but omitting from these the wounded'; the wounded military
are understood to be put back, by virtue of the two
'omissions', into the same position as the non-military ; pro-
vided, of course, that there are only two alternatives in the
case. It is just a case of x (y z) x y + z. So 'the rule
holds in Logic.



54 Symbols of classes and of operations. [CHAP.

In using this symbol we must remember the condition
necessarily implied in the performance of the operation which
it represents. As was remarked, we cannot 'except' anything
from that in which it was not included; so that x y
certainly means that y is a part of x. This is quite in
accordance with the generalized use of the symbol in mathe-
matics where it is always considered to mark the undoing of
something which had been done before. Not of course that
it must thus refer to the immediately preceding step, but that
there must be some step or steps, in the group of which
it makes a part, which it can be regarded as simply reversing.
The expression x y 4- z will be satisfactory, provided either
x or z is known to be inclusive of y, or if both combine
together to include it 1 .

III. The third logical operation, namely that of restric-
tion to the common part of two assigned classes, may be
represented by the sign of multiplication. That is, x x y t
or xy, will stand for the things which are both x and y.

The analogy here is by no means so close as in the
preceding cases, but the justification of our symbolic usage
must still be regarded as resting on a simple question of fact :
i.e. do we in the performance of the process in question, and

1 Leibnitz (Specimen demonstran- ist durch das Zeichen ( = ), das nicht

di) employed this sign, but his view durch das arithmetische Verneinungs-

of Logic being one of 'compreken- zeichen ( ) ausdriicken Kein A

sion', A-B meant, with him, the ist If ware A=-AI" (Lambert's

omission of the attribute I) from the Briefwechsel, i. 396 : This is not

notion A; not the exception of B as Lambert's own notation). There is

a class from A as a class. As al- also a suggestion in this direction in

ready remarked, the sign (-) was Leibnitz's De arte combinatorial (Erd-

frequently employed to mark logical mann, p. 23), "Quemadmodumigitur

negation, as by Maimon and Darjes. duo suut Algebraistarum et analyti-

The earliest such use that I havo corum prima signa + et - , ita dura

noted is by Lambert: "In dieser quasi copula est et non ett."
Absicht Hesse sich das Bindwortchen



II.] Symbols of classes and of. operations. 55

in the verbal statement of it, act under the same laws of
operation in each case, logical and mathematical alike? The
answer is that we do so up to a certain point, but not beyond.

For instance the commutative law 1 , as it is termed, viz. the
indifference of the order of the symbols, prevails; xy and yx,
English Protestants and Protestant English, being precisely
the same class. So does the distributive law, that every
element of the one multiplier shall combine with every
element of the other, that e.g. x (y + z) xy + xz, and
(x + y] (z + w) xz + xw + yz + yw. Of course we have to
make our way here through various grammatical obstacles,
but allowing for them it is clearly true that we do thus
combine and distribute our terms, so that ' English and French
soldiers and sailors', is the same as 'English soldiers, and
English sailors, and French soldiers, and French sailors'.
When however we say that this commutative law holds
in common speech, we must remember that the idiosyncracies
of language insist upon exceptions here as elsewhere; and
on the same ground, viz. the necessity of compressing a
quantity of implication into our sentences, in addition to the
direct assertions they contain. Thus when x implies y we al-
ways use the form yx, rather than xy. For instance one would
not direct a legacy to those old servants "who are destitute
and alive" ; but to those "who are alive and destitute".

It will be equally obvious that if x = y then zx = zy,
that is, that we may 'multiply' equivalent terms by the
same factor. For (as will be more distinctly insisted on
presently) x = y means that the individuals which go by the
names x and y are the same; consequently it comes to

1 The introduction of these techni- particular sense) were introduced by

cal terms appears to be of recent date. Servois in Gergonne's Annales v. 93

Hankel (Vorlcsungen, p. 3) says that (1814) ; and associative by Sir W. E.

distributive and commutative (in this Hamilton.



56 Symbols of classes and of operations. [CHAP.

exactly the same thing whether we select the /s from them,
under their name of x or their name of y. In each case alike
we get the same result.

Where we depart from mathematical usage, or rather
restrict the generality of its laws, is in the following respect.
As a rule, xx (or # 2 ) is different from x. If x represent
a number, then # 2 is greater or less than x, according as x is
greater or less than 1 ; if x represent a line, then x* represents
an area, and so forth. But in Logic # 2 must equal x, or
rather any number of self multiplications must leave the

significance of a term unaltered, i.e. x = xxx That this

is so as an operation, is obvious, for the selection of the
common part of two classes, when these classes happen to be
the same, is reduced to the simple repetition of this one part.
And that the same is true as regards the laws and usages of
common speech must also be admitted. Or rather, it must
be admitted under some exceptions; for there are signs
of divergence towards the mathematical rule here, when we
are dealing with comparative terms. Thus, even in English,
'great great' does not mean quite the same as 'great' alone;
and in some languages, as Italian for instance, adjectival
repetition is really almost like mathematical multiplication,
increasing or diminishing the effect according as the term is
in itself an augmentative or diminutive. If this conventional
rule prevailed universally in respect of all, or even most
terms, the logician would have to give way to the grammarian
here, with the result of having to abandon Symbolic Logic, as
a distinct formal science from Mathematics '. As it is, however,
we feel at liberty to set aside this 'comparative' usage as an

1 Any one who has read Algebra equation to the first degree. Hence

will nee the importance of this re- any equations with any number of

ntriction in the processes of logical terms are resolvable by the expendi-

calculation. It reduce* every logical ture of sufficient time and trouble.



II.] Symbols of classes and of operations. 57

irregularity, and to lay it down universally that xx shall be
considered the same as x.

It may be pointed out that this is not so much an
infringement of mathematical laws as a special restriction of
them. In fact there is one case in which the same rule does
hold in mathematics, that is when x stands for unity. In that
case xx = x obviously 1 .

So far then seems plain. It must now be enquired what,
on the suppositions thus made, will be the proper mode of
representing certain important and limiting classes with which
we shall often have to deal. How, for instance, shall we
represent "all things" : or rather, to speak more correctly,
how shall we represent the 'universe of discourse' with which
we may happen to be concerned ; about the nature of which
universe we shall have something more to say hereafter ? It
may at first occur to the reader that the most appropriate
way of representing such a huge and miscellaneous assemblage
as this would be by the sign for infinity, since mathematics has
such a sign at its disposal. We might resolve so to represent
it if we pleased 2 ; but, if we did, we should soon find that
we were acting inconsistently with the plan we have just

1 There was a much freer em- AB for "A that is B", (Erd. p. 102)

ployment of this sign for multiplica- and similar forms are of constant

tion, than of that for addition, by occurrence in Ploucquet's logical

the earlier symbolists. This was writings. But note how the latter

probably owing to two causes : partly, describes them : " Durch mb will Ich

that here the comprehensive and keine form der Multiplication sondern

extensive views come so much more eine Associirung der Ideen verstanden

nearly to the same thing; but more haben", a very different conception

to the fact that popular language had this from that of the common part

already familiarized men with the of the extension of m and b. (Samm-

juxtaposition of words to express the lung, p. 254.)

same results which the logician ex- 2 It has indeed been so represented

presses by juxtaposition of letters. by more than one symbolist. (See on,

For instance Leibnitz uses the form ch. vm.)



58 Symbols of classes and of operations. ICHAP.

adopted of indicating combined attributes or overlapping
classes by the sign of multiplication. When once that plan
is adopted we are inevitably bound in consistency to make
the symbol for unity, or 1, stand for the universe. That this is
so can very readily be shown. For what does our x stand for?
It comprises all things which are x. In other words, one way
of describing and getting at x is to say that we combine, or
take the common part of, the universe and x, just as 'English
men' may be got at by taking the common part of the classes
which are English and which are men. We do not say that
this is the only way of getting at it, but it is one way and there-
fore our results must be consistent with so reaching it. Now
the only known symbol which when combined, in accordance
with the rules for multiplication, with any symbol x, will always
still give us #, is unity. Hence the one universal class must
be represented by the sign for unity, so that 1 x # shall always
equal x.

The interpretation of the commutative law in the case of
x x 1 = 1 x a; deserves a moment's attention. If we regard it
as a process of selection rather than as the indication of a
result (which is not however the best way of regard ing it) we
should say that a; x 1 may be described as the process of
selecting from the whole universe those things which are x.
So 1 x # will be the selection from x of all those things which
belong to the universe, which of course leaves x unaffected.

On the same principles a non-existent class, or, as we
shall often find it convenient to describe it, an empty compart-
ment, will be fittingly represented by the sign for nothing, or 0.
This looks natural and plausible enough ; but it is well to
point out that it follows strictly from what has hitherto been laid
down. For if xy stands for the things which are both x and y,
then Oar, or 0, will stand for those which are both nothing and
*; in other words for the non-existent, or for any empty class.



ii. J Symbols of classes and of operations. 59

The sign of subtraction supplies us in the same way with
a suitable expression for the logical contradictory of any class
x, viz. for the class 'not-x'. For since not-# comprises every-
thing which the universe contains except what is x, we should
write it symbolically 1 , 1 x. Now combine x and 1 x, and
what do we thus indicate ? The class of things which are at
the same time x and not-x. Of course this is bound to equal
nothing ; i.e. x (1 x) = 0. This supplies a link of consistency
with some of our previous assumptions. For 'multiply out'
x (1 x), as in algebra, and we have x x 2 or x xx ; which
as just remarked, is to be equal to zero, and this necessitates
that x and xx, or x*, shall be identical with each other, which
we already know that they must be.

One or two examples may be added in illustration of the
employment of our symbols, and of the restrictions to which
they are subject. Suppose we have the expression 'English
and French Poets and Orators'. Following the indications
already given we might propose to write it down symbolically
(a + 6) (c + d), thus combining the four terms by the signs
respectively of addition and of multiplication. Multiplied
out, the latter becomes ac + ad + be + bd. Now the original
verbal statement, when fully expressed in its details, is
equivalent to English poets and English orators and French
poets and French orators, that is, the symbolic process of
multiplication gives exactly the same combinations of terms
that are obtained in common discourse ; the two corresponding
step for step, as of course they are bound to do.

In this case the condition of accurate expression previously
alluded to, about the mutually exclusive character of our
terms, does not obviously make itself perceptible, but that is
because correction is tacitly made where required, when the

1 For the sake of brevity we shall It is an abbreviation introduced and
commonly write "x for 1 - x, or not-x. frequently employed by Boole.



60 Symbols of classes and of operations. [CHAP.

symbols are translated into words. Since a and b, standing
for French and English, are already exclusive, no correction is
required. But c and d, standing for poets and orators, do
overlap, and therefore c + d does symbolically count this
common part twice. We mentally make a correction at
this stage. But working symbolically we come upon the
same repetition again in the final form be + bd. Here again
we make a correction of the true symbolic import of this
expression, without any symbolic warrant, but merely to
secure what we know to be meant ; accordingly we conclude
that the orator-poets are only to be reckoned once. Symboli-
cally we have no right to do this, for we ought to have
written our terms down in the form of mutual exclusion.
But the needed correction is so tacitly and readily made
that we are hardly conscious of requiring to appeal to it.

In other cases the necessity of making this correction,
when translating our symbols into words, is forced upon our
attention. Suppose, for instance, that instead of combining
two distinct groups, like (a + b) (c + d), we combine the same
twice over. Taking, as before c + d to stand for the poets
and orators, what is denoted by (c + d) (c + d) ? Clearly
this is only a slightly more complicated case of xx = x, viz.
(c + d) (c + d) must = (c -f d). But when multiplied out, of
course, (c + d) (c + d) yields, (on our symbolic plan) c + 2cd + d.
That is the common part cd, and the fact of its being counted
twice over, is here forced obtrusively upon our notice. We are
reminded of the fact (equally true in the former case, thougli
it escaped notice there) that we had no symbolic right to use
the expression c + d unless c and d were mutually exclusive.
In that case cd = 0, and the redundant term does not occur to
trouble us.

In order not to multiply examples needlessly we will take
one more which involves the sign of subtraction, and see how



II.] Symbols of classes and of operations. 61

that answers when worked out. Take, for instance, (a b)
(c d) and see what it might stand for in common language,
both in its present shape and as it would become when
multiplied out in detail. The following would be a fairly
corresponding verbal expression : 'Barristers (excepting
foreigners) who are graduates, but not of Dublin'. What does
this mean ? It seems quite clear that all whom we can possibly
intend to except from the barristers are foreign barristers, not
foreigners in general, and that all whom we except from the
graduates are Dublin graduates. In other words this process
of exception or subtraction always presupposes that the class
excepted is a part (formally or materially) of the class from
which it is excepted. We are therefore warranted in writing
our symbols (a ah) (c .cd) or a (1 Z>) c (1 d), and in re-
garding both this and the verbal statement as being com-
pounded of four terms, two of them positive and two of them
negative.

But the question was not so much whether the two
expressions corresponded as they thus stood, but whether
the result of multiplying out the symbols could be translated
step by step in its details and shown to correspond to the
results of ordinary thought. Those symbols, when treated
by the ordinary rules, yield the result, ac acd abc + abed.
Is there any thing in the statement about the barristers
corresponding to all this? Certainly there is, for these
symbols as they thus stand in detail, are literally translatable
into the words 'Graduate barristers, omitting Dublin graduate
barristers, omitting also foreign graduate barristers, but
adding on Foreign Dublin graduate barristers'. If we go
through this we shall see that it expresses exactly the con-
templated class, but with one important proviso. This proviso
is that the expression 'omitting' here is not to be taken
in its loose popular signification, by which its application is



(52 Symbols of classes and of operations. [CHAP.

tacitly understood to be limited to the range of the term
which precedes it, but in its strict symbolic signification in
accordance with which we may omit too much, and there-
fore find it necessary to add on a term to correct this excess.
This, of course, is the real meaning of the final term abed.
Common language left to itself, would have compendiously
expressed the details in the form ac acd abc, that is,
would have put it ' Graduate barristers, omitting those
of Dublin and the foreign ones'. The accurate language
of symbols requires us to insert a final term which common
language had rejected for the sake of brevity.

The above examples are fairly illustrative of a number
which might be offered. They will serve to explain some
of the main points of our system of Symbolic Logic. We
see that we may employ the sign of addition (+) for the
aggregation of classes, provided we make our classes mutually
exclusive (which common logic does not in general do, pre-
ferring to leave this point unexpressed, and tacitly to make
any correction and alterations which are necessary). We
may employ the sign of subtraction ( ) for the exception
of one class from another, provided the excepted class
is included in the other (common language appears so uni-
formly to take this for granted that we may consider it
as really intending this limitation, though its terms do not
formally imply it). And we may employ the sign of multi-
plication (x) for the results of selecting the common members
of two classes. (This process is so simple that common
language does not seem to have become loose or inaccurate
here, except in so far as it puts an occasional intensifying effect
upon the act of repetition of comparative terms ; a perfectly
natural intensification, but one which we find ourselves
compelled to reject.)

To one point, which has been already noticed, attention



II.] Symbols of classes and of operations. 63

must be very persistently directed, as any vagueness of
apprehension here will be fatal to the proper understanding
of symbolic reasoning. It was seen in the last example
that, in translating a common phrase into symbols, we felt
justified, ' or rather bound, to ask ourselves what exactly
that phrase must be understood to mean, instead of just
proceeding to put it into symbols as it stood. We may
say in fact that we were resolved to give an idiomatic
translation and not to resort to a bald and literal substitution
of terras. 'Lawyers excepting foreigners' was accordingly
interpreted, on this ground, to refer to what was left after
foreign lawyers, not foreigners in general, were left out
of account. Accordingly we translated it into our symbols
as a ab, and not a b, for we knew that the former was
what we meant. But in the converse case, that is when
we were translating back from the language of symbols
into that of common life, we had no right to do -anything
of this kind. Symbols have no tacit limitations or con-
ventional interpretations other than that which has been
strictly and originally assigned to them. Accordingly if I
meet the symbolic expression a b, where a stands for lawyers
and b for foreigners, I have no right to regard this as a
popular lax expression, in which only a part of b instead
of the whole is to be subducted. I must take it as signi-
fying (whatever that may mean) the subduction of all
foreigners from the lawyers; and I must look out for, and
in this case shall find, the introduction of another term
which will set matters symbolically straight, by just neutra-
lizing the surplus subduction ; common language on the
other hand sets matters straight by interpreting the subduc-
tion in its own sense, and then just neglecting the required
additional term.

It comes therefore to this. "We may translate into the



64 Symbols of classes and of operations. [CHAP.

language of symbols almost as we please, for we are the
only ultimate judges of what we really mean to signify by
our words. But having once done so, and laid down rules for
working with our symbols, our further control of them ceases.
We must translate out of these symbols into the language
of common life in exact accordance with their assigned
meaning. Otherwise we shall find that one correction and
modification after another will be called for.

IV. The only remaining mathematical symbol to which
we will at present direct attention, and almost the only
other one which we shall have occasion to adopt, is the sign
' (=) of equality. Here too we must not trust too much to
acquired associations. What this symbol generally means
in mathematics is identity (or indistinguishable similarity)
in respect of some one characteristic only in the various
things which it connects together ; this characteristic being
in most cases the number of units involved. These may
be units of space, or time, or mass, or acceleration, or what
not. Thus in the case of a falling body, v*= 2/s means that
the square of the number of units in the velocity is the
same as twice the product of the number of units of force
and space. But in logic this is not so. The sign of equality
here indicates absolute identity in all respects, except nomen-
clature, of two or more classes. The identity indeed is so
complete, that all that needs pointing out is how we can
talk of two distinguishable classes in such a case. The
answer of course is that a 'class' is merely our way of
grouping or outlining things, and is indicated and retained
by the imposition of names. The very same individuals
therefore may belong to more than one class ; in other words,
may have more than one single name or combination of
names assignable to them. It is in this sense only that
we can talk of the identity of all the members of two or



II.] Symbols of classes and of operations. 65

more classes. Such identity is what we mark by the sign
of equality ( = ).

Here, as in the case of our other symbols, we must turn
to see by what sort of contrivances popular language succeeds
in conveying its meaning. As usual it is vague, and shows
both defect and redundancy, the signification being controlled
by innumerable conventions and implications. One way of
expressing this identity is by mere predication. If in answer
to the question, What are triangles? I reply, 'triangles are
plane figures included by three straight lines' I clearly mean
that every thing referred to by the one name is also referred
to by the other ; that is, that the individuals in these classes
are identical. This is however rather a lax form of speech, for
the copula 4s' (or 'are'), by itself, properly implies nothing
more than predication. The stricter form for expressing this
identity of classes is by some such words as "consists of", or
"comprises"; and even these may sometimes need the
addition of some other clause such as "and includes no
others" in order to remove all ambiguity, and to make it
plain that we contemplate a case of identity of classes, and
not merely of the inclusion of one class within another.
Again the word "means" implies this identity, and implies
it somewhat strictly. It does not properly speaking state
it; for the two things which we thus connect in our pro-
position are not so much two classes of individuals, as
the significance of some word on the one hand and the
things it refers to on the other. But it certainly carries
with it this complete identity of classes. When I say that
'ghost' means a disembodied human spirit, I imply that
the class of things, real or imaginary, referred to by one
name is identical with that referred to by the other name.

Symbolic Logic may therefore fairly be said tQ take
an ultra-nominalistic view of this subject. The expression
V. L, 5



66 Symbols of classes and of operations. [CHAP. n.