is always successful (= diligence causes
success).
The virtuous is happy (= virtue brings happiness).
93. This relation is often expressed by using the ab-
stract names of the attributes as subject and predicate;
ex. gr. :
Knowledge is power ;
Virtue is happiness ;
Honesty is the best policy.
94. ' Must be ' is often used to express the dependence
/^^^ o^THi 'r^
\: TNIVERSITY ))
PART II.] OF SIMPLE PROPOSITIONS?^}^fOn^23
of the connotation of the subject on that of the predicate
as a condition :
A historian must be laborious ;
This patient {i.e. a patient with such symptoms)
must have been exposed to infection.
95. Propositions expressing Real Existence are of suf-
ficient importance to be classified separately in any com-
plete analysis of the import of propositions. There are,
however, but few of them that cannot be reduced to other
heads. They are such as — God exists ; the world exists ;
the soul exists.
0/ Verbal and Real, or Analytical and Synthetical y
Propositions,
96. An Analytical or Verbal Proposition is one whose
predicate is already contained in the connotation of the
subject (as part or the whole of it), as : iron is a metal ;
body is extended ; a triangle is a three- sided figure.
These only state explicitly what is involved implicitly
in the idea of the subject. They are called analytical
because they analyze the idea of the subject ; and verbal. %
because they may be regarded as being only about the
meaning of words.
97. A Synthetical or Real Proposition is one whose
predicate is not contained in the connotation of the sub-
ject ; ex.gr. iron is magnetic; all bodies are heavy ; a
triangle has its three angles together equal to two right
angles. These pr opositi ons join on something to the con-
notation of the subject, and hence they are called synthe-
tical. They are called real, as stating facts about things,
not about words only.
24 THE ELEMENTS OF LOGIC, [part ii.
98. It must be observed that the circumstance that these
propositions are, perhaps, familiar to those who know what
the subject is, ex, gr. as in the above instances, iron, tri-
angle, &c., does not constitute them analytical. For the
statement, * iron is magnetic/ is a statement of a fact in
nature which could not be discovered by analyzing our
ideas. It asserts that whatever possesses the other quali-
ties connoted by * iron ' possesses this attribute also. There
is, however, a certain latitude or indefiniteness as to the
qualities which we shall select to be connoted by the term
' iron,' or any other name of a kind of things ; and there
is implied in all propositions concerning such kinds an
assertion that a thing possessing such attributes really
exists.
99. All statements of which the subject is a proper name
are synthetical, as appears from what has been said about
such terms, viz. that they have no connotation. * London
is a great city ' is a statement of fact, and could not be
elicited from an analysis of the meaning of the word
* London.*
0/ Copulative Propositions.
100. Copulative propositions consist of several proposi-
tions asserted together. For instance : ^. A is both B and
C ; 3. A is neither B nor C ; c. Neither A is B nor C is D;
d, A is B, but C is D.
These are resolved into separate propositions connected'
by a copulative particle : ^. A is B and A is C ; 3. A is not
B and A is not C ; f . A is not B and C is not D ; t/. A is B
and C is D.
loi. Where two affirmative propositions are joined by
PART II.] OF SIMPLE PROPOSITIONS. 25
an adversative particle (although, but, yet, &c.), some
opposition is implied ; for example, * This book is learned
but interesting.' Here it is implied that most learned
books are not interesting. Again, * This task is difficult
but necessary.' Here the implied opposition is not between
difficulty and necessity, but between the understood infer-
ences from them, viz., that the thing should be done, and
that the thing should not be done. The implied proposi-
tion is then, * What is difficult is usually to be avoided,' or
the like. In general the adversative particle implies that
the propositions connected are not expected to be true
together.
102. When the first proposition is negative, the opposi-,
tion is between its predicate and that of the affirmative,
as, * He is not industrious but idle,' so that the two pro-
positions may even be synonymous.
103. An affirmative and a negative assertion may be
united in what is apparently a single statement, ex. gr.
Few men are born poets = Some men are born poets, and
Most men are not born poets.
The remarks in this section belong more properly to
grammar than to logic.
O
Of Ma^al Propositions,
104. Mjfdal propositions are those in which the asser-
tion is qualified by such words as * possible,' ' impossible,'
probable,' * certain,' * necessary,' ^contingent,' * may be,'
* must be.'
105. Of these, 'certain,' 'impossible,' 'necessary,' 'must
be,' are usually only indications that the proposition is
universal.
G
26 THE ELEMENTS OF LOGIC, [part ii.
' May,' when the subject is a common term, indicates
that the proposition is not universal, ex. gr. The best de-
vised plans may fail.
1 06. In cases not capable of this explanation the word
expressing the mode should be made the predicate, and the
proposition the subject, ex. gr. The harvest will probably
be good = That the harvest will be good is probable. These
propositions, therefore, are no exceptions to the rule that
the copula can only be ' is ' or * is not.'
Appendix to Part II.
Of the Predicahles.
107. The notion of the predicate may be related to that
of the subject in one of four ways, viz., as : Genus, Defini-
tion, Property, or Accident.
108. We have already divided propositions into verbal
(in which the connotation of the predicate is contained in
that of the subject) and real (in which it is not so con-
tained). In the former case the connotation of the predi-
cate may be either the whole connotation of the subject or
part of it. If the whole, the predicate is a Definition ; if
part only, it is a G-enus.
109. When the connotation of the predicate is not con-
tained in that of the subject, its extension or denotation
may be the same as that of the subject, or not the same.
If the extensions are the same, the predicate is a Property
{proprhwi), i. e. an attribute which belongs to the subject
alone : if they are not the same, the predicate is an Acci-
dent.
PART II.] OF THE PREDICABLES. 27
1 10. This is Aristotle's doctrine. Later logicians, influ-
enced by certain metaphysical theories, enumerated five
predicables : Genus, Species, Difference, Property, and
Accident. They regarded genera and species as having
their essences fixed by nature. The essence of the species
was expressed by the definition ; and that which was in
the essence of the species over and above the essence of
the genus they called the Difference, or Essential Differ-
ence. As * genus ' is now understood, there is no logical
distinction between it and ^ difference ' ; each is part of
the connotation of the subject.
Of the Ten Categories.
111. By Category Aristotle meant predication. The
Greek word KaTrjyop€Lv meant (in logic) 'to predicate.' The
Ten Categories were a classification of predicates of an in-
dividual thing, according to their own signification (not in
their relation to the idea of the subject as the five predi-
cables). We can say of a thing, what it is, or what sort, or
how great, or when, where, &c. These are all different
ways of filling up the blank in, * This is .' Aristotle's
enumeration of these possible predicates were as follows :
112. I. Ovo-La, — The 'whatness' of a thing, or what it
is named, called by Latin logicians 'sub-
stantia,' * substance,* as man, horse.
♦"2. TToo-ov. — Quantity, as *Two feet high.'
vj. TTotov (of what sort). — Quality, as 'white.'
. 4. Trpos Tt. — Relation (more correctly Comparison),
as ' double,' ' greater,' * less.'
5. TTouLv. — Action, as 'cuts,' 'burns.'
6. irdo-x^Lv. — Passion, as ' is being cut or burned.'
7. TTov. — Ubi, as ' here.'
8. TTore.— Quando, as 'to-day,' 'yesterday.'
9. K^la-Oai. — Situs, as ' is sitting,' ' is standing.'
10. cx^iv. — Habitus, as 'is armed.' ' Habitus' was
an incorrect Latin imitation of the Greek
€xeiv, which, used intransitively, meant to be
in a certain state.
c 2
28 THE ELEMENTS OF LOGIC, [part ii.
113. These may be arranged thus : — The predicate may
be — a, a substantive ; h, an adjective ; c, an adverb ; d, a
verb. ^If a substantive, it is a name of the kind of thing;
if an adjective, it may be one of quantity, quality, or com-
parison ; if an adverb, it may be of time or place (no
others are admissible as predicates) ; if a verb, it may be
active, passive, or neuter, or may express the result of an
action (as the Greek perfect passive).*
* The student may naturally ask how it is that adverbs of place and
time are admitted as predicates when no others can be used as such.
The explanation is that these adverbs imply existence. * Is here and
now ' means, * Is existing here and now.' What is predicated in such
cases is, therefore, Real Existence.
( 29 )
PART THIRD.
OF INFERENCES.
CHAPTER I.
1
Of Immediate Inferences,
1 14. Inference is the process by which one proposition
is derived from one or more others.
Immediate Inference is the deduction of one proposi-
tion from another, which virtually contains it.
T15. The kinds of Immediate Inference are: Subalter-
nation, Opposition, and Conversion.
Of Suhalternation,
116. Inference by Subalternation is the inference of
the Particular from the Universal. Ah universali ad par-
ticulare valet consequentia.
From the proposition, All men are mortal, we may infer,
Some men are mortal ; This man is mortal. The particular
is called the Subalterna ; the universal is the Subalternans.
117. The Subalterna of A is I ; that of E is O. The
rules of Subalternation are : —
(i). From the truth of the universal we may infer the
truth of the particular.
30 THE ELEMENTS OF LOGIC, [part hi.
(2). From the truth of the particular we cannot infer
the truth of the universal. A particulari ad universale non
valet consequentia, i. e. from a term taken particularly we
can infer nothing as to the same term taken universally.
(3). From the falsehood of the particular we may infer
the falsehood of the universal. This is a consequence of
the first rule.
(4). From the -falsehood of the universal we^cannot infer
the falsehood of the particular. This is a consequence of
the second rule.
118. From the proposition, Every A is B, I can infer,
Some A is B ; and conversely, if it is false that Some A is
B, it must be false that All A is B. But from Some A is
B we cannot infer that Every A is B ; and conversely, if it
is false that every A is B, it does not follow that Some A
is B is also false.
Of Opposition.
119. Inference by Opposition is the inference from the
truth or falsity of one proposition to the falsity or truth
of another having the same terms as subject and predi-
cate, but differing in quality.
120. There are three cases : — Contradictory Opposition
or Contradiction is between propositions each of which
asserts only what the other denies.,
121. If the subject is a singular term, the contradicto-
ries differ only in quality, as :
James is clever,
James is not clever ;
James is standing,
James is not standing.
PART III.] OF INFERENCES, 3 1
Here the predicates are contradictory, for we may regard
* not ' as belonging to the predicate ; therefore, these
come directly under the Law of Contradiction. This is
the type of Contradictory Opposition.
122. If the subject is a Common term, the Contradic-
tory propositions differ both in quantity and quality.
' Some men are black ' would not be contradicted by
' Some men are not black '; for the * some men ' may be
different, and then the propositions would not even be
inconsistent.
Now, since the ^ some men ' are not specified, it is clear
that, in order that ' black ' may be affirmed and denied of
the same man, we must deny ' black' of every man. Hence
the contradictory of, * Some men are black ' is, * No men
are black.'
123. Again, if we have a universal, ' Every man is black,'
the contradictory of this is not, 'No man is black'; for
this not only denies the former, but denies much more.
In fact these two propositions together assert and deny
* black ' of each Separate individual, and include, there-
fore, as many distinct contradictions. In order that we
may assert no more than is necessary for the denial of the
given proposition, it is only necessary to state that there
is an exception to it ; ?'. e. * Some men (one at least) are not
black.'
124. Hence contradictory propositions are — A and O,
or E and I.
N.B. — When a proposition with a singular subject is
qualified by ^ always ' or the like, its contradictory must
have some word equivalent to ' not always '; ex, gr, :
John is always busy ;
John is sometimes not busy.
32 THE ELEMENTS OF LOGIC, [part hi.
125. Of contradictories, both cannot be true, and both
, cannot be false ; hen'ce the rule of inference is : —
From the truth of either contradictory we may infer
the falsity of the other.
And vice versa :
From the falsity of either contradictory we may infer
the truth of the other.
126. Contrary Opposition is between propositions the
most remote of those having the same subject and the^
same predicate ; t. e. between a Universal Negative and
a Universal Affirmative ; ex. gr. :
All men are black-;
No men are black.
These are the most remote from each other : for, as just
observed, they include as many contradictions as there
are individuals denoted by the subject. It is obvious that
they cannot both be true. But they may both be false ;
for each asserts more than the mere negation of the other,
and this superfluous assertion may be false.
127. Thus it may be false that * All men are black,' and
therefore true that * Some men are not black.' But the
contrary asserts more than this, and the additional asser-
tion is false.
128. Hence contraries cannot both be true, but both
may be false, and the rule of inference is :
From the truth of one contrary we may infer the falsity
of the other ;
but.
From the falsity of one contrary we cannot infer the
truth of the other.
PART III.] OF INFERENCES, 33
129. These rules may be deduced from those of sub-
alternation and contradiction :
If A is true, I is true ; therefore E false.
If A is false, O is true ; thence^ :rio inference as to E.
If E is true, O is true ; therefore A false.
If E is false, I is true ; thence no inference as to A.
J 30. Subcontrary propositions are those of whicli one
affirms particularly what the other denies particularly.
They must be I and O :
Some men are black ;
Some men are not black,
are subcontraries.
131. Both may be true ; both cannot be false. For if it
is false that, * Some men are black,' its contradictory, * No
men are black,' is true ; and therefore its subalterna, * Some
men are not black,' is true. In general, if I is false, its
contradictory E is true, and therefore O is true. If I is
true, E is false, and no inference can be drawn as to O.
132. Hence the rule of inference is :
but,
From the falsity of one subcontrary we may infer
the truth of the other ;
From the truth of one subcontrary we can draw
no inference as to the other.
133. Subcontrariety is not properly called opposition,
since the subjects of the subcontraries are not the same ;
the ' some' of one not being identified with the * some* of
the other. It is therefore called apparent opposition. —
c 3
34
THE ELEMENTS OF LOGIC, [part hi.
1 34. The relations of subalternation and opposition may
be exhibited in a diagram :
A Contrariety E
y
y
/
/ 1
Q>
y
/
Subcontrariety
O
Of Conversion.
135. Conversion is the transposition of the subject
and predicate, so that the subject becomes predicate and
the predicate becomes subject.
The original is called the Convertend.
The derived is called the Converse.
136. Conversion is Logical when the Converse follows
from the Convertend.
137. Conversion is of three kinds ; simple, per accidens,
and by Contraposition.
138. Conversion is simple when the quantity of the Con-
verse is the same as the quantity of the Convertend, as :
Some men are black (I) ;
Some black things are men (I).
139. Conversion is per accidens when the Convertend
is universal and the Converse is particular, as :
All m.en are mortal (A) ;
Some mortal things are men (I).
PART III.] OF INFERENCES. 35
140. I is converted simply (into I) :
From the proposition, ' Some lawyers are historians/ we
may infer, ' Some historians are lawyers/ * Some As are
B ' is converted to ^ Some Bs are A/ Both statements
are equivalent to, * Some things are both A and B (or are
AB').
141. E is also converted simply (into E) :
From the proposition, 'No savages are philosophers/
we may infer, * No philosophers are savages.' * No A is
B' is converted to * No B is A.' Both statements express
the same thing, viz.: 'Nothing is at once A and B (or is
AB'), or the extension of A and that of B are mutually
exclusive.
142. A cannot be converted simply, but is converted per
accidens into I :
From the proposition, * Every man is mortal/ we cannot
infer, * Every mortal thing is a man.'
143. It has already been shown that the predicate of an
affirmative proposition is particular ; thus, in the proposi-
tion, * All men are mortal,' * mortal ' is taken in part of its
extension ; * Men are amongst mortal things.' Hence when
'mortal' becomes the~iubject it is still particular, other-
wise we should be reasoning from the particular to the
universal, and therefore the proposition of which it is the
subject is particular, ' Some mortal things are men.'
144. If we regard the mark of quantity, 'every,' 'all,'
' some,' as belonging to the subject, as it really does, we
shall see that it is the subject of the original proposition
that in becoming predicate loses its universality, and that,
because we do not express the quantity of the predicate.
145. If we adopt the view that the predicate of an affir-
36 THE ELEMENTS OF LOGIC, [part hi.
mative proposition may be quantified, we shall be able to
convert A simply. ' All men are mortal ' becomes ' Some
mortal things are all men,' or, in common language, ' Only
mortal things are men.'*
146. O cannot be converted while it retains its quality,
for its subject is particular ; and if made the predicate of the
converse (which would be negative), it must needs be
universal. We should thus be reasoning from part of the
extension of a term to the whole {a particulari ad univer-
sale), which is invalid. Thus, from * Some men are not
learned ' we cannot infer * Some learned beings are not
men,' nor any other proposition of which ' some learned '
is the subject. For nothing whatever has been asserted
about learned beings.f
147. In order to convert O, we must therefore treat it
as I, which we have seen we can always do by attaching
the negative particle to the predicate (62). We then con-
vert it by Contraposition.
Of Contraposition,
148, Contraposition or Conversion by Contraposition
consists in substituting for predicate and copula their
contradictories and then converting.
* It would usefully simplify the doctrine of conversion and connected
processes if quantification of the predicate were admitted in this case
at least, in which the Convertend logically leads to a universal predi-
cate in the Converse.
t Logicians who adopt the * quantification of the predicate ' convert
O thus : ''Some As are not B ' is converted to ' No B is some A';
ex, gr, ' Some men are not learned ' is converted to ' No learned
beings are some men.'
PART III.] OF INFERENCES, 37
149. A may be contraposed simply, for its equipollent
is E.
Every A is B
is equipollent with
No A is not B ;
and this is converted to
No not-B is A = Whatever is not B is not A.
150 O may also be contraposed simply, since it is equi-
pollent with I.
Some As are not B
is converted into
Some things not B are A.
Thus:
Some men of genius are not learned
becomes
Some unlearned men are men of genius.
151. E by equipollence is A, and therefore is only con-
traposed per accidens.
No A is B
becomes
Some things not B are A.
Thus:
No idler is successful
becomes
Some of the unsuccessful are idlers.
152. I by equipollence becomes O, and therefore is not
contraposed.
38 THE ELEMENTS OF LOGIC, [part hi.
Remarks on Conversion.
153. {a) A universal affirmation sometimes seems as if it
were convertible simply (A to A). All equilateral triangles
are equiangular, and, All equiangular triangles are equila-
teral. But the latter proposition is not inferred from the
form of the first, but from the matter, i, e. from something
we know, but which is not expressed.
154. There are, however, some cases, as shown above
(86), in which the predicate is expressly made universal by
the words * only,' 'alone,' etc. ; also where the proposition
takes one of the forms that express identity (89), as :
A is equal to B.
155. {b) In the case of propositions indicating depen-
dence of one attribute on another (93), the meaning of
the Convertend and that of the Converse are not identical;
ex.gr. * Virtue is happiness' is not the same as * Happiness
is virtue '; * Whatever is right is alone expedient' (or, * All
the right is all the expedient') is different from 'Whatever
is expedient is alone right ; (or, ' All the expedient is all
the right ') ; or, * Whatever is not expedient is not right.'
All these propositions affirm that right and expediency go
together ; but the question is, by which quality are we to
determine the presence of the other. One proposition
states that expediency is the test of right, the other (the
first) states that right is the test of expediency. No error
arises from this difference, because it is usually clear which
of the two attributes is supposed to be most easily recog-
nized.
156. {c) 'Converse' and 'conversely' are often used
with reference to two propositions in which, instead of
PART. III.] OF INFERENCES. 39
the Copula we have a word (or words) expressing a more
complex relation, and this word remains while the related
terms change places ; ex. gr. ^ The truth of either contra-
dictory follows from the falsity of the other, and con-
versely, the falsity of either follows from the truth of the
other.' The student must guard against confounding this
with logical conversion. It would be more correct in
such cases to use the term * vice versd.^
Of the Principle of Substitution.
157. There are some immediate inferences which, al-
though obviously valid, and of frequent occurrence, pre-
sent some difficulty when we try to bring them under any
of the preceding heads. For example : * A negro is a
man ; therefore, a sick negro is a sick man.* Or, again:
^ A horse is a sensitive animal ; therefore, he who tortures
a horse tortures a sensitive animal.'
158. The principle which is employed in these cases is
an extension of the principle of Subalternation. It is
this :
For any term used universally {i,e. distributively)
less may substituted.
Thus, wherever the name of the genus is used universally
the name of the species may be substituted. This is
obviously implied by the distribution of the term. What-
ever is true of '• every' is true of each.
159. Accordingly, in the identical proposition, *A sick
man is a sick man,' we may substitute for * man ' in the
subject, * negro,' which is part of its extension. Similarly,
in the identical proposition, ' He who tortures a sensitive
40 THE ELEMENTS OF LOGIC, [part hi.
animal tortures a sensitive animal/ we substitute for * sen-
sitive animal ' in the subject, where it is distributed, the
term ' horse,' which is part of its extension.
1 60. On the other hand, it is a principle that —
* For any term used particularly we may substitute
a term of wider extension.'
For it is clear that whatever is part of the less extension
is part of the greater,
161. In both cases of substitution it is of course as-
sumed that the words employed have a definite significa-
tion, which is the same whatever the connexion in which
they occur. This is not the case with words implying
comparison. Thus : from * a cottage is a house ' we can-
not infer * a huge cottage is a huge house ' ; or, again :
from ' a tailor is a man ' we cannot infer that * the best
of tailors is the best of men.'
CHAPTER II.
Of Mediate Reasonings or Syllogism,
162. Mediate Reasoning is the deduction of one propo-
sition from two or more. All mediate reasoning may be
reduced to Syllogism, which is the inference of one pro-
position from two, in which, taken together, it is by the
mere form of expression necessarily involved. Ex. gr. :
Logic is a Science ; ^
All Sciencels are worthy of study ;
therefore,
Logic is worthy of study.
PART III.] OF INFERENCES, 41
Here the reasoning does not depend on the meaning
of the words, but is equally valid if we substitute sym-
bols: 5^ ^ ly^-^^ • ^
^S is M,
Every M is P;
therefore, S is P.
163. The conclusion before it is proved is called the
duestion. The propositions from which it is deduced
are the Premisses {propositiones praemissae). The Subject
of the Conclusion is called the Minor Term. Its predicate
is the Major Term. Both are called Extremes.
164. Syllogisms are either simple or complex. Simple
syllogisms (otherwise called categorical) consist of simple
propositions, i. e, such as assert absolutely, that is, without
a condition.
165. In simple syllogisms, the coincidence or non-
coincidence of the extremes is ascertained by comparing
them with a third term, which is called the Middle
Term.
166. The premiss in which the major term occurs is
called the Major Premiss ; that in which the minor term
occurs is the Minor Premiss. The order of the premisses
is of no consequence ; but logicians commonly state the
major first.
167. In representing syllogisms symbolically it is usual
to employ S. (the initial of Subject) for the minor term,
which is Subject of the conclusion ; P for the major term,
which is Predicate of the conclusion, and M for the
Middle. The conclusion, therefore, always appears as
SP. The symbol .*. is used for * therefore ' (*.• is some-
times used for * because ').
42 THE ELEMENTS OF LOGIC, [part hi.
General Rules of Syllogism,
1 68. The general rules are those which are applicable
to all syllogisms : —
(i.) The middle term must be at least once universal;
in other words, it must be distributed.
For if it were undistributed (/. e. not taken universally),
it might be taken in two different parts of its extension,
and there would be really two middle terms. Thus in :
All men are animals ; All horses are animals : the parts of
the extension of ' animals * are different, and therefore we
can deduce nothing as to the relation between *men' and
* horses.'
If this rule is violated, we have the fallacy of * Un dis-_
tribute d Middle.'
169. (2) An extreme must not be taken more universally
in the conclusion than in the premisses ; in other words,
if a term is universal in the conclusion, it must have been
universal in the premisses.
For otherwise we should be arguing from the part to
the whole, or as it is called, a particidari ad imiversale.
The violation of this rule is called Illicit Process, which
may be either of the major or the minor.
170. Cor, Hence and from the preceding rule it appears
that there must be at least one more universal term in the
premisses than in the conclusion. For any term which is
universal in the conclusion must have been universal in
the premisses, and the middle term, in addition, must be
once universal.
171. (3) From two negative premisses nothing follows.
For if nothing is asserted in the premisses except that
PART III.] OF INFERENCES. 43
the extremes are both excluded from the middle term,
nothing has been implied as to their relation to each
other.
172. (4) From two affirmative premisses a negative con-
clusion cannot follow.
For if what we assert of each of the extremes is either
that it contains or is contained by the same middle (and
therefore partially coincides with it), it is not thereby im-
plied that either extreme excludes the other.
173. (5) If either premiss is negative, the conclusion is
negative.
For when what is asserted is only that one extreme
either contains or is contained by the middle, which is
excluded from the other extreme, it is not thereby implied
that either extreme contains the other ; hence an affirma-
tive conclusion cannot follow.
174. (6) From two particulars nothing follows.
They must be either II or 01 or 10. In II all the terms
are particular ; therefore the middle is undistributed, con-
trary to the first rule.
In 01 or 10 there is only one universal term, viz. the
predicate of O, which must be the middle term ; there-
fore there is no universal term in the conclusion. But as
one premiss is negative, the conclusion, if any, would be
negative, and therefore the major term in it universal.
The major term, therefore would be particular in the pre-
miss and universal in the conclusion, contrary to the second
rule.
'"""^75- (7) If either premiss is particular, the conclusion is
particular.
First Case : If both premisses are affirmative, they are
A and I. Here there is only one universal term, viz. the
44 THE ELEMENTS OF LOGIC. [part hi.
subject of A. This must be the middle term ; therefore
there can be no universal term in the conclusion, which
must be I.
Second Case : One premiss is negative. In this case
there are only two universal terms, viz. the predicate of
the negative and the subject of the universal ; therefore
there is but one in the conclusion (Ct?;'.). But as the con-
clusion must be negative, its predicate will be universal
and its subject particular, and this makes the conclusion
particular.
176. This rule and the fifth are sometimes expressed
together, thus : The conclusion follows the weaker part,
the negative being considered weaker than the affirmative,
and the particular weaker than the universal. ^ —
177. Although we cannot draw any conclusion in exten-
sion from premisses with undistributed middle, yet we may
sometimes draw an inference in Attribution. An example
will show what this means : * Men have eyes ; insects have
eyes ; therefore men have some attributes in common with
insects.' Here we reason not about the classes men and
insects having or not having individuals in common, but
about the groups of attributes. The argument may be
brought into the usual form by making the attribute the
subject of both propositions, thus : To have eyes is an attri-
bute of men ; to have eyes is an attribute of insects ; there-
. fore some attribute of men is an attribute of insects.
Of the Four Figures,
I 178. The Figure of Syllogism is the disposition of the
middle term in the premisses.
179. In the first figure the middle is subject of the major
premiss, and predicate of the minor. In this case the ex-
PART III.] OF INFERENCES. 45
tremes have the same position in the conclusion as in the
premisses,
S is M ;
M is P ;
/. S is P.
Ex, gr, Lo3ic is a Science ; ^
All Sci6iices are worthy of study ;
.*. Logic is worthy of study.
S 7D
180. The principle of this figure is: Whatever is uni-
versally predicated (affirmatively or negatively) of a com-
mon term may be similarly (J.e, affirmatively or negatively)
predicated of anything contained under that term. It may
be expressed otherwise, thus : Whatever comes under the
condition of a rule comes under the rule.
Every M is P (or not P) ;
S is M V V ^
.-.Sis P (or not P).
* 1 8 1 . In the second figure the middle is predicate of both
premisses. In this case the major term is differently placed
in the premiss and in the conclusion :
No fish breaj^he in air ;
Whiles breathe in air ;
.'.Whales are not fish.
All fish breathe in water ;
Whales do not breathe in water ;
.-.Whales are not fish.
182. The principle of this figure is : If an attribute is
Or,
46 THE ELEMENTS OF LOGIC, [part hi.
predicated affirmatively or negatively of every member of a
class, then any subject of which it cannot be so predicated
does not belong to that class :
Every P is M (or not M) ;
S is not M (or is M) ;
.-.S is not P.
183. In the third figure, the middle term is subject of
both premisses. In this case the minor term occupies a
different place in the premiss and in the conclusion ;
ex. gr, :
.Bats fly ;
HBats are not birds ;
C.'.Some things that fly are not birds.
s ^
184. The principle of this figure is: If anything which
belongs to a certain class possesses a certain attribute
(positive or negative), then that attribute is not incompa-
tible with the attributes of that class :
MisS;
M is P (or not P)
.'.Some S is P (or not P).
(This represents M as singular ; if it is a common term, of
course, in one premiss we must have 'Every M.')
185. In the fourth figure the middle term is predicate of
the major premiss, and subject of the minor. In this case
both extremes are differently placed in the premisses and
in the conclusions ; ex. gr. :
Every P is M ;
No M is S ;
.'.Seme S is Hot P.
PART III.] OF INFFjRFNCES. 47
1 86. The diagrams of the four figures are :
Fir^t. Second. Third. Fourth.
ib*< Mgr^; , . PM^ls^^W^T^E. PM
,j^^' SM fcv^oc-^^SM-m^^'; "Us'' MS
>r..- sF^^-r^-v^-SP ^ SP
Special Rules of the Four Figures,
187. Special Rules are those which are applicable only
to particular figures.
188. The Special Rules of the First Figure are:
Q / .
(i) The minor_must be affirmative. X^' -
(2) The liiajor must be universal. '
189. (i) The minor must be affirmative. For if it were
negative, the major must be affirmative and its predicate
particular ; and the conclusion would be negative and its
predicate universal ; but the predicate of the major pre-
miss and the predicate of the conclusion are the same,
viz. the major term, which would thus be particular in
the premiss and universal in the conclusion, contrary to
the second general rule.
190. (2) The major must be universal. For, since the
minor is affirmative, its predicate, which is the middle
term, is particular; it must, therefore, be universal in the
major premiss, where it is the subject, and makes that
proposition universal.
191. Note. — These rules are manifest at once from the
diagram :
MP
SM
sp"
48 THE ELEMENTS OF LOGIC, [part hi.
If P is universal in the conclusion, it must have been
universal in the premiss, i, e. if the conclusion is negative,
it was the major premiss that was negative. The second
rule is seen at once from the position of M. In this figure
the major is the general principle, and the minor brings a
case under the condition of the rule.
192. The Special Rules of the Second Figure are :
( 1 ) One of the premisses (and therefore the con-
clusion) must be negative.
(2) The major premiss must be universal.
'93- (0 One of the premisses must be negative. For
the middle term is predicate of both ; and if both were
affirmative, the middle would be undistributed.
194. (2) The major must be universal. For, since one
conclusion is negative its predicate is universal ; there-
fore, it must have been universal in the major premiss,
where it is subject, and makes that premiss universal.
195. Note. — These rules are manifest from the dia-
gram :
PM
SM
SP
196. The Special Rules of the Third Figure are :
(i) The minor must be affirmative.
(2) The conclusion must be particular.
197. (i) The minor must be affirmative. This is proved
as in the first figure, the position of the major term being
the same in these two figures.
198. (2) The conclusion is particular. For the minor
PART III.] OF INFERENCES. 49
being affirmative, its predicate, which is the minor- term,
is particular ; it is therefore particular in the conclusion,
of which it is subject, and therefore the conclusion is par-
ticular.
Note. — These rules will be seen at once from the dia-
gram:
MP
MS
sF
200. The Special Rules of the Fourth Figure are :
(i) Wlien the major is affirmative, the minor is
universal.
(2) When the minor is affirmative, the conclusion
is particular.
(3) In negative moods (?*. e. when the conclusion
is negative), the major is universal.
201. (i) When the major is affirmative the minor is uni-
versal. For if the major is affirmative, its predicate, which
,is the middle term, is particular; therefore it must be,^
universal in the minor premiss, where it is subject, and
makes that premiss universal.
202. (2) When the minor is affirmative the conclusion
is particular. For the predicate of the minor premiss is
subject of the conclusion, and if particular in the premiss
{i. e, if the minor is affirmative), it is particular in the con-
clusion (/. e, the conclusion is particular).
2*^3- (3) ^^ negative moods the major must be uni-
versal. For if the conclusion is negative, its predicate,
the major term, is universal ; therefore it must have been
universal in the premiss, where it is subject, and there-
fore that premiss is universal.
D
so THE ELEMENTS OF LOGIC, [part iii.
204. The diagram makes these rules manifest :
MP
MS
SP
Note. — These rules are all hypothetical ; because each
term being once subject and once predicate, we are com-
pelled to argue from quality to quantity, or vice versa. Of
course the rules might be converted, and we might say,
ex,gr. If the minor is particular the major must be nega-
tive, &c.
Remark,
205. It will be observed that the first figure is the only
one in which the conclusion A can be drawn. In the second
figure the conclusion must be negative ; in the third, par-
ticular, in the fourth, either negative or particular. We
may prove this directly as follows:
206. If the conclusion is A the premisses must be A A
(by the general rules). The only universal terms here are
the two subjects. Now as the minor is universal in the
conclusion A, it must have been universal in the minor
premiss ; therefore it is its subject, and the predicate
which is particular is the middle term; this term must
therefore be universal in the major premiss, and is there-
fore its subject. The syllogism then is in the first figure.
207. This is not of itself a reason for regarding the first
figure as preferable to the others. If our premisses justify
only a negative or particular conclusion, then to prefer
the first figure, because with other premisses it would give
a conclusion A, is like choosing to walk on the high road
when it is not the nearest way, because it is wide enough
for a coach and four.
208. On the other hand, if we have a premiss O we
cannot use the first figure (except by taking instead of O
its equipollent I).
209. In fact each of the three first figures has its ap-
propriate use. The first must be used when we have to
PART III.] OF INFERENCES, 5 1
establish an A conclusion, and in general when we refer a
case or class of cases to a general rule.
210. The second figure is used when we wish to prove
that certain things do not belong to a certain class, be-
cause they differ from that class either by the possession
or the want of some attribute. The name of this attribute
is naturally predicated of both extremes.
211. The third figure is useful when we wish to disprove
a proposition alleged or assumed to be universal. This
we do by establishing an exception to it. In such argu-
ments the middle is likely to be either a singular term or
the name of a definite class, and as a singular or definite
term can only be subject of a proposition, the argument
falls naturally into the third figure.
212. The fourth figure alone has no appropriate use,
being in fact a perverse mode of expressing reasoning
which falls naturally into another figure.
Of Moods.
213. Mood is the determination of the propositions of
the syllogism as to quantity and quality.
214. As each of the three propositions, if considered
by itself, may have four varieties, A, E, I, O, the number
of arrangements arithmetically possible is 4 x 4 x 4 = 64.
But most of these are logically impossible as violating the
general rules. The number of legitimate moods may be
ascertained logically as follows :
215. If the major is A the minor may be A, E, I, O. ]S^
the major is E the minor must be A or I. If the major is
I the minor must be A or E. jf the major is O the minor
must be A. Hence we have nine possible sets of pre-
misses. Now as to the conclusion : if the premisses are
universal (AA, AE, EA), the conclusion may be either
D2
52 THE ELEMENTS OF LOGIC, [part hi.
particular or universal ; in all other cases the quantity
and quality of the conclusion are determined by those of
the premisses. ^ We have, therefore, ^Iwelve legitimate
moods ; but only eleven of these are useful, for the mood
AEO is always useless (when legitimate), since as the
minor term is universal in the minor premiss, whether it
be subject or predicate, the conclusion may be E. The
mood EAO is useless in the first and second figures, as
the minor term is subject of its premiss, and therefore the
conclusion may be E. Lastly AAI is useless in the first
figure, where the premisses AA justify a conclusion A.
216. Now% let us test these moods by the special rules,
in order to see which of them are admissible in each figure.
The first figure can have as major only A or E, and as
minor A or I.
Hence we have the fourmoods,
AAA, ^ftTEAE, ElO.^^^J^f^*^
217. The second figure may have A or E as major. If
major be A the minor is E or O, and the conclusion E or O
accordingly. If major be E, minor may be A with con-
clusion E, or I with conclusion O.
Hence we have as moods of the second figure
i." ,rr^<L^ AEE, AOO, EAE, EIO. >v.t,. , v^ y- ,
218. In the third figure the minor is A or I. The major
is unrestricted, but the conclusion is particular.
Hence the moods are
AAI, All, EAO, EIO, lAI, OAp.Wr^^<^^
219. In the fourth figure the major may be A, E, or I.
With major A the minor is A or E (first special rule).
With minor A conclusion is I (second special rule) ; and
with minor E conclusion is E (O being useless).
PART III.] OF INFERENCES, 53
With major E minor is A or I; and in either case the
conclusion is O (second rule). With major I minor is A
(third rule) and conclusion I.
Hence we have in the fourth figure
AAI, AEE, EAO, EIO, lAI. mr -hj^^ i-**^
220. These moods all have settled names which are
contained in the following mnemonic lines :
Barbara^ Celarent^ Darn, Fen'oque prioris
Cesarey CamestreSy FestinOy Baroko secundae :
Tertia Daraptiy Disamis, Datisi, Felapton,
Bokardo, Ferison, habet ; quarta insuper addit
Bramantipy Camenes, Dimaris, Fesapo, Fresison,
The vowels of these names indicate the nature of the pro-
positions ; some of the consonants have a signification
which will appear presently.
Remark,
221. Strictly speaking we have only proved that these
moods do not contradict any of the rules. But that they
are also all conclusive in their proper figures appears at
once from a comparison with the rules of the figures.
222. In the first figure something is in the major predi-
cated universally of a class (A or E) ; in the minor some-
thing is asserted to belong to that class (A or I), and hence
in the conclusion we may predicate of it what we have
predicated of the class, the quality of the conclusion being
the same as of the major premiss, its quantity that of the
minor.
223. In the second figure we assert in the minor that
something possesses universally or particularly (A or I) an
attribute which a certain class lacks (E), or else that it lacks
(E or O) an attribute which the class possesses (A). Hence
we may conclude that the thing spoken of is excluded from
that class, universally or particularly, according to the
quantity of the minor.
54
THE ELEMENTS OF LOGIC, [part ni.
224. In the third figure we assert in the major that cer-
tain things possess a certain attribute, positive or negative
(A, E, I, or O), and in the minor that they belong (A or I)
to a certain class. (The distribution of the middle insures
that we speak of the same things.) Hence we may infer
that some members of this class possess the given attribute,
positive or negative (I or O).
225. In the fourth figure the moods Bramantip, Camenes,
Dimaris, are the same respectively as Barbara, Celarent,
and Darii of the first figure, only that the conclusion, in-
stead of being the direct conclusion of the first figure, is
its converse. The remaining moods of the fourth figure,
Fesapo and Fresison, are really the same as Felapton and
Ferison, respectively, in the third figure, only that instead
of the real major its simple converse is used. These five
moods then are valid.
Examples of
■ the Moods,
9
226.
Examples of these moods follow :
227.
First Figure.
Every M is P
Bar
Every M is P
Da
Every S is M
ba
Some S is M
xi
.•
. Every S is P
ra
.-. Some S is P
i
No M is P
Ce
No M is P
Fe
Every S is M
la
Some S is M
ri
.*
. No S is P
rent
.-. Some S is not P
228.
Second figure.
NoPisM* ^
Ce
Every P is M
0am
\
Every % is M r*;
sa
No S is M
es
.*
. No S is P -
re
.-.NoSisP
tres
No P is M
Fes
Every P is M"
Ba
Some S is M
ti
Some S is not M
ro
. Some S is not P
no
.*. Some S is not P
ko
PART III.]
OF INFERENCES.
5S
A
229. Third Figure
.
Every M is P
Every M is S
.-. Some S is IV
Da
rap
ti
Every M is P
Some M is S
.*. Some S is P
Da
tis
i
A
.t
No M is P \
Fe f I
No M is P
Fe
£
" Every M is S lap '
.•. Some S is not P ton ,
yy Some M is S
.-. Some S is not P
ris
on
1
Some M is P
Dis:
Some M is not P
Bok
Every M is S
.% Some S is P
am /
is T
Every M is S
.-. Some S is not P
ar
do
A
230. Fourth Figure.
.
Every P is M
Bram
Some P is M .
Dim
Every M is S
,*. Some S is P
an
tip
Every M is S
.-.Some Sis P
ar
is
/
Every P is M Cam
No -S is M e
•••
No S is I
* nes
i NoPisM
Fes
No P is M
Fres
, ' '. Every M is S ap
\\ Some S is not P
Some M is S
.-. Some Sis notP
is
on
Of Aristotle' s Dictum.
231. Aristotle (and after him other logicians) sought to
bring all mediate reasoning under a single principle, to
show, in fact, that the process is in every case essentially
one and the same.
232. According to this view all reasoning consists in
applying a general rule to a particular case or group of
cases. The. general rule is the universal major; the
56 THE ELEMENTS OF LOGIC, [part hi.
affirmative minor brings the particular case under the con-
dition of the rule, and in the conclusion it is inferred that
it comes under the rule itself. This, as will be seen pre-
sently, is not limited to simple syllogisms.
233. As applied to these, the formal principle of the
reasoning may be stated thus :
Whatever belongs to an attribute of a thing belongs
to the thing itself. '
» Or as it is stated by Aristotle —
Whatever is said of the predicate shall be said also of
the subject.''^
234 From this can be deduced the ordinary form of the
Dictum de Omni et de Nullo, ' Whatever belongs to or
contradicts the class belongs to or contradicts all the
objects contained under the class.' For the meaning of
the class name is the group of attributes belonging to the
class.f
235. This principle is directly applicable to the first
figure only; and in order to show that it applies to the other
figures we must reduce them to the first figure.
236. The problem of reduction in any given case is :
given. premisses and a conclusion, which, as they stand,
are not in the first figure, to deduce" from the given pre- ^
misses the required conclusion by a syllogism in the first
figure. The mood to be reduced is the reducend ; that to
wiiich it is reduced is the reduct.
* "Oca Korra. rov KaT7iyopovjj.4uov Xeyerai Trdvra Koi Kara rod v'jroK€i-
jxeyov ^7]dr}a'€Tai.
t The Dictum is given by Aristotle as a definition of kutol iravrhs
KarriyopeLffdai*
PART III.] OF INFERENCES. . C^lJFO^^'*^
Of Reduction.
237. Reduction is ostensive when the conclusion obtained
in the first figure is either the same as that required or
yields it by conversion.
238. Any mood which has not O as a premiss may be
readily brought into the first figure by Ostensive reduction. >
If O is a premiss it must be treated as I. /
239. In the second figure (PM, SM, SP), the premisses
do not of themselves show which term is predicate and
which subject of the conclusion. If the given minor is
affirmative, we have only to convert the major simply (it
being always universal and in this case negative). This
applies to Ce^are and Festino ; the s indicates simple con-
version of the preceding E, and the initials show that the
moods are reduced to Celarent and Ferio respectively.
Thus :
No P isM Ex. gr. No fish breathe in air :
.*•. No M is P .'. Nothing breathing in air is a fish :
S (some or all) is M Whales breathe in air :
.*. S (some or all) is not P. .*. Whales are not fish.
(N. B. — In this and other examples the propositions
which do not belong to the first figure are in italics.)
^240. If the given minor is E it cannot be a minor in the
first figure; it must, therefore, be treated as major and
converted simply. The conclusion, which will be E, will
have to be converted, in order to give the required con-
clusion. This applies to Camestres. In this name, m indi-
cates the interchange of premisses (* metathesis ') ; ^ in the
premisses the conversion (simple) of the preceding E ; s
after the conclusion, the conversion not of this conclusion
^3
58 THE ELEMENTS OF LOGIC, [part hi.
but of that obtained in the first figure. The reduct mood
is Celarent. Thus :
Every P is M Ex. gr. All fish breathe in water :
No S is M No whales breathe in water :
,\ No M is S .*. Nothing breathing in water is a whale :
,•. No P is S .'. No fish is a whale :
,', No S is P, .'. No whale is a fish,
241. This might be reduced by converting the major by
contraposition. Thus :
Every P is M All fish "breathe in water :
,\ No not-M is P .*. Nothing that does not breathe in
water is a fish :
Every S is not-M Whales do not breathe in water:
.*. No S is P. .'. Whales are not fish.
242. If the given minor is O, it cannot stand in the first
figure either as major or minor. We must, therefore,
treat it as I, and convert the major by contraposition.
Thus :
All poets are men of genius :
.', No man not a man of genius is a poet :
Some rhymesters are not men of genius :
.\ Some rhymesters are not poets.
All P is M
.-. No not-M is P
Some S is not M
.'. Some S is not P.
The reduct is in Ferio ; and if c were used to indicate con-
version by contraposition, and y to indicate the change of
O to its equipolent I, the name might be Facoyro. This
corresponds to the second mode of reducing Camestres.
PART III.] OF INFERENCES, 59
243. In the third figure (MP, MS, SP), as in the second,
the premisses do not determine which is the subject and
which the predicate of the conclusion. If the major is
universal, we have only to convert the minor, which is
always affirmative. This is the case in Darapti, Datisi
Felapton, and Ferison. The s in Datisi and Ferison indi-
cates simple conversion of the I minor ; p in Darapti and
Felapton indicates conversion of the A minor per acci-
dens.
244. If the major is particular, it cannot stand as such
in the first figure ; it must be treated as a minor, and, as
in the preceding case, converted ; if I, simply ; if O, by
contraposition ; and the conclusion will give by conversion
the conclusion of the proposed syllogism. This is the
case in Disamis and Bokardo. In order that the name of
the latter should indicate this mode of reduction it should
be Docamos, using c, as before, to indicate conversion by
contraposition. In this case the conclusion obtained in
the first figure is I, and has to be simply converted ; and
the negative, which will then be in the predicate, must be
attached to the copula, but this involves no change in
expression. Thus :
Every M is P (or not P) All ants ire invertebrate (or not
vertebrate) :
M {all or some) is S Ants {all or some) are sagacious :
.*. Some S is M .*. Some sagacious things are ants :
.*. Some S is P (or not P). .*. Some sagacious things are inver-
tebrate (or not vertebrate).
245. This example represents Darapti, Datisi, Felapton,
and Ferison. If the conclusion were, * Some invertebrates
are sagacious,' it would be Disamis.
6o THE ELEMENTS OF LOGIC, [part hi.
The following represents Disamis and Bokardo :
Some M is P Some histories are amusing :
.*. Some P is M .*. Some amusing books are histories :
Every M is S All histories are instructive ;
.'. Some P is S .*. Some amusing books are instructive :
.*. Some S is P. .*. Some instructive hooks are amusing.
By reading * not P ' for ' P,' we have Bokardo :
Some M is not P Some histories are not amusing :
.'. Some not-P is M .*. Some books not amusing are histories :•
Every M is S All histories are instructive :
.-. Some not-P is S .*. Some books not amusing are instructive :
.*. Some S is not P, .'. Some instructive books are not amusing.
The process is obviously the same in substance in all the
moods. ^
246. When the middle term is singular, or the name of
a definite group, the reduction of the third figure is awk-
ward and unnatural ; ex, gr, :
X is a recent poet :
X is a poet of the first rank ;
.*. Some recent poet is of the first rank.
In order to reduce this we must convert the minor into,
One recent poet is X. ^^' ^
The awkwardness appears in the first of the previous
examples. We really predicate of ants that they are
sagacious, not of sagacious things that they are ants.
247 In the fourth figure (PM, MS, SP), if the major is
affirmative and the minor consequently universal, they must
be treated as minor and major respectively, i. e. we make
the conclusion PS, which may be then converted into SP
(for it will not be O ; since in negative moods the given
PART III.] OF INFERENCES, 6i
major is universal, and therefore P may be universal in the
conclusion). This is the case in Bramantip, Camenes,
and Dimaris. The/ in Bramantip means that the con-
clusion A obtained in Barbara (to which this mood is
reduced) is converted per accidens.
248. If the major is negative (E) it cannot be minor in
the first figure ; in this case, therefore, both premisses are
converted. This is the case of Fesapo and Fresison.
249. It will have been observed that the initial letter of
the mood of the reducend is the same as that of the reduct ;
that s shows that the premiss preceding is to be converted
simply ; p that it is to be converted per accidens ; m that
the premisses have to be transposed, or rather that they
change names, since the order is indiff*erent. In the con-
clusion s and p refer to the conclusion in the first figure,
which has to be converted to give the required conclusion.
Of Redudio ad Impossilile,
250. The names of Baroko and Bokardo indicate a
different and more complex mode of reduction, viz. : the
contradictory of the conclusion is substituted for the O
premiss, and from this and the retained premiss A we
deduce in the first figure and mode Barbara a conclusion,
A, which contradicts the given O premiss. Since this
conclusion contradicts a premiss which is given true, it
must be false, therefore one of the premisses from which
we inferred it is false ; and, as one was given true, the
falsity must be in the substituted premiss, which contra-
dicts the required conclusion. Therefore this conclusion
is true. This reduction is called Reductio ad impossibile.
251. Stated more shortly the reasoning would stand
thus :
Baroko : Every P is M :
Some S is not M :
/. Some S is not P.
62 THE ELEMENTS OF LOGIC, [part hi.
This is reduced to,
Every P is M.
Therefore, if
Every S is P,
then (by Barbara),
Every S is M.
But this is false, for Some S is not M ; .*. it is false that
Every S is P, i. e, it is true that Some S is not P.
252. This is unsatisfactory, for we have here got a con-
ditional syllogism, the reduction of which to a categorical
form would give us back the original syllogism.
0/ the Unfigtired Syllogism,
253. Those logicians who adopt the principle of quan-
tifying the predicate escape the complexity of the figures.
For, as already shown, every proposition is on that view
an equation or a statement of inequality. These logicians
consequently found syllogism on the following axioms : —
Terms which coincide as to their extension with a
third term coincide with each other.
Terms of which one coincides "with, and the other is
excluded from, the extension of the same third term are
excluded from each other.
If two terms are both excluded from one and the same
third term, we can infer nothing as to their relation to
each other. ^
254. On this doctrine all categorical syllogisms are of
the type,
A is equal to B :
B is equal to C :
.*. A is equal to C.
PART III.] OF INFERENCES. 63
Or for negatives —
A is equal to B.
B is not equal to C :
.*. A is not equal to C.
The order of the terms in each premiss is indifferent, and
consequently there is no distinction between major and
minor.
255. It should be observed that whatever opinion is
adopted about the quantification of the predicate in gene-
ral, this is the form of the syllogism -in mathematical
reasoning, and in other cases where the premisses are
propositions in identity.
Of the Enthymeme.
256. An Enthymeme is a syllogism of which one premiss
is suppressed, as : A is B .*. A is C. The expressed premiss
is called the Antecedent, and the conclusion is called the
Consequent. If the subject of the consequent appears in
the antecedent, this is the minor, and the major is sup-
pressed ; if the predicate of the consequent appears in the
antecedent, this is the major, and the minor is suppressed.
Note. — This is the commonly accepted meaning of
' enthymeme,' but it seems to have originated in a false
etymology, as if the word were derived from Iv Ov/jlw, from
one premiss being in the mind. It is really derived from
the verb iv6vfjL€(o. Aristotle used it to mean an argument
from signs and likelihoods.
Of Sorites.
257. A Sorites is a chain of reasoning consisting of a
series of syllogisms in which each intermediate conclusion
is not expressed, but is assumed as a premiss of the suc-
ceeding syllogism.
64 ^ THE ELEMENTS OF LOGIC, [part iii.
Usually the syllogisms are in the first figure, and then
the predicate of each premiss is the subject of the next,
and the predicate of the last is in the conclusion predi-
cated of the first subject. Ex, gr. :
(Suppressed conclusions used
as minor premisses.)
AisB
Every B is C
.-. A is C
Every C is D
.-. A is D.
Every D is E
A is E.
258. The Sorites in the first figure has two Special
Bules.
The first premiss alone can be particular.
The last premiss alone can be negative.
259. The first alone can be particular. When the So-
rites is resolved into syllogisms, the first premiss is the
minor of the first syllogism, but the minor of every other
is the (suppressed) conclusion of the preceding; for the
subject of the first premiss is the subject of every conclu-
sion. All the expressed premisses after the first, being
therefore majors in the first figure, must be universal.
260. The last alone can be negative. For if any pre-
miss were negative, the conclusion of the syllogism would
be negative; but it is the minor of the succeeding one,
and therefore must be afiirmative.
PART III.] OF INFERENCES. * 65
CHAPTER III.
Of Complex Propositions,
261. Complex Propositions are those which combine
two or more simple propositions in such a way that the
truth or falsity of one is said to depend on the truth or
falsity of the other or others. They are divided into
Conditional (or Hypothetical) and Disjunctive.
262. A Conditional Proposition is one which asserts
that the truth of one proposition depends on the truth of
another.
If A is B, it is C:
If A is B,*C isD:
If A is B, either C is D or E is F.
The dependent proposition is called the Consequent, that
on which it depends is the Antecedent.
263. The truth of a Conditional proposition does not
depend on the truth of the separate propositions; it only
requires that the consequent follow from the antecedent.
264. The antecedent may be related to the consequent,"
as Reason to Consequence or as Cause to Effect. Ex. gr, :
If there is dust in the air, a beam of light passing through
it is visible (Relation of Cause to Effect).
If a beam of light is visible, there is dust in the air
(Relation of Reason and Consequence).
265. A conditional which has only three terms may often
(if these are common terms) be reduced to a simple pro-
66 THE ELEMENTS OF LOGIC [part hi.
position. * If A is B it is C ' cannot express a consequence
formally conclusive unless * Every AB is AC Thus, the
first of the preceding propositions is equivalent to, * Dusty
air makes light visible ' ; and the second to, * Dusty air
alone makes light visible*.
The converse by contraposition of 'Every AB is AC is
* Whatever is not AC is not AB', equivalent to, * If A is
not C it is not B'. This may, therefore, be called the con-
verse of the original conditional.
266. In a DisJTinctive Proposition the truth of one of
the component propositions depends on the falsity of the
other (or others) ; ex, gr, :
A is either B, or C, or D :
Either A is B, or C is D, or E is F.
.267. Disjunctives may be reduced to conditionals, thus :
* A is either B or C ' is equivalent to two conditionals, viz.,
' If A is not B it is C,' and ' If A is not C it is B'; the
latter of which is the converse of the former.
268. It is generally held by logicians that in a disjunc-
tive not only does the truth of one member depend on the
falsity of the other, but also the falsity of all but one de-
pends on the truth of that one ; so that, * A is either B
or C ' would include, besides the two propositions given
above, these two : * If A is B it is not C\ and its converse,
af Ais C it is not B\
0/ Complex Syllogisms.
269. A Complex Syllogism is one in which one or more
complex propositions occur.
As complex propositions are of two kinds. Conditional
and Disjunctive, so complex syllogisms will be of two
PART III.] OF INFERENCES. 67
kinds, conditional and disjunctive. If one premiss only is
complex, it is called the major, and the simple premiss is
the minor. If one premiss is conditional and the other
disjunctive, the conditional premiss is called the major.
Of Conditional Syllogisms.
270. The most usual form of a conditional syllogism is
where one premiss only (called the major) is conditional,
the minor being simple. There are then two legitimate
forms of reasoning. From the position, /. e, assertion, of
the antecedent to the position, i. e. assertion, of the conse-
quent ; and, From the remotion, i, e, denial, of the conse-
quent, to the remotion, 2*. e. denial, of the antecedent.
Thus, if the major is :
If A is B, C is D,
we may reason thus :
AisB; .-.C isD.
Or,
C is not D ; .'.A is not B.
Ex. gr, :
If a man is shot through the heart he dies ; X is shot
through the heart, therefore he dies. Or, X did not die,
therefore he was not shot through the heart.
271. It would be illegitimate to reason from the asser-
tion of the consequent to the assertion of the antecedent,
as : X died, therefore he was shot through the heart.
It is also illegitimate to reason from the denial of the
antecedent to the denial of the consequent, as : X was not
shot through the heart, therefore he will not die.
68 THE ELEMENTS OF LOGIC, [part hi.
For it is not asserted that the consequent may not follow
from other antecedents also.
272. If both premisses are conditional, the conclusion
will be conditional. Ex, gr, :
IfAisB, CisD:
If C isD, E is F:
.-.IfAisB, Eis F.
Of Disjunctive Syllogisms,
273. A Disjunctive Syllogism is one which has a dis-
junctive major and a simple minor. Ex, gr, :
Either A is B or C is D :
But A is not B; .-.C is D.
If the disjunctive premiss has only two members, we may
reason from the denial of either to the assertion of the
others. If there are several members, we may reason from
the denial of all but one to the assertion of that one, or
from the denial of all but two or more to the assertion of
these disjunctively. Ex, gr, : * A successful man must have
talents, industry, or good fortune \ * So-and-so, who is
successful, has neither talents nor industry; therefore he
has good fortune'. Or, * So-and-so has not talents ; there-
fore he has either industry or good fortune '.
274. Logicians generally assume, as already stated, that
the members of a disjunctive proposition are mutually
exclusive, so that no two can be together asserted of the
subject. If this is the case, we have a second valid form
of reasoning, namely, from the assertion of one member
to the denial of the rest. It follows, on this view, that if
PART III.] OF INFERENCES. 69
we do not intend to exclude the possibility of two or more
of the suppositions being true together, we must enume-
rate in the disjunctive all the possible combinations. * A
is either B (only) or C (only), or both B and C\ With three
suppositions, as in the example above given, w*e should
have seven members. ^
275. This view, however, does not agree wil^b
nary use of language ; and in the cases in whi
infer from the assertion of one member the den|
rest, it is not from the form of the expression
the terms are opposed), but from our knowledge of the
matter. t
Of the Dilemma.
276. A Dilemma is defined as a syllogism which has a
conditional major premiss, with more than one antecedent,
and a disjunctive minor. Ex. gr, :
If either A is B or C is D, X is Y :
But either A is B or C is D : '
.-.JCisY.
If AisB, C is D; and ifEis F, X is Y: ^--^
But either A is B or E is F : '-V%^^w<r-r'
/. Either C is D orXis Y.
In the latter case we might also reason thus :
If A is B, C is D ; and if E is F, X is Y :
But either C is not'D, or X is not Y :
.•. Either A is not B, or E is not F. •
277. The folloVing form is by some writers regarded as
a dilemma, but is by others viewed as a conditional syllo-
gism, the minor ndt being disjunctive. • 9
70 THE ELEMENTS OF LOGIC, [part hi.
If X is Y, either A is B or C is D :
But neither A is B nor C is D :
.-. X is not Y.
Here the minor simply denies the consequent of the
major.
Note. — There is much difference of opinion amongst
modern, logicians as to the forms which should be included
undef tiie name Dilemma.
Of the Reduction of Complex Syllogisms,
278. It has already been observed that what a condi-
tional proposition asserts is that the consequent follows
from the antecedent.
If the conditional has only three terms, as : * If A is B
it is C ' ; then, as we have already seen, if these are com-
mon terms, it is reducible to 'Every AB is AC. A syllo-
gism with such a conditional premiss may therefore at
once be reduced to a categorical form by substituting for
the conditional premiss this equivalent ; ex, gr. : * If any
bird devours grubs it is useful to the farmer. These birds
devour grubs, .*. they are useful to the farmer,' is reduced
to, ' All birds which devour grubs are useful to the farmer.
These birds devour grubs, .*. they are useful to the farmer.
Similarly, if the conditional premiss is, *If A is C, B is C,
this is equivalent to, * B is A' ; and by substituting this
equivalent, the syllogism is at once reduced to the cate-
gorical form. Ex, gr, : *If birds that devour grubs are use-
ful, then rooks are useful.' This is equivalent to * Rooks
devour grubs.'
279. If either subject is a singular term, the inference
PART III.] OF INFERENCES, 7 1
may be from something not expressed, as : * If Jones has
written, the matter is settled '. * If this quotation is found
in Cicero at all, it must be in the philosophical treatises'.
In such cases we cannot reduce the reasoning to a cate-
gorical form without further information, simply because
the actual grounds of the inference are not expressed.
280. Similarly, if there are four terms : * If A is B, C is
D'; there is no connexion between the antecedent and
the consequent. The inference, therefore, depends on
something not expressed.
281. Some logicians suggest reducing these by saying,
* The case of A being B is a case of C being D '. This is
illusory. ^ Case of* is ambiguous. It may mean 'instance
of (whether actual or imagined), and then this is only an
awkward way of stating a categorical. 'Every case (in-
stance) of A being B is a case of A being C * is equivalent
to 'Every AB is AC '. * If migratory birds breed in Ire-
land they winter farther south'. This may be stated,
' Every case of migratory birds breeding,* etc. ; but this is
exactly the same as, 'All migratory birds that breed in
Ireland winter farther south'.
282. But ' case oV may mean 'hypothesis of, and then
the proposition is still hypothetical, although disguised.
Ex. gr. : ' If this witness is to be believed, the prisoner is
innocent.' Here the prisoner's innocence is an individual
fact ; there are no ' cases,' z. e, ' instances,' of it. * Case'
here means ' hypothesis'; but the consequent is not a part
of the antecedent, nor a species of it. The conclusion is
really drawn from what the witness said, not from his
credibility.
283. The hypothetical (= conditional) form is commonly
adopted when we state a connexion of cause and effect, of
72 THE ELEMENTS OF LOGIC, [part hi.
sign and thing signified, or of reason and indirect conse-
quence, /*. e, when steps intervene which are so obvious
that they need not be stated. Ex. gr. : * If the barometer
falls there will be rain'. Here the relation is that of sign
and thing signified, both being effects of the same cause.
CHAPTER IV.
Of Probable Reasoning.
§ I. — 0/ Chams of Prohahilities,
284. By Probable Reasoning is meant reasoning from
propositions which are not certain. A proposition is usu-
ally called ' probable ' when it is more likely to be true
than not ; but the word * probability * is used to express
any degree of likelihood or unlikelihood. When we speak
of probable reasoning, we use the word probable in this
wider sense. As almost all our reasoning is connected
with probabilities, not certainties, it is important to ascer-
tain what modification in our logical theory this fact
involves. The chief points to be attended to are the fol-
lowing :
285. First, it is to be observed that, for convenience of
calculation, the degree of probability of any proposition
(or event) is represented by a fraction less than unity.
The denominator is the whole number of possible cases :
and the numerator is the number of those in which the
proposition is true (or in which the event happens). Cer-
tainty is represented by unity ; and as the proposition must
PART III.] OF INFERENCES. 73
be either true or not true, the chance (or probability) of its
falsehood plus the chance of its truth is equal to unity.
286. Now the chance that two (independent) statements
are both true together is equal to the product of the
chances of their truth separately.
Thus to take a pair of premisses in the first figure : S is
M (let the chances be f) : M is P (let the chances be i).
Then out of every 24 (= 4 x 6) S 18 are M, and out of
these 18 M five-sixths, i.e. 15, are P. Hence out of 24 S
we have 15 P; ue, the chance of any given S being Pis
3 x5»
4x6*
287. It appears from this that in a chain of reasoning
consisting of several propositions, each in itself only pro-
bable7 the probability of the conclusion is diminished by
every additional link. If the chain consisted of only three
propositions, each having a chance = t (?*. e. the odds in
its favour being 4 to i), the probability of the conclusion
would be
4x4x4 64
5x5x5 125'
i. e, barely over ■
§ 2. — Of Cumulative Probahltizes.
288. This way of combining probable propositions in a
chain must not be confounded with the accumulation of
* In an affirmative mood of the third figure the chance thus obtained
would be the chance (not that any given S is P, but) that any given M
is both S and P. ,
E
74 THE ELEMENTS OF LOGIC, [part hi.
probable proofs each separately establishing the conclu-
sion ; as, for example,
A is B (probably) :
B is C (probably) :
.*. A is C (with some degree of probability).
Also —
A is D (probably) :
D is C (probably) :
.-. again, A is C (with some probability).
289. In a case like this, if either pair of premisses holds
good, the conclusion A is C holds good ; therefore, in
order that A should not be C, both reasons must fail.
Here then it is the falsity of the conclusion that involves
a coincidence, the chance of which is the product of the
separate chances of failure of the two grounds of proof.
Suppose the chance that A is C by the first proof is f , and
by the second i ; then the chance of the first not holding
good is i, that of the second failing is i ; the product of
these, aV, is the chance that A is not C ; therefore, so far,
the chance that A is C = M-
290. If we had three reasons, each independently giving
a chance = -J, then the resulting chance of the conclusion
would be
ixixi I 124
"5>^5x5'' 125" 125'
This is the case of what is called Cumulative evidence or
testimony.
When probabilities have to be balanced, i,e, when there
are reasons or evidence on both sides, the calculation is
more difficult.
PART III.] OF INFERENCES. 75
291. It deserves to be remarked that a conditional pro-
position, which expresses only a probable consequence,
cannot be converted. From
If A is B, it is probably D,
we cannot infer,
If A is not D, it is probably not B.
For the latter is, as has been shown, the converse by con-
traposition of the former. But if the former is only pro-
bable it becomes, when put into a categorical form, I
(' Some or Most B is D'), and I cannot be converted by
contraposition. Even if it takes the form, * Most B is D ',
we could not infer that there were amongst the not-D's
any things that were not B. From * Most University
students are not honormen ' we cannot infer that there
are any honormen who are not University students.
Of Inductive Reasoning,
292. Induction proceeds from particulars to particulars,
or from particulars to generals.
It assumes two principles:
First — Whatever begins to exist has a cause.
Second— Whenever the same circumstances occur the
same result will occur, i, e. the same causes always pro-
duce the same effects, ' cause ' being understood as in-
cluding everything that may influence the effect.
293. This latter principle is the principle of Uniformity
of Nature, and is the principle on which we act every time
that we take food, or use any object, or in fact perform an
action for any purpose whatever. The converse of this
principle is obviously not true, since difl'erent causes may
produce the same effect.
E2
76 THE ELEMENTS OF LOGIC, [part iit.
294. A special case of the principle of Uniformity of
Nature is the principle of Continuity, viz., that when a
certain relation is found to exist between the variations of
the cause and those of the effect in several instances, the
same relation, or one including it, exists in all the inter-
mediate instances.
295. The following example illustrates the process of
inductive reasoning : — Having read certain works of a
novelist, and finding them witty and brilliant, we expect
to find the same qualities in other works of the same
author. In this inference we pass through an intermediate
stage. Thus we first conclude that the author of these
works possesses wit and talents, and then we infer from
this, that his other works will display the same characters.
That is to say, we reason first from the effects to the cause,
and then from the cause to the effects.
296. So again, from the past mortality of all men we
infer the mortality of men now living. Why .? First, the
mortality of men in all circumstances justifies us in con-
cluding that it is the effect of something not peculiar to
this or that man, place or circumstance, but belonging to
human nature ; and hence again we are justified in infer-
ring that wherever human nature is found mortality be-
longs to it. This conclusion is further confirmed by other
considerations ; for example, similar experience in other
animals, which enables us to say that the * something ' is
not even peculiar to human nature, and again, the obser-
vation that death is only the climax of a series of changes
always going on.
297. So also Newton, in establishing the law of gravita-
tion, first ascended to the cause of the moon's revolution
round the earth, i. e. the attractive force of the earth.
He found, deductively, the law of this force. He then
argued (from continuity) that a force which operates all
through the moon's orbit at varying distances from the
earth, according to a certain ratio, probably operates at
other distances also, according to the same ratio, down
even to the surface of the earth. It was easy to calculate
what its amount at the surface would be, and this proved
PART III.] OF INFERENCES. 77
to be exactly equal to the force which actually exists there,
and which is, like the former, directed to the earth's centre.
This coincidence was sufficient to prove the continuity of
the law of attraction, at least to a high degree of proba-
bility. And this probability became certainty when it
was found that the same law prevailed in other parts of
space.
Of the Logical Basts of Induction.
298. The principles on which Induction rests are not
themselves capable of proof, strictly so called. The prin-
ciple of Uniformity of Nature makes a general statement
as to the Unknown, viz., that in certain respects it resem-
bles the Known ; and this statement cannot be proved by
either logic or experience, without taking for granted the
very principle itself.
299. It is a question amongst logicians whether Induc-
tion can be brought under Deduction or not. If it is so,
it must be by taking as a major premiss some form of the
principle of Uniformity just stated. It comes to the same
thing if we enumerate certain cases, and then assert that
these are all the cases ; ex. gr., Mars, Jupiter, Venus, etc.,
revolve round the sun in ellipses. These are (= constitute)
all the planets ; .*. All planets revolve round the sun in
ellipses. In most cases we cannot enumerate all the in-
stances ; and the statement, ^ These are all ', means that
for the purpose of the present argument these may be
taken as representing all ; u e. that all the others may be ,
supposed to resemble these. This involves the principle
of Uniformity. The question, what condition must be ful-
filled in order that this reasoning may be valid, does not
belong to Formal but to Applied Logic.
Mr. MiWs View of the Type of Reasoning,
300. Mr. Mill considers that the true type of reasoning
is that which proceeds from particulars to particulars, and,
consequently, is not conclusive from the form of the ex-
78 THE ELEMENTS OF LOGIC, [part iii.
pression. The universal proposition interposed as a major
is, according to his view, not really a stage in the inference,
but merely a convenient memorandum of past inferences,
and a short formula for making more. When we say ^ All
men are mortal ; Jones is a man ; .'.Jones is mortal', we
do not infer the mortality of Jones from that of * All men
(including Jones'), but from the observed mortality of all
men hitherto. The major, ' All men are mortal ', means
that we are justified in inferring from our past experience
that any given man is mortal. Hence Mr. Mill regards
syllogism as a process of interpretation (of our major), not
as a process of inference. It is useful, therefore, as a test
of the correctness of the reasoning by which we established
the major.
301. This view may be applied to syllogisms in which
the reasoning consists in bringing a particular case or
group of cases under a general rule, 2*. e, to syllogisms
which fall naturally into the first figure. It does not
apply to syllogisms in the third figure with a definite
middle.
Of Analogy.
302. Analogy is an argument in which from the resem-
blance of two things in certain respects we infer their
resemblance in others ; ex. gr. : if from the fact that Mars
has many points of resemblance to the Earth we conclude
that it is probably inhabited. It is from analogy we con-
clude that vertebrate animals (or even that other persons)
feel pain as we do. Or the things compared may not be
objects, but relations of objects. In this sense analogy is
the resemblance of relations.
303. The term Analogy is, however, frequently used to
signify imperfect induction, and by older writers to signify
complete induction.
( 79 )
PART FOURTH
CHAPTER I.
Of Fallacies.
304. A fallacy is an unsound argument. The unsound-
ness maybe either — (i). In the reasoning considered in
itself without regard to anything outside it; or, (2). In
the reasoning considered in relation to a definite ques-
tion proposed — as, for instance, the refutation of a given
opinion.
305. The first class includes the fallacies in Expression.
The second includes fallacies in the Matter.
306. The former class is again subdivided into fallacies
which, in the form of the expression, violate the rules of
logic, and fallacies which covertly violate them ; that is to
say, in which the violation appears only where the mean-
ing of the terms is explained. These are called respec-
tively Logical and Semi-logical fallacies. The latter are
also called fallacies in dictione, i, e, in the wording.
Of Logical Fallacies,
307. Of strictly logical fallacies (otherwise called para-
logisms) there might be as many species as there are
80 THE ELEMENTS OF LOGIC, [part iv.
general rules of syllogism, which may be separately vio-
lated. There are, however, only three that deserve special
enumeration here, viz., Illicit Process (taking a term more
universally in the conclusion than in the premisses), Un-
distributed Middle, and Two Middle Terms.
308. Illicit Process may be either of the Major or of the
Minor.
Of the major^ as : ^^^
Men of genius are (generally) eccentric :
Xts not eccentric i^Hf ,
Therefore he is not a man of genius.
Even although the major should be given as probable (as
in this case), we could not infer that the conclusion is
probable. In order to ascertain whether this is so or not,
we should know the proportion of not-eccentric persons
who are men of genius, and this is not stated.
309. Illicit Process of the Minor, as :
A, B, and C, are polite :
A, B, and C, are Frenchmen :
.*. All Frenghmen are polite.
This is the fallacy in induction from an insufficient number
of instances.
310. An example of Undistributed Middle is:
All profound works are obscure :
This is obscure :
.*. It is profound.
311. The fallacy of Two Middle Terms escapes notice
PART iv] OF FALL A CIES. 8 1
most readily when a verb is treated as if it were the
copula, as :
■ - Pf ^ A :^3
' ^^ ^ A resembles B : c' // ;d - (^
B resembles C : -^^ /\ -z. C
.*. A resembles C.
5
A syllogism which is exactly the same in form as
A is different from B :
B is different from C :
.•. A is different from. C.
Fallacies in Complex Syllogisms,
312. Under the head of logical fallacies are to be
reckoned the two fallacious forms of inference from a
conditional proposition, viz. from the assertion (position)
of the consequent to the assertion of the antecedent, and
from the denial (remotion) of the antecedent to the denial
of the consequent.
313. A common instance of the former fallacy is, infer-
ring the truth of the premisses, or the legitimacy of the
reasoning, from the truth of the conclusion.. It is often
owing to a tacit inference of this kind that inconclusive
arguments are brought forward by persons little likely to
be deceived by a fallacy. They repeat an argument with-
out examination, because they are firmly convinced of the
truth of the conclusion.
314. They injure their cause by exposing it to the in-
fluence of the counter fallacy, which consists in inferring
the falsehood of the conclusion from the falsehood of the
premisses or the logical defect of the argument. This is
82 THE ELEMENTS OF LOGIC, [part iv.
inferring the denial of the consequent from the denial of
the. antecedent. Even when good and bad arguments are
mixed, there is a tendency to regard the bad as to a cer-
tain extent counterbalancing the good, instead of letting
them go for nothing. An unfair disputant will of course
attack the weak points only ; and, when he has exposed
these, will assume that he has refuted the conclusion.
In any particular syllogism these fallacies would resolve
themselves into illicit process, undistributed middle, or
two negative premisses.
Of Semi'logical Fallacies^ or Fallacies in Dictione —
Fallacy of Ambiguous Terms,
315. Of Semi-logical Fallacies the most important is
that of Ambiguous Terms, called the Fallacy of Equivo-
cation. The term most commonly ambiguous is the mid-
dle term, but either of the extremes may be ambiguous
likewise. An example of ambiguous middle is :
The old are more likely to be right in their
judgment than the young :
The men who wrote a thousancj years ago
are old writers :
.-. They are more likely to be right in their
judgment than those of our own day.
Here the word * old ' is used in two sensQS — in the major,
for those who have lived longer, and so have had more
experience ; and in the minor, for those who lived a long
time ago, when the world was younger.*
* * Antiquitas saeculi juventus mundi'.
PART IV. OF FALLACIES. 83
316. An important case of ambiguity is where a term is
taken in one place collectively, and in another distribu-
tively. When we reason from the distributive sense to the
collective, it is called the fallacy of Composition ; when
we reason from the collective to the distributive, it is the
fallacy of Division. Overrating the probability of an infe-
rence from probable premisses comes under the former
head. If the premisses are probably true (taken together),
the conclusion is probably true : the premisses are pro-
bably true (each separately) ; .*. the conclusion is probably
true. Underrating the probability of an inference based
on cumulative evidence comes under the head of the fal-
lacy of division ; as in the case of circumstantial evidence,
where it is argued that this point and that point, and the
third, etc., are not conclusive, 2*. e. separately, and it is
inferred that they are not conclusive when taken to-
gether.
317. These two fallacies sometimes arise from the am-
biguity of the words * all,' * some '. * All ' means either
" * every ' (distributively), or ' all together ' (collectively).
*Some' has a somewhat similar ambiguity; sometimes
meaning one or other, as when we say, * Something must
be done', * You will meet some personfr-onTthe way'; and
at other times, * Some definite^
318. Ambiguity may arise from a term being taken in
one place in an abstract sense and in another in a con-
crete ; ex, gr.y
.'. ; "P ■
Books are a solace to the weary :
Every book is either bouna or unbound :
Therefore either bound books are a solace,
or unbound books are so.
84 THE ELEMENTS OF LOGIC, [part. iv.
Again —
Food is necessary to life :
All food is either animal or vegetable :
.*. Either animal food is necessary, or vegetable
food is necessary.
319. The Fallacy of Accident (Fallacia Accidentis), or
a dido sivipliciter ad dictum secundum quid, consists in rea-
soning from a term stated without qualification to the
same term with a qualification, as :
You eat to-day what you bought in the market
yesterday :
What you bought was raw meat :
Therefore you eat raw meat to day.
Here the term ' what you bought ' is used in the first pre-
miss of the substance only, but in the second of the sub-
stance in a particular state. If the first premiss had been
stated with exactness, the^e would be palpably two middle
terms.
320. The counter fallacy is called a dicto secundum quid
ad dictum simpliciter, and consists in arguing from a state-
ment with a particular qualification to a statement without
the qualification, as :
Opium is a poison
Physicians give their patients opium :
Therefore they give their patients poison.
321. There is an important ambiguity in words and
expressions implying 'Sameness' which deserves notice in
a treatise on Logic, because sameness is sometimes that
which is implied by the form of the proposition.
PART IV.] OF FALLACIES. 85
322 Strictly speaking, a thing cannot be the same with
anything but itself. This sameness is for the sake of exact-
ness sometimes called * numerical identity/ i. e. there is
only * one ' thing. When we speak of attributes, this
meaning is modified. We call these the same when they
are precisely similar, so that the same name applies to
them. We speak of the same colour, sound, etc., mean-
ing that the impressions produced on us are precisely
similar. Similarly, when we say that two persons have
the same disease, the same symptoms, etc., what we
assert is similarity, sometimes including such similarity
of origin as belongs to plants or animals which we say are
of the *same * species. We say, for instance, that one plant
is the * same ' as another, meaning that the similarity is
such as exists between plants produced from the seed of
one identical plant. When we say that one man has the
same organs as another, or as certain animals, what we
predicate is precise simiharity of function, etc.
323. There is a still further extension of the notion when
we speak of the skull as a modified vertebra, meaning that
certain parts which we regard as essential to our concep-
tion of the plan of skull and vertebra may be described in
the same terms. Still further, an organ is sometimes said
to be a modification, development, survival, etc., of some-
thing which existed in an ancestor of the plant or animal.
There is, of course, no one thing which has passed through
the changes spoken of; but the plant or animal is treated
as if it were one and the same with its own ancestors, and
its parts numerically identical with the more or less similar
parts in them, the only real unity being in the mind's con-
ception of them. The discussion of this belongs rather to
metaphysics than to logic, and it is only referred to here
because relations of this kind are often disguised under
the form of the simple proposition A is B.
86 THE ELEMENTS OF LOGIC, [part iv.
Of Material Fallacies^ or Fallacies extra Dictionem,
324. Of Material Fallacies, the first to be noted is Igno-
ratio Elenchi, i. e. irrelevant conclusion, or proving what
is not the question. Elenchus means the refutation of an
argument, and ignoratio elenchi strictly means ignoring the
proper contradictory of the proposition to be refuted. This
would include proving a particular, instead of a univer-
sal {i.e. proving a subcontrary instead of a contradictory),
and proving an irrelevant conclusion, ex. gr. arguing that
a thing is legally right, when the question is whether it is
morally right; or, on the other hand, that it is either
desirable or equitable, when the question is whether it is
legal.
325. A third case comes under the definition, but can
hardly be called a fallacy, viz. attempting to prove a
universal when the proposition we wish to refute is uni-
versal (i. e. mistaking the contrary for the contradictory).
This is not a fallacy, since the contrary includes the con-
tradictory ; but it exposes us to the risk of being refuted
by the proof of a particular, and is therefore a serious fault
in reasoning.
326. Petitio Principii, or Begging the Question, con-
sists in taking as a premiss, without proof, a proposition
which is equivalent to, or virtually involves, the conclusion ;
ex, gr. the ancient argument to prove that the earth is the
centre of the universe. The point towards which all
heavy bodies tend is the centre of the universe ; but the
point to which all heavy bodies tend is the centre of the
earth ; therefore the centre of the earth is the centre of
the universe. Or again, the proof that a certain work >vas
not written by its reputed author, since it contains allu-
. UNIVEESIT^S ))
PART IV.] OF FALLACIF^^C4ijO%^.^,^ %^Jj
sions to a theory which (we assume) was not known in his
day.
327. This fallacy is not limited to the cases in which
the conclusion is logically contained in a single premiss.
It must be remembered, that if one premiss of a syllogism,
or all but one in a train of reasoning, be admitted, the
truth of the conclusion turns on that of the remaining
premiss. The fallacy consists in taking this for granted
without attempt at proof. Inasmuch as in a legitimate
syllogism the conclusion is contained in the premisses
(taken together), the question may be asked. How does a
legitimate syllogism diifer from a petitio principii ? The
answer is, that in legitimate reasoning the premisses are
either admitted or have been proved, whereas in the
•fallacy in question the very premiss which would be
denied by those who reject the conclusion is taken for
granted. But the actual syllogism is logically valid, the
fallacy being in the matter, viz. in the assumption that
the conclusion has been proved from admitted prin-
ciples.
328. In disjunctive reasoning petitio principii very
easily occurs, by the enumeration of alternatives in the
disjunctive premiss being incomplete. Indeed, we can
hardly ever be sure that it is complete unless the members
are contradictory, and they may even appear to be contra-
dictory when they are only contrary, as in the old dilemma
to prove motion impossible. If a body moves, it moves
either in the place where it is, or in the place where it is
not; but both these are impossible, .'.motion is impos-
sible. By * place where it is,' is meant the actual space it
ocoapies. But then, * in the place where it is ' is not con-
tradictory of * in a place where it is not', both being sub-
88 TBE ELEMENTS OF LOGIC, [part iv.
divisions of *in a place', which in the sense in which the
word is taken is contradictory of motion. The alternative
omitted, namely, ' from the place where it is to the place
where it is not', is the only possible one.*
329. The question may be begged by a single word;
for instance, if we speak of an opinion we are attacking as
* foolish', * heretical', * audacious', etc. These have been
called 'question-begging epithets', since they connote the
very thing to be proved.
330. Arguing in a circle may be regarded as a species
of petitio principii. It consists in using the conclusion to
prove the disputed premiss, from which again the con-
clusion is inferred. For instance, in the example above
given : * This work is not the production of the reputed
author, since it contains allusions to a theory not known
in his time ' ; if we are challenged to prove the assump-
tion, and attempt to do so by saying, * It is not referred to
in any genuine work of that age', we now assume that the^
work in question is not genuine. This fallacy is likely to
be committed when we attempt to prove a principle which
is incapable of strict proof, only because it is above proof.
For example, an attempt is sometimes made to prove from
experience the principle used in induction, viz. that in like
circumstances like results follow. Thus : We have hitherto
found that in like circumstances like results have followed;
therefore we are justified {a) in concluding that in like
circumstances like results will always be found. Here we
already assume at {a) the very principle which we profess
* The alternative, * partly in one place, and partly in the other*, is
inadmissible. A part of a body cannot any more than the whol«^be in
two places at once. *
PART IV.] OF FALL A CIES. 89
to prove in the conclusion, viz. that we can argue from the
known to the unknown.
331. The fallacy called a non causa pro causa consists
in inferring a certain effect from something which is not
really a cause of it ; ex. gr. : -fiz^aX^A^a^
There will be war, for a comet has appeared.
There will be a change in the weather, for there
has just been a change in the moon.
A common case of this fallacy is the assumption that
one thing is the effect of another, merely because it has
followed it ; for instance, that an increase or decrease in
the prosperity of a nation is the effect of some particular
measure, or that rain takes the cold out of the air. This
form of the fallacy is called the fallacy of reasoning, Post
hoc ergo propter hoc.
332. Another common case is, inferring that one thing
is the effect of another, because it has often followed it,
without regarding the instances in which it has not fol-
lowed it. It is thus that prophetic dreams, &c., which
appear to come to pass, are carefully noted, while those
which fail are forgotten.
333. The fallacy of Many Questions (^plurium interroga-
iionum) consists in combining two or more questions in one,
and insisting on a simple answer. A traditional example
is : Have you left off beating your father ?
334. The fallacy of False Analogy consists in inferring
from a resemblance of relations a resemblance of the
things themselves in other relations. This is often aided
by the metaphorical use of language. Thus a nation or a
city is spoken of as feminine, and depicted in the form of
a woman, solely because it is conceived as giving birth
90 THE ELEMENTS OF LOGIC, [part iv.
to its * sons ' ; but the figure has been used as if a nation
resembled a woman in feebleness and helplessness.
335. The Argumentum ad hominem is an argument
founded on premisses not supposed to be universally
admitted, but admitted by a particular opponent. The
inference then is not absolute, that the conclusion is
true, but conditional — that the person who accepts the
premisses must accept the conclusion. It becomes a
fallacy only when this conditional conclusion is assumed
to be absolute or categorical. 3
/V'
I
CHAPTER 11.
0/ Methods of Proof and Exposition,
' 336. The order of Exposition of a system of knowledge
may be either Analytic or Synthetic.
337. The Analytic method proceeds from the complex
Z' ^' i to the simple ; from results to principles. Thus in Logic
the analytic method would commence with chains of rea-
v- , soning, and proceed through syllogisms and propositions
/ to terms. Or, in Astronomy, again, it would start from the
apparent motions of the planets, and proceed to the fact
of their revolution round the sun, arriving, as a last result,
at the law of gravitation.
338. The Synthetic method proceeds from the simple
y to the complex ; from principles to results. Such is, in
Y Logic, the method usually adopted, which commences
^ with Terms and ends with complex reasonings ; and in
,(y Astronomy, the method which begins with the law of
PART. IV.] OF FALLLACIES. 91
gravitation and deduces from it the apparent motions of
the planets. The Analytic method is suited to Discovery ;
the Synthetic is suited to Instruction.
339. Methods of Proof are either a priori or a pos-
teriori.
The a prjioi^i proof is drawn from that which is logically
first, viz., from the cjiise or the general law.
The a posteriori from the effect or the actual observa-
tion of facts. Thus the difference in the rate of vibration
of pendulums of different lengths is proved a priori when
it is deduced from mechanical principles, and a posteriori
when proved by actual experiment.
340. Explanation consists in referring a particular phe-
nomenon to a general law. Thus we are said to explain
the fall of a stone when we show that it is a consequence
of the general law of gravitation. By 'consequence' is
here meant logical consequence. The law of gravitation
is not the cause of the stone falling. Gravitation may be
called the cause, but is really only a general name for a
large class of similar facts.
341. It is clear that there is a limit to explanation. We
at last reach the most general principle — one which is
incapable of explanation. This does not mean that it is
more mysterious than other facts, but that it is more
general, and that we know none more general. Isolated
facts are sometimes called mysterious when we are not
able to bring them under any law, and yet, as there is no
isolated fact in nature, we believe that there is some law
in the case.
342. Explanation is Analytic : Deduction is Synthetic.
The fewer the general principles which we require to
assume, the more perfect in form is our science.
92 THE ELEMENTS OF LOGIC, [part iv.
343. But it is clear, as was said of explanation, that
there must be some limit. There must be some princi-
ples in every science or subject which are incapable of
proof ; just as a chain cannot hang unsupported. Thus,
the principles already referred to, of Universal Causation,
and of the Uniformity of Nature, cannot be logically proved.
Yet it would be an error to suppose that they rest on a
weaker foundation than other familiar laws of nature, which
we consider as proved, for the latter ultimately depend on
these and other unproved principles, and the inference
cannot be more certain than the principle from which it
is inferred.
APPENDIX.
EXERCISES.
In stating a proposition in strictly logical form, whether for
the purpose of immediate inference or of syllogism, the
student must 'bear in mind —
( I ). That when the predicate is a verb it must be resolved
so that the copula shall be simply * is ' or * is not.'
(2). That the grammatical subject is not always the
real or logical subject ; ex, gr.^ * There was no decisive
result from this experiment \ or, * No inference can be
drawn from this experiment \ Here the real subject is,
' This experiment \ of which it is asserted that it had no
decisive result.
In order to ascertain what the real subject and predicate
are, consider what it is that an assertion is made about, and
what it is that is asserted.
(3). When any proposition is compound (copulative,
adversative, etc.) it must be resolved.
(4). Some of the arguments in the following exercises
are valid, and some invalid. Having brought the reason-
ing into apparently syllogistic form, then, if this is cate-
gorical, the student should examine — {a) whether there
are more than three terms (really and not merely in ap-
pearance) ; {b) whether the middle term is distributed ;
{c) whether any extreme which is distributed in the con-
clusion has been distributed in the premisses. If there is
a disjunctive major, care must be taken that the enumeration
94 APPENDIX.
of alternatives is complete. Of course the other rules of cor-
rect reasoning must not be overlooked, but these are the
points most likely to escape notice.
Propositions.
State the following propositions in strictly logical form,
stating, in the case of simple categorical propositions,
what is the subject, what the predicate, and what the
quantity and quality of the proposition ; and, in the case
of complex propositions, whether they are conditional or
disjunctive : —
1 . Troja fuit.
2. Humanum est errare.
3. Much study is a weariness to the flesh.
4. All is not gold that glitters.
5. Many a little makes a mickle.
6. Sapientis est providere.
7. Great is Diana of the Ephesians.
8. All is not truth that is confidently asserted.
9. The most honest statesmen are not always the most
popular.
10. Scholarship is not what it was.
11. Poeta nascitur, non fit.
12. A little learning is a dangerous thing.
13. * Books are not absolutely dead'. (Milton) (wrongly
resolved by Mr. Jevons into * Some books are living \)
14. That may be legal, but it is not equitable.
15. Things equal to the same are equal to each other.
(Observe that in this proposition we could not substitute
* every thing ' for * things ' in the subject.)
16. It was Newton that discovered the law of gravita-
tion.
17. It was not Newton that discovered the orbits of the
planets.
18. Whales are not the only marine mammalia.
19. All these duties are too much for me.
20. Few know how little they know.
APPENDIX, 95
21.' God did not make men barely two-legged creatures,
and leave it to Aristotle to make them rational' (Locke).
22. To be or not to be, that is the question '.
23. * On earth there is nothing great but man ; In man
there is nothing great but mind '.
24. 'Non tali auxilio nee defensoribus illis Tempus
eget\
25. There is something unreasonable in most men.
26. Truth is stranger than fiction.
27. Such liberty is only licence.
28. Gold is the monetary standard of Great Britain.
(We cannot assert either of * all gold' or of *some gold'
that it is the standard.)
29. Hops are the staple produce of Kent.
30. No man is wholly bad.
31. * An honest man's the noblest work of God '.
32. * Adsum qui feci '.
33. After all, a promise is a promise. (How would you
contradict this ?)
34. The wisest of men sometimes errs.
35. Books are a source both of instruction and amuse-
ment.
Conversion,
Convert the following propositions : —
1 . A is father of B.
2. Cain killed Abel.
3. Jones struck Smith.
4. All cats have been kittens.
5. All water contains air!
6. It rains. (Incorrectly converted by Jevons thus:
Something that is letting rain fall is the atmosphere.
Test this by converting the following proposition, which
is similar.)
7. It is freezing. (N.B. — These must be regarded as
propositions expressing Real Existence.)
8. To be righteous is to be happy.
9. Boys will be boys.
96 APPENDIX,
10. Judges ought to be impartial.
1 1 . Life all men hold dear.
Related Propositions.
Assign the logical relation between the propositions in
each of the following groups : —
Every A is B.
No A is B.
Some Bs are not A.
Some As are not B.
Some things not A are not B.
Some Bs are A.
Some things not A are not B.
Every B is A.
Nothing not-A is B.
2. All birds are bipeds.
Some things not birds are bipeds.
Some things not bipeds are birds.
Some bipeds are not birds.
No birds are quadrupeds.
No birds are not bipeds.
Bipeds alone are birds.
3. I believe this story: I disbelieve this story.
4. I ought to do this : I ought not to do this.
5. The Tories are always right : The Tories are always
wrong.
6. Is either of the following propositions true, and if so,
which .? —
All Englishmen who do not take snufF are to be
found among Europeans who do not use to-
bacco.
All Englishmen who do not use tobacco are to be
found among Europeans who do not take snufF
(De Morgan).
APPENDIX 97
Immediate Inferences.
Examine the following immediate inferences : if they
are correct, state the logical principle under which they
come ; if incorrect, point out the fallacy : —
1. Every old man has been a boy ; .'.Some boys will
be old men.
2. No animals are self-made = All animals are not self-
made ; .'.All self-made things are not animals ; .y Sapie
things not animals are self-made.
3. No persons of the male sex are winged; .*. No
winged persons are of the male sex ; .*. Some winged per-
sons are not of the male sex ; .*. Some persons not of the
male sex are winged.
4. A man is an animal ; .-.The head of a man is the
head of an animal.
5. Truth always triumphs; .'.Whatever opinion has
triumphed is true.
On the Moods,
1. Show that if we substitute the conclusion for the
major premiss in Barbara or Celarent we obtain legitimate
premisses in the third figure, giving as a conclusion the
subalterna of the original major.
2. If in Bramantip we substitute the conclusion for the
minor premiss we obtain premisses which in the first figure
give a conclusion, the converse per accidens of the original
minor, or in the fourth figure would give a conclusion the
subalterna of the original minor.
3. If in Cesare we substitute the conclusion for the
major, we can draw in the third figure and mood Felapton,
a conclusion which is the subalterna of the converse of the
original major.
4. If in Camenes we substitute the conclusion for the
minor, we obtain premisses which will give in the fourth
figure and mood Fesapo a conclusion, the subalterna of
the original minor.
5. If in Camestres we substitute the conclusion for the
minor premiss, we obtain premisses which in the fouilh
98 APPENDIX.
figure and mood Fesapo lead to a conclusion, the subal-
terna of the converse of the original minor.
6. Show that in no other case will the substitution of
the conclusion for a premiss furnish a legitimate pair of
premisses.
7. Show that in the third figure we may with the same
minor premiss have contradictory major premisses, and
that the conclusions will be subcontrary.
8. Show that the same is true of the fourth figure.
9. Show that in no other case can any conclusion be
drawn from one premiss and the contradictory of the
other.
Miscellaneous Examples of Arguments.
Reduce the following reasonings to strict logical form,
if possible : if the syllogism is simple, find mood and
figure ; if complex, find to what species of complex
reasoning it belongs. If the reasoning is invalid, show
where the fallacy lies: —
1. None but whites are civilized : the ancient Germans
were whites ; .*. they were civilized.
2. None but whites are civilized : the Hindoos are not
whites ; .*. they are not civilized.
3. None but civilized people are whites : the Gauls
were whites ; .*. they were civilized.
4. This disease is not infectious ; for A and B were ex-
posed to it, and did not take it.
5. All moral precepts are binding on every man: some
of the precepts of Confucius are moral ; .*. the precepts of
Confucius are, to a certain extent, binding on every man.
6. Some vertebrates are bipeds : some bipeds are birds ;
.'. some birds are vertebrates.
7. Whales are not true fishes, for they cannot breathe
in water, and, besides, they suckle their young.
8. Food is necessary to life : whatever is necessary to
life must exist in all inhabited countries; .*. there are cer-
tain kinds of food that exist in all inhabited countries.
9. Every B is A : only C is A ; .*. only C is B.
APPENDIX, 99
10. Snow is white: white is a colour; .'.Snow is a
colour.
11. The sun is insensible : the Persians worship the sun ;
.-.the Persians worship a thing insensible (Port Royal
Logic).
12. That which does not consist of parts cannot perish
by the dissolution of its parts: the soul has no parts;
.*. the soul cannot perish by the dissolution of its parts.
(There seem to be two negative premisses ; is this really
the case .?) (Port Royal Logic.)
13. He who believes himself to be always right in his
opinion lays claim to infallibility : you always believe your-
self to be in the right in your opinion (else it would not be
your opinion) ; .*. you lay claim to infallibility. (Whately).
14. If benevolence were the whole of virtue, we should
not approve of benefits done to one more than to another,
as we actually do, inasmuch as we approve of gratitude and
acts of friendship. (Butler.)
15. Our notion of man includes a certain figure, as well
as rationality, for if a parrot talked rationally we should not
call it a man. (Locke.)
1 6. If the germination called spontaneous did not depend
on external germs, it would occur in perfectly pure air ; but
it does not. (The conclusion is suppressed.) (Try to reduce
the reasoning to a categorical form.)
17. If the planets shone by their own light, Venus would
not show phases. (The conclusion and one premiss are
suppressed.)
18. If the Claimant were the person he pretended to be
he would not have forgotten his Virgil so completely.
19. If education alone made men moral, the Redpaths
and Robsons would not have been guilty of forgery.
(More than one syllogism is implied.)
20. A historian ought to be impartial: in order to be
impartial it is necessary to know what has been said on
both sides ; .*. a historian ought to know what has been
said on both sides.
21. Every good book is worth reading more than once :
few books are worth reading more than once ; .-. few books
are good books. (This seems to be valid reasoning, and
too APPENDIX.
yet it seems to be a syllogism in the second figure with
both premisses affirmative.)
22. No evil should be allowed that good may come of
it: punishment is an evil; .'.punishment should not be
allowed that good may come of it. (Whately.)
23. If, as you say, every student ought to read this book,
it would probably sell well ; but it does not, therefore some
students at least ought not to read it.
24. Unless some one else has locked the door, I must
lock it ; I find some one has done so ; therefore I must not.
25. Vaccination is no protection whatever against small-
pox ; for A, B, and C were vaccinated, and yet have taken
smallpox.
26. Exposure to cold is good for children, for all the
grown people who have been exposed to cold as children
are strong.
27. No soldiers should be brought into the field who are
not well qualified to perform their parts: none but veterans
are well qualified to perform their parts ; therefore none but
veterans should be brought into the field.
28. Truth always triumphs : the present theory has
triumphed; .-. it is true.
29. Truth always triumphs over persecution : this doc-
trine has not triumphed over persecution (never having
been persecuted) ; .'.it is not true.
30. No X is Y : no Z is X ; .*. some things not Y are
notZ.
31. No judge is infallible; no judge is unskilled in law;
.'. some persons skilled in law are infallible.
32. If competitive examinations do not tend to the selec-
tion of the best men they ought to be abolished; if they
do so tend they should be applied to every appointment ;
therefore either competitive examinations should be
abolished, orthey should be applied to every appointment.
33. 'The essences of the species of things are nothing
else but abstract ideas. For the having the essence of any
species being that which makes anything to be of that
species, and the conformity to the idea to which the name
is annexed being that which gives a right to that name,
the having the essence and the having that conformity
APPENDIX i©i
must needs be the same thing ; since to be of any species
and to have a right to the name of that species is all one.'
(Locke.)
34. The perfection of virtue consists in perfect harmony
with right, so that right is done without a struggle, and
the nearer a man approaches to this, the more perfect his
virtue is ; .-. the greater the virtue, the less the self-denial.
35. The stronger the temptations and the inclination to
wrong-doing, the greater is the virtue which, in spite of
these, adheres to the right ; .-.the greater the self-denial
the greater the virtue.
36. ' If there be no difference between inward principles
but that of strength, we can make no distinction between
the murder of a father and an act of filial duty ; . . . but
in our coolest moments must approve or disapprove them
equally : than which nothing can be reduced to a greater
absurdity.' (Butler.)
37. Men in some ages have thought usury a vice ; men
in other ages have not; .*. the standards of virtue and vice
are not invariable.
Exercises on Prohahility*
I. An urn contains 1000 balls, numbered i to looo, from which one
is drawn. A witness, whose average credibility = p, testifies that a
particular number, say 926, has been drawn. What is the credibility
of this particular assertion ?
We must make some assumption as to the witness's knowledge of
the number of balls in the urn. Let us, then, assume that he knows the
number. Then in the case given there are two possible hypotheses : —
First Hypothesis — 926 has been drawn.
Here two things coincide. No. 926 is drawn ; the chance of this
being; , and the witness speaks truth, the chance of this
^ 1000 ^
being p. Hence the chance of the coincidence is proportional to
the product, .
1000
Second Hypothesis — 926 has not been drawn.
Here three things coincide. No. 926 not drawn ; the chance of
999
this is ; the witness speaks falsely : the chance of which
1000 ^ ^
:?: I — p, and thirdly, out of the 999 balls not drawn, he names
J02 APPENDIX.
this particular number ; the chance of his doing so being — .
The chance of the coincidence is proportional to the product of
these three, /. e. to -. Therefore the chances in favour of
lOOO
his evidence being true are to those against as p to i — p. The
resulting probability is therefore p, that is to say, the same as
the witness's average credibility.
2. Suppose the witness not to know the number of balls.
3. An urn contains I black, 99 red, and 900 white balls. One is
drawn, and a witness, whose average credibility = p, announces that
black has been drawn. What is the degree of credibility of his state-
ment }
The data are insufficient . We must be told what is the number 01
possible false assertions, that is, in this case, how many different colours
the witness is liable to mention falsely, whether by mistake or otherwise.
Let us first assume that he has the choice of 13 in all.
First Hypothesis —
Black is drawn,
The witness speaks truly,
_ I
~ 1000*
p.
Product, = -^
1000
Second Hypothesis—
Black was not drawn, = -??? .
lOOO
The witness speaks falsely, = i - p.
Of the twelve colours not drawn, he specifies black, = -.
Product, = 999 (I -P) .;
12000
Here the chances in favour of his truthfulness are to those against
as I2p to 999 (i — p). The probabiHty may therefore be stated as
—, •. Had he announced white, the probability would
I2p+ 99(i-p) ^ ^
, 9 io8p io8p
nave been
io8p + (I _ p) I07P + i'
4. An event whose antecedent probability is v is announced by a
witness whose average credibility is p. Let the antecedent probability
of his announcing this event falsely be x. What are the chances that
his statement is true }
INDEX
The nmnhers refer to the Paragraphs.
Abstraction, 19
Abstract Terms, 9.
Accident, Fallacy of, 319.
Analogy, 302.
False, 334.
Analytical Propositions, 96.
Analytic Method, 336.
A posteriori Proof, 339.
A priori Proof, 339.
Argumentum ad hominem, 335.
Aristotle's Dictum, 231.
Begging the question, 326.
Categories, iii.
Categorical, 164.
Circle, arguing in, 330.
Clearness, 35.
Common Terms, 12.
Complex Propositions, 261.
Syllogisms, 269.
Comprehension, 16.
Concepts, 25.
Concrete Terms, 9.
Conditional Propositions, 262.
Syllogisms, 270.
Conjunctive Propositions ; the
name given by some logicians
to Conditional Propositions.
Connotation, 16.
Continuity, Law of, 294.
Contradiction, Law of, 31.
Contradictory Propositions, 120.
Contraposition, 148.
Contrary Terms, 33.
Propositions, 126.
Conversion, 135.
'Converse,' incorrect use of, 150.
Copula, 56.
Copulative Propositions, 100.
Cumulative Probabilities, 288.
Definition, 38.
Denotation, 16.
Dilemma, 276.
Disjunctive Propositions, 266.
Syllogisms, 273.
Fallacies in, 328.
Distinctness, 35.
Distribution, 73.
Division, 47.
Enthymeme, 256.
Equipollent, 62.
Excluded Middle, Law of, 32.
Explanation, 340.
I Extension, 16.
104
INDEX.
Fallacies, 304.
Figure, 178.
Figu-es, Uses of, 209.
Form, 2.
Genus, 20.
— '■ Summum, 24.
Higher Notions, 22.
Hypothetical Propositions, 262.
Ignoratio Elenchi, 324.
Illicit process, 169.
Induction, 292.
Inverse : ' Every A is B ' is called
the inverse of ' Ever)' B is A.'
Matter, 2.
Mill's view of Syllogism, 300.
Modal Propositions, 104.
Moods, 213.
Non-connotative Terms, 27.
Obverse : the same as ' equipol-
lent ' : see 62.
Obversion : the change of a pro-
position into its equipollent ;
by some logicians called Per-
mutation.
Opposition, 119.
Paralogism, 307.
Permutation : see Obversion.
Petitio principii, 326.
Post hoc, Ergo propter hoc, 331.
Predicables,. 107*
Predicament = * Category' . * Pre-
dicamentum ' was formed by
Latin logicians in imitation of
the Greek KaTTjyopla.
Probable Reasoning, 284.
Property, 109.
, Propositions, Import of, 88.
Quality, 60.
Quantity, 60.
Quantified, 70.
Quantification of Predicate, 82.
Real Propositions, 96.
Reduction, 236.
of Complex Syllo-
gisms, 278.
Reductio ad Impossibile, 250.
* Same,' ambiguity of, 321.
Semi-logical fallacies, 306, 315.
Singular Terms, 12.
Sorites, 257.
Species, 20.
infima, 24.
Subaltemation, 116.
Subcontrariety, 130.
Substitution, Principle of, 157.
Syllogism, 162.
Synthetical, Propositions, 96.
— '■ Method, 336.
Term, 7.
Unfigured Syllogism, 253.
Uniformity of Nature, Law of, 293.
Verbal Propositions, 96.
END.
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