every logical equation can be expressed in the form of a class
group of a peculiar kind. That is, it may be made to say
that a certain aggregate of classes' is, collectively and
individually, non-existent. Of course this needs some modifi-
cation of form and arrangement, but there is not the slightest
change or loss of significance entailed. Take, for instance,
the statement w =; xy + ~xz. When put in the form

w xy xz = 0, ,

it is a case of /(#) = 0. Now though, as it thus stands, it
does not fulfil the conditions assumed above for f(x}, for it is
not built up entirely of the sum of a number of mutually
exclusive elements, we know that it can easily be made



302 The Expressions f (1) amZ/(0). [CHAP.

to assume that form. Develop and rearrange it, and it
stands thus,

wxy + wxz + way + wxz = 0,

this being in every essential respect, as regards assertion and
denial, the exact equivalent of w = a-y + xz. The two are
really the same statement differently worded or arranged.
Hence it is clear that f(x) = is merely a particular case of
f(x) ; and everything that has been said above about the
interpretation of /(I) and /(O) and their mutual relations to
one another will hold good here also. The form, and the
detailed meaning, of a class expression, are entirely unaffected
by our having to add that it is known that no such class
is in existence, which is the only difference introduced by the
equational form.

We are now in a position to understand Boole's rule for
Elimination. This rule does not tell its own tale so clearly
as the simple plans offered in the last chapter, and in actual
performance it is unquestionably apt to prove tedious.
Symbolically however nothing can be more beautifully neat
and effective, and the penetrative originality which enabled
Boole to discover it is quite beyond all praise.

The rule is simply this. Let y(#) = be any logical
equation involving the symbol x, then /(1)/(0)=0 is
the full expression for the result of the elimination of x
from it.

This rule does not, at first sight, appear to have the
remotest connection with either of those offered in the last
chapter, but a little consideration will show that it is sub-
stantially identical with the second of them. It only differs
in fact by offering a methodical formula for a process which
we had partly left to empirical judgment. We shall soon see
this by examination of an example. Take the following,
w = xy + xz,



xiv.] The Expressions f '(1} and f '(0). 303

and suppose we want to eliminate y from it. We have
/(I) = w x xz,f(fy = w xz, so that the rule gives us

(w x xz) (w xz) = 0.
Multiplying out, this reduces to

wxz + wxz = 0,

or exactly the same result as we should have obtained by
breaking up the equation into its separate denials and se-
lecting those only amongst them which do not involve y.
(These separate denials were given on the preceding page.)

But our present object is not so much to show that the
two methods agree in their result as to explain the logical
meaning underlying this highly abstract symbolic formula.
This may readily be done as follows. We have shown that
by eliminating y is meant the selection of those elements
of denial which do not involve y or y. We were then met
by the difficulty that in the complete development every
term must involve one or other of these. True, but those
which are in effect free from yory are exactly those which
involve both y and y, for wxz wxz (y + y). Hence all that
we have to do in the complete development is to select those
elements only which involve both y and y. Thus if the com-
plete development of f(y) is Ay 4- By, then the terms really
free from y are those, and those only, which occur in both
A and B, But the way of rinding the elements common to
both A and B is simply to multiply A by B. In other words
AB, that is,y(l) f(Q), is the symbolic expression for those ele-
ments of denial which the equation can yield free from y or y.
And the statement of such denial is as usual given by equation
to zero, so that f(l)f(ty = is the precise symbolic state-
ment of that process which we have worked out logically 1 .

The same conclusion would follow as a simple corollary from

1 It is not a little strange to find Ulrici declaring that after careful
such a philosophical authority as Btudy of Boole he ' found that at



304 The Expressions /(I) and /(O). [CHAP.

the results of page 299. Thus the development of f(y) =
assuming the form Ay -f By = 0, and these being mutual
exclusives, each term must vanish separately. But the meaning
of Ay = is that none of class A is y, and the meaning
of By = Q is that all class B is y. Hence no A is B, or
AB = 0; that is, as before, /(I) /(O) = 0. (It will be re-
membered that this result was only exceptionally true when
/(I) and /(O) were derived from a simple class group in-
stead of from an equation.) This expression being certainly
free from y, and moreover all that can be obtained free from
it, is a true elimination of y.

The case just hinted at, in which /(I) and /(O) happen
to be exclusives, so that /(1)/(0), being formally =0, will
lead to no result, deserves a moment's notice. It tells us
that the term in question cannot be eliminated. Thus in

xyz + z = yz + xyz,

we have, in trying to eliminate x, /(I) /(O) = yz x yz =
from which nothing can be deduced. The meaning of this
is that every term in the ultimate development involves x or
x separately. But where every constituent element is con-
ditioned by the same term it is clear that nothing can
be inferred about the data when free from such condition. A
more obvious case of this is afforded by the statement
xy=xz. If we tried to eliminate x we should get nothing
better than $ y = {} z, from which nothing follows. The only
known relation here between y and z being given under con-
dition of x, no inference can be drawn as to what relations
there may subsist between them in absence of x.

bottom this new Logic offered no- (Zeittchrift filr Phil, und phil.

thing new, but was essentially no- Kritik, 1878). One would like to

thing else than a translation of the see the original of the translation in

old Formal, that is, Aristotelian the case of this formula /(I)/ (0)=0.
Logic, into mathematical formulae'



XIV.] The Expressions /(I) and /(O). 305

Boole generalizes this formula of elimination to embrace
the simultaneous treatment of any number of terms. Thus
the formula for eliminating the two terms x and y from.
f(x, y) = is

/(I, !)/(!, 0)/(0, 1)/(0,0) = 0,

and so on, for any number of terms. The reader who has
followed the explanation so far will without difficulty see the
requisite logical interpretation here. For instance, when the
expression 'f(x, y) is fully developed, the only elements
of the series of constituent denials which can be exhibited
free from both a> and y are those which simultaneously
involve xy, xy, xy, and xy (for the sum of these = 1). Now
the above formula of elimination is nothing but the logical
rule for selecting such elements. And so on with three,
four, or more simultaneous eliminations.

The employment of the peculiar symbol to express the
results of our elimination of a term, affords a convenient
opportunity for making a few concluding remarks upon
its significance. It is of course a logical factor, standing
therefore for a logical term, but its peculiarity is that it
stands for any term whatever and is therefore perfectly
indefinite. We did not, it will be remembered, borrow
it directly from mathematics, but found that it spontaneously
presented itself in certain cases of the performance of the
inverse operation denoted by the fractional sign. This,
I think, suggests a caution or restriction in respect of its use.
We ought not to regard it as in any way dividing any ulti-
mate class subdivision to which it is prefixed. If we had in-
vented it for ourselves, as a sort of substitute for the 'some '
of ordinary logic, it might be asked why we should not prefix
it at once to any simple class term, and therefore write down
such an expression as x. The objection to so doing is, I think,
connected with a. fundamental characteristic of our whole

V.L. 20



806 The Expressions /(I) and /(O). [CHAP.

scheme. Given a class term x, and this only, we have
no right to talk of a part of x. A class, with us, is only
divisible by another class term ; that is, given y also, we can
at once contemplate a division of ac into xy and xy r but
we need the pressure or suggestion of such another term
in order to do this. I call attention to this, because it seems
an indication that this Symbolic Logic, so far from being
specially mathematical in the sense of having anything
special to do with quantity, is in some respects less so than
the common system. The class x, say, is given to us as
undivided by any other class term. We therefore avoid
talking of some of its members, because to do so would be to
found assertions, not on mere class distinctions given purely
by class characteristics, but on some kind of distinction which
had been got at by investigating individuals without the help
of class characteristics, and which is therefore very much the
Same thing as proceeding to count them. At least it seems
one step towards proceeding so to do.

Herein lies the difficulty, I should say the impossibi-
lity, of representing the true particular proposition by aid of
or any symbolic equivalent. From Boole's vx = vy, if v be
indefinite, it is clear that nothing whatever can be obtained ;
for %x = $y has not a definite word to say upon any subject.
So this expression will not subserve the purposes of ordinary
thought. As regards Prof. Jevons' AB = AO the case is
rather different. Of course if A were really indefinite (and
this it is sometimes called) the two expressions would be
exactly equivalent; but as he generally terms his form a
'partial' or 'limited identity', this seems to point to another
signification. What we are expressing is the fact that ' the B
which is A is the same as the G which is A'. In this
expression we must assume that A is unknown, as otherwise
we merely have, not a particular, but a convertible universal



xiv.] The Expressions /(I) and /(O). 307

of the type 'All X is all Y, X and Y being complex terms.
Is then AJB = AC, where A is unknown, a fair type of the
particular proposition ? I think not, because it postulates
that the members common to B and to C have some attribute
in common, such attribute being probably connoted by the
term A. But what right have we to assume that because there
are individuals common to B and to C, that therefore these
individuals must possess some attribute in common ? One such
attribute I admit them to possess, viz. that of their common
membership in B and C, but this would clearly only yield an
identical proposition 1 , and we have no warrantfor insisting upon
any other such attribute. Hence, although AB = AC can be
read off, as 'some B is C", it is not true that every 'some B is
C" can be formulated as AB= A C. I can see no form which will
cover all particular propositions, except that which throws
them into an assertion of existence, and confines itself to
declaring that there are IPs which are C, as explained in the
seventh chapter.

After what has been already said it hardly needs repeat-
ing that $ is not the equivalent of some. Probably the best
compendious statement of its significance is that it is a con-
fession of entire ignorance in respect of the term to which it
is prefixed 2 . If it be asked why we want such a symbol, the
answer has been already given in the results of the Chapter
on Development : viz. that the comprehensiveness of our

1 If A stands for the fact of Hughlings : a little book with some
belonging to both B and C, then fair points in it, but -which hardly
AB AC becomes BCB = BCC, or answers to its designation), this
BC BC, which is clearly unmean- should not be said without a caution,
ing. The two former multiplications really

2 Though therefore it is quite true do result in known equality ; in the
to say that "either 1, 0, or $, mul- third case what is meant is that our
tiplied into itself, equals itself" ignorance is equally complete before
(The Logic of Names, an introduction and after the multiplication.

to Boole's Laws of Thought, by I. P.

202



308 The Expressions f (1) and /(O). [CHAP. xiv.

system, in which we frame a perfectly complete subdivision
and call for an answer as regards every compartment con-
tained in it, necessarily demands the equivalent of such a
symbol. A confession of perfect ignorance in respect of a
single class, taken by itself, has no significance ; but the same
confession in reference to one or more remaining classes of an
exhaustive catalogue, after we have definitely pronounced
upon all the preceding classes, has a decided significance.

Thus %x intimates nothing; and xy+$x intimates no-
thing in respect of its second term. But make these expres-
sions members of a logical equation or statement, and the
indefinite elements immediately acquire a certain significance.
Thus z $x cannot possibly tell us whether z is any actual
part of x, but assures us that it is no part of not-#, and this
is important. Similarly z = xy + %x assures us that z is no
part of not-(xy + x), which is by no means the same thing as
informing us that it is no part of not-xy. In every case the
indeterminate term represents a confession of ignorance over
its whole range, but to confine our ignorance within that
range is to yield knowledge in reference to what is outside it,
and this is done in any of these logical equations.



CHAPTER XV.

COMPLETION OF THE LOGICAL PROBLEM. THE SYLLOGISM.

WE have now reached the last step of our purely logical
analysis. "We have shpwn how to resolve any propositions,
and any group of propositions, into all their ultimate denials,
that is, into all the unconditional elements which they con-
tain. This was the first step, and was fully treated in what
was said about Equations and the Interpretation of Equations.
The next step was to show what could be done with a portion
of these elements ; that is, how near we could attain towards
giving the full force of the propositions in question by the use
of a selection only of the total number of terms involved in
them. This was treated in the Chapter on Elimination.

What we have now to do is to take a step in the way of
Synthesis. We want to investigate some rule for determining
the value of new groups of these elements in terms of the
given class symbols. The full symbolic statement of the pro-
posed step would be this; Given f 1 (x,y^,...'},f i (x,y^ y ...) t &c,
determine 1 F(x, y, #,...) in terms of any assigned selection of

1 This F (x, y, <fcc.) may involve function of the class terms, but a
any selection, of course, of the terms new function of a selection of them.
x, y, &G. It is not only a new



310 Completion of the Logical Problem. [CHAP.

the remaining symbols. A special case of this process may

be detected in that generalization of the syllogistic process

already referred to amongst the examples in Chap, xil.,

when we determined xz from xy and yz. That is, we have,-

for /(#, y y z, a, c) = 0, xy-a = 0,

for / (or, y, z, a, c) = 0, yz-c = 0.

The problem is, find F(x, z), or xz, in terms of a and c,
omitting y from the conclusion ; i.e. eliminating y.

The general solution of this problem was probably first
conceived, and almost certainly first effected, by Boole. As
a piece of formal symbolic reasoning there seems nothing to
be added to it as he left it, and it is a marvellous example of
his penetration and power of generalization. It cannot often
be the lot of any one to conceive and so completely to carry
out such a generalization in an old and well-studied subject.

We will approach this problem in the same way as we
have attacked the previous ones, that is, by first seeing what
suggestions our common logical knowledge could offer towards
the solution of it. We will then turn to Boole's method of
solution ; the real logical significance of which is by no means
easy to grasp, unless we have thus examined the matter first
in a somewhat more empirical way.

Take the simple example offered above : viz.

Given ^ k find xz in terms of a and c.
yz = c)

We should naturally begin by breaking up the given
equations into all their ultimate denials, so as to obtain the
whole materials for whatever they can affirm, deny, or leave
in doubt.

These materials are the following :

ax = 0, cy = 0,

ay = Q, cz = 0,

axy 0, cyz = 0.



XV.] Completion of the Logical Problem. 311

Now develop xz, the quantity to be determined, in terms
of the other elements, and we have

xz = xz(acy 4- acy + acy + acy + acy + acy + acy + acy)...(l).

This is, the reader will remember, a merely formal result,
an expression which must always hold good. We shall, so to
say, materialize it, that is, bring it into accordance with the
assigned data, by seeing to what it reduces on the introduction
of the above denials. Remove then from it all those ele-
ments which, in virtue of the given equations, can be shown
to vanish, and it reduces to

xz = xz(acy + acy) or xz = $(acy + acy),

or, in the form in which we want it, in which xz is expressed
only in terms of a and c, it would stand

xz = $ac + %ac (2).

It must be observed, here, that the factor g is introduced
by a double right. For one thing we want to eliminate y,
and we know that the most direct way of doing this is simply
to substitute for the term to be eliminated. Then again,
anything standing in the form X= XY is known to give an
indefinite value of X, since this is one of the alternative forms
for X = $Y. Hence even if y had not been eliminated we
should have known that we had only got a result of the form

xz = $acy + fficy.

When equation (2) is read off into words, it stands,
"All xz is either both a and c, or neither a nor c".
This answer is quite correct so far as it goes, but it must ,
be carefully observed that it does not go so far as it might.
One side of the equation is plain enough, but not the other.
We know, that is, that xz is confined to ac and ac, but we
do not know whether it contains the whole of either or both
of these terms. It represents the present state of our know-



SI 2 Completion of the Logical Problem. [CHAP.

ledge ; with further knowledge we might ascertain that one
or both of these factors $ must be converted into 1. In order
to decide this point we should, if we continued the same plan,
have to examine both the elements ac and ac in terms of xz,
in order to determine whether we could thus partially or
entirely convert the equation. As it happens, ac is very
easily determined. For, multiplying together the two origi-
nal equations, we have at once ac = xyz = xz.

Hence it is clear that the equation can be written

xz = ac + ac,

viz. that 'all ac is xz\ as well as 'all xz is ac or ac'. Again as
reards ac we, have



This is not expressible simply in terms of xz. Accordingly
we cannot convert the second term in the expression for xz,
but must leave the equation as it stands above, viz.

xz = ac + $ac ..................... (3).

(The reader will remember that this result is the same as
we obtained in a former chapter by resort to what may be
called simple geometrical considerations ; that is, by reference
to diagrams.) - .

Let us, for further illustration, vary the example by
making it a trifle more complicated. Let it be proposed to
determine the expression xz + xz from the same data as
before : that is, let F(x, z) = xz + xz. Proceeding exactly as
before, by developing each of the expressions xz and xz in
respect of the remaining three terms, and omitting those
elements which the original equations prove to be non-exis-
tent, we have

xz = xz(acy -f acy) = $ac -f- $ac,
xz = xz(acy + acy) = gac + $ac. .



XV.] Completion of the Logical Problem. 313

Gathering the two together, and remembering that the
multiplication by 2, or any other factors, of a term affected
by the indeterminate factor , still leaves it indeterminate,
we have

F(x, z)=a;z + xz = %ac + %ac -f $ac,
which is the required answer.

Here, as in the last example, the answer is not quite
complete. Of course it is possible that it may yield all that
we want, for we may have only wanted to know the constitution
of the assigned expression xz + xz, in respect of a and c.
But if it were also required to know the converse, or con-
verses, of this equation ; that is, to determine whether xz + xz
included the whole of any one or more of these classes ac, ac,
ac, we should have to take each of them in turn, and ascer-
tain whether either of the three was included in xz + xz.
We might proceed to do this by the complete method of
developing ac, ac, ac, and then striking out the terms which
are proved to be non-existent. But as it happens there is a
much shorter way, as follows :

We have a = xy, c = yz,

:. ac = xy(\ - yz]
xy xyz xy(\. z) = xyz

:. ac = $xz.
Similarly ac = %xz.

The remaining term ac does not admit of such statement.

Hence, finally, we have,

scz + xz = ac + ac + $ac,

from which it appears that the proposition may be converted
as regards the two terms ac and ac. It is fully stated in
words by saying, "The classes represented by xz and xz are
both certainly contained in the aggregate of the classes com-
prised by ac, ac, and ac ; and conversely, both ac and ac are



314 Completion of the Logical Problem. [cfiAP.

contained in the aggregate comprised by xz and xz"; or, in
somewhat more familiar language, 'Every xz and every xz is
either ac or ac or ac; and conversely, every ac and every ac
is either xz or xz '.

The process above illustrated is a perfectly general one,
and if nothing more were desired in a logical process than
the solution of the assigned problem, it would probably be
the best, that is, the most effective and convenient way of
setting to work. It may be described as follows, in a series
of perfectly intelligible logical steps : Take the given equa-
tions and analyze them into all their constituent elements,
that is, into all the ultimate denials which they involve and
which collectively make up their significance. Then take
the given function which we are told to find the value of,
and make the requisite synthesis. That is, build up succes-
sively each part of it, employing for this purpose the above-
mentioned denials. This latter stage is really one of rejec-
tion, for we begin by developing the required function into
its full complement of potential classes, and then strike out
as many of these as are shown to vanish in consequence of
the previous analysis. Having thus gone through the Ana-
lysis and the Synthesis there remains the third step, namely
that of Elimination. It may be required to express the
desired function in terms of part only of the terms involved
in the equations. If so, the elimination is of that easy kind
discussed in the earlier part of Chap, xni., in which the terms
to be eliminated entered on one side only of the explicit
equation. Substitute therefore the indefinite symbol % for
the terms to be eliminated, and the whole problem is solved,
so far as the determination of the given function in terms
of the assigned class symbols is concerned.

If we want also to go on to determine how many converse
statements can be made, that is, to determine not only of



xv.] Completion of the Logical Problem. 315

what classes the given function is composed, but also in
which cases it comprises the whole of these classes, then we
must go through the same processes in the case of each of
these classes. We must take each of them in turn and build
it up in the same way as the given function was built up.
The same list of elementary denials will serve, of course, in
each case, for we are dealing with one and the same set of
original equations.

In familiar language the process may be described by
saying that we take the given premises, break them up into
fragments, and then put these fragments, or a part of them,
together in some other arrangement in order to build up the
structure we require.

Boole's plan for attaining this end is one which would
probably seem the most natural to any mathematician who
was disposed to apply to Logic the methods found so success-
ful in his own science. He takes the assigned function
F(x, y, z) and puts it equal to, say, t ; where t is of course
simply a new symbol, the equivalent for this function. Our
equations then stand thus :

/i(i y> *} = >

/$! y, z) = o,