the former we saw that the case of coincidence might fairly
be recognized as a distinct one ; for, though the individuals
included were identical, we could conceive their being called
up under different names and by reference to different attri-
butes. Hence 'All A is all B' is not necessarily an identical
proposition. But when A and B are groups of attributes
their coincidence merges into absolute identity; for these
being results of Abstraction, every distinguishing character-
istic has been stripped off them. So we do not obtain what
we can represent as an 'All A is all B' proposition, but
rather an 'All A is all A' proposition.
When however we come to work out the scheme of. pro-
positions demanded by this way of interpreting our terms, or
rather our 'concepts', as we will call them, in order to use an
appropriate conceptualist expression, we find that the ap-
parent parallel is very far from holding throughout. The
two cases indeed of entire inclusion seem to repeat themselves
on either interpretation; for, if the concept A is a part of B,
we have the proposition 'All B is A', and conversely if B be
a part of A. But the remaining two cases exhibit a wide
divergence. For instance, if the concepts A and B are
partially the same, we shall most likely find that this gives
rise, not to a 'Some A is B', but to a 'No A is B' proposition,
for the presence of a single really differentiating attribute on
each side is sufficient to separate entirely the classes A
and B. On the other hand, even though the concepts A and
B have not a single element in common we shall not neces-
sarily obtain an ordinary universal negative, for attributes
may be quite distinct and yet compatible with each other.
In fact we might even have an 'All A is all B' in such
a case. Thus 'existent marsupial' and 'large quadruped
indigenous to Australasia ' comprise very distinct groups of
attributes but denote approximately coincident groups of
39-i Variations in symbolic procedure. [CHAP.
things; whilst 'men under thirty' and 'men over thirty' have
every essential point in common with only one divergence,
and yet denote entirely distinct groups.
It is clear then that the arrangement of propositions in
accordance with this scheme is of a somewhat peculiar kind.
It will be most easily carried out by the aid of a little
symbolic notation. Suppose then two concepts A and AB,
where the extra intension indicated by B is of course not
contradictory of A. Give these concepts the names P and
Q and we have the ordinary universal affirmative, All Q is P.
Had the names P and Q been inverted we should have had
the similar proposition All P is Q. Now suppose that
the concepts had been AB and AC, we see at once that we
cannot with certainty conclude any ordinary proposition from
this, in the absence of information as to the mutual relations
of B and C. If B and C are really contradictory, so that they
might be written pq and pq, then we conclude that No P is
Q. If B includes C, so that these might be written p andpq,
then we know that All Q is P, and similarly if C includes B.
If however B and C are really distinct in their meaning, so
that one cannot be shown, in the present state of our know-
ledge, to be a part of the other, then no proposition whatever
can with certainty be elicited out of these concepts AB and
AC. As regards concepts which are entirely distinct from
each other, say A and B, the remark just made may be
repeated verbatim. Without further analysis of A and B,
and discovery of identities or contradictions in the constituent
elements, no proposition of the ordinary kind can with confi-
dence be enunciated.
One or two remarks may be added in explanation of the
above analysis. It will be seen that the only universal pro-
positions which can thus effect an entrance are those called
verbal, necessary, analytic, or essential. Given A and AB as
xix.] Variations in symbolic procedure. 395
concepts, it is inevitable that we should conclude that 'All
AB is A\ So in the case of universal negatives. If A and
B can be analysed into say pq and pq, it is equally inevitable
that ' No A is B\ From the former by conversion, a particular
proposition can be inferred, provided of course that we under-
stand ' some' in its ordinary sense of ' some at least', for we can
infer that ' Some A is AB'. As regards those universal pro-
positions which are accidental, I cannot see that they have
any right to admission on such a scheme as this, where we
start from concepts and their intension. For example, 'All
American citizens know the name of their President' : how
can we fairly interpret this with strictness from the side of
its intension ? Surely on no admissible sense of the term can
this incident be regarded as a part of the intension of 'Ameri-
can'; nor have we any thought of making it such for the
future, and so agreeing to enlarge the intension. If the fact
were new to us the most we should do would be to take a
note of it, so as to regard this mark as an accidental one of all
I am aware that some logicians would explain that what
is meant in such a case^as the above is that we are to 'attach'
(or something equivalent to this) the new attribute to the old
group, without however regarding it as any part of the essence
or intension. I really do not know exactly what this can
mean. The sole bond of coherence in the attributes con-
sists in the fact of our keeping the members of it together in
our minds, so that if a new-comer takes up its permanent
abode amongst them, it obtains thenceforth every right of
occupancy which they possess. An attribute attached to a
group of attributes is one of the group. At least I can see
no ground for disputing this unless we admit Kant's doctrine
as to the existence of a priori synthetic judgments.
The general position adopted here will not, I hope, be
396 Variations in symbolic procedure. [CHAP.
misunderstood. I hold very decidedly that all names (with
certain trifling exceptions which need not be specified) possess
both intension and extension, and that we may conceivably
examine the mutual relations of terms, and through these
develop a system of propositions and of reasonings, from either
of these two sides. Popular speech appears to adopt an inter-
mediate course, not adhering strictly to either of these sides.
In this it has been followed, and very rightly so, by the tra-
ditional Logic, which appears to me (broadly speaking) to
prefer regarding the subject in respect of extension and the
predicate in respect of the attributes which it conveys. This
is certainly the view which I should endeavour to work out
if I were composing a systematic work upon the ordinary
When however we develop a system of Symbolic Logic we
are forced to be more consistent in our adherence to one side
or the other, for symbols do not readily tolerate an uncertain
and hesitating attitude. The question therefore is, which
view we had best adopt. In this choice I cannot hesitate for
a moment, and it must be left to the reader to judge by the
whole drift of this work how far the decision is justified. In
fact as regards the alternative view, we cannot so much say
that it has broken down on trial, as that it never has been
fairly tried at all.
We shall see this best by examining the most appropriate
usage, on this plan, of some of the now familiar signs of our
logical operations. Begin with the sign (+). If A and B are
attributes or partial concepts, we should I presume agree in
saying that A + B will signify those attributes ' taken to-
gether'. But when we go on to enquire what is meant by
taking attributes together we perceive that this is a very
different thing from taking classes together. The only con-
sistent meaning surely is that we do this when we construct
xix.] Variations in symbolic procedure. 397
a new concept which contains them both. But this is clearly
the analogue not of addition but of multiplication. Thus let
'man' be 'rational animal'; it would naturally be said that
the concept humanity is made up of the attributes rationality
and animality taken together. This clearly results in a pro-
cess of restriction in extension such as we have throughout
indicated by A x B and not by A + B. In fact ' extensive
multiplication' corresponds to 'intensive addition 1 '.
So again with the subtractive sign. There can be little
doubt that the most natural and appropriate employment of
it here would be to indicate abstraction rather than exception.
Reverting to the same example, the subduction of 'rationality'
from 'humanity' would leave ' animality', and this process is
precisely that which logicians mean by abstraction : does not
indeed the very etymology of the word suggest the taking of
one attribute from others just as we remove one material
object from a group ? Now logical abstraction, when modified
so as to bring it into accordance with our class interpretation,
was seen to correspond to the process which we mark by
A -r- B rather than A B. This is, in fact, exactly what
Leibnitz does, in a passage referred to in the Introduction,
when he makes (man) (rational) = (brute).
1 It deserves notice that the sign slight suggestion. This was appar-
of addition has actually been em- ently the only employment of these
ployed for this purpose. Thus Ha- signs which had come under Tren-
milton (whose usage however of delenburg's notice, and forms the
mathematical terms and symbols is ground of his objections to their in-
almost always an indication of some- troduction into Logic (Logische Un-
thing not to be done) has said, "The tersuclmngen I. 20).
concept as a unity, is equal to the cha- It may interest the mathema-
racters taken together, Z=a + b + c" tician to remark that 'addition' of
(Logic I. 80). And he has had attributes is like that of powers or
predecessors here, for instance Twes- logarithms; it performs 'multiplica-
ten (Logik p. 211), who has also tion ' of extension and therefore does
used the minus sign for Abstraction ; not leave any convenient opening for
but neither of these is more than a the mere addition of the same.
398 Variations in symbolic procedure. [CHAP.
On the whole then it seems that a consistent intensive
interpretation would find an opening for the symbols + and ,
but that it would use them for the purpose of expressing,
(approximately only, in the case of the latter), operations
which we have indicated by x and -=-. And those operations
which we denote by + and it would not find any place for,
nor therefore for symbols to represent them.
It need hardly be remarked that this is not the course
which has been actually adopted. Nearly all logicians, so far
as I know, have consulted convenience at the expense of
rigid consistency, and, whenever they have employed these
four simple arithmetical signs, have done so in a way more in
harmony with that adopted in this work. Their words have
been the words of the conceptualist, but their deeds have
been mostly those of the nominalist. Hence their language
is often tinged, and their usage modified, especially in
the case of the older symbolists, in a way which it would
be very difficult to understand unless we keep in mind the
characteristics of the system which they nominally professed 1 .
The reader will find plenty of illustrations in support of this
remark in the historical references in the next chapter. My
main object in fact in making the present remarks is to aid
in the intelligent study of some of those anticipations of
Whatever such writers may have said, what they have
actually done has been briefly this. They have taken the
sign +, say, and employed it, not to connect real attributes.
1 One example may be given here intensive interpretation ; whereas ire
from Lambert: "Die Redensart, 'A should proceed to represent 'A which
BO nicht in ist, ' wird BO gezeichnet is not m' extensively in some such
A . way as A - Am or A - m. Elsewhere
- , veil nun m von A als erne Modifi- T , . , . ...
m Lambert himself has used this sub-
cation von ihrer Substanz kann ab- tractive sign very nearly as we do.
Btrahirt werden." This is a purely
xix.] Variations in symbolic procedure. 399
but rather the extensions referred to by those attributes.
They may have defined in a way which would have led to the
expression ' rationality + animality', but what they have
really intended to do was, not to combine these two attributes
or concepts into a more determinate concept, but to obtain a
concept whose extent should be equal to that of the other
two together. And a similar explanation would apply to
their usage of the other three signs. This I cannot but re-
gard as an awkward way of expressing ourselves, as compared
with saying at once that A and B and our other terms stand
directly for classes of things. Moreover this want of harmony
between theory and practice, produces, as it seems to me, a
perpetually recurring confusion and inaccuracy in matters of
detail 1 .
III. The third of the symbolic variations to be here noticed
stands upon a very different footing, and, if admitted, would
involve far more serious changes than any which we have yet
contemplated. The best way of introducing the question
will be by referring to a simple example of a kind familiar to
every student of Logic. For instance, every beginner has
learnt to avoid the pitfall of attempting to syllogize from two
such premises as 'A is equal to B', 1 B is equal to C". He
sees that however certain and obvious the step may be, we
cannot avoid resorting to four terms, instead of three, in
stating these premises according to strict rule. If we adhere
to the correct logical copula, 'is', we have in one premise the
term B, and in the other premise 'an equal to B'. Accor-
1 The nearest approach, perhaps, any serious recognition of what ap-
to a consistent intensive interpreta- pear to be the insurmountable dif-
tion is to be found in Hamilton's ficulties of carrying out this inter-
attempt to translate judgments and pretation in the case of universal
reasonings in this way as well as propositions when accidental, or of
extensivsly. But I cannot find there any ordinary particular propositions.
400 Variations in symbolic procedure. [CHAP.
dingly, on the common system, we have either to reject such
examples, or else to admit them under what is little better
than a subterfuge. We have to abandon any analysis of the
process by which every mind does as a matter of fact proceed,
(viz. from B to its equal,) and to throw the real process
of reasoning into the major premise, by assuming that all
things which are equal to the same thing are equal to one
What the so called Logic of Relatives seems to aim at is
to frame a set of rules which shall deal directly with reason-
ings of this kind ; or, in other words, shall take account of
relations generally, instead of those merely which are indicated
by the ordinary logical copula 'is'. It would be quite im-
possible to treat this question in a portion of a single chapter;
so the reader must understand that I am here only making a
few remarks upon a subject which cannot well be passed
entirely by, But which would need a separate work for its
Suppose then that we have the relation assigned of
A to B, and of B to C, and that we wish to determine the
relation of A to C, how should we proceed ? Symbolically we
find no difficulty in starting. Express the relations in the
form A = L 1 B, B = L i C, and the conclusion will be repre-
sented by A = LI (LJJ) if we want A in terms of C,
or C = L~ l (L~ 1 A} if we want C in terms of A. But, as the
reader knows by this time, symbolic representation of a
desired result is by no means a performance of the desired
process, nor is it even an assurance that that process admits
of performance. Accordingly what we have to do, if we are
to take any steps whatever towards our aim, is to ascertain
the rules according to which the L l and L z of the above
formulae can be analysed and compounded. As soon as we
proceed to do this we find that instead of one simple and uni-
xix.] Variations in symbolic procedure. 401
form set of rales, as in the system hitherto expounded, we
are introduced to a most perplexing variety of them.
Begin with the simplest case, by supposing that the two
relatiojas in the premises are the same, may we conclude that
L(LA) LA, in accordance with the familiar rule of our
logic, that xx = #? Certainly not, as a general rule; for
though, as is abundantly obvious, 'equal of (equal of A)'
is equivalent to 'equal of A', and 'brother of (brother of A}'
is equivalent to 'brother of A', (provided we assume that a
man may be his own brother), yet this is a very exceptional
state of things, and accordingly one great resource in our
logical procedure has to be given up. If we attempted
to ""make any set of rules here at all, we should be forced
to subdivide, by assigning to a separate class all those cases
in which LL = L, (these are what De Morgan has termed
transitive relations). Having done this, we should find further
subdivisions awaiting us. For instance there is one kind of
relation, and for logical purposes one of extreme importance,
in which all odd powers of L are equivalent, and also all
even powers, but these are distinct from each other. Let L
represent 'contradiction of, and it is plain that L*=L*=etc. = I.
That is, if A be a statement, then LLA = A, and so on. But
after having thus distinguished these, and possibly other
similar special cases, we should be left with a bulk of
relations on our hands which admitted of no simplification of
this kind, and which had to be left in the merely symbolic
form LLA, or its equivalent' L?A, when twice repeated.
Again, we know what perpetual recourse has to be made
to the commutative law, xy yx. Does this hold of relations ?
Certainly not, as a general rule, though presumably 'father
of grandfather' is equivalent to 'grandfather of father'. But
'father of a brother' is very different from 'brother of a
father'; in fact it would not be easy to hit upon relations
V. L. 26
402 Variations in symbolic procedure. [CHAP.
which could thus be commuted at will without altering the
result. Accordingly we must be careful to distinguish 1 be-
tween LJL 2 A and L y L l A.
Once more; some relations when inverted yield a definite
result, others yield an indefinite result, and thus are in more
agreement with our ordinary logical formulae. That is,
L~ 1 LA may = A simply, or it may have A as one only out of
a possibly infinite number of solutions. Thus if A be the
husband of B, (in a monogamous country) B must be the
person of whom A is the husband; elsewhere she may only be
one of a number of women who will all equally answer the
It is this immensely extensive signification of the word
relation, and the consequent variety of symbolic procedure
which is called for if we attempt to treat it symbolically,
which seem to me to render it hopeless to establish any-
thing in the least deserving the name of a Logic of Relatives.
The attempt however to introduce relative terms into Logic
was made by some of the first who employed symbolic
notation. Lambert indeed has treated these questions more
fully perhaps than any one since his time : I cannot but
think that his attempt to introduce them into his system was
one of the principal causes of his failure to work it out more
completely and successfully than he did. He expressly dis-
cusses the question whether powers are to be admitted in
1 The earliest definite notice that remarks that WBDS is by no means
I have seen of the fact that, in cer- the same as WDBS, on the ground
tain canes, the commutative law that the position of a term is often
must be rejected, is in Semler ( Ver- an index of the relative importance
such ilber die combinatorlsche Me- to be attached to the attributes re-
tliode, 1811), but it is scarcely more ferred to. Semler, it may be re-
than a pasHing statement. Thus, marked, was acquainted with the
putting H for bequem, D for dauer- works of Lambert and Ploucquet.
haft, W for wohlfeil, S "or tchon, he
xix.] Variations in symbolic procedure. 403
symbolic notation, and decides that they are ; as indeed was
unavoidable in such a wide extent of signification as must
then be accepted. Thus in his Architectonic (i, 82) where
he has discussed this question more fully than elsewhere,
he expressly includes such various relations as 'cause and
effect, means, intention, ground, species and genus'. Thus
7 representing a genus of A we have the result ay ; a higher
genus of this will be indicated by arf, and so on. Similarly
with the relation of species.
As I have already intimated, the attempt to construct
a Logic of Relatives seems to me altogether hopeless owing
to the extreme vagueness and generality of this conception of
a Relation. Almost anything may be regarded as a relation,
and when we attempt to group them into manageable portions
we find that several distinct codes of laws are required.
Even in one of the simplest possible kinds of relation we
find that the inverse processes may be quite insoluble.
Given the relation of A to C, and of B to G, it might be
assumed that that of A to B was determinable. But a
moment's consideration shows that this is not so. If the
distance (to take one of the simplest and most definite of all
relations) of A and of B from C is exactly a mile, that of A
from B, (the relation desired), may be anything not exceed-
ing two miles : a similar result holds if that distance be
anything under one mile. But if the 'relation' were that the
distance is not less than a mile in each case, the resultant
distance of A from B might be absolutely anything whatever.
What therefore a Logic of Relatives has to do is to make
a selection out of an altogether infinite field, confining itself to
those relations only which obey certain definite symbolic
laws. Mr Murphy, for instance, following some suggestions
by De Morgan (Camb. Phil. Trans. X, 336), has proposed a
scheme of the following kind : "Relations (indicated by the
404 Variations in symbolic procedure. [CHAP. xix.
symbol L) may be divided into those which are ' transitive'
and those which are ' intransitive', i.e. according as L* is or is
not equal to L. They are also invertible or uninvertible, that
is to say, L' 1 is or is not equal to L. There are thus four
1. Transitive and invertible.
2. Transitive but uninvertible.
3. Intransitive but invertible.
4. Intransitive and uninvertible.
To the first belong equality, brotherhood, and various other
relations. To the second inclusion, causation, sequence,
greater magnitude, &c. To the third exclusion, difference,
&c. To the fourth the great majority of relations.
I certainly think that this scheme is sound as a classifica-
tion. But for the purposes of a logical system it does not
seem to me that it would add any large domain to our pro-
vince. The two last classes indeed are surely unworkable in
anything which can be called a logical system ; for, involving
powers, they would in all but the simplest cases lead to
quadratic or higher equations.
(The reader who wishes to study the little that has yet
been done on this subject may consult the following notices
and papers :
Ellis L. Mathematical and other writings, p. 391. (Also
published by Mr Harley in the Report of the Brit. Ass., 1870.)
De Morgan. Syllabus of Logic.
Trans, of Gamb. Phil. Soc. Vol. X. (In my
opinion one of the fullest and best papers yet produced on
Peirce C. S. American Acad. of Arts and Sciences. Vol. IX.
Macfarlane A. Proc. of R. Soc. Edinburgh, 1879.
Murphy J. J. Belfast Nat. Hist, and Phil. Soc., 1875.
Manchester Lit. and Phil. Soc. t Vol. vn.
(1880); do. Feb. 1881.)
I. On the various notations adopted for expressing the
common propositions of Logic.
Attention has been already called to the general fact
of the perplexing variety of symbolic forms which have
been proposed from time to time by various writers, but
probably few persons have any adequate conception of the
extent to which this license of invention has been carried.
I have therefore thought it well to put together into one list
the principal forms, so far as I have observed them, in
which one and the same proposition has 'thus been expressed 1 .
For this purpose the Universal Negative has been selected,
as being about the simplest and least ambiguous of all forms