do so. And since the simple forms of language indicative
of class aggregation are meant to be of general application,
many cases might be conceived in which they do thus
doubly reckon the common members. This has been too
hastily objected to by urging, for example that if we take all
the cattle and all the beasts of burden from a promiscuous
assemblage of animals, we do not think of counting the
common part, viz. those which fall under both designations,
twice over. Of course we do not, because the word "take" is
one which in most of its applications negatives the possibility
of repeated performance. A thing taken once may be
considered to be taken altogether. A still better instance in
support of this view, so far as material considerations are
concerned, would have been found in the proposal to kill the
members of both of these classes; for some of the beasts
of burden having been put an end to whilst we were dealing
with the cattle, would certainly not need any further atten-
tion. On the other hand it would be easy to find instances
in which the same form of class aggregation by no means
denies, but rather suggests, this double counting. Suppose,
for instance, we found, by putting, together two Acts of
Parliament, that 'all poachers and trespassers are to be fined
20 shillings': is it quite certain that poachers who trespass,
could not be fined 40 shillings? This is, I apprehend,



42 Symbols of classes and of operations. [CHAP,

a question for the lawyers to decide 1 . But the language
is, to the common understanding, certainly ambiguous, which
is sufficient for our present purpose. Or, if postmen and
parish clerks were authorized to apply for a Christmas box of
five shillings, does anyone suppose that postmen who hap-
pened to be parish clerks, would not apply for ten shillings
altogether ; and is it quite certain that their claims would be
rejected ?

On the whole therefore it seems clear that we must
at least recognize all these three varieties of simple class
aggregation as actually likely to occur, and that we must also
admit that popular language does not distinguish clearly
between them. It uses, or might use, the same form for all
three, leaving it to the context to decide which of them
is intended in any particular case.

There is, however, for logical purposes, a broad distinction
between them. The last is not strictly a logical mode of
aggregation, or rather it is one, not of Pure Logic, but of
Applied. The sorts of classes and class relations which we
require as a basis of Logic, must lend themselves to every
kind of application, qualitative as well as quantitative. But
the last mode, in which the surplus is reckoned twice, seems
to be confined to applied Logic; that is, to some kind of
numerical application of our class terms, as in the instances
given above. This mode therefore may be said to be practi-
cally rejected by all logicians, but there is a very important
difference in respect of the mode which they adopt for

1 There seems to be no general that it would probably be decided by

rule here. On enquiring of good such considerations as whether the

authorities whether, for instance, if it conditions of the two licences were the

were enacted that auctioneers and same, whether they were issued by

hawkers should take out a licence, the same authority, were imposed in

those who belonged to both classes the same Act, and so forth,
would require two licences, I was told



ii.] Symbols of classes and of operations. 43

rejecting it. Boole preferred what we will call the strict plan
of formally excluding any duplicate counting. He wrote his
alternatives (as will presently be indicated) in the form 'A and
the B which is not A.' Others, who have more or less
adopted his system, have employed the looser form of writing
'A and B,' assuming it to be understood as a matter of
course that the A which happens to be B is not to be
reckoned twice. The main grounds for preferring the former
plan cannot be assigned here, since they introduce important
questions of symbolic procedure. But simply on the grounds
now before us I should choose the former. If indeed it were
quite certain that the designation 'A and B,' never counted
AB twice over, there would be something to be said in
favour of taking this for granted in Logic, and writing
it accordingly. Inasmuch however as this does not seem
to be the case, it is decidedly better to mark our meaning by
the very form of our symbolic arrangement, and to make
it formally clear that there is to be no such double reckoning.
It involves somewhat more trouble to adopt this plan; but
saving of trouble is of very little importance in Logic,
compared with the habit of keeping necessary distinctions
prominently before us.

In order not to break the thread of connection at this
point I will just remark at once that these three meanings
may be thus indicated on the system here adopted :

A not-B + B not- A, A + B not- A, A +B.

In the first we exclude the AB members ; in the second
we simply include them, that is we count them once like
their neighbours ; in the third these AB's must be considered
to be counted twice over. But inasmuch as this last would
introduce alien considerations of a numerical character we shall
as a rule reject that form. If ever it makes its appearance, as



44 Symbols of classes and of operations. [CHAP.

representing a class, in the statement of our premises, it must
be understood that A and B are in that case so obviously
exclusive by their very signification (i. e. materially exclusive)
that such a form cannot lead to error. It is somewhat
briefer, and, if it be rather slovenly to adopt a form which
strictly counts a part twice over, at least there can be no
mistake in consequence, if it be known that such a part does
not exist.

We must now take some notice of the attempts of popular
language to express the above meanings. That it can do this,
by using words enough for the purpose, is obvious ; but there
is something almost bewildering in the laxity, in the combined
redundancy and deficiency, of our common vocabulary in this
respect. Broadly speaking we employ two conjunctions, 'and'
and 'or', for thus aggregating classes; these terms being practi-
cally synonymous in this reference, and both alike leaving it to
be decided by the context whether or not they exclude the
common part. Often there is no such part, the terms being
known to be exclusive of one another ; but if there be such,
and the context does not make our meaning plain, we often add
a clause 'including both', or 'excluding both', or something to
that effect, in order to remove all doubt. (The third of the
recently mentioned cases, being comparatively exceptional,
and hardly likely to occur except in some kind or other
of numerical inference or application, may be left out of
consideration.)

Thus, 'Lawyers are either barristers or solicitors', 'Lawyers
consist of barristers and solicitors', must be taken as being
equivalent statements. They both alike state that the class
of Lawyers is made up of, or co-extensive with, the two classes
of 'barristers' and 'solicitors'. Whether a barrister can be a
solicitor they do not give the slightest hint. Nor if such bo
the case do they unequivocally inform us whether these com-



II.] Symbols of classes and of operations. 45

mon members are to be included or not ; though as inclusion
is the far more usual case this would be very strongly assumed
in the absence of any statement to the contrary.

There is, of course, a slight difference between the signi-
fication of these two particles. The word 'lawyers', when
identified with ' barristers and solicitors', being taken some-
what more collectively, and when identified with ' barristers
or solicitors' somewhat more distributively, as logicians say.
Hence when our subject is an individual a real distinction will
be introduced by the use of one term rather than the other.
Thus to say of any person that 'he is deceiver or deceived', is
by no means the same thing as to say that 'he is deceiver and
deceived'. But the distinction here seems merely forced
upon us by the necessity of the case and not by the nature of
the grouping of the two classes. The individual cannot, like
a class, be split up into two parts ; accordingly his 'collective'
reference to two classes forces us to conclude that one person
at least must be common to both classes, that one deceiver
must be deceived; whereas his 'distributive' reference to the
two classes carries no such implication with it. This must
rank of course amongst the many perplexities and intricacies
of popular speech, but it does not seem at variance with the
statement that regarded as mere class groupings, independent
of particular applications, 'A and B', 'A or B\ must as a rule
be considered as equivalent.

In addition to these words, and and or, we have a variety
of other words and phrases at our occasional service ; such as,
'as well as', 'also', 'not excepting', and so forth, all of which
serve the same purpose of aggregating class terms together so
as to make them represent a single whole. But how they
aggregate them, in respect of this matter of iuclusiveness,
must be a matter of interpretation and of reference to the
context.



46 Symbols of classes and of operations. [CHAT.

II. The next way in which we have to consider the
mutual relation of classes is when one is excluded from another.
It is often convenient to begin by taking account of some
aggregate class, and then proceed to set aside a portion of it
and omit it from consideration. This process is the simple
inverse of that discussed above, and the difficulties to which its
expression has given rise spring from corresponding causes.
Three cases may be noticed in order.

(1) One of the classes may be included within the other
(or within one or more of the others, if the group consists of
several). In this case the omission or exclusion cannot
give rise to any perplexity, but is perfectly simple and intel-
ligible.

(2) Again, if the classes are mutually exclusive of one
another, the omission of either from the other is equally
simple, but unintelligible. The direction to omit the women
from the men in a given assembly has no meaning. At least
it will require some discussion about the interpretation of
symbolic language in order to assign a meaning to it.

(3) But if the two classes are partly inclusive and partly
exclusive of one another, we are landed in somewhat of a
difficulty. What for instance, would be meant by speaking
of "all trespassers omitting the poachers"? If we interpreted
this with the rigid stringency with which we treat symbols,
we should have to begin by deducting the common part, viz.
the poachers who trespass, from the trespassers ; and should
then be left with an unmanageable remainder, viz. the poachers
who did not trespass, and who could not therefore be omitted.

Popular language is of course intended to provide for
popular wants, and must be interpreted in accordance with
them. Hence a laxity of usage is permitted in its case which
could not be tolerated in the case of symbols. Accordingly
when we meet with such a phrase as that in question, we take



II.] Symbols of classes and of operations. 47

it for granted that any such uninterpretable remainder is to
be disregarded, and we understand that 'trespassers omitting
poachers' is to be taken to signify 'trespassers, omitting such
trespassers as poach.'

Here too we have a variety of phrases at command to
convey the desired meaning. The word most frequently used
for this purpose is perhaps 'except', as when we say 'Lawyers,
except Chancery barristers'. In this case we specify the sub-
class to be omitted, but sometimes we can express our meaning
better by specifying the portion which is not to be omitted,
as when we say 'Lawyers, provided they are Chancery
barristers', which means that we are to except all who are
not so. Besides these phrases we have a number of others
to choose from, such as, 'omitting', 'excluding', 'but not',
'only if not', and so forth.

III. Our third logical operation in dealing with classes
consists of selecting the common members from two or more
overlapping classes. This is the statement of the process in
respect of denotation, the only side of terms, as previously
remarked, with which we are properly concerned. If however
we choose to express the same thing in respect of connotation
we should say that given two sets of attributes as distinctive
of certain groups, we propose to confine ourselves to the
attributes which occur in both groups. It is obvious that such
a mode of class relation will as a rule result in a restriction or
limitation of the numbers of things taken into account. This
clearly must be so unless the two classes happen to be
coincident, owing to our having really two names for the same
group ; or unless one class is entirely included in the other, in
which case the combination of the two is equivalent to the
neglect of the wider one. In this latter case we have limited
the wider class, but have left the narrower one with its limits
unaltered.



48 Symbols of classes and of operations. [CHAP.

The way in which common language indicates this opera-
tion is very commonly by simple juxtaposition of the terms
involved. This plan is always admissible when one term
is a substantive and the other an adjective, and not unfre-
quently in other cases also ; though as the logician, when
he employs ordinary language, has to act in concert with
the grammarian, he naturally finds his freedom of expression
rather hampered.

But though the readiest way of naming the part common
to two assigned classes is by simple juxtaposition of the
respective names, as when we say 'black men' to mark
those individuals who are both black and men, it is far
from being the only way. Here, as elsewhere, the resources
of language are only too copious and varied for the logician.
We have quite a collection of popular phrases at disposal,
differing from one another not so much in what they logically
assert as in what they are conventionally understood to
imply in addition. Sometimes this implication may be
so strong that it is difficult to say where mere suggestion
ceases and actual logical predication must be considered
to begin. Thus it comes about that such words as and,
bid, accordingly, together with a host of others, are all
employed to express the process of selecting the common
part in two overlapping classes. 'Zulus are savages and
cunning', 'are savages but cunning', 'are cunning but
savages', 'being savages are cunning', must all be understood
as indicating exactly the same kind and degree of class
limitation. They all alike assert that the class 'Zulu' is
contained somewhere within the common part of the
classes 'savage' and 'cunning'; or, if we prefer so to put
it, that they possess the attributes distinctive of these
two terms. Where they differ from one another is not
in respect cf what they assert but in respect of what they



II.] Symbols of classes and of operations. 49

imply ; the first containing no further implication, the second
implying the independent proposition that 'most savages
are not cunning', the third that 'most cunning people
are not savage', and the fourth that 'most, or all, savages
are cunning'. But when we come to write them down,
we should be forced to reject all these suggestions, and
to express them each alike in the form 'All Zulus are
savage-cunning'.

Again; in common speech, adjectives are often am-
biguous by not distinguishing whether they are predicative
of a whole class or merely selective out of it. When I
read of "those who claim the black-skinned Hindoo as
a brother" I cannot be certain whether this phrase picks
out certain Hindoos only, by this characteristic, or whether
it is meant to inform us that all Hindoos possess the
characteristic. As in so many other cases we do not know
what exactly is meant to be implied beyond what is directly
stated.

The following simple sentence is one of many which
might be offered to illustrate the combined redundancy
and deficiency of common speech in the expression of these
class relations : "The proper recipients of charity are those
who are poor but honest, or sick and old, and those who
are young if they be orphans". Here we are simply treating
as an aggregate class the three classes describable as 'poor-
honest', 'sick-old', and 'young-orphan', each of these three
being the common part of two overlapping classes. But
the various words used to connect our class-terms seem
almost as if they were chosen at hazard. For the purpose
of expressing aggregation or addition we have used and
and or ; and for that of expressing the selection of a common
part we have used and, but, and if.

We have now enumerated and described the three
v. L. 4



50 Symbols of classes and of operations. [CHAP.

principal logical operations which are concerned with the
mutual relations of our class terms. They are, I think,
the only operations of the kind which would naturally
and spontaneously present themselves to the mind. There
is, it is true, a fourth operation which will have to be dis-
cussed in the next chapter, but it is not by any means
an obvious one 1 . Instead of being forced upon our notice
like the above three, not only by logical necessity but by
the requirements of daily speech and thought, it rather
comes to us by way of our symbols. Its very existence
may be said to be suggested by the wish to make our
symbolic scheme complete and symmetrical. We will there-
fore set it aside for the present, until we have discussed
the appropriate symbolic language for the three operations
which are so familiar to us.

When we look about, in order to choose our symbols,
those of elementary mathematics naturally offer themselves.
Any that we shall need are very simple and almost univer-
sally familiar, so that it seems at any rate worth while to
try if they will answer our purpose. Not of course that
we propose to use them in the same sense as that primarily
imposed upon them. On the contrary, the signification
they will have to bear has been already definitely settled
for them in the foregoing discussion, and this distinctly varies
from their natural or primary signification. Even where
the analogy is closest, as in the aggregation of classes above
described, we are not engaged in a process of addition. The
process we have to perform may be be,st described, in
familiar words, as that of throwing several compartments
into one ; but we have no thought of counting individuals

1 I allude here, not to the process cussed presently, but to the logical
of equating, indicated by ( = ) and dis- analogue of division.



II.] Symbols of classes and of operations. 51

or in any way adding up numbers. We do not 'add' together
the English, French, Germans, and so forth, in order to make
up the Europeans. Of course such a process of class ag-
gregation may be made a basis of numerical calculation, but
then so it may of many other operations with which it is
in no way to be confounded. I cannot see that we are
justified in any case in considering that there is more than an
analogy, sometimes indeed a very close one, between these
operations of Logic and those of mathematics. Certainly
the employment of the same symbols must not be construed
into an admission that this is so.

I. To begin with the operation of aggregating two
or more classes into one. This seems to be so naturally
represented by the sign for addition, that one can hardly
avoid writing down some such expression as x + y + z to
represent the class made up of x, y y and z. No other formal
or symbolic justification for it seems to be called for than these,
that the order of the terms thus connected is entirely
indifferent, and that the aggregation of two groups is equi-
valent to the aggregation of the detailed classes which com-
pose them; that is, we must accept the commutative and
associative laws. Whatever sense we put upon the sign
we must secure that x + y and y + x shall have precisely the
same signification, and that (x + y) + (z + w) is equivalent
to x + y + z + w. It is so in mathematics, and it must be
so in Logic too if the symbol is to answer its purpose. That
this condition is secured in Logic is obvious ; for the order
in which we group our terms is perfectly immaterial, and
is recognized as being so in the common usage of and and or.
Indeed common language has so thoroughly appreciated this
fact that it does not seem to have prepared any pitfalls here
against which the logician has to be on his guard. In what-
ever order we arrange the 'Jews, Turks, infidels, and heretics, '

42



52 Symbols of classes and of operations. [CHAP.

we cannot extract any difference of signification out of
the combined group 1 .

We must however clearly understand that in adopting
this sign (+) we have to some extent committed ourselves in
the matter of the mutual exclusiveness discussed a few pages
back. For wherever we meet with such a form as A +B + C
in mathematics it is always supposed that A, B, C, are
mutually exclusive, whether the symbols stand for things or
for operations ; or that, if they are not, the common parts
must be considered to have been taken twice over. It is never
tolerated that they should overlap one another and leave it to
us to make the correction by omitting what would otherwise
be doubly reckoned. Accordingly, if we wish to be quite
consistent, we ought to do the same in Logic. That is,
we must take care to ascertain that the terms which we thus
connect are mutually exclusive of one another. If we happen
to know that this is already the case, on either formal or
material grounds, then there is no harm in our writing x + y
at once. But if this be not known, then we must take care
to make them exclusive, which is readily done by expressing
it as l x + y not-#.' (Of course if we want to omit the com-
mon part altogether, we should put it 'x not-y + y not-#.')

The necessity of thus expressing ourselves has been

1 The systematic employment of ton (Log. I. 80), but this is an inter-

this sign (+) Tor this particular pur- pretation in intension. The sign was

pose of class aggregation dates, I also used by Ploucquet to mark the

apprehend, from Boole. Of course combination of assertions or premises,

the sign itself had been introduced and by Maimon and Darjes to stand

into Logic long before. Thus Leib- for affirmation in contrast with nega-

nitz (Specimen demonstrandi, Erdm. tion. Maimon however used another

p. 94) writes A + B to mark the ad- sign ( | ) in a case of exactly the

dition of attributes or notions to same kind as that which we now

form a more complex notion, in mark by ( + ), as in, a + b \ c \ d, for

which he has been followed by others, l a is either b or c or d' (Versuch

e. g. Twesten (Log. p. 25) and Hamil- einer neucn Logik, p. G9).



II.] Symbols of classes and of operations. 53

objected to on various grounds, which seem to me decidedly
insufficient, but which will be discussed in a future chapter.
It doubtless involves somewhat more trouble in writing down
our formulae. But the extra trouble thus entailed is after all
but slight, and, as pointed out in the preface, mere expedition
in the performance of the work is not by any means an
important consideration in Symbolic Logic. On the other
hand, there may be urged in its favour, the high advantage of
keeping real distinctions present to the eye by means of
the formulae we employ; the greater harmony thus secured
with the analogous steps in Mathematics; and a certain
considerable increase of symbolic power, which (as will be
shown in future chapters) we are thus enabled to acquire.

II. The deduction, omission, or subtraction, (it will
be seen that we can hardly help resorting at once to mathe-
matical terms here) of one class from another is expressed
with equal convenience by aid of the symbol ( ); so that
x y will stand for the class that remains when x has had
all the ?/'s left out of it. The only point here that seems to
call for sj'mbolic justification is the ascertainment of the fact
that the well known mathematical rule, about minus twice
repeated producing plus, is secured in Logic. That is,
we must ascertain that it is so, both in the processes we
actually perform, and in the language we use to describe
them. And this may be established by a single suitable
example. Thus if we describe the persons who may remain in
a captured town as 'all the inhabitants except the military,