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term are extraneous to the subject.) the former, and meet the difficulty by
1 The alternative which every admitting a third value v for proposi-
symboliBt has to face here may be tions which are incurably particular,
concisely expressed thus : If he ad-

viz.] Symbolic expression of ordinary propositions. 171

(1) In the first column we declare that such and such a
compartment is empty, that is, that the corresponding class is
unrepresented in our universe. This grouping of such propo-
sitions of course differs considerably from the customary one.
The first of the four is always naturally couched in a negative
form in ordinary language and logic, whereas the second and
third generally appear as affirmatives. The fourth again has
no precise logical equivalent, the exactest popular equivalent
being ' There is nothing but what is either x or y\ I call
attention the more expressly to this point as it enforces the
opinion, laid down in the introductory chapter, that there can
be no absolute arrangement of prepositional forms. The
number and grouping of our forms must depend upon the
fundamental view we take as to what should be the import
of a proposition.

(2) In the second column we declare that such and such
a compartment is occupied, that is, that the class is repre-
sented. The only trouble here is in settling the degree of
indefiniteness to be assigned to v. Fortunately for us, in
such a notation as this, there are almost no acquired as-
sociations to be attended to, so we may define freely accord-
ing to our judgment. That being so, it would seem best to
lay it down that v shall be perfectly indefinite, except that it
excludes and 1. This of course makes the second column
intermediate between the first and the third. The exact
meaning of this form of proposition is that a portion, and a
portion only, of the things in our universe are found to
belong to the class in question ; the remainder of them being
distributed in some way, we know not how, amongst the
other three conceivable classes. The particular propositions
of ordinary logic are best assigned to this class ; for, though not
always precise in their signification, yet when they are made
precise they most naturally take up the meaning here assigned.

172 Symbolic expression of ordinary propositions. [CHAP.

(3) In the third column we declare that the compart-
ment is not only occupied, but occupied to the exclusion of
all else ; that that class, and that class only, is represented.
Here we make a complete departure from the familiar forms
of ordinary logic, no one of its recognized propositions coin-
ciding with anything in this column. Popular language
would express xy 1 in the words 'Everything is both x and y ',
whilst xy = 1 would stand as 'Everything is neither x nor y\
or 'Nothing is x, and nothing is y\

We might of course easily express all three columns in
the generalized form xy = w, provided we make w perfectly
general ; that is, regard it as a general term standing, as the
case may be, for any of the values 0, v, 1. We shall have
occasion to treat it so when we come to the consideration
of the Aristotelian syllogism in a future chapter.

It must be observed that every one of the above forms
readily gives rise to a corresponding alternative. This result
follows symbolically from the fundamental formula connecting
our class terms, xy + xy + xy + xy = 1 ; or, logically, from the
fact that every existing thing must belong to one or other of
the four classes thus indicated. In this case since the prepo-
sitional forms with which we are concerned are of the
simplest character, dealing with one sub-class only, the corre-
sponding alternative will be somewhat complex, for it will
have to deal with the remaining three classes.

There is no need to go in detail through all the applica-
tions of this principle, as the reader will find it easy enough
to work them out for himself, but it will be well just to take;.
one from each column as a sample. For instance xy Q
yields the alternative xy + xy + xy = I ) the two forms being
precisely equivalent and convertible. Put into words this
amounts to saying that it is exactly the same thing to assert
that ' No x is y', or that ' Everything is either x and not y,

VII.] Symbolic expression of ordinary propositions. 173

y and not as, or neither x nor y\ which of course it is. The
second column yields a somewhat different kind of alterna-
tive. From xy = v we deduce xy + xy + xy = 1 v. Now
1 v has exactly the same significance as v, for this was
defined to be indefinite between and 1 exclusively. Hence
the alternative form might equally be written

xy + xy + xy = v.

The meaning of this is easily assignable, for just as xy = v
meant 'There are such things as xy^so the longer alternative
means ' There are such things as either xy, xy, or xy'. It is
of course a very indefinite, and not very useful, form, for it
simply assures us that one or more of three classes, we know
not which, is represented.

There only remains the corresponding alternative in the
third column. This yields a slightly different result from either
of the foregoing ; for we thus obtain, not a disjunctive of a
positive kind, but a series of categorical negatives. That -is,
if xy = \, then xy + xy + xy must equal 0, a conclusion which
as will be shown hereafter necessitates the three separate
conclusions xy = 0, xy = 0, xy = 0. Logically this is clear
enough, for if ' Everything is xy', then certainly ' Nothing is
xy, xy or xy';'ih&t is, to crowd everything into one compart-
ment is to cause every other compartment to lie empty.

This will be a convenient place for noticing the charge
which has been repeatedly brought against Boole's system,
and presumably against every analogous system of Symbolic
Logic, -that it is forced to adopt the Hamiltonian doctrine of
the Quantification of the Predicate. Thus Mr Lindsay says,
"The doctrines contained in this New Analytic of logical
forms lead directly to the theories of Boole and Jevons. A
leading characteristic of the doctrine of the Quantification of
the Predicate, and other recent theories of a similar kind, is
the attempt to assimilate all propositions to the type of

174 Symbolic expression of ordinary propositions. [CHAP.

mathematical identities... 1 "; and Prof. Jevons goes further by
declaring that "Dr Boole, employing this fundamental idea
[of Quantification] as his starting-point," arrived at such and
such results 8 .

The assertion that Boole's system is in any way founded
on the doctrine of the Quantification of the Predicate, is, in
fact, not directly hostile to that doctrine, is so astonishing
that one is inclined to suspect some lurking confusion of
meaning. So I will just remark that what I understand by
the doctrine is this: Whereas the ordinary forms of proposi-
tion leave it uncertain whether we are speaking of the whole
predicate, or part only, in affirmation, and decide that we must
be speaking of the whole predicate in negation ; we thus
leave four possibilities unrecognized : that in fact we may think
the predicate either as a whole or as a part, and must think
it as one of the two, in both affirmation and negation alike.
Moreover, since what exists in thought should be expressed
in words, a really complete scheme of propositions demands,
and is satisfied by, eight forms. There is surely no doubt that
this is the sense in which Hamilton, and his authorized ex-
ponent Prof. Baynes, understood the doctrine.

Now though it seems hard upon ordinary predicates thus to
charge them with secretly quantifying, it may be brought
against them that at least they have nowhere denied that
they do so. But with Boole's system it is otherwise. If the
should unequivocally and even ostentatiously reject this
unfortunate doctrine, what better could be found than a; = \$y

1 Translation of Ueberweg's Logic, Boole himself expressly states that he

p. 668 : I am not sure to what ex- takes the four old forms of proposi-

tcnt Mr Lindsay is responsible for tion "with little variation from the

this part, as it is actually contributed Treatises of Aldrich and Whately."

by another writer, Prof. W .B. Smith. (Math. Anal, of Logic, p. 20.)

* Sulftitution of Similars, p. 4:

Vll.] Symbolic expression of ordinary propositions. 175

for such a purpose ? So far from quantifying the predicate,
by specifying whether we take some only or all of it, we
select a form which startles the ordinary logician by the un-
customary language in which it announces that it does not at
all mean to state whether some only, or all, or even none is to
be taken. The negative equivalent, acy = 0, is just as resolute
not to commit itself on this point; whilst, as I have pointed
out, as = xy is a precisely synonymous expression. It is diffi-
cult to conjecture how these symbolic forms could be thus
connected with Hamilton's doctrine, unless by a hasty con-
clusion from the fact that both systems adopt the equational
form.

We have thus discussed categoricals with sufficient ful-
ness, and have touched incidentally upon disjunctives. It
more about the latter kind of proposition. The reason is that
on our system they do not differ from categoricals in any re-
spect of serious importance. The only difference in them
is that their subjects or predicates, or both, are composed
of two or more class terms instead of one only. To the
common logic this may involve a real distinction, but on any
class view it is of no significance, since these groups composed
of two or more classes are really classes all the same. In
xy = 1 we declare that a single class constitutes the universe,
whilst in ocy -\- xy + xy = 1, we declare that three such classes
collectively constitute it ; but this difference does not essen-
tially alter the kind of proposition.

Suppose, for instance, we are given a; + y = a + b, (as and
y, as also a and b, being supposed exclusives). Categorically
described, this declares that the members embraced col-
lectively by the two former classes are identical with those
so embraced by the two latter. It then really forms only one
proposition. Or we might express it in two paiigof ordinary

176 Symbolic expression of ordinary propositions. [CHAP.

disjunctives by saying ' Every x is either a or b', and 'Every y
every a and b being either x or y. Or we might make a more
complicated proposition with disjunctive subject and predicate,
by saying that whatever is either x or y is either a or b, and
vice versa. Some licence of phraseology is allowed on every
system ; even the common universal affirmative may be read
as ' All x is y', or ' Every a; is y'; what we do is to insist that
this licence shall be rather wide. Provided the class facts
asserted remain unchanged, we claim to make almost any

The nature of the Hypothetical Proposition and the
desirability or otherwise of assigning it a special symbolic
form will be best reserved for a future chapter.

A few examples of propositions are added, in order to
illustrate the use of our symbolic expressions, as explained in
the course of this and preceding chapters. The reader will
observe that we purposely employ sometimes one, and some-
times another, of the various alternative forms of the Univer-
sal Affirmative which were noticed at the commencement of
this chapter.

1. ' Men who are honest and pious will never fail to be
respected though poor and illiterate, provided they be self-
supporting, but not if they are paupers'. As explained, the
various particles here used must all alike be replaced by the
mere symbols of connection, so that the proposition may
be phrased as follows: All honest (a), pious (6), poor (c),
illiterate (d), self-supporting (e), are respected (/) ; and no
honest, pious, poor, illiterate, self-supporting paupers (g) are
respected.

(abcde (1 -/) = >
(abcdefg = 0.

Of course when, as here, a whole group of terms presents

VII.] Symbolic expression of ordinary propositions. 177

itself which does not demand analysis into its details, we may
substitute a single letter for the group. Thus Ave might
replace abode by a single letter.

2. ' No a; can be both a and 6 ; and, of the two c and e,
every x must be one or other only'. This may be written in
one sentence,

x = x (1 db) (ce + ce),
or in two, separately,

(xab 0,

[ac (ce + ce) = 0.

In the one case we make a single Affirmative out of the
proposition; in the other we couch this in the form of its
two constituent elements of denial.

3. ' Every a is one only of the two as and y, except when
it is z or w ; in the former of which cases it is both x and y,
and in the latter case neither of them'. This may be ex-
pressed in three sentences :

a, that is neither z nor w, is x or y only, azw = \$ (xy + xy),
a, that is z, is both x and y, az = \$ xy,

a, that is w, is neither x nor y, aw = xy,

or, in one sentence, and without the express sign of indefini-
tude, a = a [zw (xy + xy) + zwxy + zwxy}.

4. As an example of translating symbols into words,
take the following :

a + a (1 ce}.

Here ce stands for what fails to be c and fails to be e, so that
I ce stands for all that does not so fail. Hence the given
expression may be read off, ' Anything which is a ; or even
not a, provided that in this case it does not fail both to be c
and to be e'.

An alternative symbolic statement here would be 1
v.L. 12

178 Symbolic expression of ordinary propositions. [CHAP.

(since a + a = 1). It might then be read off, ' All that does
not fail to be a, c and e '.

5. * Every member of the Committee (x) is a Protes-
tant (a), and either a Conservative (c), or Liberal (e} ; except
the Home Rulers (?/), who are none of the three'.

x = x {ya (c -t- ce) + yace}.

If we are supposed to know that Conservative and Liberal
are exclusives, we may put c + e for c + ce. The best way
perhaps of interpreting the symbolic sentence here is just to
substitute the significant words, when it would stand :
Every member of the Committee is a member of the
Committee who is not a Home Ruler, (and then he is a
Protestant, and either a Conservative or a Non-Conservative
Liberal), or he is one who is a Home Rulerj and then he is
neither Protestant, Conservative, nor Liberal.

6. xy + xy + z (xy + xy).

This may be read off, ' x or y only ; or, provided there be z,
both x and y or neither of them'. An alternative symbolic
statement would be 1 z (xy + xy), which might be read,
' All excepting what is not z, but is both or neither x and y\

7. As an illustration of the symbolic signification of
particular propositions we may take the following : ' Every
ab is either x or y only, and it is known that there are some
a which are x and some 6 which are not y'.

If the latter clauses were omitted, the sentence would be
written simply :

d> = \$ (*y + y).

This would merely obliterate the two classes abxy and dbxy,
leaving the remaining 14 classes perfectly indeterminate,
subject to the formal condition that one at least of them
must be represented. Now the statement ' Some a is a;', or,
' There are such things as ax', puts a check upon the destruc-

vii.] Symbolic expression of ordinary propositions. 179

tion of ax, insisting that some one at least of its four consti-
tuents (or rather three, since abxy is gone) shall be saved.
Bat it does not tell us which of them is thus rescued.
Similarly, ' There is b which is not y ' saves some one at least
of the three surviving elements of by. On the diagrammatic
scheme this would be carried out by our taking a note, so to
say, that the whole compartments ax and by were not to
be shaded out in any case.

122

CHAPTER VIII.

THE UNIVERSE OF DISCOURSE.

WE have had repeated occasion to refer to the Logical
Universe of Discourse in the foregoing chapters, but the
present will be the best opportunity for completing what it is
necessary to say upon this point. As in other parts of our
subject, there are three main topics of enquiry before us ;
for, in trying to rearrange things in accordance with the
principles of Symbolic Logic, we cannot afford to pass over
either the conclusions of unassisted common sense, or the
rules and assumptions of the logicians.

As regards then the popular way of thinking, the question
of course is this. When we make use of names and resort
to reasonings, what limits of reference, if any, do we make ?
What is the range of subject matter about which we con-
sider ourselves to be speaking ? I think we must answer that
as regards negative terms we always make very considerable
restrictions, and that as regards positive terms we only some-
times make them, and then comparatively slight ones 1 . The

1 True negative names of the popular speech, but are mainly an
type 'not X 1 are not very frequent in invention of the logician. Still they

CHAP, viii.] The universe of discourse. 181

limits of positive terms are generally settled very readily :
we all know what is in most cases meant to be included
under the name 'black'. But what does 'not-black 'include?
Does it apply to all things without exception to which the
colour black cannot be applied ; including, say, the Geological
Glacial Period, the sources of the Nile, the claims of the
Papacy, the last letter of Clarissa Harlowe, and the wishes
of our remote posterity ? Clearly not : some kind of limit,
more or less restricted, is generally understood to be drawn ;
but where exactly it may be traced must depend upon the
nature of the subject and the associations of the speaker.

This distinction between the application of positive and
that of negative names is in great part of a comparatively
verbal character. It is not because a name is negative that
we commonly have to refer to a part only of its denotation,
but because certain classes are tolerably definite and often
have to be referred to as a whole, that we confer a positive
name upon them, the heterogeneous multitude outside falling
to the share of the corresponding negative name. But of
course, when we have got this comparatively definite name,
it does not follow that we must in every case refer to the
whole of it, especially when it is itself a somewhat extensive

do occur sometimes both in subject is indisputably true. (We must re-

and predicate, when the classes indi- member to keep clear of the Quanti-

cated by them happen to be narrower, fication dispute here. In such a

or more conveniently assigned, than proposition as ' some contractions are

those indicated by the corresponding involuntary', the question now before

affirmative names: e.g. 'What is not us is not as to how much of the

conceivable is no fit subject of instruc- whole extent of the 'involuntary'

tion '. Of course if we class with is present in thought in the propo-

these, as I think we must, such sition, but rather as to what we are

names as 'inhuman', 'unnatural', to consider is that whole extent when

and so forth, what is here said about we come to reflect upon what has

the very great restriction with which been said.)
their extent is commonly interpreted,

182 The universe of discourse. [CHAP.

one. Hence we constantly make assertions about 'all men '
without the slightest intention of being bound by our words
beyond a reference to a comparatively small selection of
mankind.

The same general question is sometimes practically raised
in another form by enquiring whether we have any pairs of
terms in our language which are strictly contradictories.
That we have plenty of formal contradictories, such as good
and not-good, human and inhuman, &c., is obvious enough ;
but what is here sought for is rather a pair of material con-
current use and application. The reply must be, as above,
that language being relative to human wants every pair of
contradictories is restricted to some tolerably well understood
universe. Such restriction is commonly more constant in the
case of material than in the case of formal contradictories;
for each of the pair being a so-called positive, and a natural
instead of an artificial term, each carries its customary
limitation of signification with it : thus British and alien are
equivalent to British and not-British, provided we under-
stand that we are talking only of human beings. Not-
British being an artificial word its range of application may
be very variable, but legal and customary usage have decided
much more rigorously to what objects the word 'alien' shall
be rightly applied.

We must now notice briefly what the ordinary logician
has to say upon the matter. The use of this word universe
was first made familiar by De Morgan ', but the conception
itself is one that is suggested at more than one point in the
traditional treatment. For instance the doctrine of a sum-
mum genus is connected with the present enquiry ; involving,

1 Camb. Phil. Tr. vin. 880. He technical term ".
speaks there of "inventing a new

viii.] The universe of discourse. 183

as it does, the necessity of some restriction upon the extent
of the class which we take into account. But where the
need of some restriction of the kind seems mostly felt is in
the discussions about the nature of 'infinite' or indefinite
terms and propositions. I have no wish to enter into that
Serbonian bog further than one not brought up in those parts
may venture with safety, and will therefore merely refer to
the form in which this doctrine was held by a very eminent
thinker who was but little restrained by traditions of the
past. Students of Kant will remember the three-fold divi-
sion of propositions which he makes, in respect of their quan-
tity, into positive, negative, and infinite. Verbally, of course,
it is easy enough to say that we must either assert that A is
B, or deny that it is B, or (couching the latter in affirmative
form) assert that A is not-5 ; and we may readily admit that
there is some conventional difference of signification between
these various cases. But is there any difference whatever, of
which logic should take account, between the last two ? On
any rigid class view of the nature of predicates it is
impossible to extract more than two divisions ; for, that to
exclude a thing from a boundary is to include it somewhere
outside that boundary, that to deny that any thing has a
given attribute is to assert that it has not that attribute,
seems indubitably clear. I suppose that the idea underlying
the distinction is this. When we deny that A is B we think
of A as a whole, and B as an attribute and therefore as a
whole, so that the judgment is finite in both terms. But
when we say that A is not-5 and try to consider this not-^
as an attribute, we have forced upon our notice the vague
amplitude of its extent; and therefore, when we do not
make appeal to a limited universe, we must recognize that
the judgment is in respect of its predicate an infinite or
indefinite one.

184 The universe of discourse. [CHAP.

Whether I am right or wrong in this last remark it will
equally serve to call attention to the view which the Symbo-
lic logician is bound to adopt. Taking, as we do, a strict
class view of the nature of propositions we meet the difficulty
by flatly denying that the class not-JT need be more 'infinite',
or in any way more extensive even, than X. The notion that
it is so is simply a survival from the traditions of common
speech and is one of which the symbolist should rid himself
as speedily as possible. Not-X is of course always the
contradictory of X, but there is no reason to suppose that
the former symbol is more appropriately applied to classes
which are essentially negative or are popularly regarded as
such 1 . There may be practical reasons of convenience for
thus assigning our symbols, but as far as any reasons of
principle are concerned we might exchange X and not-X all
through our logical processes without the slightest change of
symbolic significance. There is nothing to hinder us from
putting not-JT to stand for the few and highly specialized
members of some narrow class, and X for the innumerable
and heterogeneous individuals which do not belong to it.

When thus regarded, the conception of a universe is seen
to be strictly speaking extra-logical ; it is entirely a question
of the application of our formulae, not of their symbolic
statement. It is quite true that we always do recognize a

Online LibraryJohn VennSymbolic logic → online text (page 17 of 37)