LOGIC; INCLUDING, A GENERALISATION OF LOGICAL PROCESSES IN
THEIR APPLICATION TO COMPLEX INFERENCES, by
J. N. KEYNES, M.A., FELLOW OF PEMBROKE COLLEGE, CAMBRIDGE. London:
MACMILLAN AND CO. CamMige: PRINTED BY C. J. CLAY, M.A. & SON, AT THE
UNIVERSITY PRESS. In addition to problems worked out in detail and unsolved
problems, by means of which the philosopher may test his command over logical
processes, the following pages contain a somewhat detailed exposition of
certain portions of what may be called the “logic,” but the Grecians preferred
to refer to as “Athenian dialectic” that seduced the Romans so much. This was
necessary in the case of disputed or doubtful points in order that the working
out of the problems might be made consistent and intelligible; there were also
some points concerning which I was dissatisfied with the method of treatment
adopted in the ordinary text-books. At the same time, this volume must be
regarded, not as superseding the study of dialectic, but as supplementing it.
While certain topics are dealt with in considerable detail others have been
omitted ; e.g-., the doctrines of Definition and Division and the Predicables
are not touched upon, no definition of the Science itself is given, and no
systematic discussion of first B principles
has been introduced. For a general outline of my views on the position of
“Athenian dialectic” I may refer the reader to my essay in Mind. For several
reasons I should have been glad to rewrite and in some respects to modify this
paper; but anything like an adequate treatment of the subject would have
enlarged the book considerably beyond the limits that I had assigned to it. I
have not endeavoured to distinguish definitely between book-work and problem ;
and the unanswered exercises are not separated and placed apart at the end of
the chapters, but are introduced at the points at which the student who is
systematically working through the book will find himself in a position to
solve them. Exercises of a similar character have not been to any considerable
extent multiplied, but I believe that no kind of problem relating to the
operations of Formal Logic has been overlooked. By reference to sections 261,
262, 281 — 285, the reader will find that the ordinary syllogism admits of
problems of some complexity. In the expository portions of Parts I. il. and
III., dealing respectively with Terms, Propositions, and Syllogisms, I have in
the main followed the traditional lines, though with a few modifications ; e,
g.^ in the systematization of immediate inferences, and in some points of
detail in connection with the syllogism to which I need not make further
reference here. For purposes of illustration Euler's diagrams are employed to a
greater extent than is usual in the Oxonian manuals. In Part IV., which contains
a generalisation of logical processes in their application to complex
inferences, a somewhat new departure is taken. So far as I am aware this part
constitutes the first systematic attempt that has been made to deal with formal
reasonings of the most complicated character without the aid of mathematical
symbols and without abandoning the ordinary non-equational or predicative form
of proposition. In this attempt I have met with greater success than I had
anticipated; and I believe that the methods which I have formulated will be
found to be as easy of application and as certain in obtaining results as the
mathematical, symbolical, or diagrammatic methods of Boole, Jevons, Venn and
others. The reader may judge of this for himself by comparing with Boole's own
solutions the problems discussed in sections 368, 369, 383 — 386; or by solving
by different methods other of the problems, e.g,, the very complex one
contained in section 408. The book concludes with a general method of solution
of what PJevons called the Inverse Problem, and which he himself seemed to
regard as soluble only by a series of guesses; Of the Questions and Problems
more than half are my own composition. Of the remainder, about a hundred have
been taken from various examination papers, and about sixty are from the
published writings of Boole, De Morgan, Jevons, Solly, Venn and Whately. In the
latter case the name of the author IS appended, generally with a reference to
the work from which the example is taken. In the case of problems selected from
examination papers, a letter is added indicating their source, as follows: —
C.= University of Cambridge ; L.= University of London ; N. = J. S. Nicholson,
Professor of Political Economy in the University of Edinburgh ; O. = University
of Oxford ; R.=G. Croom Robertson, Professor of Mental Philosophy and Logic in
University College, London; V. = J. Venn, Fellow and Lecturer of Gonville and
Caius College, Cambridge; W.= J. Ward, Fellow and Assistant Tutor of Trinity
College, Cambridge. The logicians to whom I have been chiefly indebted are De
Morgan, Jevons and Venn. De Morgan's various logical writings are rendered
somewhat formidable and uninviting by reason of the multiplication of symbols
and formulae which he is never tired of introducing, and this is probably the
reason why they are little read at the present time ; they nevertheless
constitute a mine of wealth for all who are interested in the developments of
Formal Logic. With Jevons I have continually found myself in disagreement on
points of detail, and it is possible that I may give the impression of having
taken up a special position of antagonism with regard to him. This is far from
being really the case. I believe that since Mill no one else has given such an
impetus to the study of Logic, and I hold that in more than one direction he
has led the way in new developments of the science that are of great
importance. To Mr Venn I am peculiarly indebted, not merely by reason of his
published writings, especially his Logic, but also for most valuable
suggestions and criticisms given to me while this book was in progress. I am
glad to have this opportunity of expressing to him my thanks for the ungrudging
help he has afforded me. I am also under great obligation to Miss Martin of
Newnham College and to Mr Caldecott of St John's College for criticisms which I
have found very helpful. 6, Harvey Road, Cambridge, Brief definitions of word^
name, term, symbol, concept. A word is an articulate sound, or the written
equivalent of an articulate sound, which either by itself or in conjunction
with other words, constitutes a name, or forms a Sentence. A name is "a
word taken at pleasure to serve for a mark which may raise in our mind a
thought like to some thought we had before, and which being pronounced to others,
may be to them a sign of what thought the speaker had or had not before in his
mind." Hobbes, A term is a name regarded as the subject or the predicate
of a proposition. A symbolf in its widest signification^ is a sign of any kind
; narrowing our point of view, it is any written sign ; and narrowing it still
more, it is a written sign which is employed without the realization at each
step of its full signification. Thus, when symbols are used in algebraical
reasoning, it is according to certain fixed rules, without reference to or
thought of their ulterior meaning. Names may themselves be employed as symbols
in this sense. Of course, in the widest sense, all names are symbols. A concept
is defined by Sir William Hamilton as " the cognition or idea of the
general character or characters, point or points, in which a plurality of
objects coincide." In other words, a concept is the mental equivalent of a
general name. A categorematic word is one which can by itself be used as a
term, i. ^., which can stand alone as the subject or the predicate of a
proposition. A syncategorematic word is one which cannot by itself be used as a
term, but only in combination with one or more other words. Any noun
substantive in the nominative case, or any other part of speech employed as
equivalent to a noun substantive, may be used categorematically. Adjectives are
sometimes said to be used categorematically by a grammatical ellipsis. In the
examples, “The rich are happy," "Blue is an agreeable colour,"
either a substantive is understood as being qualified by the adjective, or the
adjective is used as a substantive, that is, as a mark of something, not merely
as a mark qualifying something. Any part of speech, or the inflected cases of
nouns substantive, may be used categorematically by a suppositio maierialisy
that is, by speaking of the mere word itself as a thing ; for example, "
John's is a possessive case," " Rich is an adjective," **With is
an English word." Using the word term in the sense in which it was defined
in the preceding section, it is clear that we ought not to speak of
syncategorematic terms. A general name is a name which is capable of being
truly affirmed, in the same sense, of each of an indefinite number of things,
real or imaginary. A singular name is a name which is capable of being truly
affirmed, in the same sense, of only one thing, real or imaginary. K proper
name is a singular name given merely to distinguish an individual person or
thing from others, its application after it has been once given being
independent of any special attributes that the individual may possess . Thus,
“Prime Minister” of England is a general name, since at different times it may
be applied to different individuals. We may, for example, talk about ^' the
prime ministers of England of the present century." The name is however
made singular by the prefix "/^," meaning "the present prime
minister," or " the prime minister at the time to which we are
referring." Similarly any general name may be made singular ; for example,
man, the first man ; star, the pole star. The name God is singular to a
monotheist as the name of the Deity, general to a polytheist, or as the name of
anything worshipped by anybody. Universe is A proper name might perhaps be
defined as "a non-connotative singular name." But this definition
presupposes a distinction which is best given subsequently, and it would give
rise to a controversy, that also had better be postponed. Compare section 14. 4
terms; x^art I. general in' so' far is we distinguish different kinds of
universes, ^.^., the material universe, the terrestrial universe. Sic; it is
singular if we mean ^Ae universe, Sjxice is general if we mean a particular
portion of space, singular if we mean space in the aggregate. Water is general.
PBain takes a different view here; he says, "Names of Material — earth,
stone, salt, mercury, water, flame, — are singular. They each denote the entire
collection of one species of iraterial" {Logic, Deduction, pp. 48, 49).
But when we predicate anything of these terms it is generally of any portion
(or of some particular portion) of the material in question, and not of the
entire collection of it considered as one aggregate ; thus, if we say, <*
Water is composed of oxygen and hydrogen,^' we mean any and every particle of water,
and the name has all the distinctive characters of th^ general name. Similarly
with regard to the other terms mentioned in the above quotation. It is also to
be o1> served that we distinguish different kinds of stone, salt, &c, A
name is to be regarded as general if it may be potentially affirmed of more
than one, although it accidentally happens that as a matter of fact it can be
actually affirmed of only one, e.g., King of England and Spain, We must also
note the case in which we are dealing with a name that actually is not
applicable to any individual at all; e»g>, “President of the British
Republic.” A really singular name is distinguished from these by not being even
potentially applicable to more than one individual ; ^.^., the last of the Mohicans,
the eldest son of King Edward the First^, ' It seems desirable to make the
distinction implied in this para« graph; still I am not sure that it might not
in some cases be very difficult to apply it satisfactorily. Nearly all these
divisions of names tend to give rise in the last resort to metaphysical
difficulties; but, in my opinion, these should as far as possible be avoided in
a logical treatise. “Victoria” is the name of more than one individual, and can
therefore be truly affirmed of more than one individiiaU Is it therefore
general? Mill answers this question in the negative, and rightly, on the ground
that the name is not here affirmed of the different individuals in the same
sense.: Bain brings out this distinction very clearly in his definition of a
general name: '^A general nam& is applicable to a number of things in
virtue of their being similar, or having something in common." Victoria is
then not general but singular; and it belongs to the sub-class of proper names.
Are all collective names singular? A collective name is one which is the name .
of a group of things considered as one whole ; e,g.^ regiment, nation, army. A
collective name may be singular or general It is the name of a group or
collection of things, and so far as it is capable of being truly affirmed in
the same sense of only one such group, it is singular ; e.g,^ the 29th
regi-ment of foot, “the English nation,” the Bodleian library. But if it is
capable of being truly affirmed in the same sense of each of several such
groups it is to be regarded as general; e.g,j regiment, nation, library. Bain
writes as if a name could be general and singular at the same time,— "
Collective names as nation, army, multitude, assembly, universe, are singular;
they are plurality combined into unity. But, inasmuch as there are many
nations, armieSj assemblies, the names are also general. There being but one
'universe', that term is collective and singular". I should rather say
that ^s the above stand, with the possible excepttion of universe, they are not
singular at all. . Mill and others imply that there is a distinction between
collective and general names* The real distinction however is between the
colledtve and distributive use of names. A collective name such as nation, or
any name in the plural number, is the name of a collection or group of things.
These we may regard as one whole, and something may be predicated of them that
is true of them only as a whole; in this case the name is used collectively. On
the other hand, the group may be regarded as a series of units, and something
may be predicated of these which is true of them only taken individually; in
this case the name is used distributively. Also, when anything is predicated of
a series of such groups the name is used distributively. The above distinction
may be illustrated by the propositions, — All the angles of a triangle are
equal to two right angles. All the angles of a triangle are less than two right
angles. The subject term is the same in both these cases, but in the first case
the predication is true only of the angles all taken together, while in the
second it is true only of each of them taken separately; in the first case
therefore the term is used collectively, in the second distributively. The
peculiarity, then, of a collective name is that it can be used collectively in
the singular number, while other names can be used collectively only in the
plural number ; compare, for example, the names * clergyman' and 'the Clergy.'
Collective names in the plural number may themselves be used distributively,
and it is therefore not correct to say that all collective names are singular.
It may indeed be held that, while this is true, still when a name is used
collectively, it is equivalent to a singular name. For example. The whole army was
annihilated, The mob filled the square. But I am doubtful whether even this is
true in such a case as the following, — In all cases all the angles of a
triangle are equal to two right angles. Please select the terms that are used
collectively in the following propositions; also classify the terms contained
in these propositions according as they are collective, singular, and general
respectively, and find in what way these classes overlap one another: “The
Conservatives are in the majority in the House of Lords, All the tribes
combined. The nations of the earth rejoiced. Crowds filled all the churches.
One generation passeth away and another generation Cometh. Your boxes weigh 140
lbs. The volunteers mustered in considerable numbers. Time flies. True poets
are rare. Those who succeeded were few in number. The mob was soon dispersed.
Our armies swore terribly in Flanders, The multitude is always in the wrong.
Mill defines abstract and concrete names as follows : — "A concrete name
is a name which stands for a thing; an abstract name is a name which stands for
an attribute of a thing" {JiogiCy i. p. 29)^ In many cases, this
distinction is of easy application; for example, triangle is the name of
something that possesses the attribute of being bounded by 1 The references are
to the ninth edition of Mill's Logic, three straight lines, and it is a
concrete name; triangularity is the name of this distinctive attribute of
triangles, and it is an abstract name. But there aire other cases to which the
application of the distinction is difficult; and an attempt at more precise
definition is liable to involve us in metaphysical discussions such as the
logician should if possible avoid The first question that arises is what
precisely is meant by the word things when it is said that a concrete name is
the name of a thing. By a thing, we may mean anything that exists ; but we
cannot mean that here, since "attributes" exist, and the distinction
between concrete and abstract name^ would vanish. Again, by a thing we may mean
a substance ; but substances are contrasted with feelings as well as with
attributes^ and this threefold division would make names of feelings neither
abstract nor concrete^ which can hardly be intended. With regard to the proper
place of names of states of consciousness it would be generally agreed to call
them concrete. Thus, while sensibility, the faculty of experiencing sensation,
is an abstract name, the name of a sensation itself should be regarded as
concrete, being the name of something which possesses attributes, for example,
of being pleasurable or painful, of being a sensation of touch or one of
hearing. But here again a difficulty arises, since, as pointed out by Mill, in
many cases " feelings have no other name than that of the attribute which
is grounded on them." For example, by colour we may mean sensations of
blue, red, green, &c., or we may mean the attribute which all coloured
objects possess in common. In the former case, colour is a concrete name, in
the latter an abstract name. Sounds again, is concrete, in so far as it is the
name of a sensation, e,g,, "the same sound is in my ears which in those
days I heard"; but in the following cases, it should rather be regarded as
abstract, — " a tale full of sound and fury," " a name harsh in
sound." The matter is still further complicated if Mill's view is taken,
and attributes are analysed into sensations, "the distinction which we
verbally make between the properties of things and the sensations we receive
from them, originating in the convenience of discourse rather than in the
nature of what is signified by the terms." For logical purposes however we
certainly need not pursue the analysis so far as this. But still another
difficulty arises from the fact that we sometimes speak of attributes themselves
as having attributes; and so far as this is permissible, we must agree with
Professor Jevons that ** abstractness becomes a question of degree." It
may be said that civilization is abstract regarded as an attribute of a given
state of society, but that it becomes concrete regarded as itself possessing
the attribute of progressiveness or the attribute of stationariness. Besides
all the above, we have to notice that terms originally abstract are very liable
to come to be used as concrete, and this may create further confusion. Thus,
Professor Jevons remarks,^-" -^^/chap.il
] terms. 15 arily a subject;" (and it was said to
connote or signify secondarily the subject). Thus •* white" was regarded
as con- notative, whilst the original substances or attributes, as
"man" or " whiteness " were called absolute; the former
signifying primarily a subject, the latter not signifying a subject at alL Only
adjectives and participles therefore (words called by Professor Fowler "
attributives ") are connotative in this sense. Mill {LogiCy I. p. 42,
note) says that the schoolmen used it in his own sense, though some of their
expressions are vague. He quotes James Mill as using it more nearly in the
sense ascribed by Mansel to the schoolmen. (iii) Professor Fowler uses the term
connotative in a sense different from that of Mill. "A term may be said to
denote or designate individuals or groups of individuals, o connote or mean
attributes or groups of attributes." In this sense, general names are both
connotative and deno- tative; abstract names are connotative but not
denotative', (whereas, according to Mill, they are generally speaking
denotative but not connotative). This use of the term avoids some difficulties,
and I am inclined to regard it as preferable to Mill's. Indeed Mill himself
seems to suggest it in one place. He says that James Mill "describes
abstract names as being properly concrete names with their connotation dropped
: whereas, in his own view, it is the deiioXsXion which would be said to be
dropped, what was previously connoted becoming the whole signification "
{Logic^ I. p. 42 note). As far as we can I think we should speak merely of the
" denotation " and " connotation " of names, rather than of
** denotative " and " connotative names ". ^ Fowler, Deductive
Logic^ p. 19. i6 .TERMS. [parti.18.Is' every property possessed by a class-
con'- noted by the class-name ? Unfortunately we do not find complete agreement
among logicians with regard to the answer that should be given to this question
; and I am inclined to think that in discussing points connected with **
connotation " writers sometimes misunderstand each other, because they do
not apprehend that there is fundamental disagreement between them upon this
point. I will first give Mill's answer to the question, an answer with which I
should myself concur. By the connotation of a class-name he does not mean a/i
the properties that may be possessed in common by the class, but only those on
account of the possession of which any individual is placed in the class, or
called by the name. In other words, we include in the connotation of a class-
name only those attributes upon which the classification is founded, and in the
absence of any of which we should not regard the name as applicable. For
example, although all equilateral triangles are equiangular we should not
include equiangularity in the connotation of equilateral triangle; although all
kangaroos may happen to be Australian kan- garoos, this is not part of what we
mean to imply when we use the name, — an animal subsequently found in the
interior of New Guinea, but otherwise possessing all the
propertiesofkangarooswould not have the name kangaroo denied to it; although
all ruminant animals are cloven- hoofed, we cannot regard cloven -hoofed as
part of the meaning of ruminant, and we may say with Mill that were n animal to
be discovered which chews the cud, but has ts feet undivided, it would
certainly still be called ru- minant. The above meaning of connotation is that
to which in CHAP. II.] TERMS. 17 my opinion we should strictly adhere. It is of
course open to any one to say that he will include in the connotation of a
class name all the properties possessed in common by all members of the class ;
but this is simply to use the term in adifferent sense. It is used in this
sense by a writer in a recent number of Mind, ** On the connotative side a name
means, to usy all those qualities common to the class named with which we are
acquainted; — all those properties that are said to be 'involved in our idea'
of the thing named. These are the properties that we ascribe to an object when
we call it by the name. But, just as the word * man,' for example, denotes
every creature, or class of creatures having the attributes of humanity,
whether we know him or not, so does the word properly connote the whole of the
pro- perties common to the class, whether we know them or not. Many of the
facts, known to physiologistsandanatomistsaboutthe constitution of man's brain,
for example, are not involved in most men's idea of the brain : the possession
of a brain precisely so constituted does not, therefore, form any part of their
meaning of the word *man.' Yet surely this is properly connoted by the
word" (E. C. Benecke, in Mind, 1881, p. 532). Professor Jevons also uses
the term in the same sense. " A term taken in intent (connotation) has for
its meaning the whole infinite series of qualities and circum- stances which a
thing possesses. Of these qualities or cir- cumstances some may be known and
form the description or definition of the meaning; the infinite remainder are
unknown" {Pure Logic, p. 4). Professor Bain appears to use the term in an
intermediate sense, including in the connotation of a class-name not all the
attributes common to the class but all the independent attributes, that is, all
that cannot be derived or inferred from others. It ought to be made very clear
in any discussion con- K. L. 2 i8 TERMS. [parti. ccrning the connotation of
names in which of these several senses we are using the term **
connotation" itself. It may be said that to use the term in Mill's sense,
and to make connotation depend on what is intended to be implied by the mere
use of the name, is to make it vary with every different speaker. By the same
nametwopeople may mean to imply different things, that is, the attributes they
would include in the connotation of the name would be different; and not
unfrequently some of us may be unable to say precisely what is the meaning that
we our- selves attach to the words we use. This is a fact which it is most
important to recognise. But for the purposes of formal logic we may assume that
every name has a fixed and definite connotation. The object of the definition
of names already in use is just to give this ; and in the case of an ideal
language properly employed every name would have the same fixed and precise
meaning for everybody. 14, Are proper names connotative or non- connotative }
On the question here raised Mill speaks decisively, — **The only names of
objects which connote nothing are proper names ; and these have, strictly
speaking, no signifi- cation" (Logic^ I. p. 36); and most logicians are in
agree- ment with him. An opposite view is however taken by Jevons, and some
others (r.^., F. H. Bradley, T, Shedden), In one or two places I am inclined to
think that Jevons tends somewhat to obscure the point at issue. Thus with
reference to Mill he says, — "Logicians have erroneously asserted, as it
seems to me, that singular terms are devoid of meaning in intension, the fact
being that they exceed all ther terms in that kindofmeaning" (PrindpUs of
Science^ I. pp* 3a, 33i with a reference to Mill in the foot-note). CHAP. II.]
TERMS. 19 But Mil] distinctly says that some singular names are connotative,
e,g,^ the sun, the first emperor of Rome (Logic, I. pp. 34, 5). Again, Jevons
says, — " There would be an impossible, breach of continuity in supposing
that after narrowing the extension of * thing* successively down to nimal,
vertebrate, mammalian, mati, Englishman, educated at Cambridge, mathematician,
great logician, and so forth, thus increasing the intension all the time, the
single remain- ing step of adding Augustus de Morgan, Professor in Uni- versity
College, London, could remove all the connotation, nstead of increasing it to
the utmost point" (Studies in Deductive Logic, pp. 2, 3). But every one
would allow that we may narrow down the extension of a term till it becomes
individualised without destroying its intension or connota- tion ; "the
present Professor of Pure Mathematics in Uni- versity College, London" is
a singular term, — ^we cannot diminish the extension any further, — ^but it is
certainly connotative. We must then clearly understand that the only contro-
versy is with regard to what are strictly /r^^r names. Even yet there is a
possible source of ambiguity that should be cleared up. If by the connotation of
a name we meanailthettributes possessed by the individuals denoted by the name,
or even all the independent attributes, Professor Jevons's view may be correct.
This does appear to be what Jevons himself means, but it is distinctly not what
Mill means, — he means only those attributes which are implied by the name
itself. Jevons puts his case as ollows : — "Any proper name, such as John
Smith, is almost ithout meaning until we know the John Smith in question. It is
true that the name alone connotes the fact that he is a Teuton, and is a male;
but, so soon as we know the exact individual it denotes, the name surely
implies, also, 2 — 2 20 TERMS. [part I. the peculiar features, form, and
character, of that individuaL In fact, as it is only by the peculiar qualities,
features, or circumstances of a thing, that we can ever recognise it, no name
could have any fixed meaning unless we attached to it, mentally at least, such
a definition of the kind of thing denoted by it, that we should know whether
any given thing was denoted by it or not. If the name John Smith does not
suggest to my mind the qualities of John Smith, how shall I know him when I
meet him ? for he certainly does not bear his name written upon his brow ''
(Elementary Lessons in Logic, p. 43). A wrong criterion of connotation in
Mill's sense is here taken. The connotation of a name isnotthequality or
qualities by which I or any one else may happen to recognise the class which it
denotes. For example, I may recognise an Englishman abroad by the cut of his
clothes^ or a Frenchman by his pronunciation, or a proctor by his bands, or a
barrister by his wig ; but I do not mean any of these things by these names,
nor do they (in Mill's sense) form any part of the connotation of the names.
Compare two such names as ** John Duke Coleridge" and "the Lord Chief
Justice of England." They denote the same individual, and I should
recognise John Duke Coleridge, and the Lord Chief Justice of England by the
same attributes ; but the names are not equivalent, — the one is given as a
mere mark of a certain individual to distinguish him from others, and it has no
further signification; the other is given on account of the performance of
certain functions, which ceasing the name would cease to apply. Surely there is
a distinction here, and one which it is important that we should not overlook.
Nor is it true that such a name as "John Smith*' connotes "Teuton,
male, &c." John Smith might be a race-horse, or a negro, or the
pseudonym of a woman, as in CHAP. II.] TERMS, 21 the case of George Eliot. In
none of these cases could a name be said to be misapplied as it would if a
horse were called a man, or a negro a Teuton,orawoman a male. But it may fairly
be said that in a certain sense many proper names do suggest something, that at
any rate they were chosen in the first instance for a special reason. For
example, Strongi'th'arm, Smith, Jungfrau. Such names however even if in a
certain sense connotative when first imposed soon cease to be connotative in
the way in which other names are connotative. Their application is in no way
dependent on the continuance of the attribute with reference to which they were
originally given. As Mill puts it, ^^the name once given is independent of the
reason,^* Thus, a man may in his youth have been strong, but we should not
continue to call him strong when he is in his dotage ; whilst the name
Strongi'th'arm once given would not be taken from him. The name
"Smith" may in the first instance have been given because a man plied
a certain handicraft, but he would still be called by the same name if he
changed his trade, and his descendants continue to be called Smiths whatever
their occupations may be. Nor can it be said that the name necessarily implies
ancestors of the same name. Proper names of course become connotative when they
are used to designate a certain type of person ; for example, a Diogenes, a
Thomas, a Don Quixote, a Paul Pry, a Benedick, a Socrates. But, when so used,
such names have reallyceasedtobeproper names at all; and they have come to
possess all the characters of general names. 15. Discuss the question whether
the following terms are respectively connotative or non-connota- 22 TERMS.
[part I. tive : — ^Westminster Abbey, the Mikado of Japan,Barmouth. [L.] 16.
Enquire whether the following names are respectively connotative or
non-connotative: — Caesar, Czar, Lord Beaconsfield, the highest mountain in
Europe, Mont Blanc, the Weisshorn, Greenland, the Claimant, the pole star,
Homer, a Daniel come to judgment. 17. Can any abstract names possess both deno-
tation and connotation.^ In Fowler's use of the term all abstract names are
con- notative, that is, they at once suggest or imply attributes ; while none
are denotative, that is, they do not denote individuals or groups of
individuals. Professor Fowler himself admits that it sounds paradoxical to say
that Abstract names are not denotative, but he is of opinion that the
employment of the expressions in his sense would simplify the statement and
explanation of many logical difficulties. I am inclined to think that the
present is a case in point. Mill holds that while most abstract names are
non-con- notative, still "even abstract names, though the names only of
attributes, may in some instances be justly considered as connotativeforattributesthemselvesmayhaveattributes
ascribed to them j and a word which denotes attributes may connote an attribute
of those attributes" {Logic, i. p. 33). I have some difficulty in
interpreting this passage. Suppose that we have a connotative abstract name
denoting the attri- bute A and connoting the attribute B; now a connotative
name is always defined by means of its connotation, and we shall therefore
define our term by saying that it connotes BCHAP. 11.] TERMS, 23 without any
reference whatever to A. What then will dis- tinguish it from the concrete term
denoting whatever pos- sesses B ? The solution of the difficulty seems to be
that when we talk of one attribute having another ascribed to it, the term
denoting it becomes concrete rather than abstract Comparing Mill's definitions
of an abstract name and of a connotative name, I fail to understand how the
same name can be both \ 18. Explain and discuss the statement: — "In a
series of common terms arranged in regular sub- ordination to one another, the
denotation and con- notation vary inversely." 19. Explain the following
statements : — (a) If a term be abstract, its denotation is the same as the
connotation of the corresponding con- crete ? (b) Of the denotation and
connotation of a term, one may, both cannot, be arbitrary. {c) Names
withindeterminateonnotationare npt to be confounded with names which have more
than one connotation. 20. Verbal and Real Propositions. A Verbal Proposition is
one in which the connotation Mr Killick
in his Handbook of Milts Logic makes Mill include in the class of connotative
names such abstract names as are the names of groups of attributes {e,g.^
humanity). I do not think that MiU himself intended this, nor do I think that
the view is a correct one (?.^., accord- ing to Mill's own usage of terms). If
an abstract name has both deno- tation and connotation because it is the name
of a group of attributes, on what principle shall we distinguish between the
attributes that it denotes and those that it connotes ? 24 TERMS. [part I. of the
predicate is a part or the whole of the connotation of the subject Bain
describes the verbal proposition as "the notion under the guise of the
proposition"; and it is certainly convenient to discuss verbal
propositions in con- nection with the connotation of names or the intension of
concepts. The most important class of verbal propositions are definitions, the
essential function of which is to analyse the connotation of names \ The least
important class are absolutely tautologous or identical propositions, ^.^., all
A is A, a man is a man. Real Propositions, on the other hand, "predicate
of a thing some fact not involved in the signification ofthename by which the
proposition speaks of it ; some attribute not connoted by that name." The
same distinction is also expressed by the pairs of terms, analytic* and
synthetic, explicative* and ampliative, essential" and accidental. ^
Besides propositions giving such an analysis more or less com- plete, the
following classes of propositions are frequently included under the head of
verbal propositions : where the subject and predicate are both proper names,
e.g.y Tully is Cicero; where they are dictionary synonyms, e.g., wealth is
riches, a story is a tale, charity is love. All such propositions however can
hardly be brought under the head of verbal propositions as defined in the text.
At any rate if we have decided that a proper name is not connotative, it is
clear that in no proposition having a proper name for its subject can the
predicate be any part of the connotation of the subject. To include these
classes we must define a verbal proposition as a proposition which is wholly
concerned with the meaning or application of names, a real proposition as one
which is concerned with things or qualities. Even with these definitions,
however, while it is a verbal proposition to say that Tully is Cicero (/.^.,
that these names have the same appli- cation), it is a real proposition to say
that Tully is an individual who is also denoted by thenameCicero.^Itshouldbe
carefully observed that while the term verbal is some- CHAP. II.] TERMS. 25 21.
Which of the following propositions should you regard as Real, and why ? Homer
wrote the Iliad, nstinct is untaught ability, Instinct is hereditary
experience. [c] "Homer wrote the Iliad" is regarded by Bain as a
verbal predication, " We know nothing about Homer except the authorship of
the Iliad. We have not a meaning to attach to the subject of the proposition, *
Homer', apart from the predicate, * wrote the Iliad.' The affirmation is nothing
more than that the author of the Iliad was called Homer" {Logic,
Deduction, p. 67). Taking the definition of verbal proposition given in the
text, and holding that no proper name is connotative, this view must clearly be
ejected. If however by a verbal proposition we mean one that relates in any way
to the application of names, (/>., taking the definition given in the note),
there may be some- thing to say for it. But is it true that we attach nothing
more to "Homer" than ** wrote the Iliad"? Do we not, for
example, attach to "Homer" the authorship of other poems, and also an
individuality * ? If it is the fact that the Iliad was the work of various
authors, as has been times stretched so as to include such a proposition as
"Tully is Cicero," this is never the case with the
termsanalytic,explicative,essential. These terms are strictly limited to
propositions which give no informa- tion whatever (even with regard to the
application of names) to any one who is fully acquainted with the connotation
or intension of the subject ^ I do not of course mean that this is the
connotation of ** Homer," for I hold that no proper names are connotative.
I mean that Homer denotes for me a certain individual who was a Greek, who
lived prior to a certain date, and who was the author of certain poems other
than the Iliad. 26 TERMS. [part i. asserted, would not the proposition become
false ? Still, we should perhaps admit that we have here a limiting case. Some
light may be thrown on the point thus raised by an answer once sent in by an
examinee ; " The accepted opinion is that the Iliad was not written by
Homer, but by another man of the same name." "Instinct is untaught
ability" and "Instinct is here- ditary experience" may be
regarded as verbal and real espectively. 22. Is it a verbal proposition to say
that it is hotter in summer than in winter ? Examine the following statements:
A free in- stitution is a contradiction in terms ; so is a perfect creature.
[v.] 23. If all ;r is j/, and some x is ^, and / is the name of those z's which
are ;r; is it a verbal pro- position to say that all/ is^ } [v.] 24. Give one
example of each of the following, — (i) a collective generalname,(ii)a singular
abstract name, (iii) a connotative abstract name, (iv) a con- otative singular
name ; or, if you deny the possi- bility of any of these combinations, state
clearly your reasons. CHAPTER III. POSITIVE AND NEGATIVE NAMES. RELATIVE NAMES.
25. Positive and Negative Terms. The essential distinction between positive and
negative names as ordinarily understood may be expressed as follows : — a
positive name implies the preserue of certain definite attributes; a negative
name implies the absence of one or other of certain definite attributes.
"Every name," as remarked by De Morgan, "applies to everything
positively or negatively " ; for example, every- thing either is or is not
a horse. Every name then divides all things in the universe into two classes.
Of one of these it is itself the name; and a corresponding name can be framed
to denote the other. This pair of names, which between them denote the whole
universe, are respectively positive and negative. But which is which ? Which is
the negative name, since each {positively denotes a certain class of objects?
The distinction lies in the manner in which the class is determined. We may say
that in a certain sense a strictly negative name has not an independent
connotation of its own ; its denotation is determined by the connotation of
thecorrespondingpositivename.Itdenotesanindefinite and unknown class outside a
definite and limited class. In other words, by means of its connotation 28
TERMS. [part I. we first mark off the class denoted by the positive name, and
then the negative name denotes what is left. The fact that its denotation is
thus determined is the distinctive characteristic of the negative name. We have
here supposed that between them the positive name and the corresponding
negative name exhaust the whole universe. But something different from this is
often meant by a negatiye name. Thus De Morgan considers hat parallel and alien
are negative names.' "In the formation of language, a great many names
are, as to their original signification, of a purely negative character : thus,
parallels are only lines which do not meet, aliens are men who are not Britons
(i,e.^ in our country)" {Formal Logic^ p. 37). But these names clearly
have not the thorough- going negative character that I have just been ascribing
to negative names. The difference will be found to consist in this, that in the
sense in which alien is a negative name, the positive and negative names
(Briton and alien) do not between them exhaust the entire universe, but only a
limited universe, namely, in the given case, that constituted by the
inhabitants of Great Britain, We may perhaps distinguish between names
absolutely negative^wherethereferenceistothe entire universe; and names
relatively negative, where the reference is only to some limited universe. Now
it will be seen that in the use of such a term as not-white there is a possible
ambiguity; we must decide whether in any given instance the name is to be
regarded as absolutely or only as relatively negative. Mill chooses the former
alternative ; " not-white," he says, '* denotes all things whatever
except white things." De Morgan and Bain however consider that in such a
case the reference is not to the whole universe but to some particular universe
only. Thus, in contrasting white and not-white we are CHAP. III.] TERMS. 29
referring solely to the universe of colour; not-white does not include
everything in nature except white things, but only things that are black, red,
green, yellow, &c, that is, all toloured things except such as are white \
Whately and Jevons agree with Mill ; and from a logical point of view I think
they are right. Or rather I would say that two such terms as S and not-.S' must
between them exhaust the universe of discourse, whatever that may be ; and we
must not be precluded from making this, if we care to do so, the entire
universe of existence. That is, not-5 may be called upon to assume the
absolutely negative character*. For if we are unable to denote by not-5 all
things whatsoever except iS, it is difficult toseein what way we shall be able
to denote these when we have occasion to refer to them. On the other hand, we
must also be empowered to indicate a limitation to a particular universe where
that is intended. By not-*S then referred to without qualification expressed or
implied by the context I would understand the absolute negative of S\ but I
should be quite prepared to find a limitation to some more restricted universe
in any particular instance. It should be noted that in the case of a limited
uni- verse it is sometimes difficult to say which of the pair of contrasted
names is really to be regarded as the negative name. For example, De Morgan
says that parallel is a negative name, since parallel lines are simply lines
that do not meet But we might also define them as lines such that ^ Thus, on
Bain's view it would be incorrect to say that an im- material entity such as
honesty was not-white. • On this view, "not- white'* might be used to
denote not merely coloured things that are not white, but also things that are
not coloured at all. It would for example be correct to say that honesty was
not- white. 30 TERMS. [part i. if another line be drawn cutting them both, the
alternate angles are equal to one another; and then the name appears as a
positive name. Similarly in the universe of property, as pointed out by De
Morgan, persoruU and realarerespectivelythe negatives of each other ; but if we
are to call one positive and the other negative, it is not quite clear which
should be which. For a suggestion of Mr Monck's as to the definition of
negative terms, see section 29. 26. Privative Names, To the distinction between
positive and negative names, MDl adds a class of names called privative. "
A privative name is equivalent in its signification to a positive and a
negative name taken together; being the name of some- thing which has once had
a particular attribute, or for some other reason might have been expected to
have it, but which has it not Such is the word dlindy which is not equivalent
to nof seeing, or to not capable of seeing, for it would not, except by a
poetical or rhetorical figure, be applied to stocks and stones" {Logic, i.
p. 44). Perhaps also idle, which Mill gives as a negative, should rather be
regarded as a privative term. It does not mean merely "not-working,"
but "not-working where there is the capacity to work." We should
hardly speak of a stone as being **idle.'' The distinction here indicated does
not appear to be of logical importance. 27. How far is it true that, as
ordinarily under- stood, negative terms have a definite connotation, while in
Logic they have not ? So far as it is true, how would you explain the fact }
[w.] CHAP.III.]TERMS.2128.Contradictoryandcontraryterms. A positive term and
its corresponding negative term are called contradictories, A pair of
contradictory terms are so related that between them they exhaust the entire universe
o which reference is made, whilst in that universe there is no individual of
which both can be at the same time affirmed. The nature of this relation is
expressed in the two laws of Contradiction and Excluded Middle. Nothing is at
the same time both X and not- A"; Everything is X or not-X For the
application of the above to complex terms, see Part iv. The contrary of a term
is usually defined as the term denoting that which is furthest removed from it
in some particular universe ; e.g., black and white, wise and foolish. Two
contraries may in some cases happen to make up between them the whole of the
universe in question, e.g., Briton and alien ; but this is not necessary, e.g.,
black and white. It follows that although two contraries cannot both be true of
the same thing at the same time, they may both be false. The above may be
called the material contrary. In the case of complex terms, we may also assign
a formal con- trary, as is shewn in Part iv. 29. Illustrate Mill's statement
that " names which are positive in form are often negative in reality, and
others are really positive though their form isnegative."Thefact that a
really positive term is sometimes negative in form results from the
circumstance that the negative pre-^ fix is sometimes given to the contrary of
a term. But we have seen that a term and its contrary may both be positive. 32
TERMS. [part i. For example, pleasant and unpleasant; "the word un-
pleasant, notwithstanding its negative form, does not con- note the mere
absence of pleasantness, but a less degree of what is signified by the word
painful^ which, it is hardly necessary to say, is positive." On the other
hand, some names positive in form may be regarded as relatively nega- tive,
PART II. PROPOSITIONS. CHAPTER I. KINDS OF PROPOSITIONS. THE
QUANTITY AND QUALITY OF PROPOSITIONS. 36. Categorical, Hypothetical and
Disjunctive Propositions. For logical purposes, a Proposition may be defined as
" a sentence indicative or assertory,*'' (as distinguished, for example,
from sentences imperative or exclamatory); in other words, a proposition is a
sentence making an affirma- tion or denial, as — All 5is /*, No vicious man is
happy. A proposition is Categorical if the affirmation or denial is absolute, as
in the above examples. It is Hypothetical if made under a condition, as — If
A\& B^ Cis D; Where ignoranceisbliss,'tisfolly to be wise. It is
Disjunctive if made with an alternative, as — Either P is ^, or X is Y\ He is
either a knave or a fool*. ^ It should be observed that in a disjunctive
proposition there may be two distinct subjects asinthefirstoftheabove examples,
or only one as in the second. Disjunctive propositions in which there is only
one distinct subject are the more amenable to logical treatment. 3— a 36
PROPOSITIONS. [part ii. [The above threefold division is adopted by Mansel. It
is perhaps more usual to commence with a twofold division, the second member of
which is again subdivided, the term Hypothetical being employed sometimes in a
wider and sometimes in a narrower sense. To prevent confusion, it may be
helpful to give the following table of the usage of one or two modern logicians
with regard to this division. Whately, Mill and Bain : — 1. Categorical. 2.
Hypothetical, lypotnencai, r Conditional, or Compound, -I) ! . i- _. 1 1(2)
Disjunctive, or Coniolex. V ' •' or Complex Hamilton and Thomson : — 1.
Categorical. X-. jv 1 (M Hypothetical 2. Conditional, i) : t^. . ,. ((2)
Disjunctive. Fowler (following Boethius) : — 1. Categorical. 2. Conditional
((i) Conjunctive. or Hypothetical. ((2) Disjunctive. Mansel, as I have already
remarked, gives at once a
threefolddivision,1.Categorical2.HypotheticalrConditional.3.Disjunctive.Hestateshisreasons
for his own choice of terms as follows : — " Nothing can be more clumsy
than the employ- ment of the word cofiditional in a specific sense, while its
Greek equivalent, hypothetical^ is used generically. In Boe- thius, both terms
are properly used as synonymous, and generic ; the two species being called conjunctivi^
conjuncti^ CHAP. I.] PROPOSITIONS. 37 or connexiy and disjuncHvi or disjunctu
With reference to modern usage, however, it will be better to contract the
Greek word than to extend the Latin one. Hypothetical in the following notes,
will be used as synonymous with con- dttionar (Hansel's edition oi Aldrich^ p.
103).] 36. A logical analysis of the Categorical Pro- position. In logical
analysis, the categorical proposition always consists of three parts, namely,
two terms which are united by means of a copula. The subject is that term about
which affirmation or denial is made ; it represents some notion already
partially deter- mined in our mind, and which it is our aim further to
determine. The predicate is that term which is affirmed or denied of the
subject; it enables us further to determine the subject, />., to enlarge our
knowledge with regard to it. The copula is the link of connection
betweenthesubjectandthepredicate,andconsistsofthewordsisoris not ccording as we
affirm or deny the latter of the former. In attempting to apply the above
analysis to such a proposition as "All that love virtue love
angling," we find that, as it stands, the copula is not separately
expressed. It may however be written, — subj. cop. pred. All lovers of virtue |
are | lovers of angling; and in this form the three diflferent elements of the
logical proposition are made distinct. This analysis should always be performed
in the case of any proposition that may at first present itself in an abnormal
form. A difficulty that may sometimes arise in discriminating the subject 38
PROPOSITIONS. [PART ii. and the predicate is dealt with subsequently,-^ompare
section 50. The older logicians distinguished propositions secundi adjacentiSf
and propositions tertii adjacentis. In the former, the copula and the predicate
are not separated ; ^.^., The man runs. All that love virtue love angling. In
the latter, the copula and the predicate are made distinct ; e.g,^ The man is
running, All lovers of virtue are lovers of angling. A categorical proposition,
therefore, when expressed in exact logical form, is tertii adjacentis. 37.
Exponible, capulative^ exclusive^ exceptive pro- posiions. Propositions that
are resolvable into more propositions than one have been called exponible^ in
consequence of theirsusceptibilityofanalysis.Copulativepropositionsareformedby
a direct combination of simple propositions, e,g.^ P is both Q and R (/>., P
is Q, P is ^), A is neither JB nor C (/./., A is not Bf A is not C)\ they form
one class of exponibles. Exclusive propositions contain some such word as
" only," thereby limiting the predicate to the sub- ject ; e.g,^ Only
S is F» This may be resolved into S is F^ and P is S. Propositions of this kind
also are therefore exponibles. Exceptive propositions limit the subject by such
a word as "unless" or "except"; e.g.^ A is X^ unless it
happens to be B. These too may perhaps be regarded as exponible propositions.
38. The Quantity and Quality of Propositions. The Quality of a proposition is
determined by the copula, being affirmative or negative according as the copula
is of the form '*is" or ''is not." Propositions are also divided into
universal and partir CHAP. I.] PROPOSITIONS, 39 culavy according as the
affirmation or denial is made of the whole or only of a part of the subject.
This division of Propositions is said to be according to their Quantity,
Combining the two principles of division, we get four fundamental forms of
propositions : — (i)
theuniversalaffirmative^AllSisPyusuallydenotedbythesymbolA;(2)theparticularffirmative^SomeSisPyusuallyde^notedbythesymbolI;(3)theuniversalnegativeyNoSisPyusuallydenotedbythesybolE
; (4) the particular negativCy Some S is not Py usually denoted by the symbol
O. These symbols A, I and E, O are taken from the Latin words affirmo and nego,
the affirmative symbols being the first two vowels of the former, and the
negative s)mibols the two vowels of the latter. Besides these symbols, it will
also be found convenient sometimes to use the following, — SaP=K\\S\sP', SiP =
Some S is P; SeP=NoS isP; SoP= Some S is not P. The above are useful when we
wish that the symbol which is used to denote the proposition as a whole should
also indicate what symbols have been chosen for the subject and the predicate
respectively. Thus, MaP=Al\AfisP; PoQ= Some P is not Q. The universal negative
should be written in the form No S is Py not All S is not P^ for the latter
would usually 40 PROPOSITIONS. [part ii. be understood to be merely particular.
Thus, All that glitters is not gold is really an O proposition, and is equi-
valent to — Some things that glitter | are not | gold. 39. Indefinite
Propositions. According to Quantity, Propositions have
sometimesbeendividedinto(i)Universal,(2)Particular,(3)Singular,(4)Indefinite.Singularpropositionsarediscussedinthefollowing
section. By an Indefinite Proposition is meant one " in which the Quantity
is not explicidy declared by one of the designatory terms «//, every ^ some,
many, &c" We may perhaps say with Hamilton that indesignate or
preindesignate would be a better term to employ. There can be no doubt that, as
Mansel remarks, " The true indefinite proposition is in fact the
particular; the statement *some A'\& B^ being applicable to an uncertain
number of instances, from the whole class down to any portion of it. For this
reason particular pro- positions were called indefinite by Theophrastus "
{Aidrich, p. 49). Some indesignate propositions are no doubt intended to be
understood as universals, e.g.. Comets are subject to the law of gravitation ;
but in such cases before we deal with the proposition logically it is better
that the word all should be explicitly prefixed to it. If we are really in
doubt with regard to the quantity of the proposition it must logically be
regarded as particular. Other designations of quantity besides all and some,
e.g, most, are discussed in section 41. The term indefinite has also been
applied to propositions in another sense. According to Quality, instead of the
two- folddivisiongivenintheprecedingexample,athreefolddivisionis sometimes
adopted, namely into affirmative, CHAP. I.] PROPOSITIONS. 41 negative, and
infinite or indefinite. For further explanation, see section 44. 40. Singular
Propositions. By a Singular or Individual Proposition is meant a pro- position
of which the subject is a singular term, one there- fore in which the
affirmation or denial is made but of a single specified individual; e.g,y
Brutus is an honourable man; Much Ado about Nothing is a play of Shakespeare's
; My boat is on the shore. Singular propositions may usually be regarded as
forming a sub-class of Universal propositions, since in every singular
proposition the affirmation or denial is of the whole of the subject. Such
propositions have however certain pecu- liarities of their own, as we shall
note subsequently; e.g,^ they have not like other universal propositions a
contrary distinct from their contradictory. Hamilton distinguishes between
Universal and Singular Propositions, the predication being in the former case
of a Whole Undividedy and in the latter case of a Unit Indivisible, This
separation is sometimes useful ; but I think it better not to make it absolute.
A singular proposition may without risk of confusion be denoted by one of the
symbols A or E ; and in syllogistic inferences, asingularmayalwaysberegardedasequivalent
to a universal proposition. The use of independent symbols for affirmative and
negative singular propositions would introduce considerable additional com-
plexity into the treatment of the Syllogism ; and for this reason alone it
seems desirable as a rule to include par- ticulars under universals. We may
however divide universal propositions into General and Singular^ and we shall
then have terms whereby to call attention to the distinction wherever it may be
necessary or useful to do so. 42 PROPOSITIONS. [PART II. There is a certain
class of propositions with regard to which there is some difference of opinion
as to whether they should be regarded as singular or particular ; for example,
such as the following : A certain man had two sons ; A great statesman was
present. Mansel {Aldrich, p. 49) decides that they should be dealt with as
particulars, and I think rightly, on the ground that if we have two such
propositions, " a certain man '' or " a great statesman " being
the subject of each, we cannot be sure that the same individual is referred to
in both cases. Sometimes however the context may enable us to decide the case
differently. There are propositions of another kind with a singular term for
subject about which a few words may be said; namely,
suchpropositionsasBrowningissometimesobscure;Thatboyissometimesfirstinhisclass.Thesepropositionsmaybetreatedas
universal with a somewhat complex predicate, (and it should be noted that in
bringing propositions into logical form we are frequently compelled to use very
complex predicates) ; thus : — Browning ] is | a poet who is sometimes obscure.
That oy | is | a boy who is sometimes first in his class. By a certain
transformation however these propositions may also be dealt with as particulars,
and such transforma- tion may sometimes be convenient; thus, Some of Browning's
writings are obscure, Some of the boy's places in his class are the first
places. But when the proposition is thus modi- fied, the subject is no longer a
singular term. 41. The logical signification of the words some^ mostyfeiv, all,
any. Some may mean merely " some at least," />., not none, or it
may carry the further implication, " some at most,'' /., it has the same
indefinite character which we logically ascribe to "some"; since the
antecedent condition is satisfied if a single A is B. The proposition might
indeed be written — If one or more A is B, C is JD. 42. Examine the logical
signification of the itali- cised words in the following propositions : — Some
are born great- Few Sire chosen^ All is not lost. All men are created equal.
All that a man hath will he give for his life. If some A is By some C is D, If
any A is By any C is D, IfallAisByallCisD. 48 PROPOSITIONS. [part li. The
student must be warned against . treating such a proposition as "If any A
is By some C is Z>*' as par- ticular'. Regarded separately the antecedent
and the con- sequent in this example are both particular ; but the con- nection
between them is affirmed universally, the proposition asserting that " in
all cases in which any A is By some C It should be observed that in a
considerable number of cases, the hypothetical is of the nature of a singular
pro- position, the event referred to in the antecedent being in the nature of
things one which can happen but once ; e,g,j If I perish in the attempt, I
shall not die unavenged.TotheDisjunctivePropositionwe are unable to apply
distinctions of Quality. The proposition. Neither F \s Q nor X i& Y states
no alternative, and is therefore not dis- junctive at all. Distinctions of
Quantity are however still applicable. Thus, Universal, — Either F\s ^ or -Y is
K Particular, — In some cases either jP is ^ or -ST is Y. It is again to be
observed that frequently the dis- junctive proposition is of the nature of a
singular proposi- tion, the reference being but to a single occasion on which
it is asserted that one of the alternatives will hold good. 46. Determine the
Quantity and Quality of the following propositions, stating precisely what you
regard as the subject and predicate, or in the case ^ I cannot agree with
Hamilton (Logic^ i. p. 248), in regarding the following as a particular
hypothetical — If some Dodo is, then some animal is. The proposition is a
little hard to interpret, but it seems to mean that if there is such a thing as
a Dodo, then there is such a thing as an animal; and we must consider that a
imiversal connection is here affirmed. CHAP. I.] PROPOSITIONS. 49 of hypothetical
propositions, the antecedent and consequent of each : — (i) All men think all
men mortal but them- selves. (2) Not to know me argues thyself unknown. (3) To
bear is to conquer
ourfate.(4)Berkeley,agreatphilosopher,deniedtheexistenceofMatter.(5)Agreatphilosopher
has denied the existence of Matter. (6) The virtuous alone are happy. (7) None
but Irish were in the artillery. (8) Not every tale we hear is to be believed,
(9) Great is Diana of the Ephesians ! (10) All sentences are not propositions.
(11) Where there's a will there's a way, (12) Some men are always in the wrong.
(13) Facts are stubborn things. (14) He that increaseth knowledge increaseth
sorrow. (15) None think the great unhappy, but the great. (16) He can't be
wrong, whose life is in the right. (17) Nothing is expedient which is unjust.
(18) Mercy but murders, pardoning those that kill. (19) If virtue is
involuntary, so is vice. (20) Who spareth the rod, hateth his child. 47.
Analyse the following propositions, i,e,, ex- press them in one or more of the
strict categorical forms admitted in Logic : — K. L. A 5Q PROPOSITIONS. [part
ii. (i) No one can be rich and happy unless he is also temperate and prudent,
and not always then. (ii) No child ever fails to be troublesome if ill taught
and spoilt (iii) It would be equally false to assert that the rich alone are
happy, or that they alone are not. [v.](i) contains two statements which may be
reduced to the following forms, — All who are rich and happy | are | temperate
and pudent A. Some who are temperateandprudent| are not I rich and happy. O.
(ii) may be written, All ill-taught and spoilt children are troublesome. A.
iii) Here two statements are given false^ namely, the rich alone are happy ;
the rich alone are not happy. We may reduce these false statements to the
following, — all who are happy are rich ; all who are not happy are rich. And
this gives us these true statements, — Some who are happy are not rich O. Some
who are not happy are not rich. O. The original proposition is expressed
therefore by means of these two particular negative propositions48. The
Distribution of Terms in a Proposition. A term is said to be distributed when
reference is made to all the individuals denoted by it ; it is said to be
undis- tributed when they are only referred to partially^ />., in- formation
is given with regard to a portion of the class denoted by the term, but we are
left in ignorance with regard to the remainder of the class. It follows
immediately CHAP, I.J PROPOSITIONS. 51' from this definition that the subject
is distributed in a universal, and undistributed in a particular, proposition.
It can further be shewn that the predicate is distributed in a negative, and
undistributed in an affirmative proposition. Thus, if I say. All Sis P, I imply
that at any rate some P is 6", but I make no implication with regard to
the whole of P, I leave it an open question as to whether there is or is not
any P outside the class 5. Similarly if I say. Some S is P, But if I say. No S
is Py in excluding the whole of S from /*, I am also excluding the whole of P
from 6", and therefore P as well as S is distributed. Again, if I say,
Some S is not Py although I make an assertion with regard to a part only of Sy
I exclude this part from the whole of Py and therefore the whole of P from it.
In this case, then, the predicate is distributed, although the subject is not.
Summing up our results we find that A distributes its subject only, I
distributes neither its subject nor its predicate,E distributes both its
subject and its predicate, O distributes its predicate only, ^y 49. How does
the Quality of a Proposition aflfect its Quantity 1 Is the relation a necessary
one ? [L.] By the Quantity of a Proposition must here be meant the Quantity of
its Predicate, and we have shewn in the preceding section that this is
determined by its Quality. The predicate is distributed in negative,
undistributed in_ a ffirmativ e^ proposition s. The latter part of the above
question refers to Hamilton's doctrine of the Quantification of the Predicate. According
to this doctrine, the predicate of an affirmative proposition is sometimes
expressly distributed, while the predicate of a 4—2 52 PROPOSITIONS. [PART II.
negative proposition is sometimes given undistributed. For example, the
following forms are introduced : — Some S is all P, No S is some -P. This
doctrine is discussed and illustrated in Part in. chapter 9. 50. In doubtful
cases how should you decide which is the subject and which the predicate of a
proposition ? [v.] The nature of the distinction between the subject and the
predicate of a proposition may be expressed by saying that the su bject is that
of whichsomething is affirmed o r denied, the predicate is that which is
affirmed or denied ofthe subject; or perhaps still better, the subject is that
which we think of as the determined or qualified notion, the predicate that
which we think of as the determining or qualifying notion. Now, can we say that
the subject always precedes the copula, and that the predicate always follows
it? In other words, can we consider the order of the terms to suffice as a
criterion? If the proposition is reduced to an equation, as in the doctrine of
the quantification of the predicate, I do not see what other criterion we can
take; or we might rather say that in this case the distinction between subject
and predicate itself fails to hold good. The two are placed on an equality, and
we have nothing left by which to distinguish them except the order in which
they are stated. This view is indicated by Professor Baynes in his Essay on the
New Analytic of Logical Forms. In such a proposition, for example, as
"Great is Diana of the CHAP. I.] PROPOSITIONS. 53 Ephesians/' he would
call " great " the subject, reading the proposition, however, "
(Some) great is (all) Diana of the Ephesians." But leaving this view on
one side, we cannot say that the order of terms is always a sufficient
criterion. In the proposition just quoted, " Diana of the Ephesians "
would generallybeacceptedasthesubject. What further criterion then can be given
? In the case of E and I propositions, (propositions, as will be shewn, which
can be simply con- verted), we must appeal to the context or to the question to
which the proposition is an answer. If one term clearly conveys information
regarding the other term, it is the predicate. It is also more usual that the
subject should be read in extension and the predicate in intension. If none of
these considerations are decisive, then I should admit that the order of the
terms must suffice. In the case of A and O propositions, (propositions, as will
be shewn, which cannot be simply converted), a further criterion may be added.
From the rules relating to the distribution of terms in a proposition it
follows that in affirmative pro- positions the distributed term, (if either
term is distributed), is the subject ; whilst in negative propositions, if only
one term is distributed, it is the predicate. I am not sure that the inversion
of terms ever occurs in the case of an O pro- position ; but in A propositions
it is not infrequent. Ap- plying the above to such a proposition as *"*
Workers of miracles were the apostles," it is clear that the latter term
is distributed while the former is not. The latter term is therefore the
subject. A corollary from the rule is that in an affirmative proposition if one
andonlyoneterm is singular that is the subject, since a singular is equivalent
to a distributed term. This decides such a case as " Great is Diana of the
Ephesians." 54 PROPOSITIONS. [part ll. 61. What do you consider to be
respectively the subject and the predicate of the following sentences, and why
?(i) Few men attain celebrity. (2) Blessed are the peacemakers.(3) It is mostly
the boastful who fail.(4) Clematis is Traveller's Joy. [v.] 62. What do you
consider to be the essential distinction between the Subject and Predicate of a
proposition ? Apply your answer to the following : — (i) From thence thy
warrant is thy sword. (2) That is exactly what I wanted. [v.]CHAPTER II. THE
OPPOSITION OF PROPOSITIONS. 53. The Opposition of Categorical Propositions. Two
propositions are said to be opposed to each other when they have the same
subject and predicate respectively, but differ in quantity or quality or both I
Taking the propositions SaP^^ SiPy SePySoP,inpairswe find that there are four
possible kinds of relation between them. (i) The pair of propositions may be
such that they cannot both be true, and they cannot both be false. This is
called contradictory opposition, and subsists between SaP and SoP^ and between
SeP and SiP, ^ This definition is given by Aldrich (p. 53 in Mansel's edition).
Ueberweg however defines Opposition in such a way as to include only
contradiction and contrariety (translation by Lindsay, p. 328); and Mansel
remarks that '* Subalterns are improperly classed as opposed propositions"
(Aldrich^ p. 59). Professor Fowler follows Aldrich's definition (Deductive
Logic^ p. 74), and I think wisely. We want some term to signify this general
relation between propositions ; and though it might be possible to find a more
convenient term, I do not think hat any confusion is likely to result from the
use of the term opposition if the student is careful to notice that it is here
used in a technical PROPOSITIONS. [part II. (2) They may be such that they
cannot both be true, but they may both be false. This is called contrary oppo-
sition. SaP and SeP. (3) They may be such that they cannot both be false, but
they may both be true. Subcontrary opposition. StP and SoP, (4) From a given
universal proposition, the truth of the particular having the same quality
follows, but not vice versa.Thisis subaltern opposition^ the universal being
called the subaltemant, and the particular the subaltemate or the subaltern,
SaP and SiP. SeP and SoP. All these relations are indicated clearly in the
ancient square of opposition. A Contraries I Subcontraries O Propositions mu^t
of course be brought to such a form that they have the same subject and the
same predicate before we can apply the terms of opposition to them ; for
example, All *S is jP and Some P is not S are not contra- dictories. CHAP. II.]
PROPOSITIONS* 57 54. On the common view of the opposition of propositions what
are the inferences to be drawn (i) from the truth, (2) from the falsity, of
each of the four categorical propositions ? [L.] 66. Explain the nature of the
opposition between each pair of the following propositions : None but Liberals
voted against the motion. Amongst those who voted against the motion were some
Liberals. It is untrue that those who voted against the motion were all
Liberals. 66. Give the contradictory and the contrary of the following
propositions : — (i) A stitch in time saves nine. (2) None but the brave
de^^^he fair. (3) He can't be wrong whd^^H^is in the right. (4) The virtuous
alone are happy. (i) A stitch in time saves nine. This is to be regarded as a
universal affirmative proposition, and we therefore have Contradictory y Some
stitches in time do not save nine. I. ~ Contrary^ No stitch in time saves nine.
E. (2) None but the brave deserve the fair, = None who are not brave deserve
the fair. E. Contradictoryy Some who are not brave deserve the fair. I.
Contrary y All who are not brave deserve the fair. A. 8 PROPOSITIONS. [part il
(3) He can't be wrong whose life is in the right. E. Contradictory^ Some may be
wrong whose lives are in the right. I. Contrary^ All are wrong whose lives are
in the right. A. (4) The virtuous alone are happy, = No one who is not virtuous
is happy. E. Contradictory^ Some who are not virtuous are happy. I. Contrary^
All who are not virtuous are happy. A. 67. Give the contrary, contradictory,
and sub- altern of the following propositions : — (i) All B.A.'s of the
University of London have assed three examinations.(2) All men are sometimes
thoughtless. (3) Uneasy lies the head that wears a crown. (4) The whole is
greater than any of its parts. (5) None but solid bodies are crystals./ (6) He
who has been bitten by a serpent is afraid of a rope. (7) He who tries to say
that which has never been said before him will probably say that which will
never be repeated after him. [Jevons, Studies in Deductive Logic, p. 58.] 68.
Explain the technical terms " contrary " and
"contradictory," applying them to the following pro- positions : —
(i) Few 5 are P.(2)Atany rate, he was not the only one who cheated. (3)
Two-thirds of the army are abroad. [v.] CHAP. II.] PROPOSITIONS. 59 It is the
same thing to deny the truth of a proposition and to affirm the truth of its
contradictory ; and vice versa. The criterion of contradictory opposition is
that of the two propositions^ one must he true and the other must be false ;
they cannot be true together, but on the other hand no mean is possible between
them. The relation between two ontradictories is mutual ; it does not matter
which is given true or false, we know that the other is false or true ac-
cordingly. Every proposition has its contradictory, which may however be more
or less complex in form. The contrary of a given proposition goes beyond mere
denial, and sets up a further assertion as far as possible removed from the
original assertion. It declares not merely the falsity of the original
proposition taken as a whole, but the falsity of every part of it. It follows
that if we cannot go beyond the simple denial of the truth of a proposition,
then it has no contrary distinct from its contradictory. For example, in order
simply to deny the truth of " some S is jP,'' it is necessary to affirm
that " no *S is jP,'^ and it is impossible to go further than this in
opposition to the given proposition. "Some *S is P" has therefore no
contrary as distinguishedfromitscontradictory. We may now apply the terms in
question to the given propositions : — ( I ) " Few S are P" = "
Most S are not P',' and we might hastily be inclined to say that the
contradictory is " Most S are jP." Both these propositions would
however be false in the case in which exactly one half S was P. The true
contradictory therefore is " At least one half *S' is /I" The ontrary
is "All S is P'' Similarly the contradictory of " Most S are P'' is "
At least one half S is not P"; and its contrary " No ^S is -P."
These examples shew that if we once travel outside the 6a PROPOSITIONS. [part
li. limits set by the old logic, and recognise the signs of quantity most and
few as well as all and some, we soon become involved in numerical statements.
Propositions of the above kind are therefore usually relegated to what has been
called numerical logic, a topic discussed at length by De Morgan and to some
extent by Jevons. (2) "At any rate, he was not the only one who
cheated." A question of interpretation is naturally raised here ; does the
statement assert that he cheated, or is this left an open question? We may I
think choose the latter alternative. What the speaker intends to lay stress
upon is that some others cheated at any rate, whatever may have been the case
with him. The contradictory then becomes " No others heated";andwe
have no distinct contrary. (3) "Two-thirds of the army are abroad."
This may mean "At least two -thirds of the army are abroad," or **
Exactly two-thirds of the army are abroad." On the first interpretation,
the contradictory is "Less than two-thirds of the army are abroad";
and the contrary " None of the army are abroad." On the second
interpretation, the contradictory is " Not exactly two- thirds of the army
are abroad," /. ^., " Either more or less than two -thirds of the
army are abroad." With regard to the contrary we are in a certain
difficulty ; for we may as it were proceed in two directions, and take our choice
between " All the army are abroad " and " None of the army are
abroad." I hardly see on what principle we are to choose between these.
Fortunately, contrary opposition, unlike contradictory opposition, is of very
little logical importance. 59. The Opposition of Singular Propositions. Take
the proposition, Socrates is wise. The contra- CHAP. II.] PROPOSITIONS. 6i
dictory is — Socrates is not wise ; and so long as we keep to the same terms,
we cannot go beyond this simple denial. We have therefore no contrary distinct
from the contra- dictory. This opposition of singulars has been called
secondary contradiction (ManseFs Aidrich, p. 56). There are indeed two methods
of treatment according to which we might findadistinctcontrarynd contradictory
in the case of singular propositions, but I think that the above treatment
according to which they are not distinguished is preferable to either. (i) We
might introduce the material contrary of the pre- dicate instead of its mere
contradictory, (compare section 28). Thus we should have — Original
proposition, Socrates is wise ; Contradictory, Socrates is not wise ; Contrary,
Socrates has not a grain of sense. This might be called the material contrary
of the given proposition ^ A fresh term is introduced that could not be
formally obtained out of the given proposition. It still remains true that the
singular proposition has no formal contrary distinct from its contradictory.
(2) Some principle of separation into parts might be introduced according to
which the subject would be no longer a whole indivisible ; for example,
Socrates might be regarded as having different characteristics at different
times or under different conditions. The original proposition would then be
read Socrates is always wise, and the contra- dictory would be Socrates is
sometimes not wise, while the trary would be Socrates is never wise. Treated in
this manner, however, the proposition hardly remains a really singular
proposition. ^ The same distinction might be applied to general propositions.
62 PROPOSITIONS. [PART n. 60. Can the ordinary doctrine of the opposition of
propositions be applied to hypothetical and dis- junctive propositions ? It has
been already shewn that the ordinary distinctions of quantity and quality may
be applied to Hypothetical Propositions, and it follows that the ordinary
doctrine of opposition will also apply to them. We have UAisB, CisD. A. In some
cases in which A is B, C is D. I. If^ is^, CisnotZ>. E. In some cases in
which A is B, C is not JD. O. Then, as in the case of Categoricals, — A and I,
E and O are subalterns. A and E are contraries. A and O, E and I are
contradictories. I and O are subcontraries. There is more danger of
contradictories being confused with contraries in the case of Hypotheticals
than there is in the case of Categoricals. 7f A is By C is not D is very liable
to be given as the contradictory oi If A is Bj C is jD. But it clearly is not
its contradictory, so far as they are general propositions^ since both may be
false. For ex- ample, the two statements, — If the Times says one thing, the
Pall Mall says another; If the Times says one thing, the Pall Mall says the
same, /.^., does not say another, — are both false : the two papers are
sometimes in agreement and sometimes not If however the Hypothetical
proposition is of the nature of a Singular, that is, if the thing referred to
in the ante- cedent can happen but once; then as in the case of Singular
Categorical propositions, the Contradictory and the Contrary are not to be
distinguished. Taking the proposition — If I CHAP. II.] PROPOSITIONS. 63 perish
in the attempt, I shall not die unavenged; its con- tradictory may fairly be
stated — If I perish in theattempt, I shall die unavenged. We cannotapply
distinctions of quality to Disjunctives, and therefore the ordinary doctrineof
opposition cannot be applied to them. We may however, find the contradictory
and the contrary of a disjunctive proposition, such as A is either B or C Its
Contradictory is — In some cases A is neither B nor C; its Contrary — A is
neither B nor C We observe then that the contradictory and contrary of a
disjunctive are not themselves disjunctive. What has been said with regard to
Singular Hypotheticals also appUes mutatismutandis to what may be called
Singular Dis- junctives. A point to which our attention is called by the above
is that the relation of reciprocity that holds between contra- dictories does
not always hold between contraries. If the proposition ^ is the contradictory
of the proposition a, then a is also the contradictory of p ; but if 8 is the
contrary of a, it does not necessarily follow that a is the contrary of 8.
Thus, we have seen that the contrary of ^^A is either B or C" is ''A is
neither B nor C" The contrary of the latter however is " A is both B
and C," which is not the original proposition over again \ 61. How would
you apply the terms contradictory and contrary to the case of complex
propositions : e.g., He was certainly stupid ; and, if not mad, either
miserably trained, or misled by bad companions } [v.] The criterion of contradictories
given in section 58, may be applied to the case of complex propositions. For
example, take the complex proposition X is both A and B^ ^ Cf. also the
Examples given in section §8. 64 PROPOSITIONS. [part ii. (where J^ is a
singular term). Regarded as a whole, this statement is evidently false if X
fails to be either one or the other of A and B. It is also clear that it must
either be both of them or it must fail to be at least one of them. We have then
this pair of contradictories, — \X is both A and B ; \X is either not A or not
B, Thus, what we may perhaps call a conjunctive is contra- dicted by a
disjunctive, and vice versa. Next take the rather more complex proposition — X
is A, and either B or C\ Its contradictory, following the above rule, is X is
either not A or neither B nor C Next taketheproposition X is Yy and if it is
not Z, // is either Qor R^. It may be reduced to X is y ; and either Z, Qot R,
and we at once get the contradictory J^is either not Yox neither Z, Q nor R, It
will be noticed that the last example chosen is equi- valent to the one given
in the question, the terms of the latter being translated into symbols. The
required contra- dictory is therefore — Either he was not stupid, or he was
neither mad, miser- ably trained nor misled by bad companions. The application
of the term contrary to complex pro- positions is of less interest. We may
however consider that we have the contrary of such a proposition when we deny
every part of the statement. Thus the contrary of " X is ^ I still assume
that the subject of the proposition is a singular term. CHAP, n.] PROPOSITIONS.
65 both A sndB" is ''X is neither A nor jB''; of ''Xis A and either B or
C" "^is neither A^ B nor C"; and of the given proposition,
"He was neither stupid nor mad nor miserably trained nor misled by bad
companions."62. What is the precise meaning of the assertion that a
proposition — say "All grasses are edible" — is false ? [Jevons,
Studies in Deductive Logic^ p. 116.] Professor Jevons discusses at some length
the point here raised, but I find myself quite unable to agree with what he
says in connection with it. He commences by givingananswer, which may be called
the orthodox one, and which I should certainly hold to be the correct one. When
I assert that a proposition is false, I mean that its contradictory is true.
The given proposition is of the form A^ and its contradictory is the
corresponding O proposition, — Some grasses are not edible. When, there- fore,
I say that it is false that all grasses are edible, I mean that some grasses
are not edible. Professor Jevons however continues, " But it does not seem
to have occurred to logi- cians in general to inquire how far similar relations
could be detected in the case of disjunctive and other more com- plicated kinds
of propositions. Take, for instance, ^ the assertion that *all endogens are all
parallel-leaved plants.' If this be false, what is true ? Apparently that one
or more endogens are not parallel-leaved plants, or else that one or more
parallel-leaved plants are not endogens. But it may also happen that no endogen
is a parallel-leaved plant at all. There are three alternatives, and the simple
falsity of the original does not shew which of the possible contra- dictories
is true." In this statement, there appear to me to be two errors. In the
first place, in saying that one or more endogens areK. L. 5 66 PROPOSITIONS.
[PART ll. not parallel-leaved plants, we do not mean to exclude the possibility
that no endogen is a parallel-leavedatall.S3rmbolically, Some S is not P does
not exclude No S is P. The three alternatives are therefore at any rate reduced
to the two first given. But in the second place, I think Professor Jevons is in
error in regarding each of these alternatives by itself as a contradictory of
the original proposition. The true logical contradictory is the affirma- tion
of the truth of one or other of these alternatives. If the original complex
proposition is false we certainly know that the new complex proposition
limiting us to such alternatives istrue.Thepointat issue may be further
illustrated by taking the proposition in question in a symbolic form. All S is
all jP is a complex proposition, resolvable into the form, All S is Fy and all
P is S. In my view, it has but one contradictory, namely, Either some S is not
P, or some P is not S.^ If either of these alternatives holds good, the
original statement must in its entirety be false ; and on the other hand, if
the latter is false, one at least of these alternatives must be true. Professor
Jevons speaks as if Some S is not P were by itself a contradictory of All S is
all P. But it is merely inconsistent with it. They may both be false. No doubt
in ordinary speech contradictory frequently implies no more than
"inconsistent with," and if Professor Jevons means that we should
also use the term contradictory in this sense in Logic, the question becomes a
verbal one. But he means more than this; he seems to mean that in some cases we
can find no proposition that must be true when a given proposition is false. And
here I hold that he is rong. ^ The contradictory of " AU .S* is all
i"* may also be expressed ** S and P are not coextensive." CHAP. II.]
PROPOSITIONS, ^-j If the original proposition is complex, its contradictory
will in general be complex too, and possibly still more complex ; but that
might naturally be expected. Compare the two preceding sections, where several
cases are worked out in detail. The above will I think indicate how misleading
is Professor Jevons's further statement, — "It will be shewn in a
subsequent chapter that a proposition of moderate complexity has an almost
unlimited number of contradic- tory propositions, which are more or less in
conflict with the original. The truth of any one or more of these con-
tradictories establishes the falsity of the original, but the falsity of the
original does not establish the truth of any one or more of its
contradictories." No doubt a complex proposition may yield an indefinite
number of other propo- sitions the truth of any one of which is inconsistent
with its own. But it has only one logical contradictory^ which con- tradictory
as suggested above is likely to be a still more complex proposition affirming a
number of alternatives one or other of which must hold if the original
proposition is "With the point here raised Professor Jevons mixes up
another, with regard to which his view is almost more mis- leading. He says,
"But the question arises whether there is not confusion of ideas in the
usual treatment of this ancient doctrine of opposition, and whether a
contradictory of a proposition is not any proposition which involves the
falsity of the original, but is not the sole condition of it I apprehend that
any assertion is false which is made with- out sufficient grounds. It is false
to assert that the hidden side of the moon is covered with mountains, not
because we can prove the contradictory, but because we know that the assertor
must have made the assertion without evidence. PROPOSITIONS. [PART II. person
ignorant of mathematics were to assert that nvolutes are transcendeotal
curves,' he would be making se assertion, because, whether they are so or not,
he lot know it." Surely in Logic we cannot regard the 1 or falsity of a
proposition as depending upon the dedge of the person who affirms it, so that
the same osition would now be true, now false. The question lat is truth?"
may be an enormously difficult one to 'er absolutely, and I need not say that I
shall not npt to deal with it here ; but unless we are allowed to eed from the
falsity of "All S is J"' to the truth of [ne ■$ is not P," I do
not think we can go far in Logic. 13. Analyse all that Is implied in the
assertion le falsity of each of the following propositions : — ■ [l) Roger
Bacon was a giant. {2) Descartes died efore Newton was born. (3) Bare assertion
is not necessarily the naked (4) All kinds of grasses except one or two species
are not poisonous. [Jevons, Studies, p. 124.] 64 Assign precisely the meaning
of the assertion that it is false to say that some English soldiers did not
behave discreditably in South Africa. [l.] 66. Examine in the case of each of
the follow- ing propositions the precise meaning of the assertion that the
proposition is false : — (i) Some electricity is generated by friction. CHAP.
II.] PROPOSITIONS. 69 (ii) Oxygen and nitrogen are constituents of the air we
breathe. (iii) If a straight line falling upon two other straight lines make
the alternate angles equal to eachother, these two straight lines shall be
parallel. (iv) Actions are either good, bad, or indifferent. /^ tii?/^ ^'^- ^^/
*/;^.f>/*nw^ h ^^pf^yy/)r*^ji i'?f iiss^y y/j^j:i^ ^/Vm' W///;/'^-44 f^^,'A«
'/< V^^ i/Ht^^^/n nay \^ i^n^A i^ f//;y/i)f^f *^«i/J M M»*/ }/*:
<.^^^yj^v.j^/^ frtt^% UM4^,iijiUmf p* f^4) 4^^f^*t4ff9 Owr«r*k« ihu^.
PROPOSITIONS.Thus^ given a proposition having S for its subjectand P for its
predicate, we seek to obtain by immediate inference anew proposition having P
for its subject and S for its predicate; and applying this rule to the four
fundamental forms of proposition, we get thefollowing table : — Original Proposition*
Converse. mS'i&F. A. ^omtPisS. I. Some S is P. I. Some -Pis5. I. No S is P.
E. No P is S. E. Some S is not P. 0. (None.) 67. Simple Conversion, and
Conversion per ac- cidens. It will be observed that in the case of I and E, the
converse is of exactly the same form as the original pro- position (or
convertend) ; we do not lose any part of the information given us by the
convertend, and we can pass back to it by re-conversion of the converse. The
convertend and its converse are equivalent propositions. The con- veifeion in
both these cases is said to be simple. In the case of A, it is different;
although we start with a universal proposition, we obtain by conversion a
particular one only, and by no means of operating upon the converse can we
regain the original proposition. The convertend and its converse are not
equivalent propositions. This is called conversion /^r accidens^, or conversion
by limitation, ^ The conversion of A is said by Mansel to be caUed conversion
72 PROPOSITIONS. [PART II. 68. Particular negfative propositions do not admit
of ordinary conversion. It is clear that if the converse is to be a legitimate
formal inference from the originalproposition (or convert- end), it must
distribute no term that was not distributed in the convertend From this it
follows immediately that Some S is not P does not admit of ordinary conversion
; for S which is undistributed in the convertend would be- come the predicate
of a negative proposition in the converse, and would therefore be distributed.
(I may remind the reader that in what I have called ordinary conversion, with
which alone we are now dealing, we do not admit the ontradictory of either the
original subject or the original predicate as one of the terms of our
converse.) I cannot understand why Professor Jevons should say that the fact
that the particular negative proposition is in- capable of ordinary conversion
" constitutes a blot in the ancient logic" (Studies in Dedtcctive
Logic^ p. 37). We shall find subsequently that just as much can be inferred
from the particular negative as from the particular affirmative, (since the
latter unlike the former does not admit of con- traposition). Less can be
inferred from either of them than can be inferred from the corresponding
universal proposition, and this is obviously because the latter gives all the
informa- per accidens " because it is not a conversion of the universal
per se, but by reason of its containing the particular. For the proposition *
Some B is A* is primarily the converse of* Some ^ is ^,' secondarily of * All A
is B^" (Mansel's Aldrich, p. 61). Professor Baynes seems to deny that this
is the correct explanation of the use of the term {New Analytic of Logical
Forms, p. 29) ; but however this may be, I do not think that we can really regard
the converse of ^ as obtained through its subaltern. We proceed directly from
**A11 A is B" to ** Some Bis A" without the intervention of ** Some A
is -ff." CHAP. HI.] PROPOSITIONS. Jz tion given by the particular
proposition and more beside. No logic, symbolic or other, can actually obtain
more from the given information than the ancient logic does. 69. Give the
converse of the following pro- positions : — (i) A stitch in time saves nine.
(2) None but the brave deserve the fair. (3) He can*t be wrong whose life is in
the right. (4) The virtuous alone are happy. No difficulty can be found in
converting or performing other immediate inferences upon any given proposition
if it is once brought into logical form, its quantity and quality being
determined, and its subject, copula and predicate being definitely
distinguished from one another. If this rule is neglected, the most absurd
results may be elicited. For example, amongst several curious converses of the
first of the above propositions I have had seriously given, — Nine stitches
save a stitch in time. Here it is of course entirely overlooked that
"save'' cannot be a logical copula. The proposition may be written, All
stitches in time I are | things that save nine stitches. This being an A
proposition is only convertible per aaidens^ thus, Some things that save nine
stitches are stitches in time. The following is wrong, — ^The means of saving
nine stitches is a stitch in time ; since there may be other ways of saving
" None but the brave deserve the fair." For the converse of this I
have had, — The fair deserve none but the brave; and, again, No one ugly
deserves the brave. Logically the proposition may be written, No one who is not
brave is deserving of the fair. This, being an E proposition, may 74 PROPOSITIONS.
[PARTII. be converted simply, giving, No one deserving of the fair is not
brave. ** He can't be wrong whose life is in the right" Written in strict
logical form, this proposition becomes, — No one whose life is in the right is
able to be in the wrong ; and therefore its converse is, — No one who is able
to be in the wrong is one whose life is in the right This proposition may now
be written in the more natural but not strictly logical form. His life cannot
be in the right who can him- self be wrong. " The virtuous alone are
happy." In logical form this may be written either, No one who is not
virtuous is happy, or All who are happy arevirtuous.Taking it in the first
form, the converse is — No one who is happy is not virtuous ; and from this we
may again get the second form by changing its quality* — All who are happy are
virtuous. The converse of this is, — Some who are virtuous are happy. 70. State
in logical form and convert the follow- ing propositions : — (i) There's not a
joy the world can give like that it takes away. (2) He jests at scars who never
felt a wound. (3) Axioms are self-evident. (4) Natives alone can stand the
climate of Africa. (5) Not one of the Greeks at Thermopylae escaped. (6) All
that glitters is not gold. [c] 71. Give all the logical opposites of the pro-
position : — Some rich men are virtuous ; and also the * Cf. section 73. CHAP.
III.] PROPOSITIONS. 75 converse of the contrary of its contradictory. How is
the latter directly related to the given proposition } 72. Point out any
possible ambiguities in the following propositions, and shew the importance of
clearing up such ambiguities for logical purposes : — (i) Some of the
candidates have been successful. (ii) Either some gross deception was practised
or the doctrine of spiritualism is true. (iii) All are not happy that seem so.
(iv) All the fish weighed five pounds. Give the contradictory and (where
possible) the converse of each of these propositions. CHAPTER IV. THE OBVERSION
AND CONTRAPOSITION OF PROPOSITIONS. 73. The Obversion of Propositions,
Obversion is the process of changing the quality of a pro- position without
altering its meaning. This change of quality may always be made if at the same
time 7ve substitute for the predicate its contradictory. Applying this rule, we
have the following table : — Original Proposition, Obverse, All 6" is ^.
A. No S is not-^. E. Some S is P, I. Some S is not not--P. 0. No S is P, E. All
S is not-^. A. Some S is not P, 0. Some S is not-/*. I. The term Obversion is
used by Professor Bain, and it is a convenient one. The process is also called
Permutation (Fowler), Aequipollence (Ueberweg), Infinitaiion (Bo wen).
Immediate Inference by Privative Conception (Jevons), Contra- version (De
Morgan), Contraposition (Spalding). CHAP. IV,] PROPOSITIONS. 'J^ Obversion
depends on the supposition that two negatives make an affirmative. De Morgan
(Formal Logic, pp. 3, 4) points out that in ordinary speech this is not always
strictly true. For example, **not unable" is scarcely used as strictly equivalent
to "able," but is understoodto imply a some- what lower degree of
ability. " John is able to translate Virgil" is taken to mean that he
can translate it well; "Thomas is not unable to translate Virgil" is
taken to mean that he can translate it —indifferently. This distinc- tion,
however, depends agood deal on the accentuation of the sentence; and it is not
one of which Logic can take account. Logically, "-<4 -="" -p="" 0.="" 32="" 38="" 5="" 6-="" 61="" 6="" 74.="" 76.="" 77.="" 78.="" 78="" 79.="" 79="" 8.="" 80.="" 82.="" 82="" 83.="" 83="" 8l="" :-="" :="" a.="" a="" above="" abp.="" according="" accurate="" acquire="" admit="" agreeable.="" agreeable="" aldrich="" all="" allanimals="" alone="" also.="" also="" altered="" always="" an-="" an="" and="" angles="" animals="" annth="" another="" answered="" antecedent.="" antecedent="" any="" anything="" appeal="" appear="" appears="" application="" applying="" apuleius="" are="" aristotle="" arrive="" arrived="" as="" associations="" at.="" at="" attached="" attempt="" b="" bad.="" bain="" base="" be="" becomes="" been="" before="" between="" boethius="" book="" both="" brave="" briefly="" but="" by="" c="" caiegorico="" call="" called="" can="" capella="" carbon.="" carbon="" careful="" case="" cases="" cate="" categorical="" chap.="" cisdy="" ckd="" clusions="" cold="" com-="" con-="" consequent="" consequently="" contain="" containing="" contra-="" contradictory="" contraposition.="" contraposition="" contrapositionthe="" contrapositive.="" contrapositive="" contrapositives="" conver-="" converse="" conversion="" convert="" correspond="" de="" deduce="" defined="" definition="" denial="" denoting="" depending="" describe="" deserve="" dicate="" difference="" difficulties.="" difficulty="" diflference="" dis-="" discover="" discussed="" distinguished="" do="" doctrines="" does="" done="" doubt="" e.="" e="" each="" easy="" educated="" either="" elementary="" employ="" employed="" equal.="" equally="" equivalent.="" establish="" euclid="" every="" evil="" examination="" example="" examples="" expanded.="" explain="" explained="" expression="" f="" failed="" fair.="" feed.="" feed="" ferences="" first="" follow="" followed="" following="" follows="" for="" force="" form="" formal="" forms="" found="" from="" gations="" generalise="" geometrical="" gets="" give="" given="" gives="" good.="" good="" granted="" happy.="" has="" have="" having="" he="" helpful="" here="" higher="" his="" hispanus.="" hiswritings:="" how="" however="" hypothetical="" i.="" i.some="" i="" identity="" if="" igno-="" ignorant="" ii.="" ii="" iii="" immediate="" immediately="" in-="" in="" inclined="" independent="" infer="" inference="" inferences="" inferred="" inferrible="" inorganic="" into="" investi-="" investigation="" is="" isnotp="Some" isosceles="" it="" its="" iv.="" iv="" jevons="" justified="" k.="" kind="" knowledge="" l.="" left="" legitimate="" lessons="" life="" likely="" ll="" logic.="" logic="" logical="" logically="" logicians="" losing="" maintain="" make="" makes="" making="" mansel="" mate-="" material="" matter="" may="" meaning="" means="" meant="" merely="" might="" more="" morgan:="" most="" must="" n="" name.="" nature="" need="" negation.="" negation="" negative="" negatives="" never="" new="" news.="" news="" nine.="" no="" noi-a="" noi-b="" noi-p="" none="" not--="" not--p="" not--y="" not-5.="" not-5="Some" not-5by="" not-="" not-b="" not-i="" not-p="All" not-pby="" not-s.="" not="" note="" notice="" o="" observed="" obtain="" obtainable.="" obtained.="" obtained="" obtaining="" obverse="" obversion.="" obversion="" obvert="" obverted="" of="" ofthe="" ogic="" ognise.="" old="" ome="" on="" one.="" one="" only="" onversion="" open="" or="" orators="" order="" ordinary="" organic="" original="" originally="" other="" others="" otherwise.="" our="" out="" p-="" p.="" p="" pall="" part="" particular="" peace="" petnis="" pis="" plants="" plicated="" point="" pointed="" position.="" position="" positions="" practice="" pre-="" precbe="" preceding="" predi-="" predicate.="" predicate="" presently="" previous="" principle="" pro-="" problem="" proceeding="" process="" productive="" professor="" proof="" proposi-="" proposition.="" proposition:="" proposition="" propositions.="" propositions.in="" propositions:="" propositions="" purposes.="" quality="" question="" quite="" quoted="" r="" rance="" re-="" reducing="" reference="" regard="" regarded="" related="" relations="" remain="" remembered="" require="" respectively="" rest="" result.="" results="" rial="" right.="" riginal="" rule="" s="Some" same="" satisfactorily="" saves="" says="" section="" seek="" separable.="" shall="" shew="" shewn="" should="" simpler="" simplest="" simply="" sion="" solved="" some="" sometimes="" special="" state="" statesmen.="" stitch="" strictly="" subject.="" subject.the="" subject="" substances="" succeed="" such="" sum="" syllogisms.="" syllogistno="" symbols="" table="" taken="" taking="" tbe="" temperature="" text-books="" that="" the="" thefollowing="" their="" them.="" then="" theproposition="" there="" therefore="" these="" they="" think="" this="" those="" though="" thus="" time="" tion="" tions="" tjun="" to="" top.="" transform="" transformation="" traposition="" trapositive="" treated="" triangle="" uaisb="" unchanged.="" unchanged="" uneducated="" up="" us="" usage="" use="" uses="" utility="" v.="" verbal="" vert="" very="" virtuous="" volcanoes="" wallis="" war="" warmth="" way="" we="" well="" what="" whately="" whatever="" when="" whether="" which="" who="" whose="" will="" with="" without="" would="" wrong="" y4="" yield="" yllabus="" z="">, A is not B, E. In some cases
in which AisB, C is £>. I. In some cases in which Cis£>, A is B, L None.
If A is B, C is not Z?. E. If C is n, A is not . E. In some c^ses in which C is
not Z>, A is B, I. In some cases in which ^is^, CisnotZ>. 0. None. In
some cases in which C is not A ^ is ^. I. 6—2 84 PROPOSITIONS. [part ii. It
must be remembered that we regard the quality of a hypothetical proposition as
determined by the quality of the consequent. The obverse of a hypothetical
proposition is usually awkward to express. We may however find it if required ;
e,g.y the obverse of " If A is B, Cis Z> " is "If A is B, C
is not not-Z>." 84. Give the converse and the contrapositive of "
If a straight line falling upon two other straight lines make the alternate
angles equal to one another, these two straight lines shall be parallel.'* [l.]
The application of the doctrines of Conversion and Contraposition to
Hypothetical Propositions may be illus- trated by means of the above
proposition. We must note carefully that it is a universal affirmative, and is
therefore only convertible per accidens. This is a point particularly liable to
be overlooked where a universal converse can be legitimately inferred (as in
the case of the above proposition), though not as an immediate inference. We
are in no danger of saying, All men are animals, therefore, all animals are
men; but we may be in danger of saying, All equilateral triangles are
equiangular, therefore, all equiangular triangles are equilateral. From the
point of view however of Formal Logic the latter inference is as erroneous as
the former. So far as thgiven proposition is concerned, we have — Converse^ In
some cases in which two straight lines are parallel, a straight line falling
upon them sKall make the alternate angles equal to one another. Contrapositive^
If two straight lines are not parallel, then a straight line falling upon them
shall make the alternate angles not equal to one another. CHAP. IV.]
PROPOSITIONS. 85 86. Give the contradictory, the contrary, the converse, and
the contrapositive of the following propositions : (i) Things equal to the same
thing are equal to oneanother. (2) No one is a hero to his valet. (3) If there
is no rain the harvest is never good. (4) None think the great unhappy but the
great. (5) Fain would I climb but that I fear to fall. 86. Name the form of
each of the following propositions ; and, where possible, give the converse and
the contrapositive of each : — (i) Some death is better than some life. {ii)
The candidates in each class are not arrangedin order of merit. (iii) Honesty
is the best policy.(iv) Not all that tempts your wandering eyes And heedless
hearts is lawful prize. (v) If an import duty is a means ofrevenue, it does not
afford protection. (vi) Great is Diana of the Ephesians. (vii) All these claims
upon my time overpower me. CHAPTER V. THE INVERSION OF PROPOSITIONS. 87. In
what cases can we obtain by immediate inference from a given proposition a new
proposition having the contradictory of the original subject for its subject,
and the original predicate for its predi- A new form of immediate inference is
here indicated, by which given a proposition having ^S" for its subject
and P for its predicate, we seek to obtain a new proposition having
not-^S" for its subject and P for its predicate. If such a proposition can
be obtained at all, it will be by a certain combination of the elementary
processes of ordinary conversion and obversion. We will take each of the
fundamental forms of proposition and see what can be obtained (i) by first
converting it, and then performing alternately the operations of obversion and
conversion; (2) by first obverting it, and then performing alternately the
operations of conversion and obversion. We shall find that in each case we can
go on till we reach a particular negative proposition whose turn it is to be
converted. (i) The results of performing the processes of con- version and
obversion alternately, commencing with the former, are as follows : — CHAP, v.]
PROPOSITIONS. 87 ' (i) AWSisP, therefore (by conversion), Some P is ^S",
therefore (by obversibn), Some P is not not-S. Here comes the turn for
conversion;but we have an O proposition, and can therefore proceed no further.
(ii) Some S is F, therefore (by conversion), Some P is 5, therefore (by ob
version), Some Pis not not-.S; and we can get no further, (iii) No S is P,
therefore (by conversion), No P is 5, herefore (by obversion), All P is not-5,
therefore (by conversion). Some not-S is /*, therefore (by obversion), Some
not-»S is not not-P. In this case the proposition in italics is the immediate
inference that was sought. (iv) Some ^S" is not P, In this case we are not
able even to commence our series of operations. (2) The results of performing
the processes of Con- version and obversion alternately, commencing with the
latter^ are as follows : — (i) All 5 is P, therefore (by obversion), No S is
not-P, therefore (by conversion), No not-/* is S^ therefore (by obversion), All
not-P is not-5, therefore (by conversion). Some not-^ is not-P^ therefore (by
obversion). Some notS is not P, Here again we have obtained the desired form.
(ii) Some 6* is P, therefore (by obversion), Some S is not not-jP. SS
PROPOSITIONS. [part ii. (iii) TSioSisP, therefore (by obversion), All S is
not-P, therefore (by conversion), Some not-jP is S, therefore (by obversion),
Some not-P is not not-5. (iv) Some iS is not i', therefore (by obversion), Some
S is not-jP, therefore (by conversion), Some not-P is S,
therefore(byobversion). Some not-/* is not not-5. We can now answer the
question with which we com- menced this enquiry. The required proposition can
be obtained only if the given proposition is universal; we then have, according
as it is affirmative or negative, — All S is Pp therefore. Some not-iS is not
P; No S is P, therefore. Some not-iS is P\ It must be observed that in the case
of the former of these we commenced with obversion in order to get the new
form, in the latter we commenced with conversion. This form of immediate
inference has been more or less casually recognised by various logicians; but I
do not remember that it has ever received any distinctive name. Sometimes it
has been vaguely classed under contraposition, (compare Jevons, Elementary
Lessons in Logic, pp. 185, 6), but it is really as far removed from the process
to which that designation has been given as the latter is from ordinary
conversion. I venture to suggest the terms Inversion and Inverse*. Thus,
Inversion is a process of immediate inference 1 For assumptions respecting
"existence" involved in these in- ferences, see chapter 8. *
Professor Jevons (carrying out a suggestion of Professor Robert- son's) has
introduced the term Inverse in a different sense. I do not however think that
for logical purposes we want any new term in the sense in which he uses it ;
and I havebeenunable to think of any other equally suitable term for my own
purpose, for which a new term really CHAP, v.] PROPOSITIONS. 89 in which from a
given proposition we infer another proposition having the contradictory of the
original subject for its subject, is needed, if the scheme of immediate
inferences by means of conver- sion and obversion is to be made scientifically
complete. The term contraverse has occurred to me, but I do not like it so well
; and this again has been appropriated by De Morgan in another sense. Professor
Jevons'snomenclature is explained in the following passage from his Studies in
Deductive Logic, p. 32 : — *' It appears to be indis- pensable to endeavour to
introduce some fixed nomenclature for the relations of propositions involving
two terms. Professor Alexander Bain has already made an innovation by using the
term obverse, and Professor Hirst, Professor Henrici and other reformers of the
teaching of geometry have begun to use the terms converse and obverse in
meanings inconsistent with those attached to them in logical science {Mind,
1876, p. 147). It seems needful, therefore, to state in the most explicit way
the nomenclature here proposed to be adopted with the concurrence of ProfessoV
Robertson. Taking as the original proposition * All A are B,* the following are
what we may call the related propositions — Inferrible, Converse, Some B are A,
Obverse, No A are not B, Contrapositive, No not B are A, or, all not B are not
A, Non-Inferrible. Inverse, All B are A. Reciprocal. All not A are not B, It
must be observed that the converse, obverse, and contrapositive are all true if
the original proposition is true. The same is not neces- sarily the case with
the inverse and reciprocal. These latter two names are adopted from the excellent
work of Delboeuf, ProUgomhus Fhilo- sophiques de la GSomitrie, pp. 88 — 91, at
the suggestion of Professor Croom Robertson {Mind, 1876, p. 425)." In this
scheme what I propose to call the Inverse is not recognised at all. On the
other hand, I hardly see why the non-inferrible forms need such a distinct
logical recognition as is implied by giving them distinct names ; while except
in books on Logic I anticip>ate that the term converse is likely still to be
used in its non-logical sense, {i,e,, "All B are ^ " is likely still
to be spoken of as the converse of ** All A are and the original predicate for
its predicate. In other words, given a proposition having S for subject and P
for predicate, we obtain by inversion a new proposition having not-S for subject
and P for predicate. We may now sum up the results that have been obtained with
regard to immediate inferences. Given two terms iS and Py and admitting their
contradictories not-^S" and not-/*, we have eight possible forms of
proposition as shewn in the following scheme: — B "). It may be noted that
in Jevons's use of terms, the inverse would be the same as the converse in the
case of E and I propositions. I imagine also that in consistency there should
be yet another term to express the relation of "No not-^ is not-^" or
"All not-^ is A^' to "No A is -5"; it is, in the sense in which
Jevons uses these terms, neither the Converse, Obverse, Contrapositive, Inverse
nor Reciprocal. CHAP, v.] PROPOSITIONS. 91 These propositions may be designated
respectively: — (i) The original proposition, (ii) The obverse, (iii) The
converse, (iv) The obverted converse, (v) The contrapositive, (vi) The obverted
contrapositive, (vii) The inverse, (viii) The obverted inverse. It has been
shewn, in sections 66^ 73, 76, and in the above, that if the original
proposition is universal, we can infer from it propositions of all the
remaining seven forms ; but if it is particular, we can infer only three
others. Working out the different cases in detail we have : — A. (i) Original
proposition, A/l S is P, (ii) Obverse, M? S is not-P, (iii) Converse, Some P is
S, (iv) Obverted converse. Some P is not noi-S. (v) Contrapositive, No not-P is
S, (vi) Obverted Contrapositive, All not-P is n^f-S. (vii) Inverse, Some not-S
is not P, (viii) Obverted Inverse, Some not-S is not-P, I. (i) Original
proposition, Some S is P. (ii) Obverse, Some S is not not-P. (iii) Converse,
Some P is S, (iv) Obverted Converse, Some P is not not-S (v) Contrapositive,
none can be inferred, (vi) Obverted Contrapositive, none, (vii) Inverse, none,
(viii) Obverted Inverse, none. E. (i) Original proposition, JVo S is P, (ii)
Obverse, All S is not-P. PROPOSITIONS. (in) Converse, No P is S, (iv) Obverted
Converse, All P is not-S. (v) Contrapositive, Some not-P is S. (vi) Obverted
Contrapositive, Some not-P is not not-S. (vii) Inverse, Some not-S is P. (viii)
Obverted Inverse, Some not-S is not not-P, O. (i) Original proposition, Some S
is not P, (ii) Obverse, Some S is not-P. iii) Converse, none can be inferred,
(iv) Obverted Converse, none, (v) Contapositive, Some not-P is S. (vi) Obverted
Contrapositive, Some not-P is not not-S. (vii) Inverse, none, (viii) Obverted
Inverse, none. All the above is summed up in the following Table (using the
symbols described in section 38, and denoting not-S by S\ not-P by P'):—
Original PropositionConverse Obverted Converse Contrapositive Obverted
Contrapositive . . . Obverted Inverse CHAP, v.] PROPOSITIONS. 93 It is worth
noticingthatwe can infer the same number of propositions from E as from A (7),
from O as from I (3), and the same number of universal propositions from E as
from A (3);also in two cases we can get no more from A than from I, and no more
from E than from O. 88. Give the inverse of the following proposi- t(i) A
stitch in time saves nine. (2) None but the brave deser\'e the fair. (3) He
can't be wrong whose life is in the right. (4) The virtuous alone are happy.
89. Assuming that no organic beings are devoid of Carbon, what can we thence
infer respectively about beings which are not organic, and things which are not
devoid of carbon ? [L.] 90. Make as many Immediate Inferences as you can from
the following propositions : — (i) Civilization and Christianity are
coextensive. (2) Uneasy lies the head that wears a crown, (3) Your money or
your life ! [l.] 91. Write out all the propositions that must be true, and all
that must be false, if we grant that (a) A straight line is the shortest
distance be- tween two points ; (j8) All the angles of a triangle are equal to
two right angles ; (7) Not all the great are happy. [c.] 94 PROPOSITIONS. [part
ii. 92. De Morgan says (Fourth Memoir on the Syllogism^ p. 5) of the Laws of
Thought : " Every transgression of these laws is an invalid inference ;
every valid inference is not a transgression of these laws. But I cannot admit
that everything which is not a transgression of these laws is a valid
inference.*' Investigate the logical relations between these three assertions.
[Jevons, Studies^ p. 301.] 93. Assign the logical relation, if any, between
each pair of the following propositions : — (i) All crystals are solids. (2)
Some solids are not crystals. (3) Some not crystals are not solids. (4) No
crystals are not solids. (5) Some solids are crystals. (6) Some not solids are
not crystals. (7) All solids are crystals. [l.] 94. . " All that love
virtue love angling." Arrange the following propositions in the four
following groups : — (a) Those which can be inferred from the above proposition
; (/8) Those from which it can be inferred ; (7) Those which do not contradict
it, but which cannot be inferred from it ; (S) Those which contradict it. CHAP,
v.] PROPOSITIONS. 95 (i) None that love not virtue love angling. (li) All that
love angling love virtue, (iii) All that love not angling love virtue, (iv) None
that love not angling love virtue. (v) Some that love not virtue love angling,
(vi) Some that love not virtue love not angling, (vii) Some that love not
angling love virtue, (viii) Some that love not angling love not virtue. ^
CHAPTER VI. THE DIAGRAMMATIC REPRESENTATION OF PROPOSITIONS. 96. Methods of
illustrating the ordinary processes of Formal Logic by means of Diagrams.
Representing the individuals included in any class, or denoted by any name, by
a circle, it will be obvious that the five following diagrams represent all
possible relations between any two classes : — The force of the different
prepositional forms is to ex- clude one or more of these possibilities*. ^ The
method of interpreting a proposition by what it excludes or negatives is discussed
in more detail in chapter VIII. CHAP. VI.] PROPOSITIONS, All S is P limits us
to ( «/» j orSome Sis P to one of the four NoSis P to Some Sis Mi P to one of
the three To represent All ^S is i* by a single diagram, thus 98 PROPOSITIONS.
[part il. or Some 5 is i' by a single diagram, thus is most misleading; since
in each case the proposition really leaves us with other alternatives. This
method of employ- ing the diagrams is however adopted by most logicians who
have used them, including Sir "William Hamilton {Logic, i. p. 255), and
Professor Jevons {Elementary Lessons in Logic, pp. 72 — 75); and the attempt at
such simplification has brought their use into undeserved disrepute. Thus, Mr
Venn remarks, "The common practice, adopted in so many manuals, of appealing
to these diagrams, — Eulerian diagrams as they are often called, — seems to me
very questionable. The old four propositions A, E, I, O, do not exactly corre-
spond to the five diagrams, and consequently none of the moods in the syllogism
can in strict propriety be represented by these diagrams V {Symbolic Logic, p.
15, compare also pp. 424, 425). This is undoubtedly sound as against the use of
Euler's circles by Hamilton and Jevons; but I do not admit its force as against
their use in the manner described above \ Many of the operations of Formal
Logic can be satisfactorily illustrated by their aid; though it is true that
they become somewhat cumbrous in relation to the Syllogism. Thus, they may be
employed, — (i) To illustrate the distri- bution of the predicate in a
proposition. In the case of each of the four fundamental propositions we may
shade the part of the predicate concerning which knowledge is given us. We then
have, — ^ They are used correctly by Ueberweg. Cf. Lindsay's translation of
Ueberweg's System of Logic ^ pp. a 16 — 218. CHAP. VI.] PROPOSITIONS. The
result is that with A and I there are cases in which only part of jP is shaded;
whereas with E and O, the whole of F is in every case shaded; and it is made
clear that negative propositions distribute, while affirmative proposi- tions
do not distribute theirpredicates.(2) To illustrate the Opposition of
Propositions, Com- paring two contradictory propositions, e,g,y A and O, we see
hat they have no case in common, but that between them they exhaust all
possible cases. Hence the truth, that two contradictory propositions cannot be
true together but that one of them must be true, is brought home to us under a
new aspect. Again, comparing two subaltern propositions, e.g., A and I, we
notice that the former gives us all the information given by the latter and
something more, since it still further limits the possibilities. To make this
point the more clear the following table is appended:-— CHAP. VL] PROPOSITIONS.
(3) To illustrate the Conversion of Propositions. Thus, it is made quite clear
how it is that A admits only of Con- version per accidens. All S is P limits us
to one or other of the following The problem of Conversion is — ^What do we
know of P in either case? In the first, we have All P is 5, but in the second
Some P is *S; /.^., taking the cases indifferently, we have Some P\% S and
nothing more. Again, it is made clear how it is that O is inconvertible. Some S
is not P limits us to one or other of the following, — What then do we know concerning
P} The three cases give us respectively (i) No T' is 5, (ii) Some P is S, and
Some P is not 5, (iii) KWPisS. (i) and (iii) are contraries, and (ii) is
contradictory to both of them. Hence nothing can be affirmed of P that is true
in all three cases indifferently. (4) To illustrate the more complicated forms
of imme- diate inference. Taking, for example, the proposition All S is Py we
may ask, What does this enable us to assert about 102 PROPOSITIONS. [PARTii:
not-/* and not-^S* respectively? We have one or other of these casesWith regard
to not-/*, these yield respectively, to prove conversions (i) by
meanslof syllogisms, (2) by means of thethree laws of identity, contradiction
and ex- cluded middle. ■ Bain writes as follows,-— " When we examine
carefully* the various processes in Logic, we find them to be material to the
very core. Take Conversion, How do we know that, if No -AT is PJ No y is -Y? By
examining cases in^ (Jetail, and finding the equivalence t6 be true. Obvious as
the inference seems on the mere fonnal ground, we do not; content ourselves
with the formal aspect. If we did, we sjiould be as likely to say, All JY is Y
gives All Y is X\ Wfe are prevented from this leap merely by the examination '
of cases" {Logic, Deduction, p. 251). The implication here made that the
proof of rules of conversion is a kind of inductive proof seems to me
unwarranted. The justification of conversion that I should myself give is that
in the case of each of the four fundamental ' forms of proposition, its
conversion (or in the case of an O ^ proposition, the impossibility of
converting it) is self-evident; and that we cannot go beyond this simple
statement. Thus, taking an E proposition, I should say that it is self-evident
. that if one class is entirely excluded from another class, this second class
is entirely excluded from the first. In the case of an A proportion it is clear
on reflection that the statement All S y& P may include^ one or other of
the two relations of classes, — either S andPcoincident, or P con- - taining S
and moire besides, — ^but that these are the only two possible relations to
which it can be applied. It is self-evident that in each of these cases some P
\% S\ and hence the inference by conversion from an A proposition is shewn to
be justified \ In the case of an O proposition, ^ Compare section 95, where
these inferences are illustrated by the aid of the Eulerian diagrams. CHAP,
VII.] PROPOSITIONS. IIS if we consider all the relationships of classes in
which it holds good, we find that nothing is true of P in terms of S in all of
them. Hence O is inconvertible*, I may add that I do not see that in the above
reasoning we should be assisted by any explicit reference to the three laws of
thought; nor that the application of the three laws of thought alone would-be
sufficient to give us our results. 103. Without assuming Conversion, how would
you logically justify the process of Contraposition ? [c] ^ Again, compare
section 95. CHAPTER VIIL PREDICATION AND "EXISTENCE*." 104. Are
assuniptions with regard to *' existence " involved in any of the
processes of immediate infer-» As pointed out by Mr Venn {Symbolic Logic^ pp.
127, 128), a discussion about "existence" need not in this con- nection
involve us in any kind of metaphysical enquiry. " As to the nature of this
existence, or what may really be meant by it, we have hardly any need to
trouble ourselves, for almost any possible sense in which the logician can
understand it will involve precisely the same difficulties and call for the
same solution of them. We may leave it to any one to define the existence as he
pleases, but when he has done this it will always be reasonable to enquire
whether there is anything existing corresponding to the X or Y which constitute
our subject and predicate. There can in fact be no fixed tests for this
existence, for it will vary widely according to the nature of the
subject-matter with which we are concerned in our reasonings^ For in- stance,
we may happen to be speaking of ordinary pheno- menal existence, and at the
time present; by the distinction ^ It may perhaps be advisable for students, on
a first readings to omit this chaptert CAP. VIII.] PROPOSITIONS. ir; in
question is then meant nothing more and nothing deeper than what is meant by
saying that there are such things as antelopes and elephants in existence, but
not such things as unicorns or mastodons. If again we are referring to the
sum-total of all that is conceivable, whether real or imagi- nary, then we
should mean what is meant by saying that everything must be regarded as
existent which does not involve a contradiction in terms, and nothing which
does. Or if we were concerned with Wonderland and its occu- pants we need not
go deeper down than they do who tell us that March hares exist there. In other
words, the inter- pretation of the distinction will vary very widely in
different cases, and consequently the tests by which it would have in the last
resort to be verified ; but it must always exist as a real distinction, and
there is a sufficient identity of sense and application pervading its various
significations to enable us to talk of it in common terms." Now, several
views may be taken as to what implication with regard to existence, if any, is
involved in any given proposition. (i) It may be held that every proposition
implies the existence of its Subject, since there is no use in giving infor-
mation with regard to a non-existent subject. (2) It may be held that although
such existence is gene- rally implied, still it is not so necessarily ; and
that at any rate in Formal Logic we ought to leave entirely on one side the
question of the existence or the non-existence of the subjects of our
propositions. (3) The view is taken by Mr Venn that for purposes of Symbolic
Logic, universal propositions should not be regarded as implying the existence
of their subjects, but \ki2X particular propositions should be regarded as
doing so. This view might be extended to ordinary Formal Logic ^la PROPOSITIONS,
[part II.Withoutat ohce deciding' which of these' views is to be pr:efeiTed, we
may briefly investigate the consequences which follow from them respectively so
far as immediate inferences, are concerned. Fir sty we may take the supposition
that every proposition implies the existence of its subject. Thiis, All S\% P
implies the existence of 5, and it follows that it also implies the existence
of P. No Sv&P implies the existence of 5, and since by the law of excluded
middle every S is either Pot not-/', it follows that it also implies the
existence of noXrP. But now if from All 6" is /* we are to be allowed to
obtain the ordinary immediate inferences, — if, for example, we may infer AH
not-/' is not-*S, — the existence of not-/* and nbt-iSare also involved. Similarly,
the conversion of No *S is P requires that we posit the existence of P and
not-5. On this supposition, then, we find that propositions are not amenable to
the ordinary logical operations^ except on the assumption of the existence of
classes corresponding not merely to the terms directly involved but also to
their contradictories, De Morgan practically adopts this alternative. "By
the universe (of a proposition) is meant the collection of all objects which
are contemplated as objects about which assertion or denial may take place. Let
every name which belongs to the whole universe be excluded as needless: this
must be particularly remembered. Let every object which has not the name X {of
which there are always some) be conceived as therefore marked with the name x
meaning not-^y" (Syllabus^ pp. 12, 13). Compare also Jevons, Pure Logic,
pp. 64, 65; Studies in Deductive Logic, p. i8i. Secondly, we may take the
supposition that no proposition logically implies the existence of its subject.
On this view, the proposition All -S is /* may be read. All S, if there is any
S, or, when there is any S, is P] and its full implication CHAP* VIII.]
PROPOSITIONS. 119 with regard to existence may be expressed by saying that it
denies the existence of. any thing that is at the same time S and not F. In Mr
Venn's words, "/^ burden of implication of existence is shifted from the
affirmative to the negative form'; that is, it is not the existence of the
subject or the predicate (in affirmation) which is implied, but the non-existence
of any subject which does not possess the predicate" {Symbolic Logic, p.
141). Similarly,onthis view, No S is F implies the existence neither of S nor
of F, but merely denies the existence of anything that is both S and F. Some S
is F (or is not F) may be read Some 5, if there is any 5, is F (or is not F).
Here we do not even negative or deny the existence of any class absolutely; the
sum total of what we affirm with regard to existence is that if any S exists,
then some F (or not-F) also exists. Now having got rid of the implication of
the existence of the subject in the case of all propositions, we might
naturally suppose that in no case in which we make an immediate inference need
we trouble ourselves with any question of " existence " at all. On
further enquiry, however, we shall find that so far as particulars are
obtained, assump- tions with regard to existence are still involved in somQ
processes of immediate inference. All S is F2X any rate implies that if there
is any S there is also some F, whilst on our present view it does not re- quire
that if there is any F there is also some S, But the converse of the given
proposition, — Some F is 5, — does imply this. ** If the predicate exists then
also the subject exists " must therefore be regarded as an assumption
which is involved in the conversion of an A proposition; similarly, in the
conversion of an I, and in the contraposition of an £ or of an O proposition.
It follows also that in passing from All ^ is /* to Some not-5 is not-/*, we have
to assume 12b Propositions. [part h. that if there is any not-S there is also
some not-P. It does not appear th^t there is any similar assumption in the
conversion of an K proposition ; nor do I think that there is any in the
obversion either of A or E, or in the contra- position of A. It might indeed at
first sight seem that in passing from No iS is Z' to All S is not-P, we have to
assume that if there is any S there is also some hot-P. " But, even on our
present supposition, this is necessarily implied in the proposition No S is P
itself. If there is any S it is by the law of excluded middle either P or not
P; therefore, given that No S is P, it follows immediately that if there is any
S there is some not-P. It can also be shewn that since No S is P denies the
existence of anything that is both S and Py it implies by itself that if there
is any P there is some not-^S; and that since the proposition All S is P denies
the existence of anything that is both S and not- P,it implies by itself that
if there is any not-P there is some The given supposition then provides for the
obversion and contraposition of A, and for the obversion and con- version of E,
without any further implication with regard to existence than is contained in
the propositions them- selves. But the conversion or inversion of A involves
further assumptions, as shewn above ; and the same is true of the
contraposition or inversion of E, the conversion of I and the contraposition of
O. Now it will be observed that in the first set of cases we obtain by our
immediate inference a universal propo- sition ; in the second set a particular
one. We may there- fore generalise our results as follows, — On the supposi-
tion that no proposition logically implies the existence of its subject we do
not require to make any assumption with regard to existence in any process of
immediate inference provided that CHAP. VIII.] PROPOSITIONS. 121 // yidds a
universal conclusion ; but it is generally other- wise in cases that yield only
a particular conclusion. In other words, whenever we are left with a universal
con- clusion we need not be afraid that any assumption with regard to existence
has been introduced unawares; but whenever We are left with a particular
conclusion such an assumption may have been made, and if we find that it is so,
this should be explicitly stated TTiirdly, taking Mr Venn's view, which is the
same as the preceding so far as universal propositions are concerned, but which
regards particular propositions as implpng the existence of their subjects, the
result just obtained hardly requires to be modified It must however be observed
that on this supposition we cannot even pass from All 5 is F to Some S is Py
except under the condition that the existence of S is granted. 106. Shew that
in some processes of conversion assumptions as to the existence of classes in
nature have to be made; and illustrate by examining whether any such
assumptions are involved in the inference that if All 5 is P, therefore Some
not-5 is not-P. Concerning this question, Professor Jevons remarks that it
"must have been asked under some misapprehension. The inferences of Formal
Logic have nothing whatever to do with real existence; that is, occurrence
under the con- ditions of time and space *' {Studies in Deductive Logic, p. 55).
The question is doubtless somewhat unguarded with regard to the nature of the
existence implied, but I think that in any case the discussion in the preceding
section shews that it does not admit of being so summarily dis- missed*. Even
granting that the formal logician may say ^ What follows is to some extent a
repetition of what has been I2a PROPOSITIONS* [PAKT ii> that given the
proposition AH .S is F, it is "no concern of his whether or not there are
any individuals actually belong- ing to the classes S and -P, nevertheless he
must admit that the proposition at least involves that if therie are any ^S*
there must be some P, while it does not involve that if there are any F there
must be some S. But now convert the propo- sition. We obtain Some F is S, and
this does involve that if there are any F there roust be some S, I do not
there- fore see how in converting the given proposition this as- sumption can
be avoided Thus, from **A11 dragons are serpents", we may infer by
conversion **Some serpents are dragons,'' and this proposition implies that if
there are; serpents there are also dragons. Similarly, in passing from All 5 is
/^ to Some noi-S is not-/*, it must at least be assumed that if S does not
constitute the entire universe of discourse, neither does F do so. If we make
immediate inferences from hypothetical propositions, the necessity of a similar
assumption seems still more obvious. For example, from the true statement that
if Governor Musgrave's econo- mic doctrines are correct, Mr Mill makes mistakes
in his Political Economy, we can hardly without qualification infer that in
some cases in which Mr Mill makes mistakes in his Political Economy, Governor
Musgrave's doctrines are cor- rect, since Mr Mill might be sometimes wrong, and
never- theless Governor Musgrave always so. In another place {Studies in
Deductive Logic, p. 141) Jevons remarks, " I do not see how there is in
deductive logic any question about existence"; and with reference to the
opposite view taken by De Morgan, he says, " This is One of the few points
in which it is possible to suspect him given in the preceding section. The view
that I am here especially combating however is that Formal Logic cannot
possibly have any concern with questions relating to *' existence." CHAP.
VIII.] PROPOSITIONS. 123 of iinsoundnisss." I can however attach no
meanibg to Jevons's own " Criterion of Consistency '* {Studies in DeiiU-
HveLogiCy p. 181) unless it has some reference to "existence," "
It is assumed as a necessary law that every term mu5t have its negative. This
was calledthe Law of Infinity in* my first logical essay {Pure Logic ^ P- 65;
see also p. 45); but as pointed out by Mr A. J. Ellis, it is assumed by De Mor-
gan, in his Syllabus^ Article 16. Thence arises what I pro- pose to call the Criterion
of Consistency^ stated as follows : — Any two or more propositions are
contrculictory 7vhen, and ofily when, after ail possible substitutions are
made, they ouasion the total disappearance of any term, positive or negch live,
from the LogiccU Alphabet J' What can this mean but that although we may deny
the existence of the combination AB, we cannot without contradiction deny the
existence of A itself, or not--^, or B, or not-.5? Indeed, in reference to
Jevons's equational logic generally, what can negativing a combination mean but
denying its existence ? For example, I take the following quite at random, —
" There remain four combinations, ABC, aBC, abC, and abc* But these do not
stand on the same logical footing, because if we were to remove ABC, there
would be no such thing as A left; and if we were to remove abc there would be
no such thing as c left. Now it is the Criterion or condition of logical con-
sistency that every separate term and its negative shall remain. Hence there
must exist some things which are described by ABC, and other things described
by abc^^ {Studies in Deductive Logic, p. 216). With regard to Jevons's
criterion of consistency itself, I am hardly prepared to admit it. If I am not
allowed to negative X, why should I be allowed to negative AB"^ There is
nothing to prevent X from being itself a complex term. In certain combinations
indeed it may be convenient124 PROPOSITIONS. [PARTIL to substitute X for AB, or
vice versa. It would appear then that what is contradictory when we use a
certain set of symbols may not be contradictory when we use another set of
symbols. I should say that Jevons's criterion is some- times a convenient
assumption to make, but nothing more than this; and it is I think an assumption
that should always be explicitly referred to when made. 106. Is a categorical
proposition to be regarded as logically implying the existence of its subject }
Our answer to this question must depend to some extent on popular usage, and to
some extent on logical conveni- ence. So f?ir as universal propositions are
concerned, I should be inclined on both grounds to answer it in the In the
first place, I do not think that in ordinary speech we always imply the
existence of the subjects of our pro- positions. No doubt we usually regard
them as existing; but as Mr Venn shews there are undoubtedly exceptions to this
rule. ** For instance, assertions about the future do not carry any such
positive presumption with them, though the logician would commonly throw them
into precisely the same * All X is y type of categorical assertion. * Those who
pass this examination are lucky men ' would certainly be tacitly supplemented
by the clause *if any such there be.' So too, in most circumstances of our
ordinary life, wherever weare clearly talking of an ideal * Perfectly
conscientious men think but little of law and rule,* has a sense without
implying that there are any such men to be found^ " {Symbolic Logic^ pp.
130, 131). Again, a mathe- * Theaboveseems to me an answer to such a statement as
the following: — "In an ordinary proposition the subject is necessarily
admitted to exist, either in the real or in s6me imaginary world assumed HAP.
VIII.] PROPOSITIONS. J25 matician might assert that a rectilinear figure having
a mil- lion equal sides and inscribable in a circle has a million equal angles,
without intending to imply the actua) exist- ence of such a figure ; or if I
know that A is X, B is K,C is Z, I may affirm that ABC is XVZ without wishing
to commit myself to the view that the combination ABC does ever really occur \
Taking complex subjects, and limiting our conception ofexistence as we not
unfrequently do to some particular universe, cases of this kind might be
multiplied indefinitely. But if it is granted that in ordinary thought the existence
of the subject of the proposition sometimes is and some- times is not implied,
it follows that since the logician cannotdiscriminate between these cases, he
had best content him* self with leaving the question open, that is, he should
regard such existence as not necessarily or logicallyimplied. And, further, to
adopt this alternative is logically more convenient, since SiO far as the
obtaining universal propo- sitions by immediate inference is concerned, we do
not on this supposition require any further assumptions with regard to
existence in order that such immediate inference may be legitimate. On the
other hand, if we take the other alter- for the nonce When we say No stone is
alive, or All men are mortal^ we presuppose the existence of stones or of men.
Nobody would trouble himself about the possible properties of purely prob-
lematic men or stones" {Mind, 1876, pp. 290, 291). But the conclu- sions,
" Those who pass this examination are lucky men," " Perfectly
conscientious men think but little of law and rule '* may certainly be worth
obtaining, although in the' universe to which reference is made, (and in both
the cases in question this would be the actual material universe), the subjects
of these propositions might be non-existent. ^ Is it not sometimes the case
that in order to disprove the existence of some combination, say AB, we
establish a self-contradictory pro-r position of the form AB is both C and
not-C ? 126 PROPOSITIONS. [PARTir. native and regard categorical propositions
as always im- plying the existende of their subjects, we have shewn in section
104 that we require to assume the existence not merely of the actual terms
involved in any given proposi- tion, but also of their contradictories. ' •'
The importance of the question here raised is mor^ i)articularly manifest when
we are dealing' with very complex propositions?, as is shewn by Mr Venn. '• We
say then that logically All S is F implies oiily the non-existence of anything
that is both S and not-/*; No S is P implies only the non-existence of anything
that is both The case oi particular propositions still remains; and here again
I am inclined to agree with the view taken by Mr Venn in his Symbolic Logic,
namely that such proposi- tions should be regarded as implying the existence of
their subjects. The chief grounds for adopting this view is that "•'an
assertion confined to 'some' of a class generally resti upon observation or
testimony rather than on reasoning or imagination, and therefore almost
necessarily postulates existent ata, though the nature of this bbservation and
consequent existence is, as already remarked, a perfectly open question"
{Symbolic Logic, p. 131). I doubt whether in ordinary speech we ever predicate
anything of a non- existent subject unless we do so universally. The principal
objection to this view is perhaps the paradox which follows from it, namely
that we are not without qualification justified in inferring from All S is F
that Some S is F, (since the latter proposition implies the existence of S,
while the former does not). It may even be said that this view practically
banishes the particular proposition from Logic altogether. Possibly if it were
so, it would be no very serious matter. But I do not think that it is so. We
have CftAl>. VUI.] l>ROPOSITIONS, 6xily to be careful in using such
propositions to note the assumption -involved in their use. The principal value
of particulars is in their relation of contradiction to universals of different
quality. But their use in this respect is entirely consistent with the above.
We have taken the view that the import of All S is P is to deny that there is
any S that is not-/*; we are now taking the view that the import of Some S is
not F is to iiffirm that there is some S that is not-P. This clearly brings out
the contradictory character of the two propositions. Similarlywith I and E. One
interesting point ta notice here is that if there is no implication of the
existence of the subject in universal pro^. positions we are not actually
precluded from asserting to- gether two contraries. We may say All S is Pajid
No S is P; but this virtually is to deny the existence of S. ^// S /V P
excludes JVb S tsV e;ccludes But these are all possible cases. In other
respects, this investigation if pursued might somewhat modify accepted logical
doctrines ; but I feel convinced that we should be ultimately left with a
consistent whole. The truth is, as Mr Venn has remarked, that most English
logicians have made no critical examination at all 12^ PROPOSITIONS. [part II.
of the question here raised It may be desirable to return to it briefly in
connection with the syllogism. Compare sections 273 — 277. [The above view,
which is taken by Mr Venn in respect to Symbolic Logic, and which I have
attempted to apply to ordinary Formal Logic, is practically identical with that
somewhat recently put forward in a more paradoxical form by Professor Brentano.
Compare Mind^ 1876, pp. 289 — 292. *^ Where we say Some man is sick^ Brentano
gives as a sulv stitute, There is a sick man. Instead of No stone is aiive^ he
puts There is not a live stone. Some man is not learned becomes There is an
unlearned man^ Finally, All men are mortal is to be expressed in his system
There is not an im- mortal manP'\ 107. Discuss the relation between the
propositions All 5 is Z' and All not-5 IS Z'. This is an interesting case to
notice in connection with the discussion raised in the preceding sections. All
S is P= No S is not-i'= No not-i' is S. All not-^S" is i'= No not-5 is
not--P= No not-P is not-5 = All not-i' is S. ' The given propositions come out
therefore as contraries. (i) On the view that we ought not to enter into any
discussion concerning "existence" in connection with im- mediate
inference, we must I suppose rest content with this statement of the case. It
seems however sufficiently curious to demand further investigation and
explanation. (2) On the view that propositions imply the existence of their
subjects, we have shewn in section 104, that we are not justified in passing
from All not-^* is P to All not-/* is S unless we assume the existence of
nolt-P. But it will CHAP, viii.] PROPOSITIONS. 129 be observed that in the case
before us, the given propo- sitions make such an assumption unjustifiable.
Since All S is P and All not-S is P, and ever3rthing is either S or not-^ by
the law of excluded middle, it follows that nothing is In reducing the given
propositions therefore to such a form that they appear as contraries, (and
therefore as in- consistent with each other), we assume theverything that taken
together they really deny. (3) On the view that at any rate universal
propositions do not imply the existence of their subjects, we have shewn in the
preceding section, that the propositions No not-/* is S, All not-/* is S, are
either inconsistent or else they express the fact that /* constitutes the
entire universe of discourse. But thisfact is the verything that is given us by
the propo- sitions in their original form. On either of the views (2) or (3),
then, the result obtained is satisfactorily accounted for and explained.
CHAPTER IX. HYPOTHETICAL AND DISJUNCTIVE PROPOSITIONS. 108. The nature of the
logical distinction between Categorical and Hypothetical Propositions. Are the
propositions "All B is C" and "If any- thing is B, it is C"
equivalent.? or can either be inferred from the other ? Mr Venn holds that the
real differentia of Hypothetical Propositions is *'to express human doubt"
{Mind, 1879, p. 42). I should myself prefer to express the import of
Hypothetical Propositions by saying that they affirm a connection between
certain events, whenever they happen or if they ever happen, whilst leaving the
question en- tirely open whether or not they do ever happen. The doubt which
they imply is rather incidental, than the fundamental or differentiating
characteristic belonging to them. Materially indeed I think that they do
sometimes imply the actual occurrence of their antecedents. When- ever the
connection between the antecedent and the con- sequent in a hypothetical
proposition can be inferred from the nature of the antecedent independently of
specific experience, (and this may be the more usual case), then the actual
happening of the antecedent is not in any sense in- CHAP. IX.] PROPOSITIONS.
131 voived; but if our knowledge of the connection does depend on specific
experience, (as it sometimes may), and could not have been otherwise obtained,
then such actual happening would appear to be materially involved. For example,
the statement, "If we descend into the earth, the temperature increases at
a nearly uniform rate of i® Fahr. for every 50 feet of descent down to almost a
mile," requires that actual descents into the earth should have been made,
for otherwise the truth of the statement could not have been known. It may,
however, be replied that the doubt applies to the actual occurrence of the
antecedent in a given instance. When I say " If the glass falls, it will
rain," I imply doubt as to whether it actually will fall on the occasion
to which I am referring. (Compare Venn, Symbolic Logic, pp. 331 — 333.) But may
not this be the case also with categorical propo- sitions ? For example, if I
am in doubt whether a given plant is an orchid, I may apply the proposition
"All orchids have opposite leaves*' in order to resolve my doubt. We have
such a case as this whenever categorical propositions are used in the process
of diagnosis, and it can hardly be said that wc never doemploycategorical
propositions in this mStill, it is clear that the hypothetical proposition does
not necessarily imply the actual occurrence of its antecedent; and therefore,
if the view is taken that the categorical pro- position does necessarily imply
the actual existence of its subject, (compare sections 104, 106), we have a
marked distinctionbetween the two kinds of propositions. "If anything is By
it is C" cannot be resolved into "All B is C", since thelatter
implies the existence of B while the former does not. Another vie^r with regard
to categorical propositions, 132 PROPOSITIONS. [PART 11. and the one for which
I have expressed a preference, is that they do not necessarily imply, (and
therefore do not logically imply), tlie existence of their subjects. On this
view, I do not see that we have any logical distinction between hypo- thetical
and categorical propositions, except a distinction of form ; that is, they may
be resolved into one another. We may say indifferently " All B '\% C or
" If anything is B it is C"; " If AisB, C is Z>" or
" All cases of A being B are cases of C being £>" Kant denies that
we can reduce the hypothetical judg- ment to the categorical form on the
following ground : " In categorical judgments nothing is problematical,
but every- thing assertative; in hypothetical it is merely the connection
between the antecedent and the consequent that is assertative. Hence here we may
combine two false judgments." This view has I think been virtually
discussed in what I have already said. If the categorical judgment is regarded
as affirming not merely a connection between the subject and the predicate but
also the existence of the subject, then I admit the force of the above
argument, and allow that the hypothetical judgment cannot be reduced to the
categorical form. But if the categorical judgment is not regarded as affirming
the existence of the subject^ it (like the hypothetical judgment) asserts no
more than a connection ; it is no more assertative than the hypothetical
judgment, and just as problematic. The non-existence of the subject of the
cate- gorical corresponds exactly to the falsity of the antecedent of the
hypothetical ; and if in the latter we may combine two false judgments, in the
former we may combine two non-existent entities. I may say, If A is By C is
Z>, although A is B IS di false judgment ; but similarly I may say any case
of A being ^ is a case of C being Dy although the case of A being B is z.
non-existent, case. I cannot CHAP. IX.] PROPOSITIONS. 133 see that in the
latter of these statements I have committed myself to anything whatever that is
not contained in the former. Hamilton also {Logic, i. p. 239) holds that a
hypothetical judgment cannot be converted into a categorical. "The
thought, A is through B, is wholly different from the thought, A is in B, The
judgment, — If God is righteous, then will the wicked be punished, and the
judgment, — A righteous God punishes the wicked, are very different, although
the matter of thought is the same. In the former judgment, the punishment of
the wicked is viewed as a consequent of the righteousness of God ; whereas the
latter considers it as an attribute of a righteous God. But as the conse- quent
is regarded as something dependent from, — the at- tribute, on the contrary, as
somethipg inhering in, it is from two wholly different points of view that the
two judgments are formed.'* Now it must certainly be admitted that in any given
instance there are reasons why we choose the h)T)othetical mode of expression
rather than the cate- gorical, or vice versa ; but the only question that
concerns us from a logical point of view is whether precisely the same meaning
cannot be expressed in either form. Hamilton would appear to deny not merely
that a hypothetical judgment can be converted into a categorical, but also that
a categorical can be converted into a hypothetical. But, (leaving on one side
the question of the existence of the subject in a categorical proposition,
which has already been discussed), can any one who allows that "all
orchids have opposite leaves'' deny that "if this plant is an orchid it
has opposite leaves"? Can any one who allows that "if there are
shg.rpers in the company we ought not to gamble," deny that "all
cases in which there are sharpers in the company are cases in which we ought
not to gamble"? 134 PROPOSITIONS. [part II. If this is admitted, the
logical question is to my mind dis- posed of ^ No doubt hypothetical
propositions will frequently look awkward when expressed in the categorical
form, but in some cases logical error is more likely to be avoided if we reduce
them to this form before manipulating them; and I cannot see how we lose
anything, or, (on the view now taken with regard to the existential import of
categorical propositions), imply anything that we should not imply, in so
dealing with them. I have given examples shewing that the doctrines of
opposition and immediate inference may be applied to hypothetical. We shall
find that the same is true of the doctrine of syllogism, though it may be
useful to frame special rules when we are dealing with propositions expressed
in this form. 109. The interpretation of Disjunctive Proposi- tions. There is a
difference of opinion among logicians as to ^ Hansel's view upon this question
{Aldrich^ pp. 103, 104) is not easy to understand. He admits however that *'^\{
A is B, C is Z? " implies that ** Every case of A being -5 is a case of C
being Z?." He even goes so far as to resolve ** If all A is B^ all A is C*
into ** All B is C," which is clearly erroneous. His whole treatment of
hypo- theticals is puzzling. For example, he says, **The judgment, *If A is By
C is />,' asserts the existence of a consequence necessitated by laws other
than those of thought, and consequently out of the province of Logic"
{Aldrichf p. 236; Prolegomena Logica, p. 230). But «imilarly a categorical
proposition may assert a connection not neces- sitated by laws of thought ; and
I do not see that we have here any reason for subjecting hypothetical
propositions to a peculiar treatment. I am inclined to think that what makes
Hansel's discussion of hypo- thetical propositions so difficult is that he
attempts to apply to them the strict conceptualist view of Logic, which it is
impossible to apply consistently throughout without divesting Logic of all
content what- soever.CHAP. IX.] PROPOSITIONS. 135 whether the alternatives in a
disjunctive proposition should be regarded as mutually exclusive. For example,
in the proposition A is either B or C, there is not general agree- ment as to
whether it is logically implied that A cannot be both B and C \ There are at
least two questions involved which should be distinguished. (i) In ordinary
speech do weintend that the alter- natives in a disjunctive proposition should
be necessarily understood as excluding one another ? A very few instances will
I think enable us to answer this question in the negative. "Take,for
instance, the proposition — 'A peer is either a duke, or a marquis, or an earl,
or a viscount, or a baron '...Yet many peers do possess two or more titles, and
the Prince of Wales is Duke of Cornwall, Earl of Chester, Baron Renfrew,
&c....In the sentence — 'Repent- ance is not a single act, but a habit or
virtue,' it cannot be implied that a virtue is not a habit... Milton has the
ex- pression in one of his Sonnets — * Unstained by gold or fee,* where it is
obvious that if the fee is not always gold, the gold is a fee or bribe.
Tennyson has the expression * wreath or anadem,' Most readers would be quite
un- certain whether a wreath may be an anadem, or an anadem a wreath, or
whether they are quite distinct or quite the same'* (Jevons, Furg Logic, pp.
76, 77).(2) But this does not absolutely settle the question. It may be said: —
Granted that in common speech the alternatives of a disjunction may or may not
be mutually exclusive, still in Logic we should be more precise, and * Whately,
Mansel, Mill, and Jevons would answer this question in the negative ; Kant,
Hamilton, Thomson, Boole, Bain, and Fowler in the affirmative. 136
PROPOSITIONS. [part ii. thie statement ^^A is either B or C^ (where it may be
both) should be written " A is either B or C or both." This is a
question of interpretation or method, and I, do not apprehend that any burning
principle is involved in the answer that we may give. For my own part I do not
find any reason for diverging from the usage of everyday language. On the other
hand, I think that if Logic is to be of practical utility, the less logical
forms diverge from those of ordinary speech the better. And further, it
conduces to clearness if we make a logical proposition express as little as
possible. " A is either B or C, it can- not be both" is best given as
two distinct propositions ^ ^ A view strongly opposed to that adopted in the
text is taken in a recently published work on the Principles of Logic by Mr
Bradley of Merton College, Oxford. His argument is as follows : — *• The com-
monest way of regarding disjunction is to take it as a combination
ofh)rpotheses. This view in itself is somewhat superficial, and it is possible
even to state it incorrectly. * Either A\& B ox C\% D"^ means, we are
told, that if A is not B then C is Z?, and if C is not D then A is B, But a
moment's reflection shews us that here two cases are omitted. Supposing, in the
one case, that A is Bi and supposing, in the other, that C is Z>, are we
able in these cases to say nothing at all? Our * either — or' can certainly
assure us that, if A is B^ C — D must be false, and that, if C is Z>, then A
— B is false. We have not exhausted the disjunctive statement, until we have
provided for four possibilities, B and not --5, C and not-C" (Principles
of Logic ^ p. I2i). The question raised is really one of interpretation, as I
have indicated above; but this is what Mr Bradley will not admit. In my view,
it is open to a logician to choose either of the two ways of interpreting a
disjunctive proposition, provided that he makes it (juite clear which he has selected
; but I can see no good in dogmatising as in the following passage, — **Our
slovenly habits of expression and thought are no real evidence s^ainst the
exclusive character of dis- junction. M is ^ or ^' does strictly exclude *^A is
both b and r.' When a speaker asserts that a given person is a fool or a rogue,
hemay not mean to deny that he is both. But, having qo interest in CHAP. IX,]
PROPOSITIONS. 137 Professor Fowler indicates this view in his statement that
"it is the object of Logic not to state our thoughts in a condensed form
but to analyse them into their simplest elements" (Deductive LogiCy p.
32); though he does not apply it to the case before us. Mansel arguingin favour
of the view that I have taken remarks, — "But let us grant for a moment
the opposite view, and allow that the proposition, * All C is either A or B^
implies, as a condition of its truth, * No C can be both.' Thus viewed, it is
in reality a complex proposition, contain- ing two distinct assertions, each of
which may be the ground of two distinct processes of reasoning, governed by two
opposite laws. Surely it is essential to all clear thinking, that the two
should be separated from each other, and not confounded under one form by
assuming the Law of Ex- cluded Middle to be, what it is not, a complex of those
of Identity and Contradiction" {Prolegomena Logica, p. 238). Of course if
the alternatives are logical contradictories they are logically exclusive, but
otherwise in thetreatment of disjunctive propositions in the following pages I
donot regard them as being so. If in any case they happen to be materially
incompatible, this must be separately stated. 110. From the statement that
blood-vessels are either veins or arteries, does it follow logically that a
blood-vessel, if it be a vein, is not an artery ? Give your reasons. [L.]
shewing that he is both, being perfectly satisfied provided he is one, either b
or c, the speaker has not the possibility be in his mind. Ig- noring it as
irrelevant, he argues as if it did not exist. And thushemaypractically be right
in what he says, though formally his statement is downright false : for he has
excluded the alternative bc"*^ (p. 124). 138 PROPOSITIONS. [part ii. 111.
Put, if you can, the whole meaning of a dis- junctive proposition (such as,
Either A is 5 or C is D) in the form of a single and simple Hypothetical, and
prove your expression to be sufficient. [R.] Adopting the view that in a
disjunctive proposition the alternatives are not to be regarded as necessarily
excluding one another, such a disjunctive proposition as the above is primarily
reducible to two hypotheticals, namely, If ^ is not B, C is Z>, and If C is
not Z>, A is B. But each of these is the contrapositive of the other, and
may therefore be in- ferred from it. Hence the full meaning of the disjunctive
is expressed by means oi either oi \h&&Q hypotheticals ^ Professor
Croom Robertson called attention to this point in Mind, 1877, p. 266, —
"The other form of propo- stion ranged by logicians with the Hypothetical,
namely the Disjunctive, may be shewn to be as simple as the pure Hypothetical
being in fact a special case of it. The com- mon view is that it involves at
least two hypothetical propo- sitions, or, as some say, even four. Thus *
Either A is B or C is Z> ' is resolved by some into the four hypotheticals —
* Mr Bradley {Principles of Logic, p. 121), lays it down that ** disjunctive
judgments cannot really be reduced to hypotheticals " at all ; but I
hardly care to disagree with him since he admits all that I should contend for.
He distinctly resolves **^ is b or ^" into hypotheticals (p. 130); but, he
adds, although the meaning of dis- junctives can thus **be given hypothetically
; we must not go on to argue from this that they trr^ hypothetical" (p.
121). They "declare a fact without any supposition " (p. 122). But so
does the hypothetical itself, namely, the connection between the antecedent and
the conse- quent. Further, "A combination of hypotheticals surely does not
lie in the hypotheticals themselves" (p. 122). Undoubtedly, by means of a
combination of hypotheticals, we may make a most categorical state- ment; e.g.,
l(AisS,CisD; and ii A \& not S, Cis D. CHAP. IX.] PROPOSITIONS. 139 KAisBy
CisnotZ)(i), If A is not B, CisD (2), If Cis D, A is not B (3), If C is not I?,
AisB (4), — but the first and third of these are rejected by others, and with
reason, because they are in fact implied only when the alternatives are logical
opposites. The remaining propo- sitions (2) and (4) are, however, the logical
contrapositives of one another; and this amounts to saying that either of them
dy itself is a full and adequate expression of the original disjunctive."
PART III. SYLLOGISMS. CHAPTER I. THE RULES OF THE SYLLOGISM. 112. The Terms
oftheSyllogism.Areasoningconsistingof three categorical propositions (of which
one is the conclusion), and containing three and only three terms, is called a
Categorical Syllogism. Every categorical syllogism then contains three and only
three terms, of which two appear in the conclusion and also in one or other of
the premisses, and one in the premisses only. That which appears as the
predicate of the conclusion, and in one of the premisses, is called the major
term; that which appears as the subject of the conclusion, and in one of the
premisses, is called the minor term; and that which appears in both the
premisses, but not in the conclusion, (being that term by their relations to
which the mutual relation of the two other terms is determined), is called the
middle term. HAP. I.] SYLLOGISMS. 141 Thus, in the syllogism, — All M is P, All
S is M, therefore. All S\s F-y P is the major term, S is the minor term, and M
is the middle tenn. [These respective designations of the terms of a syllogism
resulted from such a syllogism as, — All M is P, All S is M, therefore, All S
is /*, being taken as the type of syllogism. With the exception of the somewhat
rare case in which the terms of a propo- sition are coextensive, such a
syllogism as the above may be represented by the followingdiagram.Here clearly
the major term is the largest in extent, and the minor the
smallest,whilethemiddle occupies an intermediate position. But we have no
guarantee that the same relation between the terms of a syllogism will hold,
when one of the pre- misses is a negative or a particular proposition; d^., the
following syllogism, — No^isP, A115isJf, therefore, No S is P^ 42 SYLLOGISMS.
[part iil gives as one case where the major term may be the smallest in extent,
and the middle the largest Again, the following syllogism, — No M'\%P, Some S
is J/, therefore, Some S is not P^ gives as one case where the major term may
be the smallest in extent and the minor the largest. With regard to the middle
term, however, we may note that although it is not always a middle term in
extent, it is always a middle term in the sense that by its means the two other
terms are connected, and their mutual relation deter- mined.] 113. The
Propositions of the Syllogism. Every categorical syllogism consists of three
propositions. Of these one is the conclusion. The premisses are called the
major premiss and the minor premiss according as they contain the major term or
the minor term respectively. CHAP. I.] SYLLOGISMS. 143 Thus, All M is P, (major
premiss), All S is J/, (minor premiss), therefore, All S is F, (conclusion). It
is usual, (as in the above syllogism), to state the major premiss first and the
conclusion last. 114. The Rules of the Syllogism; and the Deduc- tion of the
Corollaries, The rules of the Syllogism as usually stated are as follows ; —
(i) Every syllogism contains three and only three terms. (2) Every syllogism
consists of three and only three pro- positions. It may be observed that these
are not so much rules, as a general description of the nature of the syllogism.
A reasoning which does not fulfil these conditions may be formally valid, but
we should not call it a syllogism*. The four following rules are really rules
in the sense that if, when we have got the reasoning into the form of a syl-
logism, they are not fulfilled, then the reasoning is invalid. (3) No one of
the three terms of the syllogism must be used ambiguously ; and the middle term
must be distributed once at least in the preinisses. This rule is frequently
given in the form : " The middle term must be distributed once at least,
and must not be ambiguous," (e.g.^ in Jevons, Elementary Lessons, p. 127).
* For example, B is greater than C, • A is greater than B, therefore, A is
greater than C. Here there are four terms, since the predicate of the second
premiss is "greater than ^," and this is not the same as the subject
of the first premiss **^.'* . 144 SYLLOGISMS. [part hi. But it is obviousthat
we must guard against ambiguous major andambiguous minor as well as against
ambiguous middle. If the middle term is distributed in neither of the pre-
misses, the syllogism is said to be subject to the fallacy of undistributed
middle, (4) No term must he distributed in the conclusion which was not
distributed in one of the premisses. The breach of this rule is called illicit
process of the major y orillicit process of the minor, as the case may be; or,
more briefly, illicit major or illicit minor. (5) From tutonegativepremisses
nothing can be inferred, (6) If one premiss is negative, the conclusion must be
nega- tive; and to prove a negative conclusion, one of the premisses must be
negative. From these rules, three corollaries may be deduced: — (i) From two
particular premisses nothing can be in- ferred. Two particular premisses must
be either (a) bothnegative, or ()8) both affirmative, or (y) one negative and
one affirmative. But in case (a), no conclusion follows by rule 5. In case
()8), since no term can be distributed in two particular affirmative
propositions, the middle term cannot be distributed, and therefore no
conclusion follows byrule 3. In case (y), if we can have a conclusion it must
be nega- tive (rule 6). Themajor term therefore will be distributed in the
conclusion ; and hence we must have two terms dis- tributed in the premisses,
namely, the middle and the major (rules 3, 4). But a particular negative
proposition and a CHAP. I.] SYLLOGISMS. X45 particular affirmative proposition
between them distribute only one term. Therefore, no conclusion can be
obtained. [De Morgan {Formal Logic, p. 14) proves this corollary as follows : —
" Since both premisses are particular in form, the middle term can only
enter one of them universally by being the predicate of a negative proposition
; consequently the other premiss must be affirmative, and, being particular,
neither of its terms is universal. Consequently both the terms as to which the
conclusion is to be drawn enter partially, and theconclusioncan only be a
particular affir- mative proposition. But if one of the premisses be negative,
the conclusion must be negative* This contradiction shews that the supposition
of particular premisses producing a legitimate result is inadmissible."]
(ii) If one premiss is particular, so must be the conclusion^. We must have
either (a) two negative premisses, but this case is rejected by rule 5 ; or {P)
two affirmative premisses; or (y) one affirmative and one negative. In case {P)
the premisses, being both affirmative and one of them particular, can
distribute but one term between them, This must be the middle term by rule 3.
The minor term is therefore undistributed in the premisses, and the conclusion
must be particular by rule 4. In case (y) the premisses will between them
distribute two and only two terms. These must be the middle by * This and the
sixth rule are sometimes combined into the one rule, Concltisio sequitur partem
deteriorenty — i.e,^ the conclusion follows the worse or weaker premiss both in
quality and in quantity ; a negative being considered weaker than an
affirmative, and a particular than a universal* K. L. 10 146 SYLLOGISMS; [part hi.
rule 3, and the major by rule 4, (since we have a negative premiss,
necessitating a negative conclusion by rule 6, and therefore the distribution
of the major term in the conclusion). Again, therefore, the minor cannot be
dis- tributed in the premisses, and the conclusion must bepar- ticular by rule
4. [De Morgan {Formal Logic, '^. 14) gives the following very ingenious proof
of this corollary: — " If two propositions F and Q, together prove a
third, F, it is plain that F and the denial of F, prove the denial of Q. For F
and Q can- not be true together without F, Now if possible, let F (a
particular) and Q (a universal) prove F (a universal). Then F (particular) and
the denial of F (particular) prove the denial of Q, But two particulars can
prove nothing."] (iii) From a particular major and a negative minor
nothing can be inferred. Since the minor premiss is given negative, the major
premiss must by rule 5 beaffirmative. But it is also particular, and it
therefore follows that the major term cannot be distri- buted in it. Hence, by
rule 4, it must be undistributed in the conclusion,/.^., the conclusion must be
affirmative. But also by rule 6, since we have a negative premiss, it must be
negative. This contradiction establishes the corollary that under the supposed
circumstances no conclusion is possible. 115. Shew by aid of the. syllogistic
rules that thepremisses of a syllogism must contain one more distributed term
than the conclusion ; also, that there is always thesame number of distributed
terms in the predicates of the premisses taken together as in the predicate of
the conclusion. Hence deduce CHAP. I.] SYLLOGISMS. 147 the three corollaries.
[Cf. Monck, Introduction to Logic, pp. 40, 4l] 116. " When one of the
premisses is Particular, the conclusion must be Particular. The transgression
of this rule is a symptom of illicit process of the minor." Spalding,
Logic, p. 209. Is it the case that we cannot infer a universal conclusion from
a parti- cular premiss without committing the fallacy of illicit minor } 117.
Illustrate De Morgan's statement that any case which falls under the rule
that" from premisses both negative no conclusion can be inferred" may
be reduced to a breach of one of the preceding rules. De Morgan {Formal Logic,
p. 13) takes two universal negative premisses E, E, In whatever figure they
are, they can be reduced by conversion to, — NoPis Jf, No S is M. Then by
obversion they become, (without losing any of their force), — All P is not-
Jt/; All S is XiOi-M\ and we have undistributed middle. Hence rule 5 is ex-
hibited as a corollary from rule 3. An objection may perhaps be taken to the
above on the ground that the premisses might also be reduced to,— All M is
not-i', Allil/isnot-^; where the middle term is distributed in both premisses.
Here however it is to be noted that we have no longer a middle 10 — 2 148
SYLLOGISMS, [part hi. term connecting S andP at all. We shall teturn
subsequently to this method of dealing with two negative premisses. The case in
which one of the premisses is particular is dealt with by De Morgan {Formal
Logtc^ p. 14) as follows : — "Again, No yis X, Some Fs are not Zs, may be
converted into Every X is (a thing which is not F), Some (things which are not
Zs) are Fs, in which there is no middle term." This is not quite satisfactory,
since we may often exhibit a valid syllogism in such a form that there appear
to be four terms ; ^.^., I might say, " All M is F^ All S is M, may be
converted into All M isFy No S is not-J^, in which there is no middle
term." The case in question may however be disposed of by saying that if
we can infer nothing from two universal negative premisses, a fortiori we
cannot from two negative premisses, one of which is particular, 118. The rule
that "if one premiss is negative, the conclusion must be negative,"
may be established as a corollary from the rule that " from two negative
premisses nothing can be inferred.'* The following has been suggested to me by
DeMorgan's deduction of corollary ii., (cf. section 114): — If two pro- positions
F and Q together prove a third R^ it is plain that F and the denial of R prove
the denial of Q, For F and Q cannot be true togetherwithoutR. Now if possible
let F (a negative) and Q (an affirmative) prove R (an affirmative). Then F (a
negative) and the denial of R (a negative) prove the denial of Q, But two
negatives prove nothing. ■vsscsver^^sr-^^nM^^BVS CHAP. I.] SYLLOGISMS. 149 119.
Simplification of the Rules of the Syllogism. It would now seem as if the six
rules of the syllogism might be simplified Rules i and 2 may be treated as a
description of the syllogism rather than as rules for its validity. The part of
rule 3 relating to ambiguity may be regarded as contained in the proviso that
there shall be only three terms, {t,e,, ifone ofthe terms is ambiguous, we have
not really a syllogism according to our definition of syllogism). Rule 5 has
been exhibited in section 117 as a corollary from rule 3; and the first part of
rule 6 has been shewn in section ii8 to be a corollary from rule 5. We are left
then with only three independent rules, — (a) The middle term must be
distributed once at least in the premisses ; {P) No term must be distributed in
the conclusion un- less it has been distributed in the premisses ; (y) A
negative conclusion cannot be inferred from two afiirmative premisses* 120. In
reference to the syllogism, it has been urged that the old rule that negative
premisses yield no conclusion does not hold true universally, as in the
example. Whatever is not metallic is not capable of powerful magnetic
influence, carbon is not metallic, therefore, carbon is not capable of powerful
magnetic influence. Examine this criticism. [c] Professor Jevons gives this
case in his Principles of Science (ist edition, vol. i., p. 76; 2nd edition, p.
6$\ and he states that "the syllogistic rule is actually falsified in its
bare and general statement." Professor Croom Robertson has however
conclusively shewn (in Mind^ 1876, p. i 19, note) that this apparent ex- I50
SYLLOGISMS. [part hi. ception is no real exception*. *' There zxtfour terms in
the example, and thus no syllogism, if the premisses are taken as negative
propositions ; while the minor premiss is an affir- mative proposition, if the
terms are made of the requisite number three.*' Mr Bradley {Principles of
LogiCy p. 254) returns to the position taken by Professor Jevons. In reference
to the example given in the above question, he says, " This argu- ment no
doubt has qimternio terminorum and is vicious technically, but the fact remains
that from two denials you somehow have proved a further denial. * A is not B^
what is not B is not (7, therefore A is not C7'; the premisses are surely
negative to start with, and it appears pedantic either to urge on one side that
^A is not--^ ' is simply positive, or on the other that B and not-.^ afford no
junction. If from negative premisses I can get my conclusion, it seems idle to
object that I have first transformed one premiss ; for that objection does not
shew that the premisses are not negative, and it does not shew that I have
failed to get my con- clusion." This is somewhat beside the mark ; and if
the points on both sides are clearly stated there appears no room for further
controversy. On the one hand, it is implicitly admitted both by Professor
Jevons (Studies in Deductive Logic, p. 89), and by Mr Bradley, that two
negative premisses invalidate a syllogism , i.e., understanding by a syllogism
a mediate reasoning containing three and only three terms. Oh the other hand,
everyone would allow that from two propositions which may both be regarded as ^
Mr Venn, also, (in the Academy^ Oct. 3, 1874), — "The reply clearly is,
that if * not metallic' is to be regarded as the predicate of the minor, then
the minor i^ affirmative; if 'metallic' is predicate, then there are four
terms." CHAP. I.] SYLLOGISMS. 151 negative, a conclusion may sometimes be
obtained; for example, the propositions which constitute the premisses of a
syllogism in Barbara^ may be written in a negative form, thus, No M is not-P,
No S is not-Jtf, and no doubt the con- clusion — All S is P — still follows. We
must not, however, attach undue importance to the distinction between positive
and negative propositions. By means of the process of Ob- version, the logician
may at will regard any given propo- sition as either positive or negative. [A
similar case to that given in the question is dealt with in the Port Royal
Logic (Professor Ba)mes's translation, p. 211) as follows ; — ** There are many
reasonings, of which all the pro- positions appear negative, and which are,
nevertheless, very good, because there is in them one which is negative only in
appearance, and in reality affirmative, as we have already shewn, and as we may
still further see by this example : That which has no parts cannot perish by
the dissoltdion of its parts; The soul has no parts; Therefore^ the soul cannot
perish by the dissolution of its parts. There are several who advance such
syllogisms to shew that we haveno right to maintain unconditionally this axiom
of logic, Nothing can be inferred from pure negatives; but they have not
observed that, in sense, the minor of this and such other syllogisms is
affirmative, since the middle, which is the subject of the major, is in it the
attribute. Now the subject of the major isnot that which has parts, but All S
is M, therefore, All S\% P, Cf. section 158. 152 SYLLOGISMS* [part hi. that
which has not parts, and thus the sense of the minor is, TIu soul is a thing
without partSy which is a proposition affirmative of a negative
attribute."] 121. By what means can we obtain a conclusion from the two
negative premisses, — No M is P, No J/ is 5.? By obverting the premisses, we
have—, All J/ is not-/'. All Mh not-5, therefore, Some not-^S is not-/'\ 122.
Take an apparent syllogism subject to the fallacy of negative premisses, and
enquire whether you can correct the reasoning by converting one or both of the
premisses into the affirmative form. [Je- vons, Studies in Deductive Logic, p.
84.] Both in the Studies and in the Principles of Science (Vol. I., p. 75),
Professor Jevons appears to answer this question in the negative. It is certainly
not put in an unexceptionable form, but apparently reference is made to the
case given in the preeding section. No A is By No A is Cy may be transformed
into, — All A IS not-^, All A is not-C; * But this does not invalidate the
syllogistic rule that from two nega- tive premisses nothing can be inferred,
since so long as both the pre- misses remain negative we have more than three
terms and therefore not a syllogism at all. CHAP. I.] SYLLOGISMS. IS3 yielding
a conclusion, — Some not- C is not--fi^. [In Jevons's system, this would
become, — A »Ac; yielding a conclusion, — Alf = Ac. (Cf. Principles of Science^
voL i., p. 71 j 2nd ed., p. 59).] 123. Given (i) All P is M, (ii) All 5 is J/,
(iii) M does not constitute the entire universe of discourse. What conclusion
can we infer ? Exhibit the reasoning in the form of an Aristote- lian
syllogism. Is the third premiss necessary in order that the conclusion may be
obtained ? Make any comments that occur to you in connection with this point
From (i) we can obtain by immediate inference. All tiO\.-M\s noXrPy and from
(ii) All not- J/ is not-»S ; and these premisses yield the conclusion, — Some
not-.^ is not-/*. The reasoning is here exhibited in the form of an Aristote-
lian syllogism. Or, we might reason as follows : — Since S and P are both
entirely included in M, there must be outside M some not-»S and some not-/*
that are coincident ; and this is the same conclusion as before.Now in the
latter form of the reasoning it would seem that we have assumed that there is some
not-M^ /.^,, that M 154 SYLLOGISMS. [part iii. does not constitute the entire
universe of discourse. But the necessity of this assumption was not apparent in
our first method of treatment, according to which by a simple process of
immediate inference we obtained a perfectly valid syllogism \ The truth appears
to be thathere at any rate we have an illustration ofDeMorgan'sview (Formal
Logic, p. 112) that in all syllogisms the existence of the middle term is a
datum. From the premisses All M is P, All M is 5, we cannot obtain the
conclusion Some S\%P without implicitly assuming the existence of M. Take as an
example, — All witches ride through the air on broomsticks; All witches are old
women; therefore, Some old women ride through the air onbroomsticks. This point
is further discussed in sections 273—277. We may note that the reasoning, — All
P is M, hWSviM, therefore, Some not-5 is not-i', does not invalidate the
syllogistic rule that the middle term must be distributed once at least in the
premisses, since as it stands it contains more than three terms and is
therefore not a syllogism. 124. Examine the following assertion: ''In no way
can a syllogism with two singular premisses be viewed as a genuine syllogistic
or deductive inference.^' This assertion is made by Professor Bain, and he
illus- trates it {Logic J Deduction, p. 159) by reference to the fol- lowing
syllogism : ^ Cofflpare, however, lection 104. CHAP. I.] SYLLOGISMS. 155
Socrates fought at Delium, Socrates was the master of Plato, therefore, The
master of Plato fought at Delium. But "the proposition * Socrates was the
master of Plato and fought at Delium *, compounded out of the two pre- misses
is nothing more than a grammatical abbreviation " ; and the step hence to
the conclusion is a mere omission of something that had previously been said.
" Now, we never consider that we have made a real inference, a step in ad-
vance, when we repeat less than we are entitled to say, or drop from a complex
statement some portion not desired at the moment. Such an operation keeps
strictly within the domain of Equivalence or Immediate Inference. In no way,
therefore, can a syllogism with two singular premisses be viewed as a genuine
syllogistic or deductive inference." The above leads up to some very interesting
considera- tions, but it proves too much. In the following syllogisms the
premises may be similarly compounded together, — all men are mortal, ) ,, . 1 j
^' t ,, . , V all men are mortal and rational: ''all men are rational,)
therefore, some rational beings are mortal. all men are mortal,)' ,, • 1 j- 1 •
^1 , ,, . \ all men including kings are mortal : all kings are men, j 00
therefore, all kings are mortal*. 1 With the above, compare the following
syllogism, baving two singular premisses : — The Lord Chancellor receives a
higher salary than the Prime Minister, Lord Selborne is the Lord Chancellor,
therefore, Lord Selborne receives a higher salary than the Prime Minister. The
premisses here would similarly, I suppose, be compounded by Professor Bain into
"The Lord Chancellor, Lord Selborne, receives a higher salary than the
Prime Minister." 156 syllogisms; [part in. Do not Bain's criticisms apply
to these syllogisms as much as to the syllogism with two singular premisses ?
The method of treatment adopted is indeed particularly ap- plicable to
syllogisms in which the middle term is subject in both premisses *; but in any
case it is true that the con- clusion of a syllogism contains a part of^ and
only a part of, the information contained in the two premisses taken to-
gether. Also, we may always combine the two premisses in a single statement;
and thus we may always get Bain's result. In other words, in the conclusion of
every syllogism "we repeat less than we are entitled to say,'* or, if we
care to put it so, **drop from a complex statement some portion not desired at
the moment." It may be worth while here to refer to the charge of
incompleteness which Professor Jevons (Principles of Science^ i. p» 71) has
brought against the ordinary syllogistic conclusion. *' Potassium floats on
water, Potassium is a metal," yield, according to him, the conclusion,
"Potassium metal is potassium floating on water;" But "Aristotlp
would have inferred that some metals float on water. Hence Aristotle's conclusion
simply leaves out some of the informa- tion afforded in the premisses; it even
leaves us open to interpret the some metals in a wider sense than we are
warranted in doing." In reply to this it may be remarked : first, that the
Aris- totelian conclusion does not profess to sum up the whole of the
information contained in the premisses of the syl- logism; secondly, that some
in Logic means merely **not none", "one at least". The
conclusion of the above syllo- gism might perhaps better be written "some
metal floats on water," or "some metal or metals, &c."
Compare Mr Venn", ^ i,e,^ to syllogisms in Figure 5. Cf. section 143. ' '^
Surely, as the old expression * discursive thought' implie3, we CHAP. I.]
SYLLOGISMS. 157 in the Academy^ Oct. 3, 1874; also, Professor Croom Robert- son
in Mind^ ^876, p. 219. 126. How far does the conclusion of an Aristo- telian
syllogism fall short of giving all the informa- tion contained in the
premisses? [Jevons, Studies^ P- 215.]126. The connection between the Dictum de
omni et nulla and the ordinary rules of syllogism. The Dictum de omni et nulla
was given by Aristotle as the axiom on which all syllogistic inference is
based. It; applies directly, however, to those syllogisms only in which the
major terai is predicate in the major premiss, and the minor term subject in
the minor premiss, (/>., to what are called syllogisms in Figure i). The
rules of syllogism, on the other hand, apply independently of the position of
the terms in the premisses. Nevertheless, it is interesting to trace the connection
between them. We shall find all the rules implicitly contained in the Dictum^
but some of them in a less general form, in consequence of the distinction
pointed out above. The Dictum may be stated as follows: — "Whatever is
predicated, whether affirmatively or negatively, of a term distributed may be
predicated in like manner of everything contained under it" designedly
pass on from premisses to conclusion, and then drop the premisses from sight.
If we want to keep them in sight we can perfectly well retain them as premisses
; if not, if all that we want is the final fact, it is no use to burden our
minds or paper with premisses as well as con- clusion. All reasoning is derived
from data which under conceivable circumstances might be useful again, but which
we are satisfied to recover when we want them." 158 SYLLOGISMS. [part hi.
(i) The Dictum provides for three and only three terms; namely, (i) a certain
term which must be distributed, (ii) something predicated of this term, (iii)
something contained under it. These terms are respectively the middle, major,
and minor. We may consider the rule relating to the ambiguity of terms also
contained here, since if any term is ambiguous we have practically more than
three terms. (2) The Dictum provides for three and only three pro- positions;
namely, (i) a proposition predicating something of a term distributed, (ii) a
proposition declaring something to be contained under this term, (iii) a
proposition making the original predication of the contained term. These pro-
positions constitute respectively the major premiss, the minor premiss, and the
conclusion of the syllogism. (3) The Dictum prescribes not merely that the
middle term shall be distributed once at least in the premisses, but more
explicitly that it shall be distributed in the major premiss, — "Whatever
is predicated of a term distributed^ [This is really another form of what we
shall find to be a special rule of Figure i, namely that the major premiss must
be universal. Cf. section 144.] (4) The proposition declaring that something is
con- tained under the term distributed must necessarily be an affirmative
proposition. The Dictum provides therefore that the premisses shall not be both
negative. [It really provides that the minor premiss shall be affirmative,
which again is one of the special rules of Figure i.] (5) The words "in
like manner" clearly provide against a breach of rule 6, namely that if
one premiss is negative, the conclusion must be negative, and vice versa, (6)
Illicit process of the major is provided against indi- rectly. We can commit
this fallacy only if we have a nega- CHAP. I.] SYLLOGISMS, 159 tive conclusion,
but the words "in like manner" declare that if we have a negative
conclusion, we must have a nega- tive major premiss, and since in any syllogism
to which the Dictum directly applies, the major term is predicate of this
premiss, it likewise will be distributed. Illicit process of the minor is
simply provided against inasmuch as we are warranted to make our predication in
theconclusion only of what has been shewn in the minor premiss to be contained
under the middle term. 127. Can the Syllogism be based exclusively on the laws
of Identity, Contradiction and Excluded Middle } Mansel answers this question
in the affirmative and main- tains {Prolegomena Logica^ p. 222) that "the
Principle of Identity is immediately applicable to affirmative moods in any
figure, and the Principle of Contradiction to negatives." In order to shew
this, he commences by quantifying the predicate (cf. section 217), and taking
as an example the syllogism,— All M is some P^ ll S is some M, therefore, All S
is some P^ he reads it thus,— " the minor term aU S is identical with a
part of My and consequently with a part of that which is given as identical with
all J/*, namely some i'." He then takes the syllogism, — All M is some P,
Some S is some My therefore. Some S is some Py and, treating it similarly,
finds that "the principle immedi- ately applicable to both is the axiom,
that what is given as identical with the whole or a part of any concept, must
be i6o SYLLOGISMS, [1>ART ill. identical with the whole or a part of that
which is identical with the same concept" Passing by the inaccuracy of
speaking of the concepts as being identical*, I cannot see that the above axiom
is the same as the Principle of Identity, "Every A is AJ^ The syllogism is
something riiore than mere subaltern inference ; it involves a passage of
thought through a middle term; and it is just this that the Law of Identity as
expressed in the formula " Every A is A" ap- pears to me unable to
provide for. This. law may tell us that if all Mis P, then some ^is J^; but
does it tell us that if all Mis P, therefore S is jP, because it is M'i The
Dictum de omni et nullo clearly enunciates the principle involved in
syllogistic reasoning; the Law of Identity, if it does so at all, does so less
satis- factorily. Or rather I would say that if the Law of Identity is to cover
this principle, then it is inadequately expressed in the formula Every A is A\
Similar remarks apply to the attempt to bring syllogisms with negative
conclusions under the Principle of Contradiction, "No A is hot-A" ^
It is really the extension of the one concept that is identical with the whole
or a part of the extension of the other; and although the comprehension of a
concept is practically the concept itself, it is clear that the same is not
true of its extension. It has always seemed to me rather curious that the
doctrine of the Quantification of the Predicate should have been introduced by
writers like Hamilton and Mansel, who lay so much stress on concepts, * I
should say the same in reference to ManseVs remark (Prole^ gomena Logica, p.
103), that the Axiom "things that are equal to the same are equal to one
another" is only another statement of the Principle of Identity. CHAPTER
11. IMPLE EXERCISES ON THE SYLLOGISM. 128. Explain what is meant by a
Syllogism; and put the following argument into syllogistic form : — ** We have
no right to treat heat as a substance, for it may be transformed into something
which is not heat, and is certainly not a substance at all, namely, mechanical
work." [N.] 129. Put the following argument into syllogistic t ^ ^\^ \
form: — How can any one maintain that pain is always an evil, who admits that remorse
involves pain, and yet may sometimes be a real good } [v.] 130. It has been
pointed out by Ohm that reasoning to the following effect occurs in some works
on mathematics: — "A magnitude required for the solution of a problem must
satisfy a particular equation, and as the magnitude x satisfies this equa-
tion, it is therefore the magnitude required." Examine the logical
validity of this argument, [c] 131. If P is a mark of the presence of Q, and R
of that of S, and if P and R are never found together, K. L. II i62 SYLLOGISMS.
[part hi. . am I right in inferring that Q and S sometimes exist separately ?
[v.] The premisses may be stated, — All P is Q, All jR is S, No-Pisi?j and in
order to establish the desired conclusion we must be able to infer at least one
of the following, — Some Q is not S, Some S is not Q. But neither of these
propositions can be inferred, since they distribute respectively S and Q,
whilst neither of these terms is distributed in the given premisses. The
question is therefore to be answered in the negative. 132. If it is false that
the attribute B is ever found coexisting with A, and not less false that the
attribute C is sometimes found absent from A, can you assert anything about B
in terms of C? [c] 133. Enumerate the cases in which no valid con- clusion can
be drawn from two premisses. 134. Shew that (i) If both premisses of a
syllogism are affirma- tive, and one but only one of them universal, they will
between them distribute only one term ; (ii) If both premisses are affirmative
and both universal, they will between them distribute two terms; (iii) If one
but only one premiss is negative, CHAP. II.] SYLLOGISMS. 163 and one but only
one premiss universal, they will between them distribute two terms; (iv) If one
but only one premiss is negative, and both premisses are universal, they will
between them distribute three terms, 136. Ascertain how many distributed terms
there may be in the premisses of a syllogism more than in the conclusion. [L.]
136. Prove that, when the minor term is predicate in its premiss, the
conclusion cannot be A. [l.] 137. If the major term of a syllogism be the
predicate of the major premiss, what do we know about the minor premiss ? [l.]
138. How much can you tell about a valid syllogism if you know, — (i) that only
the middle term is distributed; (2) that only the middle and minor terms are
distributed ; (3) that all three terms are distributed ? [w.] 139. If it be
known concerning a syllogism in the Aristotelian system that the middle term is
dis- tributed in both premisses, what can we infer as to the conclusion ? [c]
If both premisses are affirmative, they can between them distribute only two
terms; but by h)rpothesis the middle term is distributed twice in the
premisses, the minor term cannot therefore be distributed, and it follows that
the conclusion must be particular. j64 syllogisms. [paRt lii. If one of the
premisses is negative, we may have three terms distributed in the premisses ;
these must, however, be the middle term twice (by hypothesis), and the major
term (since the conclusion must now be negative and the major term will
therefore be distributed in it); hence the minor term cannot be distributed in
the premisses, and it again follows that the conclusion must be particular. But
either both premisses willbeaffirmative, or one affirmative and the other
negative ; in any case, therefore, we can infer that the conclusion will be
particular. [This proof seems preferable to that given by JevonSj Studies in
Deductive Logic^ p. 83.] 140. Shew that if the conclusion of a syllogism be a
universal proposition, the middle term can be but once distributed in the
premisses. [L.] As pointed oiit by Professor Jevons {Studies in Deductive
LogiCy p. 85), this proposition is the contrapositive of the result obtained in
the preceding section. 141. Shew directly in how many ways it is pos- ible to
prove the conclusions SaPy SeP\ point out those that conform immediately to the
Dictum de omni et nullo ; and exhibit the equivalence between these and the
remainder. [w.] (i) To i[>TOve At/ S is P. Both premisses must be
affirmative, and both must be universal. S being distributed in the conclusion,
must be distri- buted in the minor premiss, which must therefore be All S is MM
not being distributed in the minor must be distri- buted in the major which
must therefore be All M is P, CHAP. II.] SYLLOGISMS. i6s SaF can therefore be
proved in only one way, namely, All M is F, All SisM, therefore, All S is F-,
and this syllogism conforms immediately to the Dictum, (2) To prove No S is F.
Both premisses must be universal, and one must be negative while the other is
affirmative, />., one premiss must be E and the other A, First, let the
major be E, />., either No M is F ox No F is M. In each case the minor must
be affirmative and must dis- tribute S ; therefore, it will be All S is M,
Secondly^ let the minor be E, /.isJf, All 6* is P, yielding the conclusion All Si^ M. {4) Some S is not
M (2), and All iS is Jl[f (4) are therefore true together; but this is
self-evidently absurd, since they are contradictories. Hence it has been shewn
that the consequence of sup- posing Some S is not F false is a
self-contradiction ; and we may therefore infer that it is true. It will be
observed that the only explicit syllogism that has been made use of in the
above is in Figure i * ; and the ^ Solly (Syllabus of Logic y p. 104) maintains
that a full analysis of the reasoning will shew that three distinct syllogisms
are really in- volved, — " Let A and B represent the premisses, and C the
conclusion of any syllogism. In order to prove C by the indirect method, we
commence with assuming that C is not true. The three syllogisms may be then
stated as follows : CHAP, iv.] SYLLOGISMS. i8i process is therefore regarded as
a reduction of the reasoning to Figure i. This method of reduction is called
Reductio ad impossibiky or Reductio per impossibtle^ , ox De duetto ad
impossibiky oxDe- ductio ad absurdum. It is the only way of reducing AOO
(Figure 2), or OAO (Figure 3), to Figure i, unless we make use of negative
terms (as in obversion and contraposition) ; and it was adopted by the old
writers in consequence of their objection to negative terms. 158. The mnemonic
lines Barbara^ Celarent, &c. The mnemonic verses, (which are spoken of by
De Mor- gan as "the magic words by which the different moods have been
denoted for many centuries, words which I take to be more full of meaning than
any that ever were made "), are usually given as follows, — Barbara^
Celarent^ Darii^ J^erioquQ prioris: CesarCy Camestres^ Festino^ Baroco^
secundae : Tertia, Darapti^ Disamis, Datisij Felaptofty Bocardo, Ferison^ habet
: Quarta insuper addit Bramantipj CameneSy DimariSy Fesapo^ Fresison, Each
valid mood in every figure, unless it be a subaltern First syllogism : * ^ is ;
C is not ; therefore B is not*. Second syllogism : * If ^ is, and C is not, it
follows that B is not ; but B is; therefore it is false that Ais and G is
not*" Third syllogism : * Either both propositions A is and C is not are
false, or else one of them is false; but that A is is not false; therefore that
C is not is false, («. ^., C is).' ** I do not see any flaw in this analysis;
at any rate it must be admitted that the reasoning involved in Indirect
Reduction is highly complex, and since the two moods to which.it is generally
applied can also be reduced directly (compare section 159), some modern
logicians are inclined to banish it entirely from their treatment of the
syllogism. 1 Cf. Hansel'sAldrich,pp. 88^, 89. i82 SYLLOGISMS. [part hi. tnood,
is here represented by a separate word ; and in the case of a mood in any of the
so-called imperfect figures, (;>., Figures 2, 3, 4), the mnemonic gives full
information for its reduction to Figure i, the so-called perfect figure. The
only meaningless letters are b (not initial), d, /, n, r, t \ the signification
of the remainder is as follows : — The vaivels give the quality and quantity of
the propo- sitions of which the syllogism 19 composed ; and therefore really
give the syllogism itself. Thus, Camenes being in Figure 4, represents the
syllogism, — NoJ/is^S, therefore. No S Is P. The initial letters in the case of
Figures 2, 3, 4 shew to which of the moods of Figure i the given mood is to be
reduced, namely to that which has the same initial letter. [The letters B, C,
2>, F were chosen for the moods of Figure i as being the first four
consonants in the alphabet.] Thus, Camestres is reduced to Celarenty — All /'is
J/, v^ 'NoMisS, No hM, ^ All P is M, therefore, No S is P. therefore, No P is
S, therefore. No *S is P. s (in the middle of a word) indicates that in the
process of reduction the preceding proposition is to be simply con- verted.
Thus, in reducing Camestres to Celarenty as shewn above, the minor premiss is
simply converted. s (at the end of a word*) shews that the conclusion of the
new syllogism has to be simply converted in order to ^ This slight difference
in the signification ois and / when they are fined letters is frequently
overlooked* CHAP. IV.] YLLOGISMS. 183 obtain the given conclusion. This again
is illustrated in the reduction of Camestres, The final s does not affect the
con- clusion of Camestres itself, but the conclusion of Celarent to which it is
reduced. / (in the middle of a word) signifies that the preceding proposition
is to be converted per accidens. Thus, in the reduction of Darapti to Darii^ —
KWMv&P, KWM'^P, All M\& Sy Some S is M, therefore, Some S is F»
therefore, Some S is P. p (at the end of a word*) implies that the conclusion
obtained by reduction is to be converted per accidens. Thus, in Bramantip^ the
p obviously cannot affect the / conclusion of the mood itself; it really
affects the A conclusion of the syllogism in Barbara which is given by
reduction. Thus, — h\\P\%M, .. AllJlfisS, AUil/is^, ^ AW Pis M, therefore, Some
SisP, therefore. All P is S, therefore. Some S is P. m indicates that in reduction
the premisses have to be transposed, {Metathesis prcemissarum) ; as just shewn
in the case of Bramantip, c signifies that the mood is to be reduced indirectly
y (/.^., by reductio per impossibile in the manner indicated in the preceding
section) ; and the position of the letter indicates that in this process of
indirect reduction the first step is to omit the premiss preceding it, /.^j,
the other premiss is to be combined with the contradictory of the conclusion,
(Con- versio syllogismi, or ductio per CoHtradictoriani ptopositionem sive per
impossibile), c is by some Writers replaced by ^, thus Baroko and Bokardo
instead of Baroco and Bdcairdo, ^ See note on the preceding pag0i i84
SYLLOGISMS. [part hi. The following lines are sometimes added to the Verses
given above, in order to meet the case of the subaltern moods: — Quinque
Subalterni, totidem Generalibus orti, Nomen habent nullum, nee, si bene
coUigis, usum\ 169. The direct reduction of Baroco and Bocardo. Mnemonics
representing the direct reduction of these moods. ^ The mnemonics have been
written in variouft forms.Those given above are from Aldrich, and they are the
onesthat are in general use in England. Wallis in his Institutip Logica (1687)
gives for Figure 4, Balani^ Cadere^ Digami, Fegano^ Fedibo, P. vanMusschenbroek
in his Instiiutiones Logica (1748) gives Barbaric Calenies, Dibatis^ Fes- pamos
Fresisom, This variety of forms fpr the moods of Figure 4 was no doubt due to
the fact that the recognitipn of this figure at all was quite exceptional until
comparatively recently. Compare section 173. According to Ueberweg, the
mnemonics run,— Barbara^ Celarent primce, Darii Ferioo^t, CesarCt Camestres^
Fesiino, Baroco secv)ndse. Tertia grande sonans recitat Daraptiy felapton^
Disatnis^ Datisi^ Bocardo^ Ferison, Quaftte Sunt Bamcdip, Calemes, Dimatis^
Fesapo^ Fresison* Mr Carveth Read (Mind^ 1883, p. 440) suggests an ingenious
mbdification of the verses, so as to make each mnemonic immediately suggest the
figure to which the mood it represents belongs, at the same time abolishing all
the unmeaning letters. He takes / as the sign of the first figure, n of the
second, r of the third, and / of the fourth. The lines then run Ballcda^
Celallel^ Dalits Fe/ioque prioris. Cisane, Catnesnts^ FesinoHi Banoco secundse.
Tertia Darapri, Dnsamis, Darisi^ Feraprc, BocarOf Ferisor habet. Quarta insuper
addit BamaiiPt Canutes^ Dimaiis^ FesaptOy FtsistoL Mr Read also suggests
mnemonics to indicate the direct reduction of Baroco and Bocardo. Compare the
following section. CHAP. IV.] SYLLOGISMS, 185 Baroco : — AU P is M, Some S is
not M^ therefore, Some .S" is not F^ may be reduced to Ferio by
contrapositing the major pre miss, and obverting the minor premiss, thus, — No
not- J/ is Fy Some S is not-JS/", therefore, Some S is not F, Professor
Croom Robertson has suggested Faksoko to represent this method of reduction, k
denoting obversion, so that ks denotes obversion followed by conversion, (/.^.,
con- traposition). Whately's word Fakoro (Elements of Logic, p. 97) does not indicate
the obversion of the minor premiss (r being with him an unmeaning letter).
Bocardo : — Some M is not F, All M\%S, therefore. Some S is not Fy may be
reduced to Darii by contrapositing the major premiss and transposing the
premisses, thus, All M\%Sy Some TiOXrF is My ■— - - ..I . ■ ■ - — ■ ■ ^■. -.11
- therefore, Some not-/* is S, We have first to convert and then to obvert this
conclu- sion, however, in order to get the original conclusion. This process
may be indicated by Doksamosky (which again is obviously preferable to Dokamo
suggested by Whately, 186 SYLLOGISMS, [part hi. since this word would make it
appear as if we immediately obtained the original conclusion in DartT), 160.
Shew how to reduce Bramantip by the indirect method. Just as Bocardo and Baroco
which are usually reduced indirectly may be reduced directly, so other moods
which are usually reduced directly may be reduced indirectly. Bramantip : — All
P is M, All M\% S, therefore, Some SS& P\ for, if not, then No 5 is /*; and
combining this with the given minor premiss we have a syllogism in Celarenty —
No S is P, therefore. No Jfis -P, which yields by conversion No P is M. But
this is the contrary of the original major premiss All P is M^ and it is
impossible that they should be true together. Hence we infer the truth of the
original conclusion. 161. Assuming that any syllogistic reasoning can be
expressed in the first Figure, prove that, (omitting the subaltern moods), it
can be expressed, directly or indirectly, in any given mood of that Figure. ^
Mr Carveth Read (Mind, i88i, p. 441) uses the letters k and s as above ; but
his mnemonics are required also to indicate the figure to which the moods
belong (compare the preceding note) ; and he there- fore arrives at Faksnoko
and Doksamrosk, Spalding {Logic ^ p. 135) suggests Facoco and Docamoc\ but the
processes here indicated by the letter c are not in all cases the same, and
thesemnemonics are therefore unsatisfactory. p WJ ■■ I W ■ ■ ■ ■
I^U^B^^^W^^^^PI CHAP, iv.l SYLLOGISMS. 1S7 We may extend the doctrine of
reduction, and shew not merely that any syllogism may be reduced to Figure i,
but also that it may be reduced to any given mood of that figure, provided it
is not a subaltern mood. This position will obviously be established if we can
shew that Barbara^ Celarent, Darii and Ferio are mutually reducible to one
another. Barbara may be reduced to Celarent by obverting the major premiss and
also the new conclusion which is thereby obtained. Thus, All M is /> All S
is M, therefore, All S is /*, becomes No M is not-/*, All 5 is J/, therefore,
No S is not--P, therefore, All S \% P, Conversely, Celarent is reducible to
Barbara; and in a similar manner by obversion of major premiss and con- .
elusion Darii and Ferio are reducible to each other. It will now suffice if we
can shew that Barbara and Darii are mutually reducible to each other. Obviously
the only method possible here is the indirect method. Take Barbara^ MaPy SaM, SaP;
for, if not, then we have SoP; and MaP^ SaM^ SoP must be true together. From
SoP by first obverting and then converting, (and denoting not-P by P\ we get
P'iS, and combining this with SaM we have a syllogism in Darii^ — i88
SYLLOGISMS. [part liu SaM, FiM. P'iM by conversion and obversion becpmes MoP\
and therefore MaP and MoP are true together ; but this is im- possible, since
they are contradictories. Therefore, SoP cannot be true, /.^., the truth of SaP
is established. Similarly, Darii may be indirectly reduced to Bar2>ara^,
MaP, (i) SiM, (ii) SiP, (iii) The contradictory of (iii) is SeP^ from which we
obtain PaS\ Combining with (i), we have — PaS MaP,MaS' in Barbara, But from
this conclusion we may obtain SeM^ which is the contradictory of (ii)*. 162. Some
logicians have .asserted that all the moods of the syllogismare reducible to
the form of Barbara. Inquire into the truth of thisassertion, [l.] 163. Making
use of any legitimate methods of immediate inference that may be serviceable,
shew ^ It has also been maintained, that this reduction is unnecessary, and
that, to all intents and purposes, Darii is Barbara, since the *' some S"
in the minor is, and is known to be, the same some as in the conclusion. * It
would now seem that the Dictum de omni et nulla might if we pleased be reduced
to a Dictum de omni ; but it would be vain to pre- tend that any real
simpU6cation would be introduced thereby.CHAP. IV.] SYLLOGISMS. 189 how
Barbara, Baroco and Be car do may be reduced ostensively to Figure 4. 164. Reduce
Ferio to Figure 2, Festino to Figure 3, Felapton to Figure 4. 165. Prove that
any mood may be reduced to any other mood provided that the latter contains
neither a strengthened premiss nor a weakened con- clusion. 166. Examine the
following statement of De Morgan's : — ** There are but six distinct
syllogisms. All others are made from them by strengthening one of the
premisses, or converting one or both of the premisses, where such conversion is
allowable; or else by first making the conversion, and then strengthening one
of the premisses." 167. How can you apply the Dictum de omni et nullo to
the following syllogism : — Some M is not P, All M is Sy therefore. Some 5 is
not P ? 168. How would you apply the Dictum de omni et nullo to the following
reasonings } (i) The life of St Paul proves the falsity of the conclusion that
only the rich are happy. (2) His weakness might have been foretold from his
proneness to favourites, for all weak princes have that failing. [v.] 169.
Dicta for the second and third Figures of syllogism corresponding to the Dictum
of the first. Thomson i^Laws of Thought, p. 173), and Bowen {Logic^ p. 196),
give for Figure 12, a dictum de diverso^ — "If one 190 SYLLOGISMS. [part
hi. term is contained in, and another excluded from, a third term, they are
mutually excluded"; and for Figure 3, a Dictum de exemplo^ — "Two
terms which contain a common part, partly agree, or if one one contains a part
which the other does not, they partly differ." The former of these is at
least expressed loosely since it would appear to warrant a universal conclusion
in Festino and Baroco* Mansel {Aldrtchj p. 86) puts this Dictum in a more
satis-, factory form: — "If a certain attribute can be predicated,
affirmatively or negatively, of every member of a class, any subject of which
it cannot be so predicated, does not belong to the class," This
proposition may claim to be axiomatic, and it can be applied directly to any
syllogism in Figure 2. The Dictum de eocemplo again as stated above is open to
exception. The proposition, " If one term contains a part which another
does not they partly differ," applied to No M is Py All M is 5, would
appear to justify Some P is not S just as much as Some S is not P, Mansel's
amendment here is to give two principles for Figure 3, the Dictum de exemplOj —
"If a certain attribute can be affirmed of any portion of the members of a
class, it is not incompatible with the distinctive attributes of that class
" ; and the Dictum de exceptOy — " If a certain attribute can be
denied of any portion of the members of a class, it is not inseparable from the
distinctive attributes of that class." But is it essential that in the
minor premiss we should be predicating the distinctive attributes of the class
as is here implied ? This appears to be a fatal objection to Mansel's dicta for
Figure 3. More- over, granted that P is not incompatible with .S", are we
there- fore justified in saying Some S is P} I would suggest the following
axioms, — " If two terms are both affirmatively predicated of a common
third, and one at least of them universally so, they may be par- CHAP, iv;]
SYLLOGISMS. 191 tially predicated of each other " ; " If one term is
denied while another is affirmed of a common third term, either the denial or
the affirmation being universal, the former may be partially denied of the
latter." These will I think be found to apply respectively to the
affirmative and negative moods of Figure 3, and they may be regarded as
axiomatic ; but they are certainly somewhat laboured. 170. Is Redtictio7i an
essential part of the doctrine of the syllogism ? According to the original
theory of Reduction, the object of the process was to be sure that the
conclusion was a valid inference from the premisses. Given a syllogism in
Figure i, we are able to test its validity by reference to the Dictum de omni
et nullo; but we have no such means of dealing directly with syllogisms in any
other figure. Thus, Whately says, — "As it is on the Dictum de omni et
nullo that all Reasoning ultimately depends, so, all argun^ents may be in one
way or other brought into some one of the four Moods in the First Figure: and a
Syllogism is, in that case, said to be reduced'' {Elements of Logic, p. 93).
Professor Fowler puts the same position in a more guarded manner, — "As we
have adopted no canon for the 2nd, 3rd, and 4th figures, we have as yet no
positive proof that the six moods remaining in each of those figures are valid;
we merely know that they do not offend against any of the syllogistic rules.
But if we can reduce them, /.., a line representing an
undistributed term is partly dotted Thus, in the case of All S is P, — the
diagram indicates that all S is contained under P, but that we are uncertain as
to whether there is or is not any P which is not S,CHAP, v.] SYLLOGISMS, 209 In
the case of Some S is not P^ — the diagram indicates that there is S which is
not P^ but that we are in ignorance as to the existence of any S that is P.
181. The application of Lambert's diagrammatic scheme to syllogistic
reasonings. As applied to syllogisms, the method indicated in the preceding
section is much less cumbrous than the Eulerian diagrams*. We may take the following
examples : — Barbara P M Baroco ^ Mr Venn (SymbolicLogic^ p. 432) remarks, —
"As a whole Lam- bert's scheme seems to me distinctly inferior to the
scheme of Euler, and has in consequence been very little employed by other
logicians." MrVenn's criticism is chiefly directed against Lambert's
representation of the particular affirmative proposition, namely, — The
modification, however, which I have here introduced, and which is suggested by
Mr Venn himself, meets the objections raised on this ground. K. L.
142IODatisSYLLOGISMS.[PART III.182. . Represent the moods Darii^ Cesare,
Daraptiyand Fesapo in Lambert'sscheme.183. Take the premisses of an ordinary
syllogism in Barbara, e.g,, all X is F, all Y is Z \ determine precisely and
exhaustively what those proposiaffirm, what they deny, and what they leave in
doubt, concerning the relations of the terms X, F, Z, [l.] This question can be
very well answered by the aid of any of the three diagrammatic schemes which we
have just been discussing. Compare also Jevons, Stipes in Deduc- tive Logic^ p.
216. ^J ^ ^1 J iMINlV^^VWIBHipMipiKSr CHAPTER VI. IRREGULAR AND COMPOUND
SYLLOGISMS. 184. The Enthymeme. By the Enthymeme^ Aristotle meant what has been
called the "rhetorical syllogism" as opposed to the apodeictic, demonstrative,
theoretical syllogism. The following is from Hansel's notes to Aldrich (pp. 209
— 211): "The Enthy- meme is defined by Aristotle, trvAAoytcr/tos l^
ciK(^a>v ^ injfieifiiv. The cikos and crrjfi€iov themselves are Propositions
; the former stating a general probability^ the latter a facty which is known
to be an indication, more or less certain, of the truth of some further
statement, whether of a single fact or of a general behef. The former is a
proposition nearly, though not quite, universal ; as *Most men who envy hate':
the latter is a singular proposition, which however is not regarded as a sign,
except relatively to some other propo- sition,which it is supposed may be
inferred from it. The ctKos, when employed in an Enthymeme, will form the major
premiss of a Syllogism such as the following : Most men who envy hate, This man
envies, therefore. This man (probably) hates. 14 — 2 212 SYLLOGISMS. [part hi.
The reasoning is logically faulty; for, the major premiss not being absolutely
universal, the middle term is not dis- tributed The (njfUiov will form one
premiss of a Syllogism which may be in any of the three figures, as in the
following ex- amples : F^re I. All ambitious men are liberal, Pittacus is
ambitious, Therefore, Pittacus is liberal. Figure 2. All ambitious men are
liberal, Pittacus is liberal, Therefore, Pittacus is ambitious. Figure 3,
Pittacus is liberal, Pittacus is ambitious, Therefore, All ambitious men are
liberal. The syllogism in the first figure is alone logically valid. In the
second, there is an undistributed middle term : in the third, an illicit
process of the minor." On this subject the student may be referred to the
remainder of the note from which the above extract is taken, and to Hamilton,
Discussions, pp. 152 — 156. An enthymeme is now usually defined as a syllogism
incompletely stated, one of the premisses or the conclusion being understood
but not expi;essed. As has been firequently pointed out, the arguments of
everyday life are for the most part enthymematic The same may be said of
fallacious arguments, which are seldom completely stated, or their want of
cogency would be more quickly recognised. An enthymeme is said to be of the
first order when the major premiss is suppressed ; of the second order when the
minor premiss is suppressed ; and of the third order when the conclusion is
suppressed. _V^mm.vm^ f,mi,fi^ I— jr" -. .i^-Tl* -^ ^.;^Vip^lWrV '^'^
^< a.li. JP
"^l^^amMi^^nSQP^yf^^ CHAP. VI.] SYLLOGISMS. 213Thus, '^Balbus is
avaricious,andtherefore, he is un- happy,'' is an enthymeme of the first order;
"All avaricious persons are unhappy, and therefore, Balbus is unhappy '^
is an enthymeme of the second order; "All avaricious persons are unhappy,
and Balbus is avaricious'' is an enthymeme of the third order. 186. The
Polysyllogism ; and the Epicheirema. A chain of syllogisms, that is, a series
of syllogisms so linked together that the conclusion of one becomes a pre- miss
of another, is called 2i polysyllogtsm. In a polysyllogism, any individual
syllogism the conclusion of which becomes the premiss of a succeeding one is
called a prosy Uogism ; . any individual syllogism one of the premisses of
which is the conclusion of a preceding syllogism is called an epi- syllogism. Thus,
—All C'v&DA All BisCA Prosyllogism, herefore. All B h Dyl but. All A isbX
episyllogism. therefore. All -4 is Z>. ) The same syllogism may of course be
both an episyllo- gism and a prosyllogism, as would be the case with the above
episyllogism if the chain were continued further. An epicheirema is a
polysyllogism with one or more prosyllogisms briefly indicated only. That is,
one or more of the syllogisms of which the polysyllogism is composed is
enthymematic. Whately {Logicy p. 117) calls it accord- ingly an enthymematic
sentence. The following is an example, £ is D, because it is C, ^ is ^,
therefore, A is D, 214 YLLOGISMS. [PART IIL 186. The Sorites. A Sorites is a
polysyllogism in which all the conclusions are omitted except the final one;
for example, ^is^, ^is (7, CisZ>, £> is E, therefore, ^ is ^. 187. The
ordinary Sorites, and the Goclenian Sorites. In the ordinary Sorites, the
premiss which contains the subject of the conclusion is stated first; in the
Goclenian Sorites it is stated last. Thus, — Ordinary Sorites^ — A is By Bis C,
DisE, therefore, A is E, Goclenian Sorites^ — D is Ej CisI?, Bis C, A is B,
therefore, A is E, I If, in the case of the ordinary sorites, the argument were
drawn out in full, the suppressed conclusions would appear as minor premisses
in successive syllogisms. Thus, the ordinary sorites given above may be
analysed into the three i following syllogisms, — (i) B is C, ^is^, therefore,
A is C; CHAP. VI.] SYLLOGISMS. 215 (2) C is A ^is (7, therefore, A is D; (3) ^
is ^, ^isZ>,therefore, -^ is j£. Here the suppressed conclusion of (i) is
seen to be the minor premiss of (2), that of (2) the minor premiss of (3); and
so it would go on if the number of propositions con- stituting the Sorites were
increased. In the Goclenian Sorites, the premisses are the same, but their
order is reversed, and the result of this is that the suppressed conclusions
become mq/or premisses in successive syllogisms. Thus the Sorites, — I? is JS,
CisD, BisC, A is By therefore, A is JS,may be analysed into the following three
syllogisms, — (i) D is £, CisI?,therefore, C is ^ ; (2) C is E, Bis C,
therefore, B is E; (3) ^ is ^> ^ is ^, therefore, A is JS, Here the
conclusion of (i) becomes the major premiss of (2)3 and so on. 2i6 SYLLOGISMS.
[part hi. The ordinary Sorites^ is that which is most usually discussed; but it
may be noted that the order of premisses in the Godenian form is that which
really corresponds to the customary order of premisses in a simple syllogism.
188. The special rules of the ordinary Sorites. The special rules of the
ordinary sorites are,— (i) Only one premiss can be negative; and if one is
negative, it must be the last. (2) Only one premiss can be particular; and if
one is particular, it must be the first Any ordinary sorites may be represented
in skeleton form, the quantity and quality of the premisses bdng left
imdetermined, as follows : — ^ What I have called the ordinary Sorites is
frequently spoken of as the Aristotelian Sorites ; for example, by
ArchbishopThomson {Laws of Thought^ p. 201), and Spalding (Logic, p. 302).
Hamilton howeverremarks,— **The name Sorites does not occur in any logical
treatise of Aristotle ; nor, as for as I have been able to discover, is there,
except in one vague and cursoryallusion, any referenceto what the name is now
employed to express"{Lectures on Logic, I. p. 375)* The term Sorites (from
cFriday, May 22, 2020
H. P. Grice, "Notes on J. N. Keynes"
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