John Venn.

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than the former was. The complaint here would presumably
be, not that we were misinterpreting statements, but that we
were displaying a pedantic and lawyer-like 1 determination to
insist upon an interpretation of a statement which was
absurdly put.
They are both principles however which
are certainly legitimate. No one can say that they actually
contradict or defy any known law of thought, or any express
enactment of Logic (the latter of them indeed errs, if at all,
from excessive adherence to the law). Some assumption or
understanding therefore is necessary in reference to both, and
it will be found, I think, that no other than that here
adopted will prove capable of working in a really general
symbolic system. Anyhow we shall make use of them both,
and since it is well to do nothing unusual without warning,
attention is here prominently directed to them.

But it will be readily seen that the principle we have
thus invoked is of wider application. It is not merely true
that the statement xy = 0, when x and y are exclusives,
leads to x = 0, y = 0. The same holds good when such
mutual exclusives are combined by way either of addition or

1 Not I presume that the lawyers premium was under five pounds ster-

would so determine such a question. ling: the Court of King's Bench

De Morgan (Syllabus, p. 12) gives held that the exemption did not

the following example on a somewhat" apply when there was no premium at

analogous point: "An Act was once all, because 'no premium at all' is

passed exempting indentures [of ap- not 'a premium under five pounds.' "
prenticeship] from duty when the

V. L. 15

226 Logical statements or equations. [CHAP.

subtraction, and when each of them i-s multiplied by any
numerical factors whatever. That is to say, such an expres-
sion as ax by cz &c. = 0, (so long as x, y, z, &c. are
independent class terms) necessarily leads to x = 0, y 0, z = 0.
This follows from the very meaning and employment of our
class terms. Really exclusive or independent classes are
absolutely powerless upon one another ; it is not possible for
one to neutralize another and thus to offer an alternative
for the common extinction of all. We must not be misled by
the analogy of ordinary mathematics, where we add and
subtract magnitudes, and where in consequence there are
many different ways of adjusting the contributions of the
separate items in a total, whether that total be zero or
anything else. If any analogy is sought in that direction it
must be in the equation to zero of the sum of a number of
squares, which involves the separate equation to zero of each
element of the total 1 . In Logic, if any kind of aggregate, of
the addition and subtraction kind, is declared equal to
nothing, the only way out of the difficulty is in the declara-
tion that each separate element is also nothing : provided, as
remarked, that these elements are mutually exclusive 8 .

The reader must carefully observe that this is no case of
arbitrary assumption or definition on our part. All that
we are now doing is to insist upon a stringent interpretation

1 A still more apposite analogy neons or disparate things in t , j, k.
is offered here, as indeed in some 2 The addition (when equated to
other directions, in the Science of zero) of a number of class terms,
Quaternions. The single equation it will be observed, would also lead
xi + yj + z k = { i + rjj '< + f /c does not there to the same result whether they were
lead to the indeterminateness of an exclusive or not. But this is a mat-
ordinary algebraic equation, but no- tor of procedure merely, for we could
cessitates separately ; at once put them into such a form :

*=, y=ij, = e.g. x + y = is equivalent to
And the reason is the same, viz. 2xy + xy+xy=Q.

that we are comparing heteroge-

x.] Logical statements or equations. 227

of the definitions and rules formerly laid down. Indeed we
may say more than this ; that we are only insisting upon such
interpretation as must be admitted to be obligatory according
to the forms and regulations of common speech, though
it would naturally be complained of there as pedantic and
over fastidious. If one were to say that on adding the
Englishmen in an assembly to twice the Frenchmen, and
subducting the Dutch three times over, there was nobody
left, could any other conclusion be reached than that there
were neither Englishmen, Frenchmen nor Dutch there ?
The original statement was expressed in a very absurd
manner, and we should probably be charged with over-
refining in attempting to make anything whatever out
is to be interpreted logically, this, and nothing else, is
the conclusion. (It is merely a concrete instance of
x + 2y 3z = 0, where x, y, and z are known to be exclusives :
this leads to x = 0, y = 0, z 0.)

It will now be seen therefore, that in order to analyze
a logical statement, and to extract from it the sum-total
of what it has to tell us, all that is necessary is to break
it up into a series of mutually exclusive terms the sum
of which is declared equal to zero. If this can be done
without injury or loss of significance, then the information
yielded by the statement can be read off at once in all
its details, in the form of a number of separate denials.
This, it may be pointed out, is the full analytical process ; the
subsequent synthetic process, which seeks to build up these
details into new forms and thus fully to interpret them,
will have to be discussed in a future chapter. The full
importance, from a theoretic point of view, of the Kule
of Development explained in the last chapter will there-
fore be apparent. The desired result, viz. of securing that

152

228 Logical statements or equations. [CHAP.

some complicated expression shall be broken up into all
its ultimate and consequently mutually exclusive elements, is
precisely that for which this Poile is devised.

We might begin then at once by appealing to this Rule,
and this would be the most complete and perfect form of
solution. But an intelligent appreciation of the nature and
significance of the step we have to take will better be
secured by beginning somewhat more in detail, and with
more direct appeal to familiar logical considerations.

I. Take then the simple case of an explicit statement,
by which is here meant one in which we have only a single
term standing by itself on one side, this being equated,
on the other side, to a group of terms. We must suppose
that the component elements of the group which constitutes
one side are arranged in their due and suitable form, in
which they appear as a sum or aggregate of a number of
mutually exclusive terms. In fact this is demanded, for
there is no meaning in declaring a single class to be
identical with a group wherein some of the components
are counted several times over.

Take as a specimen,

(1).

When this is put into words what it amounts to is simply
that we here have a description, definition, or synonyme
of any kind (neglecting, as we do, all but the denotative
import of our terms, these various expressions are regarded
by us as equivalent) of w in terms of x, y, and z. The
individuals referred to by w are identical with the aggregate
of the individuals referred to by the three terms equated
to w. Of course as regards the expression or extension of our
knowledge by such a statement, various views may be taken.
If all the three terms on the right hand are known in

X.] Logical statements or equations. 229

themselves, or through their component elements, then w
may be known thereby. If w were known, then we have
one condition assigned by which to determine the other
elements. If we happen to be equally familiar with both,
then the equation may be regarded as a statement of
our knowledge. Our particular personal relation to any of
our statements is not indeed a matter with which Logic need
be supposed to trouble itself.

As regards the verbal statement of such a proposition the
reader will observe how entirely it is a matter of our own
choice whether we throw it into the categorical, hypothetical
or disjunctive form. There may be no such things really as
w, xyz, xy, &c.; or some only of these classes may be missing.
If we want to avoid any reference to such a contingency,
as common logic mostly does, then we should put it either
collectively by saying, 'the w's are identical with the sum of
the things which are xyz, or y and not-#, or x and neither y
nor z\ or distributively 'every w is either xyz, y and not-#, or
x and neither y nor z\ If we wish to call attention to the fact
that our terms only hold their life, so to say, subject to the
conditions entailed by other propositions as well, then we
might say 'if there be any w it must be either xyz, &c.' The
popular expressions which thus cover the ground of a single
symbolic statement are very various.

So much for the affirmative interpretations of our equa-
tion or proposition; let us now look at the negative interpre-
tation. We begin, as usual, with examining the question on
purely logical grounds, before looking at it through the
medium of mere symbols. What the statement said was
that the whole of w was confined to a certain number of
compartments, and conversely , that every class of things
occupying any of these compartments must be a w. Now, on
the ' system of making a perfectly exhaustive scheme of

230 Logical statements or equations. [CHAP.

classification out of our class terms, it is clear that to say w
is within certain compartments is precisely equivalent to
saying that it is without certain others. Hence it follows
that an alternative or disjunctive affirmative can be broken
up into a number of independent negatives. This of course is
in no way peculiar to our system; for every one knows that
the proposition 'All X is either Y or Z' may be phrased
'no X is (neither Y nor Z) '. What is characteristic of this
symbolic Logic is the symmetry and generality with which
this procedure is carried out all through.

Now look at this symbolically. What we have to do is to
break up the given statement into a series of separate state-
ments each expressing that such and such a combination = 0.
There are a variety of ways of doing this. Perhaps one of
the most methodical is the following : Suppose we had the
very simple statement x = y we should (as was formerly
pointed out) throw this into the two negations x (1 y) = 0,
y (1 x) = 0. Now the same holds true for aggregates of
class terms as well as for simple ones, for such aggregation
does not destroy or in any way affect their class character-
istics. Hence the equation w xyz + xy + xyz, may be
fully expressed negatively by the two

w (xyz + xy + xyz) =
w (1 xyz xy xyz) =

The upper of these is already in the form of a sum of ne-
gations, and therefore breaks up at once into three separate
negations. The lower as it stands has one positive and three
negative terms, but may be easily put into a form composed
of positive terms only. For the three negative terms are
collectively a part of 'all', or unity; hence if unity be de-
veloped in respect of x, y, z, some of the eight elements
thus resulting will be neutralized by these three negative

x.] Logical statements or equations. 231

terms, and none but positive ones will remain. The result
is then

w (xyz + xyz + xyz + xyz) = 0. ,..

When this is added to the similar result in the former
equation (those three, by the resolution of xy into xyz, xyz,
yield four ultimate terms) we have the eight following ele-
mentary negations:

wxyz = wxyz =

wxyz = wxyz =

wxyz = wxyz =

wayz = Q wxyz = Q.

I shall go more fully hereafter into the explanation of
these eight denials, that is, into the various verbal forms in
which their joint force may be expressed. But the reader
should understand at once that these eight denials contain
amongst them every particle of information yielded by the
original statement, or in any way deducible from it.

II. Now consider the case of implicit equations, by
which we understand those in which we do not find a single
term standing by itself on one side, and declared equal to a
certain group of terms on the other side. The following will
serve as a simple illustration of what is meant,

xy + xz = xy + yz.

A very little observation will show that the two classes in
question can only be made equal to one another upon the
conditions expressed by xyz = 0, xyz = 0. On any other
supposition it would be equivalent to the declaration of the
identity of the money income with the acreage or family.
This is more obvious on subdividing the terms on each side,
when they stand

xyz + xyz + xyz + xyz = xyz + xyz + xyz + xyz
or xyz = xyz, which of course, as already described, demands

232 Logical statements or equations. [CHAP.

the destruction of each of these two classes. Hence the full
interpretation of the given statement, in so far as analysis is
concerned, is given by these two elements,

xyz = 01

xy~z OJ

III. In the two classes of cases hitherto discussed, all the
terms entered definitely into the equations or propositions
which involved them. We must now discuss the case in
which one or more of the terms are affected by the indefinite
sign \$ . Begin with the simplest of cases, in which a definite
class is equated to an indefinite one, for instance

This form was examined in Chap. vn. We showed that
it is, with reserves and explanations, the best accurate
symbolic equivalent for the somewhat ambiguous 'all w is xy" 1
of ordinary Logic, viz. for a form of the ordinary Universal
Affirmative. We also considered its negative form in the
same chapter, but we must now compare it somewhat more
fully with the definite statements above considered. In
those cases what we did was to exactly identify one class
group with another, which gave rise to two negations : for we
could deny of each group that it had any members outside
the other. In the present case we merely say more vaguely
that one group is at most contained somewhere within
the other, which only gives rise to the single denial that the
definite group has members outside the extreme limits of the
indefinite one. Hence all that can be elicited from such a
form as the one now before us is,

w(\ xy} = 0.

For ay is the extreme limit of the indefinite class \$xy,
when \$ becomes = 1. Hence we can assert unconditionally no
more than that there can be no w which lies outside xy. Of
course, if we like to do so, we can break this up into the

x.] Logical statements or equations. 233

constituent members of which 1 xy is composed, viz.
xy + xy + xy. Then the equation resolves itself into
the three denials given by

wxy = ]

wxy = I .

ivxy = ]

Now take a somewhat less simple example involving
indefinite terms. Suppose we have

w = xyz + xy + ^xyz.

First as regards its significance. This is not, like the
examples we began with, the identification of two class groups
with one another, for the term xyz comes in to prevent such
identification. It cannot therefore be regarded as expressive
of a definition or description of w. What exactly it tells us
is this : That w certainly comprises the whole of the two
classes xyz + xy and that it may or may not take in the
class xyz. The indeterminate sign therefore is a sort of
"look out" to us to be prepared for individuals from the class
so affected. The whole of the class may be wanted ; or, if it
be subdivided, a part of it only, or possibly none at all may
be wanted. This is left altogether indeterminate.

It is clear therefore that we cannot here make quite such
a simple double negation as we did in the former cases.
What we have to do instead is to take account of one limit of
\$ (viz. 1) in one denial, and of the other limit (viz. 0) in
the other denial.

We may say with certainty that there is no w which lies
outside xyz + xy + xyz, for this represents the extreme limit
of the admissible indefiniteness ; and we may say with
similar certainty that there is no xyz 4- xy which lies outside
w, for this represents the extreme limit of the indefiniteness
in the opposite direction. These two statements yield us
a pair of negations which do not quite so accurately balance

234 Logical statements or equations. [CHAP.

one another as was the case when we were concerned with
definite terms only, for one is less extensive than the
other. Put into symbols they stand,

w (1 xyz xy xyz) = 0)
w (xyz + xy) = 0.
We might then proceed, by expanding 1 in terms of x, y, and z,
to convert the former into positive terms only, and should
thus be finally left with a string of separate negations as
in the former cases ; the only distinction being that owing to
the occurrence of the term we get fewer of these uncon-
ditional negations than we should otherwise obtain, and
therefore our materials of information are less abundant.

The difference thus marked in the symbols is equally
noticeable in the verbal expression : that is, as regards our
powers of conversion and contraposition where these indeter-
minate terms occur. What we ought to say is on the one
hand, 'All w is made up of xyz, xy, and (possibly) xyz\ and
on the other, 'All xyz and all xy are w\ What we may
be tempted to say, however, is 'All w is made up of xyz, xy,
and some xyz\ thus omitting the full indefiniteness of \$xyz,
or introducing confusion and ambiguity by this word some.

Inasmuch as it is always well to examine limiting cases,
since principles so often lurk concealed in such holes and
corners, it will be well to see what these superior and
inferior limits of negation become when we have none
but indefinite terms on one side. Recur to the example

The limit of \$ in one direction is 1 ; thus giving us the
negation w(l xy) = 0, viz. that 'No w can lie outside xy\
But the limit on the other side is 0. In this case the whole
of the right side of the equation vanishes, and we can make
no denial by means of this inferior limit ; or rather, in formal
strictness, such denial assumes the form 'No Oxy lies outside w\

x.] Logical statements or equations. 235

which tells us nothing whatever. "We are thus reminded
again of the distinction between this really indefinite factor ,
and both the some, and the undistributed predicate, of ordinary
logic. These latter exclude the value 0, therefore we can
always make something out of the statement 'All w is xy' in
both directions. In one direction the result agrees with our
symbolic expression w (1 xy} = Q, viz. 'No w is not-ory'. In
the other it is generally stated positively, in the form ' Some
(i. e. not none) xy is w\ The validity of such conversion has
been already discussed in a former chapter.

We will now look at a more general method for ex-
amining the significance of any logical statement. Here,
as in every other case, I am more concerned to discuss
the question in a way calculated to throw light upon the
actual logical meaning of the processes we perform, than
merely to bring forward convenient or powerful symbolic
devices for reaching our conclusions.

Suppose then that we have any logical equation whatever
involving the class terms x, y, z, &c. anyhow combined. It is
assumed that what we want to do is to examine the full
significance of this equation, that is, to resolve it into all the
elementary denials which can be extracted from it.

This process of resolution of an equation into its elemen-
tary denials is, as already remarked, a work of Analysis.
What can be done afterwards from these results by way
of Synthesis, that is, by putting them together into the form
of affirmative assertions, will have to be considered further on.

Perform the following processes upon the equation :

1. Bring all the terms ever to one side, so as to reduce
it to the form /(#, y, z,...}= 0.

2. Develop every one of these terms into all the sub-
divisions attainable by taking all these class terms into account.

3. Equate separately to zero every class term which

236 Logical statements or equations. [CHAP.

finally remains after the development ; omitting any that
may be affected by the symbol \$, for of these nothing can

(1) With regard to the first process hardly anything
need be said. It is best to adopt it in order to prevent
confusion by the same terms appearing on both sides of the
equation.

(2) One or two points seem to deserve notice in respect
to this second step. For one thing it must be observed that
none but combinations of xyz... can possibly occur in our
result. That is, every term which could occur in the original
equation will resolve itself into such combinations. For
instance no numerical factor can appear there. If our
original equation had been xy + ~xy = \ we should write
it xy + xy 1 = 0. Now since unity is a logical term, like
any other, it must equally undergo expansion into the elements
xy + xy -f xy + xy. Hence. the final result of the equation is
xy + xy = 0. If our equations or statements had been
pi'operly drawn out at first, (and we must in fairness assume
so much as this), they could contain nothing but 1 and other
class terms. Consequently, after development, none but
logical class terms can be found in them.

But though nothing else than these logical class terms
can be found in our equation after it has thus undergone
development, it does not follow that all our possible class
terms will be represented in it. On the contrary one or more
terms must be missing from it, as otherwise we should be
landed in a direct contradiction in terms.

(3) The reason why some of the possible terms must
thus disappear is connected with the grounds of this third
the terms, as finally arranged, being mutually exclusive, it is
impossible for any one of them to cancel another. Conse-

x.] Logical statements or equations. 237

quently when a group of them is equated to zero this
can only be brought about by each separate term being equal
to zero, just as when in algebra we get the sum of a number
of squares = 0. But since our alternatives are collectively
exhaustive as well as mutually exclusive, it is a contradiction
in terms to suppose them all to vanish: this, it will be
noticed, being our generalized form corresponding to the
so-called Law of Excluded Middle.

Suppose for instance, just for illustration, that we write
down such a form as this,

Axy + Bxy + Cxy + Dxy =* 0,

one or more of the four factors A, B, C, D, must be supposed
= 0, in order to avoid contradiction. Suppose that B and C
thus vanish, whilst A and D do not. We then have

Axy + Dxy = 0.

Since it is impossible for these terms to neutralize one
another, and by supposition A and D do- not = 0, the other,
,or logical class terms, must vanish. That is we conclude

^/ = 0{

xy = Oj '

these ultimate denials containing, as before, the full
information yielded by the original statement.

It may be asked here, but what if all the four factors
A, B, C, D, above, had vanished: what cotdd we then
conclude ? The answer is that in that case nothing whatever
can be concluded. The vanishing of any term from our
equation is an indication that our data give us no information
about it, and the vanishing of every term is an indication
that no information whatever is obtainable. As every mathe-
matician knows, this is the usual resource of an equation
under the circumstances. When there is no information
yielded by the data, they will not unfrequently save them-
selves from misstatement by just reducing to the unmeaning

238 Logical statements or equations. [CHAP.

form = 0. Of course we could not begin by writing down
such a form as this, in which all the terms should vanish ; but
it might quite well happen (as we shall see when we come to
the study of Elimination) that we should deduce such an
equation as an inference. We must then take its collapse
and disappearance as an intimation that we were trying
to extort from it information which it was not in its power to
give us.

One case still remains for notice. We have so far sup-
posed that the factors (A, B, C, D, &c.) of our class terms
were either or 1, that is that they either vanish entirely or
present themselves simply and singly. But there is, as we
know, another recognized class factor, viz. \$ ; what is to be
made of terms which happen to be thus affected ? The
answer is that we can make nothing of them when they
thus corne out as members of a series which is collectively
made = 0. For instance if we met with such an expression
asary-f \$cy = 0, the indefinite term must just be let alone,

Online LibraryJohn VennSymbolic logic → online text (page 21 of 37)