not equally clear in the case of the syllogism is that the pre-
mises are given to us separately instead of being combined
into one. That is, in the former case we say '2 is xy, there-
fore (more vaguely) z is y or x' ; whilst in the latter we say
'z is x, and x is y, therefore z is y\ Of course the distinction
between mediate and immediate inference is on various
grounds important, both of speculation and of logical pro-
cedure; and nothing here said is meant to gloss over that
distinction in its due place. All that is now asserted is that
in each case alike, whether there be one premise or two, the
full determination of z was xy, and that consequently the
statement that it is simply y (in other words, the elimination
of x) is so far a vaguer and less exact determination of it than
could be given by the retention of that term.
This loss of precision in the process of elimination is the
general result, but a certain narrow class of exceptions can be
pointed out. When x and y are coextensive the substitution
of one for the other leaves the extension unaltered. Thus in
the elimination of y from 'all x is all y, all y is all z' the de-
termination of x as z is just as narrow as that given by calling
it yz. But of course such a case as this can but rarely occur.
It is not generally easy, except when we are dealing with
definitions, to find two terms thus coextensive ; and the oc-
currence of three such must be very rare indeed.
It will be understood that this loss of precision is no valid
objection to the process of elimination. It is one of the many
characteristic distinctions of the class- explanation of proposi-
288 Elimination. [CHAP.
tions to call attention to the fact that there is any such loss
at all. On the common explanation we only think of the
major term in its capacity of a predicate, and we want to
know whether or not it is to be attached to the subject. The
middle term is used presumably merely as a means toward
deciding this fact, and when it has answered its purpose it is
very properly dropped from notice. We only wanted to prove,
say, that x is z ; to insist upon it that x is yz, though very
true, may be a needless trouble. The very words Discourse
and Discursive reasoning seem to point to this. We let the
mind run from one thing to another ; and we only dwell
finally upon, and put into our conclusion, the particular fact
or facts which we happen to need. We distinctly want to get
rid of the middle terms, and not to carry all our knowledge
about in our predicate. It is of the essence, on the other
hand, of the Symbolic system, to keep prominently before us
every one of the classes represented by all our terms and
their contradictories. Accordingly the distinction between
yz and z, and the fact that the latter must generally speaking
be a broader and looser determination, is much more promi-
nently set before us here.
The fact is that here, as in various other directions, the
associations derived from the mathematical employment of
the term are apt to be somewhat misleading. For one thing
we are accustomed to believe that there must be some con-
nexion between the number of equations set before us, and
the number of terms involved in them, for the purpose of eli-
mination ; so that one term demands for its elimination two
equations, and so forth. In Logic on the other hand we know
that the number of statements into which we throw our data
is very much of our own choosing, a single logical equation
admitting of equally ready statement in the form of a group
of several. Accordingly the number of equations at our com-
XIIL] Elimination. 289
mand in no way affects the question of the possibility of elim-
ination here. Then again, as regards the loss of precision,
in mathematics it is rather the other way. If we have three
equations connecting x, y, z, each of these may be conceived
to represent a surface, which is satisfied by a doubly inde-
finite number of values of these variables. But if I eliminate
y and z, I obtain one or more determinate values of x corres-
ponding to the particular points where the surfaces intersect.
We have gained, in the process of elimination, an increase of
definiteness which must be estimated as one of kind rather
than of degree.
What we have now to do is to see how the logical process
which has been illustrated in one or two simple instances
can be generalized. As an easy example begin with the
w = xy + xz
and suppose we are asked to eliminate y from it. As an
equation its significance is plain enough. It is nothing else
than a definition or description of w in terms of x, y, and z.
Any one therefore who knows the meaning of these terms, or
the limits of the classes for which they stand, will have all
the information which they can furnish him as to the meaning
and limits of w. Assuming that we are confined to the use
of the three terms x, y, z, then w is as precisely determined
as circumstances permit.
This being so, what could be meant by 'eliminating' y
from the equation ? If we are not to retain it there, and are
not to introduce some new equivalent for it, the only remain-
ing course is to do as well as we can without it. But it
cannot be simply omitted ; for this would be inaccurate,
unless we took care to indicate somehow that we had dis-
pensed with it. Apparently, therefore, all that is left for us
to do is to take refuge in the vague, and to substitute for y,
290 Elimination. [CHAP.
wherever it occurs, some such word as 'some'. If we did this
we should just write the equation in the form
or w = $-x + xz ..................... (1).
This method of elimination has at least the merit of
frankness. It points out where we have let go some of the
determining elements, and it indicates exactly the nature and
amount of the oonsequent introduction of vagueness. There
is another equivalent form to this without the same merit of
straightforwardness, which consists in writing the altered
equation thus :
w = wx + xz.
The objections to this form have been already noticed. It
disguises the real vagueness under a show of information ;
and offers us an implicit equation involving w, for the explicit
description of w with which we started, and which we wish
as far as possible to retain.
Nothing could better show the nature of logical elimina-
tion than this simple example. The term y was one of the
elements employed in the determination of w; hence its
abandonment will necessarily entail some loss of precision.
If we were dealing with real equations of the mathematical
type such loss would generally be fatal to the value of our
conclusion. But what we have to do with in Logic is rather
the subdivision of classes by other class terms, and the iden-
tification of a group of individuals under different class
designations. Hence the letting slip of one of our class terms
will only refer us to a somewhat vaguer and wider class than
that with which we started. The relinquishment of y does
not destroy all our knowledge of.w, but it certainly destroys
a part of it.
If the only logical statements with which we were concerned
were of this simple type in which we have a term standing
xiii.] Elimination. 291
by itself on one side, with a description or definition of it on
the other, no other plan of elimination would be necessary.
But, as the reader knows, we have to encounter much more
complicated statements than this, viz. those in which every
term is implicitly involved. So we must look out for some
more general mode of elimination. We should best seek for
it in the alternative negative interpretation of our propositions.
We know that every logical equation can be, thrown into the
shape of a number of distinct and peremptory denials, without
any loss whatever or variation in its significance.
Take then, to begin with, the same statement as before
and look at it, for comparison, in the light of what it denies.
Adopting the plans described in former chapters, we find that
it may be thrown into the form of the five following denials ;
wxy\ wxy\ = Q
wxz f = 0. wxz)
(At least this is what we should get by resorting to the plan
of looking out for suitable factors, described in the last
chapter, and which is generally the simplest plan in practice.
As they thus stand they are not in their ultimate form, nor
are they perfectly independent ; but of this more presently.)
Now setting before us the same aim as in the preceding
example, viz. of determining w as well as we can without
making use of y, the course we should naturally adopt would
be this; we should just omit those amongst the above
denials which involve y or y, retaining only the remainder
which do not. If we did so. the result of the elimination
would stand thus ;
That this form is the precise equivalent of (1) is easily
seen. For multiplying both sides of (1) by xz and w, it
292 Elimination. [CHAP.
resolves into these denials and into no other unconditional
ones. Similarly on combining and developing (2) we have
w = xz + %x as in (1).
These two simple methods of elimination are therefore
precisely equivalent, at any rate in this instance. The only
difference between them is, that, whereas the original deter-
mination of w was complete and accurate, No. (1) retains as
far as possible the same form, marking plainly the position
and limits of the lacunae of information caused by the loss of
y, whilst No. (2) contents- itself by giving the materials for
this latter statement, leaving their solution to be worked
Take again such a case as the following :
w = xyz -f- sryz + xyz
in which w is declared to comprise 'those things which
possess two, and two only, of the attributes denoted respec-
tively by x, y, z '. Let it be required to eliminate y, a term
(be it observed) which enters into every element of w.
Form No. (1) would express the result off-hand 1 as
w = %xz + %xz -f %xz
or w = $x-r $xz (1)
a result in which we have had to depart a good way from the
original determination. All that we can substitute for that
determination without appeal to y, being, that 'w is contained
somewhere within the limits of a? and z' : that is, there is cer-
tainly no w outside that boundary.
The other form would have led to the denials involved in
the equation, and the selection of those amongst them which
1 In accordance with what has are tho same, so that 1$ is the same
been pointed out more than once we as J when used as a symbol of in-
must write I for y as well as for. dofinitencss. Each is entirely un-
y. For the limits of indefiniteness certain between and 1.
of any term and its contradictory
XIIL] Elimination. 293
do not make use of y. There is a very simple way of effect-
ing this in practice. Instead of using all the factors which
will disintegrate the given equation, and then selecting only
the elements we want, we had better only make use of those
factors which we see will produce these latter. Thus, here,
x and z would make xyz vanish but not the other two terms ;
x and ~z will make the second vanish, and x and ~z the third.
The only factors therefore that will make all the three
y-terms disappear will be xz. Hence the only elementary
denial which can be found without involving y will be
vixz =0 ........................... (2)
and this is the required elimination. As in the former case
it may readily be shown that these two results are precisely
equivalent and deductively interchangeable.
Before pointing out the practical or theoretical defects of
these methods it will be worth while to apply them to a group
of statements. As already insisted on, there is no distinction
of principle between the information conveyed by one, and by
a number of statements, but the striking difference in this
respect between the logical and the mathematical calculus
deserves emphatic notice.
Take the following :
(Every w is either x and y, or z and not x.
(Every w is either x, y, and z, or neither x nor z.
In symbols they stand thus
Let it be required to eliminate y from these two state-
The simplest plan would be to multiply the two together,
when we get w ^xyz. Substitute $ for y, and remember
that x = , (for no such multiplication can alter the range
294 Elimination. [CHAP.
of indefiniteness of the symbol) and we have w with the
w = $xz.
If we had begun by eliminating separately from each, we
should have had
= $(# + xz)
= %(xz + xz)
the combination of which would lead to the same result as
Or had we broken them up into their respective denials,
and added these together, we should have been led to the
following (omitting those which involve y)
wxz -+ wxz + wxz
which leads again to
w = $xz.
If all examples resembled the simple ones discussed above
we should need no other methods of elimination than those
just described. But the former method is only properly
available when we are dealing with equations of an explicit
kind; so that if our statements were not originally in that
shape we should have to reduce them to it. As regards the
second there is a cause of possible failure, unless we are on
our guard, which will deserve a little closer notice. It is
beautifully provided against in Boole's symbolic formula to
be explained in the next chapter.
The difficulty arises as follows. We have given directions
to break up the equation into its ultimate denials, and then
to select those amongst them which do not involve the term
to be eliminated. And, in the examples which we took, such
terms presented themselves at once. But it is easily seen
that none such may be found ; in fact, if we have developed
every element to the utmost extent, none such can be found,
for every term will then have been subdivided into its y and
xiii.] Elimination. 295
not-y parts. Thus in the example on p. 291 we had wxz = 0,
wxz = 0, which did not involve y, and we chose them accord-
ingly. But if these had presented themselves, as they might,
in the forms wxzy = 0, wxzy = 0, wxyz = 0, wxyz = 0, there
would have been apparently no terms free from y.
We have therefore to amend our rule. "We must say that
the complete results of the elimination of any term from a
given equation are obtained by breaking it up into a series of
independent denials, and then selecting from amoDgst these
all which either do not involve the term in question, or which
by grouping together can be made not to involve it. Of course
this latter enquiry may sometimes involve a little trouble,
when the elements in question are numerous, but generally
we can see our way through it easily enough. So understood,
the rule for elimination in Logic seems complete. I have
preferred to begin by discussing a rule the logical meaning of
which is clear at every step, but in the next chapter we shall
examine a rule of very remarkable symbolic neatness and
ingenuity, invented by Boole.
THE EXPRESSIONS /(I) AND /(0).
THE expressions /(I) and /(O) are presumably, to the bulk
of logicians, the most puzzling and deterrent of all the
various mathematical adaptations of which Boole has made
use in his system. They play far too prominent a part
however in that system to admit of neglect ; and indeed on
their own account they deserve careful study, as the effort to
detect the rational logical significance of such very abstract
symbolic generalizations as these seems to me one of the
most useful mental exercises which the subject can afford.
And that these peculiar expressions are really nothing more
than generalizations of very simple logical processes will soon
be made manifest.
Wo will examine these expressions under two heads;
firstly as mere class symbols, and secondly as representative
of logical equations. That is, we will begin by taking
them as derivatives of / (#), and secondly as derivatives of
1 The merely logical reader will essential to the comprehension of
not find the study of this chapter those which follow.
xiv.] The Expressions f (1) and /(O). 297
I. In order to put an interpretation upon /(I) and/(0)
we need only recall what the more general expression f (x},
of which these are merely specialized or altered forms, stands
for. In Logic, at any rate, we have always insisted upon
certain restrictions as to its form and significance. With us
it always stands for a class, actual or potential, directly or
inversely determined ; and wp recognize no form of it incon-
sistent with such interpretation. Moreover as a class expres-
sion, we may insist upon its possessing certain characteristics
which have often been described. These amount to the
condition that in whatever shape it might originally present
itself it must be capable of being arranged as the sum or
aggregation of a series of mutually exclusive terms, for thus
only will it strictly represent a logical class. If we had
ourselves written down this f(x) as the symbolic translation
of a verbal description, then we ought to have taken care so
to express it; whilst if it had been got at by logical processes,
such as development, from legitimate data, it must certainly
retain those characteristics. Accordingly we shall assume
thaty(^) is, or may be exhibited as, the sum of a series of
mutually exclusive logical class terms.
Take then, as an expression, fulfilling these conditions,
xz -f xy + wyz
and calling this f (#), examine the significance of /"(I) and
Symbolically, of course, the answer is prompt enough.
Write 1 for x all through, and we have / (1) = z + wyz ;
write for x all through, and we have f(0) = y + wyz.
But what we want is the logical interpretation. To obtain
this it is only necessary to remember that xz means l z re-
stricted by a;', and xy means 'y restricted by not-#'. But 1
means "the whole of", so that the substitution of 1 for # is
merely the direction to take the whole of z instead of merely
298 The Expressions /(I) and /(O). [CHAP.
the axpart of it 1 . Similarly meaning "none of", the substi-
tution of 1 for x, in xy, tells us to take no y instead of the
not-x part of it. Hence the exchange of /(I) for f(x) is
only the highly generalized symbolic direction : Go through
the given class expression ; and of every element of it which
is limited by x take the whole, and of every element limited
by not-# take none, and let the terms which do not involve x
or x remain unaltered.
This indicates the equally simple logical explanation of
/(O). The latter expression is in a way the exact parallel
of /(I); that is, whatever we there did with # we here do
with not-#, and vice versa. Hence what we are directed to
do is to secure that every term that involves x shall just
be dropped out, that every term which involves not-#
shall be taken in its full extent instead of under this
restriction, and that every term independent of x shall be
simply let alone.
In the above explanation we have implied that there may
be terms in our expression which do not involve either x or
not-tf. But this need not be so, and if the expression were
fully developed it could not be so, for every term would then
be divided respectively into its x and its not-# part. The
term wyz is so put for simplicity merely, it is really equiva-
lent to vyyz (x -f x). When therefore any logical class expres-
sion is completely broken up into its ultimate elements all
these will fall into two ranks, those of x and not-#. The
verbal statements of /(I) and /(O) then become simplified.
The former says, Take the ^-members unconditionally, and
1 The reader will remember the restriction from xz and we have x:
distinction between this mere taking but the inverse to xz, i.e. the class
off the restriction of x, by turning xz which will become xz when the con-
into z, and the true inverse operation dition of x is imposed upon it, is
to its imposition. Restrict x by z of course xz + %xz.
and we have simply xz : take off this x-
xiv.] The Expressions f (I) andf(0). 299
discard those which are not-# : the latter says, Take the
a;-members unconditionally, and discard those which are x.
We will now proceed to examine the relation of these
expressions, /(I) and/(0), to one another. It might perhaps
be hastily inferred that they are complementary to one
another, so that they should be mutually exclusive and
should together make up f(x); that is, that
/(I) +/(0) =/(*), and /(1)/(0) = 0.
Closer observation however will show that neither of these
results need be the case, and that the former certainly
will not; but it will be well to discuss these relations in
detail, as they have an important bearing upon the problem
Suppose then that f(x) takes the form A x + Bx, where
A and B are combinations involving y, z, and the other class
terms which enter into the given expression. We know that
f(x) must be representable thus, for when expanded fully it
can only yield x and not-x terms, and the factors of these
terms can only be composed of various combinations of the
other class terms. Then /(I) =A, and f(Q]=B; that is,
f(i) is the whole class which we have to restrict by x,
and /(O) the whole class which we have to restrict by
What then are the limits as to class extension of A and
B ? None necessarily, except that they must conform to the
fundamental law of logical classes, viz. that they cannot
either of them exceed unity. In the extreme limiting case
of A and B being both equal to 1, we should then have
taken the whole of x and the whole of not-ar, so that our
f(x) would have itself to equal 1. That is, when /(I) and
/(O) each equal 1, then/(#) = 1. (Thus let
300 The Expressions f (1} and /(O). [CHAP.
then /(I) = y -f y = 1, /(O) = y + y = 1.) Au intermediate
case is when A + B = 1, that is, when the aggregate of the
factors of x and x together make up the universe. In this
case, since /(I) +/(0) = !,/(!) and/(0) are the contradictory
opposites of one another. (Thus let
f(x) = x(yz + yz} + x(yz + yz} ;
then f(l}=yz + yz and / (0) = T/.Z + 1/2, so that
We gather then, that given f(x} as a correct class group,
f(l} and y*(0) may, as regards the extent of ground they
occupy, just make up between them the whole universe.
But they may also in one direction shrink both of them to
zero (in which case f(x) = 0), or in the other direction
extend till both are unity (when/(#) = 1). What they do in
this way depends entirely upon the aggregate extent of those
class groups which were limited respectively by x and x in
order to produce f (x).
Similarly as regards the product of these expressions, viz.
/(I) /(O). Its value depends upon the mutual relation of
these aggregate class factors of x and x in the subdivision of
f(x). In the intermediate case, when A +B= 1, of course
AB = 0. (Thus when/(;r) = a (yz + yz} + x (yz + yz), clearly
/ ( 1 ) f (0) = (yz + yz) (yz + yz)= 0.) More generally, if A and
B are entirely composed of mutually exclusive elements,
whether or not these make up the universe between them,
that is, if all those in A are exclusive of all those in B, then
AB, or /(I) /(O), must =0; i.e. /(I) and /(O) aro classes
which are mutually exclusive of one another.
We may sum up therefore by saying that, given f(x) as
a true logical expression for a class group, then /(I) and/(0)
will also represent class groups. As regards their mutual
relations to one another and to the original /(#), we may lay
xiv.] The Expressions f (1) and f '(0). 301
down the following conclusions (omitting various limiting
cases which the reader will readily work out with the help of
examples): Each of these expressions, /(I) and /(O), will
omit a portion of what was included in ./(#), but will also
contain a portion of what was not included in it. And as the
portions of f(x) which they thus omit will not be the same
portion (one being an x and the other a not-a; portion) it is
certain that between them they must at least cover the
whole of /(#), besides including something else. That is,
/(I) consists of the whole of A, and /(O) of the whole of B.
But /(#), or Ax + Bx, comprises only a part (the #-part) of
A, and only a part (the not-#-part) of B ; so that /(I) and
f(Q) must between them cover all f(x), and may cover any
part of it twice, besides covering once or twice any part of
what is not /(#). On the other hatrid it is equally possible
that /(I) and /(O) may be classes entirely exclusive of one
another, as in the example in the preceding paragraph.
II. We now turn to the far more important case of
logical equations. Nearly all that has been said above,
however, will hold good in this case, for, as the reader knows,