any other numerical factor as well, according to the degree
and nature of their defective statement. The only complica-
tion that is thus produced is that the final elements or
subdivisions of such an ill-expressed fractional form may
possess other numerical factors, positive or negative, besides
the true typical four 1, 0, , . We shall have to take some
notice of such irregular results as these hereafter. At present
I will pass them over, for they may generally be avoided
by strict attention to the phrasing of our formulae in the first
instance.

If it be asked here, as it very fairly may, why the two
peculiar factors $ and thus come into existence when we



IX.] On development or expansion. 211

develop fractional forms, but not when we develop what may be
called, by contrast, integral forms, the answer will readily be
found by examining into the process by which they are
obtained. (In saying this we are supposed to be asking for a
logical explanation, since the mere symbolic answer is obvious
at a glance: for clearly fractions like and i can only be
obtained from a fractional expression, by the process of 1 and
substitution.)

The answer is this: When we are developing an integral
form, that is, a mere group of class terms, there can be no
indefiniteness about this proceeding. Now $ is the sign
of entire indefmiteness, and cannot therefore be required
here. Of course if there was an indefinite term in the
original class group it will reappear in the subdivision:
thus, for instance, x + %y would clearly yield terms with this
indefinite symbol prefixed to them : but if the original
group was definite it can only be divisible into one set of
ultimate class elements, every one of which must necessarily
appear. There can clearly be no opening for uncertainty
about a true process of dichotomy. Again, as regards -J-.
This we saw implied a proposition or statement. It gave
an independent relation between the class elements, declaring
that such a class was non-existent. For anything of this sort
also there can be no opening in a mere class or group of
classes.

With fractional forms the case is of course different. In

Afl

whatever way - be reached; whether as a direct proposal

7

indicated in that shape, or as obtained indirectly from such a
statement as zyx\-^^ it makes a postulate. It accepts
x and y under a condition, and this condition can of course
be stated in a proposition. But every proposition is a
material limitation of the formal possibilities of expansion; it

142



212 On development or expansion. [CHAP.

extinguishes one or more of the classes yielded by mere
dichotomy or subdivision. Thus though xy is a mere class,

/*

- is a class with a condition attached ; and it is this condition

y

which, leading to the abolition of one or more classes, gives
an opening for the directive symbol which orders the suppres-
sion of such classes.

ST

So with the other symbol . Our fractional form -

i/

indicates an inverse operation, and one which, as customary
in such cases, does not lead to a single definite answer. The
reply may very likely be that, within assigned limits, any
class will answer our purpose. Such an indefinite symbol is
therefore distinctly necessary.

It should be noticed that in exceptional cases no demand
will be felt for either of these peculiar symbols, the conditions
of the problem not giving occasion to them. Thus take the

expression -* * . On expansion this yields xy -f xyz + #,

sc

but the usual condition (corresponding to xy = Q, in the

/*%

development of the simpler fractional expression -) gives here

*/

(xy + xyz) (1 x) = 0, which is identically true. That is, no
condition of relation between the symbols is demanded for the
performance of the operation in question ; it can always
be done, whatever relation exists between them. The logical
explanation of this is easy to detect; for, x includes xy + xyz,
formally and without further postulate. Hence it must
always be possible to find a class such that its limitation
by x shall reduce it to xy + xyz, which is the inverse opera-
tion called for. If it be known that X includes Y, we
can always find a multiplier (that is, in logical language, some
principle of selection,) which shall reduce X to Y.



IX.] On development or expansion. 213

Again, as regards the possible absence of terms affected
by the indefinite sign g, take the following example:

xy + xy + xy

xy + xy + xy '

Expand it, and it becomes simply xy + xy, the term which
would have been affected by $ vanishing formally. The full
logical explanation of this had better be deferred until we
have adequately discussed the meaning and interpretation of
logical equations, but the symbolic conditions for its occurrence

X

are easily seen. The indefinite factor in y is of course %XY ;

it will therefore vanish whenever XY = 0. That is, if X and Y
together make up, or more than make up, the total universe
(for this is the meaning of X Y= 0) then the inverse operation
ceases to be indefinite l .

This peculiar symbol $, perhaps the most distinctive
feature in the Symbolic Logic, has been much objected to.
We shall have to discuss it again on several future occasions,
but enough has now been said to show that it is very far from
being borrowed in needless affectation from the mathema-
ticiaus. If we chose to reject it we should still have to
invent some symbol to take its place, and where else is such
a symbol to be found which shall really express the full
indefiniteuess we need ? There is no such word in the
ordinary logical vocabulary, for 'some' in both its senses
('some, it may be all', and 'some, but not all') excludes none,
and therefore will not answer our purpose. We saw indeed,
in Chap, viz., that x xy is really, though not very obviously,

1 In which case, indeed, this in- .Y

,. , ,. or T=-i which is identically true

verse operation loses all peculiar 1

significance. For, if XY=- 0, as well as whatever may be the value of A'.
XY=0, then F=l; so that=XF



214 On development or expansion. [CHAP.

the precise equivalent of # = y ; but we could not conveni-
ently work with this equivalent in our developments, for
it demands the repetition of the whole of one side of the
equation on the other side. If we tried to employ it as

a substitute for in the development of - , viz. xy + %xy, we

J

QJ SC

should have to write it - = xy + - x xy, when the impropriety

/ J'

of substituting an implicit equation instead of an explicit
becomes glaring. If we object to make use of g in Logic we
shall simply have to invent another symbol and define it in
precisely the same sense. At least we should have to do this
if we undertook to work the same problems with the same
generality.

The reason why we require this symbol, whereas the
ordinary Logic manages to do without it, must be assigned to
the completeness of the Symbolic System. The ordinary Logic
answers what it can, and simply lets alone what it cannot
answer, whilst what the more general system aims at is
to specify the direction and amount of our ignorance, in
relation to the given data, as explicitly as that of our
knowledge. We take into consideration every class which
any combination of the terms in question can yield, and
enquire what information the data will furnish in reference to
it. As Prof. G. B. Halsted has well said, "The problem may
be very compactly stated : v . It is: Given any assertions, to
determine precisely what they affirm, precisely what they
deny, and precisely what they leave in doubt, separately arid
jointly" 1 .

We are now in a position finally to explain the various

1 Journal of Speculative Philoso- in each of which there are some
phy, Jan. 1878. See also the same valuable critical remarks by the same
journal for Oct. 1878, and Jan. 1879, author on Boole's system.



ix.] On development or expansion. 215

elements which make their appearance in the development of
one of these fractional forms which demand the performance
of an inverse operation. If our data have been properly
expressed in mutually exclusive terms at the outset, our
result must necessarily be comprised in the following form :



where X, A, B, C, D are aggregates of class elements
composed of the various combinations of the class terms
x, y, z, &c., and their contradictories.

Here then X, on the left side, is a class term. The
succession of expressions on the right side may be taken as a
description or definition of X given us in terms of x, y, z, &c.
or whatever the symbols of the classes may be. Now what
we are told in this equation is that, in order fully to
determine X as desired, we must ;

1. Take the whole of A; A consisting of one or more
class elements. Hence conversely, the whole of A is included
in X. Ordinary Logic would express this by saying, 'Some X
is A', and'AllJ. is X'.

2. Exclude from X the whole of the class B: or, to put
it in ordinary language, ' No X is B'.

3. As to the entry of the class C into X\ we can, from
the given data, determine nothing whatever. All (7, or some
only, or none of it, may go to the making up of X. To
decide this more explicitly, we should require fresh infor-
mation.

4. As to D, this class is not merely excluded from X, as
B was, but may be proved from the given data to be an
impossible or non-existent class. The compartments referred
to in D have of course to be taken notice of, but our data
pronounce them by implication to be empty.

There are two questions which may fairly be raised here,
and which demand a few moments' consideration.



216 On development or expansion. [CHAP.

I. It may possibly be objected that though we have
thus determined the relation of X to the four classes
A, B, C, D and to all the sub-classes which they contain
under them, there may yet be other classes, of a similar kind
to these, of which we have taken no account, for the possible
combinations of x, y, z, &c. and their contradictories are
numerous. The reply is that there are no other classes
possible, for however numerous they may be they are
all comprised in A, B, C and D. The veiy nature of Develop-
ment insures that every possible combination (logically) of
our class terms shall be taken due notice of.

II. But, again, it may be asked, What about the existence
of these classes referred to in A, B, and C\ Must there be
things corresponding to these various class terms ? The
answer to this question is contained in the results of a former
chapter (Chap, vi.) and is briefly this. Negation is positive
and final ; therefore every one of the classes which go to make
up D is certainly abolished: there can be no such things
as these. Moreover we know for certain (on formal grounds)
that some one class at least of those included in A, B, and C,
must exist. But beyond this we can only speak hypothetically,
as in all positive assertion. If there are any A's then
the same things must also be X, thus establishing that there
must then be X. The existence of a B, however, or a
C, proves nothing about there being X. Conversely, if there
be any X, the things which are X must be either A or C,
so that one of these is proved to exist, but we cannot tell
which.

Illustrations and Examples.

I append a few pages of illustration, purposely choosing
for the most part extreme or limiting cases in the application



ix.] On development or expansion. 217

of the simplest formulae, as these subserve the purpose of
explaining the principles of this Development better than
others.

Begin with one class term only, x, and see how many
forms we can obtain with our /'(a:) =/(!) sc +./(0) x. Of what
we have called integral forms there are the following four :
1, 0, x, x; which yield the developments,



They stand in need of no explanation. Logically, they
represent the results of subdividing (or, we should rather say,
attempting to subdivide) the four classes, 'everything',
'nothing', 'x ', and 'not-#', into their x and not-# parts respec-
tively. Of course only 'everything' can really be so divided.

Of fractional forms with one class term, we have sixteen,
since any one of the above four may stand as either numerator
or denominator. Qf these however the four with unity
as denominator are identical with the above integral forms,
thus leaving twelve. They should all be carefully worked
through for the purpose of obtaining perfect command of the
logical meaning and symbolic usage of our various forms.
They are as follows:

1 - = x r~ " - = x <>x

x

2. - = # + &. 6. n = $x + 7j-

V II v V



3. = o# + $c. 7. =

X \j

4. =0* + z. 8. =



218 On development or expansion. [CHAP.

+ x. 11. =

x

10. ^r=}>x + Qx. 12. -?- =

a; a;

We will briefly examine a few of these results in turn ;
sometimes in accordance with one, sometimes in accordance
with the other, of the two ways already pointed out for
approaching these fractional forms ; that is, either by taking
them as immediate demands for the performance of an inverse
process, or as stating a conditional proposition which leads to
such a demand.

1. This asks us the question, What is that class which,
when restricted by taking only the ic-part of it, will yield
the class 'all' ? A little reflection will show that the only
class of which this can be said is 'all'; viz. this is the limiting
case in which the imposition of a restriction is merely, as it
happens, leaving the class unaltered. (The formula indicates
this by the term ^x; reminding us that x = 0, viz. 1 x =
or x = 1.)

2. This is in reality a very familiar old friend of every
logician, for it is nothing else than 'accidental conversion'
done up in a new dress, and, I would add, far more accurately
expressed. No doubt it sounds very unfamiliar, even when
put out of symbols into words, if we phrase it as ' What is the
most general expression of that class which, restricted by
taking only the a-part of it, will coincide with xV But
we shall soon recognize its features if we look at it this
way : Suppose we had given us that yx x t and were asked,



.7,-



What then is y in general ? This would have led to y - , as



x



above, which is asking for y in terms of x. Now it was
shown in Chap. VII. that the expression, or rather an ex-
pression, for the universal proposition 'All x is y' is x = xy.



ix.] On development or expansion. 219

Hence it is clear that to ask for y in terms of x, under this
condition (as is done symbolically above), and to ask ' If all x
is y, what is y?' are really one and the same question.

Now compare our solution here with the common solution.
We say that y = x + $ x, whilst the common answer says
simply 'Some y is x\ The advantage of the former seems to
be that it so prominently forces on our attention (by the em-
ployment of the peculiar symbol $) several possible cases
which the common answer rather tends to obscure from sight.
We are reminded for instance that x may be the whole of y
(if happens to = 0) ; that it may be a part only (if g is
intermediate between and 1); or that y itself may be 'all'
(if & = 1). These three alternatives may of course be deduced
from the common form of conclusion, but they certainly do
not appear very prominently in it, especially considering the
ambiguity of the word some.

I would also call the reader's attention to the very
decisive way in which all those troublesome and perplexing
questions, as to what is implicated in the way of the existence
of our x and y, are avoided by this way of regarding the
subject. Once understand that 'all x is- y' is only uncon-
ditional in what it denies (i.e. in denying that there is any
xy}; and employ the truly indefinite symbol instead of
some, and a proposition and its converse will fit in har-
moniously with any number of other propositions without
inconsistency or demand for fresh assumption. For instance :
start with 'all x is y' in the form xy = Q, thus blotting out
one class. Now elicit x from this, and we get x=%y (the

reader will easily verify this conclusion as follows: # = -,

therefore by development x = % y}. The implication is clear
and decisive, and in perfect harmony with the unconditional
negation above. We see that if there be x there must be y,



220 On development or expansion. [CHAP.

and that if there be y there may be x ; but that there may be
neither one nor the other. Then go on to convert xy = 0, or
x = %y, and we get, as above, y = x + x. Here the same im-
plications meet us as clearly as before: if there be any x
there must be y, but if there be any y there need not be x
(since the y may be contributed by the term $ x). And there
need not be either x or y.

As just remarked, the two assumptions upon which this
explanation rests are (1) the unconditional negative interpre-
tation of the universal affirmative, and (2) the employment of
the perfectly indeterminate symbol $ in place of the ordinary
some.

3. This result is best interpreted as follows. Suppose
#y = 0, viz. 'No x is y\ What do we know about y? The
equation y = $ x tells us that y, if it exist, must lie outside x,
but that there may be no y at all (if = 0) ; that there may
be some y\ or that the y may be the whole of x (if = ] ), in
which case x and y are (materially) contradictory opposites.

4. This development must have come about from such a
statement as that l xy is the same as not-a;'. We can save this
from being nonsense or contradiction (on the principles laid
down in Chap. VI.), only by supposing that there is no such
class as y, and that x is 'all', for 'all nothing' is the same as
' no all '. The development expresses this condition.

C. This is the development of an entirely indefinite class,
and is therefore itself entirely indefinite. The nearest verbal
equivalent would be the direction to subdivide 'something'
into its x and not-# portions. We could but say that this
would yield some x and some not-#. After what has been
already said in explanation of this symbol $ the reader will
hardly need to be reminded that we regard it as a strictly
interpretable expression. It is the determination (if we may



ix.] On development or expansion. 221

use the word) of a class which proves to be strictly indeter-
minable.

7. This is of course an instance of gross misapplication of
our formula. The universe being 1, the expression ^ is
logically unmeaning as a class term. The demand which
it makes is that we shall find some class such that when no
part of it is taken we shall still have, as a result, all ; (0^=1).
Our formula resents being told to work upon such an
impossible class as this, in the way that formulae generally
do resent such a call, viz. by just talking nonsense. It
responds to the demand by declaring that there is neither a?
nor not-# : as good a reply as any other under the circum-
stances.

As the remaining examples do not raise any interesting
questions except what are contained in such as have been
already noticed we will not go into them any further. The
reader is however recommended to work through them all in
order to familiarize himself with the interpretation of general
formulae in critical and limiting cases.



CHAPTER X.

LOGICAL STATEMENTS OH EQUATIONS.

HAVING thus considered the nature of Development or
Subdivision, which may be considered an introduction to the
central subject of Logic, we must now go on to consider this
main subject itself under the heads of Logical Equations, and
Interpretation and Solution of Logical Equations. This latter
division may be said, roughly speaking, to correspond to that
between Propositions and Reasonings in ordinary Logic,
though here as elsewhere our arrangement of the subject is
very far from coinciding with the traditional one. We shall
devote this and the following chapters to the consideration of
these topics.

There are two main principles of interpretation to which
we shall have to appeal in the course of this discussion. As
neither of them is distinctly recognized in the ordinary Logic,
and both are in some respects decidedly alien to the ways of
common thought and speech, it will be well to begin by
calling prominent attention to them.

(1) The first of these is involved in the view of the
Import of Propositions explained and insisted on in the
sixth chapter. The comparative novelty of that view as a



CHA.P. x.] Logical statements or equations. 223

systematic doctrine, and its extreme importance for our
present purpose, must be the excuse for once more recalling
the reader's attention to it.

It was laid down then that propositions must be regarded
as having, generally speaking, an affirmative interpretation
of a conditional and somewhat complex character, and a
negative interpretation which is unconditional and compa-
ratively simple. That is, what they assert can only be
accepted under hypotheses and provisionally, in so far as the
existence of the objects is concerned, whereas what they can
be made to deny is denied absolutely. This contrast presented
itself even in the case of the simple propositions of the common
Logic, but when we come to the complicated systems of
propositions which we must be prepared to grapple with in
Symbolic Logic, it appears to me that without this explana-
tion we can make no way at all.

It must be frankly admitted that this is not the sense in
which the popular mind accepts and interprets propositions.
Nor is it, I presume, in accordance with the canons of
the common Logic; and very reasonably not so, on the
part of the latter; for using, as this does, forms which are but
little removed from those of common speech, it cannot risk so
complete a breach with convention as we may freely do who
deal mostly with symbols. I speak with reserve however on
this point from really not knowing what the law may be.

(2) The other principle to which we shall have to resort
may be conveniently introduced by the following question.
If any one were to declare to us that his annual income and
the acreage of his estate, taken together, amounted to
precisely 500 and his daughters, could we charge him on
the face of the matter with either falsity o? nonsense ? He
has adopted an unusual way of speaking, but a solecism need
not be without meaning. If we insisted on translating



224 Logical statements or equations. [CHAP.

his words strictly, what sort of construction should we have
to put upon them, neglecting all merely conventional impli-
cations ? We should have to say that, since it is impossible to
equate heterogeneous things, the only solution which will
avoid actual nonsense must be found in the conclusion that
his income is 500, and that he has no acres and no
daughters.

The second of these principles involves of course an
appeal to the first. Admitting that the employment of a
logical term does not necessarily carry with it the existence
of any corresponding class, we say that these are circumstances
under which this admission has to be put in force. Disparate
things can only be equated by the assumption that both are
then and there non-existent.

There is a slightly different way of looking at the same
thing which may make it a little more acceptable. Instead
of starting, as above, with the statement x = y, where x and y
are heterogeneous, put it in the form that if y be taken from
x nothing is left. Suppose I say, in reference to some given
assemblage of people, Take all the rogues from amongst the
lawyers and nobody is left, it is quite certain that this
identifies the two classes. This is a necessity of thought or
of things; but to this necessity common usage couples an
assumption, which ordinary Logic doubtfully accepts, viz.
that we must not so speak unless we mean to imply that
there are certainly some people present who belong to
both categories. Symbolic Logic distinctly rejecting this
assumption need not hesitate to accept at the same time the
proposition that no lawyers are rogues. In this case, since
the subduction of a rogue can no longer remove a lawyer the
statement can only hold good on the supposition that there
is none of either class present. In other words, whereas
the logical equation, x y = 0, necessarily and always implies



X.] Logical statements or equations. 225

the identity of x and y, all that we are now doing is to claim
the right of extending this to the limiting case in which
x and y are both = ; and consequently of inferring that
if x and y are known to be mutually exclusive then this
Limiting case is the only possible one.

It can hardly be maintained that this system of interpre-
tation is much more in accordance with popular convention