John Venn.

# Symbolic logic online

. (page 31 of 37)
Online LibraryJohn VennSymbolic logic → online text (page 31 of 37)
Font size
of contradictories of a certain kind, subject to certain laws,
and that it will therefore cover any set of contradictories
which fulfil the same requirements.

As we do not at present propose to admit anything that
can be called a structural change in our symbolic machine, by

1 In accordance with what is the Permanence of Equivalent
sometimes called the Principle of Forms.

xviii.] Class symbols as denoting propositions. 369

modifying the laws of operation 1 , all that we have to do is to
see how many of the tacit or customary restrictions upon the
nature of a 'class' can be laid aside without affecting the
validity of our symbolic operations. Now a moment's consi-
deration will show that such restrictions as those noticed
towards the commencement of this chapter are of no signifi-
cance whatever. Whether xy and xy have any attributes in
common ; whether they stand for indefinitely numerous
groups, for definite groups, or for individuals ; is indifferent.
Accordingly we may generalize without hesitation up to this
extent.

But a good deal more than this can be done by our dis-
carding any obligation on the part of the symbols to represent
material objects at all. Let #, for instance, denote the truth
of a given proposition and x its falsity. This assumption (as
Avill presently be shown in more detail) will fit in excellently
with all our requirements. We should then have to interpret
the expression scy 4- Hey as representing the alternative of the
truth of one, and one only, of the two propositions referred to
by x and y. Again, we might understand by x something
abstract, but still more complex, by regarding it as represent-
ing the validity of some whole train of argument. In this
case x stands for the invalidity of the same argument, that is,
for its failure to establish its conclusion, whether that conclu-
sion in itself be true or false. Combine these symbols with
others of a like significance, and we have at once expressions
for the combination of arguments sound and unsound. Thus
xyz might stand for the validity of the two arguments indi-
cated by x and z, and the invalidity of that indicated by y.

Again, we may set the symbols to stand for the trust-
worthiness of assigned witnesses. Thus if x stand for some
man's trustworthiness, x will stand for the contradiction of

1 This question will have to be Discussed in the next chapter.
V.L. 24

370 Class symbols as denoting propositions. [CHAP.

this, viz. for his untrustworthiness, and therefore indicates
that we have so far no grounds of opinion either for or against
the fact testified. A distinct case from this last is produced
l>y letting x stand for the truthfulness of the witness, that is,
for the fact that his statement is actually true. This leads to
some difference of interpretation throughout our use of the
pair of contradictories x and x. The antithesis now is not
merely between the fact of a thing being established and not
being established, but between its having actually happened
and not having happened. Hence combinations admissible
under the former interpretation may become inadmissible
under the latter. To say that the witness is untrue implies
that his statement is false ; to say that he is untrustworthy
leaves a way open to either the truth or falsehood of his
statement. Problems belonging to these latter classes play a
large part, under numerical treatment, in the Theory of
Probability 1 , but do not deserve any special notice in a
purely logical treatise.

It must not be assumed that the interpretations above
indicated represent the only ones available for us, or even the
only suitable kinds. It would be the height of temerity
to maintain that a symbolic apparatus will only do those
kinds of work for which we may have happened at present to
formal antithesis between ' is' and ' is not', and the perma-
nency of the same laws of operation, throughout the whole
field of available application, whatever its extent may be.
It must be clearly understood that none of these various
interpretations can be regarded as giving rise to anything
deserving the name of a distinct scheme or system. Symbo-
lically, the scheme is absolutely one and the same through-

1 A very good account of the distinction in question is to be found
meaning and consequences of the in De Morgan's FormaJ io<7/c (p. 191).

XVIII.] Class symbols as denoting propositions. 371

out. Indeed, if we abstract sufficiently, we can also say that
there is only one and the same interpretation throughout, for
in every case we deal with the various combinations in which
'presences' and 'absences' may be arranged. To the one
formal framework built up out of x and x and the like, cor-
responds the verbal framework which may be built up out of
Yes and No variously combined and applied. Symbolically
and verbally, alike, there is in each case a primary and
necessary scheme of possibilities, which is limited by the
accidental and material conditions afforded by the data.

Amongst all the various interpretations which are thus
opened out to us in the use of our symbols, there is one
which deserves particular notice, both from the generality of
its possible application and the special nomenclature to which
it leads. It is that in which the symbols stand for proposi-
tions, i.e. for their truth or falsehood. That this interpretation
could be imposed upon the symbolic processes did not escape
the penetration of Boole 1 . He seems to have regarded it
however as yielding a perfectly distinct set of meanings which,
so to say, happened to fit in with his method, by being subject
to the same set of laws. He devotes a large part of his
volume to their consideration, under the head of ' secondary

1 In a certain sense the employ- judgment 'all a is b' as a single

ment of single letters to denote whole element, we should put the letter a,

propositions has often been adopted say, to stand for it, and no. to stand

in Logic. Some writers have carried for its contradictory. Some of the

this out very systematically, with a deductions on this mode of notation

special notation to denote the con- will be familiar to readers of De

tradictory of any proposition. For Morgan and other modern symbolists:

instance, Haass (GrundrissderLogik, e.g. 'when /3 follows from o, and is

1793) has used Arabic letters to stand false, then a must be false too; when

for ordinary subjects and predicates /3 follows from a, y, 5... together, and

and Greek letters to stand for whole is false, then o, 7, 5... cannot all be

judgments or propositions. Thus, if true, and so on' (p. 126), .
we wished to represent the whole

242

372

Class symbols as denoting propositions. [CHAP.

propositions', and discusses the definitions and deduces the
laws of operation independently for them, as he did at the
outset for the more familiar application to classes of things.
This mode of treatment, combined with his rather far-fetched
and unnecessary doctrine (as it seems to me) that propositions
of this secondary kind were somehow interpretable in terms
of time, has probably contributed to make some writers for-
get how completely Boole had grasped the possibility of this
interpretation of his symbolic methods 1 . There is much in

1 This seems to me to be the case
as regards the scheme of Mr H.
Maccoll, as described in the Proc. of
the London Math. Society (Vols. ix.,
x.), and in Hind, No. xvi. After a
careful study of it, aided by a long
correspondence with the author, I
am unable to find much more in it
than the introduction of one more
scheme of notation to express cer-
tain modifications and simplifications
of a part of Boole's system: to which
however one exception should be
made in respect of a real improve-
ment in notation to be presently
noticed. That he should have worked
it out, as he assures us is the case,
in comparative ignorance of what had
been already done in this way, is, no
doubt, proof of considerable sagacity
and originality. (The general view
which he adopts as to the nature of
propositions in their symbolic treat-
ment is that which may be called the
implication view as contrasted with
the equation view. Some account
of this distinction it given in the
concluding chapter.)

On this particular point the case

seems to me to be as follows. Boole
gave two distinct interpretations to
his symbols ; in one of these x is re-
garded as a class of things, and in the
other as a proposition. Those who
have followed in his steps have,
selves to one only of these interpre-
tations (or rather to a part only of
one, for they have not admitted any
special signs for inverse operations).
Thus Mr Maccoll makes it a "cardi-
nal point" of distinction in his scheme
that "every single letter, as well as
every combination of letters, always
denotes a statement", and this he
somehow conceives to "necessitate
an essentially different treatment of
the whole subject". Prof. Jevons,
on the other hand, though not ex-
pressly defining his position here,
invariably confines his interpretation,
unless I am mistaken, to what may
be called the material class view of
the symbols.

My only complaint against Boole
on this point is that even he did not
generalize sufficiently. Doubtless he
directed attention to what are far

xviii.] Class symbols as denoting propositions. 373

his treatment of this part of the subject which invites criti-
cism, but as we are here concerned with Symbolic Logic
itself rather than with the particular opinions held by its
main originator, I pass these by wherever it is possible.

It may be well to go a little more into the details of this
interpretation, starting with the most general symbolic state-
ment where two terms are combined, viz. xy+xy+xy+xy=I.
As there are supposed to be two propositions involved
we may at once, for the sake of distinction and brevity,
call them the propositions x and y. "We might have described
the propositions as A and B, or as I and II, respectively, so
as to employ different symbols for the proposition itself and
for the fact of its being true ; but the two-fold application of
the same symbol cannot possibly lead to confusion in the pro-
cesses of working. We say then that the symbol x standing
for the truth of the one proposition, and y standing for that
of the other, the left side of the symbolic equation represents
the only four possibilities. Its four component elements
stand for the truth "of both propositions (xy}, the truth of #
and falsity of y (xy), the truth of y and falsity of x (xy), and
the falsity of both (xy).

What then is here the signification of unity, or 1 ? In the
general formulas as hitherto interpreted, unity stands for the
sum-total of all the individuals composing all the possible
classes, in other words for our universe. The best verbal
account perhaps, in the present application, is given by say-
ing that it stands for the sum-total of possibilities or possible
cases. The formula asserts that the four cases above enume-
rated, as to the truth and falsity of x and y, exhaust the
whole field of possibilities. Of course the actual limits of

the most important interpretations, is potentially an indefinitely wide
but I prefer to regard even these as field of application,
constituting only a portion of what

G7-A Class symbols as denoting propositions. [CHAP.

this field, that is, of this universe, have to be assigned on the
same principles as in the more familiar case. It may be that
the propositions are necessarily true or false, in which case
the universe is altogether unlimited. It may be that they
are true only under certain conditions, conditions of time, of
place, of circumstance, or what not, the universe of possi-
bilities is then limited by these considerations. The deter-
mination of this question really ranks as part of our data ; it
is one of the elements of information which we must assume
to be contributed to the solution of the problem ; for clearly
the general symbolic laws can no more throw light upon the
material limits within which they are to be held to apply,
than they can upon the special signification which it is pro-
posed to assign to the terms.

So with the sign for nothing, or 0. Just as it is elsewhere
interpreted 'no individual', so here it means 'no possibility*.
At least this seems the best verbal translation ; though, as
the symbolist has to reckon with. the grammarian, he is often
somewhat put to it in trying to express himself unambiguous-
ly and yet without reproach. We may say then that x
indicates that within our determinate field, that is, subject to
our determinate conditions, there is no possibility for the pro-
position x\ that that proposition is false. So x= 1 indicates
that there is nothing but such possibility, that that proposi-
tion is true. It will be seen therefore that the question
whether the proposition is, in common parlance, ' necessarily '
true is of very little importance. If we like to interpret the
symbol 1, in x 1, as meaning 'everywhere and always', then
x may be regarded presumably as a necessary proposition ;
if we interpret it as meaning ' at some specified or definite
time or place,' then x is merely empirically true.

The reader will be able without much difficulty to work
out this view into all its details. He must bear in mind that

XVIIL] Class symbols as denoting propositions. 375

this is not a process of original legislation, but simply one of
interpretation. We are not recurring to the business of
settling the general employment of our terms and operations;
this was done once for all at the outset ; but merely con^
sidering now what is the most suitable phraseology for trans*
lating and describing our results in one well-marked and
distinctive group of cases. We see at once, for instance, that
x + x = 1 should be interpreted as saying that either a propo-
sition or its contradictory must certainly be true, and that
xx = asserts that a proposition and its contradictory cannot
both be true. Again xy = 1 (implying, as it does, x = 1, y = 1)
necessarily leads to the truth of both propositions. But xy=0
only insures the denial of the particular compound xy ; that
is, it tells us that both propositions cannot be true, so that it
leaves three possibilities open ; viz. that either of the two, or
both together, may be false. We see this more plainly
by looking at the alternative or disjunctive side. From
1 = xy + xy + xy + xy blot out xy, and we have as remainder
1 = xy + xy + xy. That is, if we deny that both the proposi-
tions are true, we are reduced to the three alternatives, that
one at least, or both, must be false. Blot out xy and xy and
we have 1 =xy + xy. That is, if we can deny that ' x and y
are both true', and also that ' x is true and y false', we are
led to the conclusion that either x is false and y true, or else
that x is false and y false. In other words x must be false,
and'y doubtful, so far as this information goes.

In proposing 'All x is y' and 'If a; is true y is true',
respectively, as the most natural pair of interpretations of the
formula xy = Q (or its equivalents x = xy, x = \$ y} according
as we submit it to the class explanation or the proposition
explanation, we know that we have to face a good deal of ad-
verse association. The two interpretations meet the shock
at somewhat different points ; the former when it maintains

376 Class symbols as denoting propositions. [CHAP.

that 'All x is y' covers the case of there being no x or no y,
and the latter in maintaining that 'If cc is time y is true'
covers the case of y having no connexion whatever with x
provided only y is known to be true. That is, the former has
a difficulty in carrying the extreme values x = 0, y = 0, and
the latter in carrying the extreme value y = 1.

The general propriety of admitting, or rather the necessity
of insisting upon the admission of, both these extreme values,
has been fully discussed already (see chh. VI., XVI.) and needs
no further justification. But the language in which they are
sometimes expressed, upon the special propositional interpre-
tation now under review, needs notice, since it seems to me
to involve a needless violence to popular association. We
have seen that we can stretch the language of hypothesis in
the phrase ' If x then y ' to reach the case of y being certain :
I think it will just reach this. But is it not going even
further to try to do the same with the word implication ?
The phrase 'x implies y 1 surely 'implies' that the facts con-
cerned are known to be connected, or that the one proposition
is formally inferrible from the other.

It is of course solely to the word 'implication' that I am
objecting, the results themselves being a matter of symbolic
necessity. As regards the particular notation employed by
Mr Maccoll (the writer here principally referred 'to) one ad-
vantage must be admitted, though I think that it is far out-
weighed by the general advantages of the system mainly
introduced by Boole. He expresses 'x implies y' by a; :y, and
employs an accent to mark negation, so that x : y means ' a?
implies the contradiction of y\ Under the peculiar interpre-
tation of 'implication' just adverted to, the formula x :y be-
comes the precise equivalent of iry = (and its substitutes
x = xy, x = \$y); for the meaning of any one of these formula)
is exhausted when we have said that it excludes x not-y, and

xviii.] Class symbols as denoting propositions. 377

consequently admits the three remaining alternatives. The
one merit of Mr Maccoll's notation is, I think, in the sym-
metry \vith which it expresses compound implications ; thus
(x:y]-.(z: w) would be read off ' the fact that x implies y
implies the fact that z implies w', with of course the same
peculiar interpretation of 'implication' all through 1 . It
would be a mistake to suppose that the expression of com-
pound implications of this kind is in any way restricted to
this style of notation : as has been already shown, our
expression of the same relation between the symbols in
question is (1 xy} zw = 0. In this case the implicational
mode of expression certainly tells its tale more simply and
obviously ; but this is not always the case. For instance, the
condition (x : y) (y : z) : (x : z] is read off ' the combination of
the implication of y by x, and z by y, leads to the implication
of z by x\ On our plan of expression this would stand
xz = xy . z + as . yz. This expression is intuitively obvious
(remembering that y + y = l) and shows the dependence of
xz=Q upon xy = and yz = Q better, to my thinking, than
the other rendering does.

It was intimated at the outset of this chapter that every
statement could be thrown into the form of the truth of some
corresponding proposition. Whether it will be convenient or
not to adopt this plan depends upon circumstances. Some-
times it seems to be a matter of pure indifference, amounting

1 The earliest attempt at the ex- (Versucli einer neuen Logik, p. 69),

pression of compound hypothetical but he does not work this out any

propositions, in this kind of way, further. The first to attempt it

that I have seen, is by Maimon: systematically was probably K. Grass-

"Hypothetische Satze konnen durch mann (Begriffslehre}, but there are

das algebraische Yerhaltnisszeichen decided defects in his procedure. The

(:) angedeutet werden. " Thus, with neat and effective rendering of a

his employment of + to mark affirma- number of compound implications of

tion, the proposition 'If a is b then this description is the best feature iu

c is d ' is represented by a + b : c + d Mr Maccoll's papers.

378 Class symbols as denoting propositions. [CHAP.

at most to a trifling verbal change, sometimes such a render-
ing represents a distinct economy of language and labour,
sometimes the reverse. As an instance of the class of cases
in which either mode of interpretation does equally well, we
might take the following :

It will either blow a gale to-morrow or the mail will start

He will go if the mail starts.

/. If he does not go it will blow a gale. i

This may be written out symbolically, thus,

.'. xz = or = x.

(This may be worked out as follows : substitute the
value of y, viz. y = \$z, obtained by the second equation, in
the first, and we have x + % xz = 1, or x = \$ xz, .'. xz = Q.)

Here it surely represents such a trifling difference as
hardly to be worth notice in a system of Logic, whether we
say, Let x represent the blowing of a gale, and y the starting of
the mail ; or say, Let x represent the proposition that it will
blow a gale and y the corresponding proposition about the
starting of the mail.

Now consider a case in which the proposition explanation
seems decidedly less appropriate. Take the familiar syllo-
gism ;

No B is C,

All A is B,

:. No A is C.

The most natural course here, upon the prepositional ex-
planation now under discussion, would be to put a single
symbol for each of these propositions. Let them be repre-
sented respectively by a, y, z ; so that x = 1 is to assert the
truth of ' No B is C', y=l that of ' All A is B\ and z = 1
that of ' No A is C\ It is clear that we are at once brought
to a complete standstill symbolically, for nothing having been

xviii.] Class symbols as denoting propositions. 379

assigned by way of connexion between x, y, and z, no conclu-
sion about one of them can be elicited from the other two.
If from x = and y = we could infer directly that ~z = 0, we
should have produced a great simplification ; but clearly the
only way of deducing this is to go through the syllogistic pro-
cess. If we did this we should be found to be taking a rather
roundabout course by introducing any special symbols for the
entire propositions. The simplest plan would of course be to
write our propositions down fully in the ordinary way, putting
x, y, and z, for A, B, C themselves directly instead of for pro-
= 0,

= Q, ..xz = Q-yz Q.
For these represent the real relations between the proposi-
tions, and they must therefore be introduced somehow if any
conclusion has to be drawn.

Accordingly those who adhere to the uniform prepositional
rendering of our symbols have to state the syllogism some-
what as follows ; Let x stand for the statement, .made in
reference to any object, 'it is A', y for 'it is B\ z for 'it is C\
The premise ' No B is C' then takes the form yz = Q or one
of its equivalents ; ' All A is J3' becomes xy = 0, and so on.
In fact this rendering simply gives us the ordinary symbolic
statement over again, with a slightly different and, as it seems
to me, more cumbrous interpretation attached to it.

The cases in which this particular interpretation of our
symbols is distinctly the most appropriate, seem to me to be
almost confined to questions of Probability, and therefore I
pass them by without special illustration. In the works of
De Morgan and Boole a variety of examples will be found
which afford convenient exemplifications of the modes of
combining statements, arguments, and reports of witnesses.
It was indeed for the purpose of improving the Calculus of
Probabilities that, Book mainly worked out his system, ... .^vi

CHAPTER XIX.

VARIATIONS IN SYMBOLIC PROCEDURE.

IN the system of Symbolic Logic, as we have throughout
expounded it, nothing amounting to what may be called
structural variation has been admitted into our laws of
procedure. Two chapters indeed have been largely devoted to
explaining certain extensions of the interpretation of our
terms, but the laws which govern the combination and
arrangement of the terms have been preserved unchanged.
But having done this it will be well now to direct attention
to some other ways of regarding and expounding the subject,
in which changes of the latter kind are introduced. These
grow out of three different causes which will have to be dis-
cussed separately : (1) the plan of writing alternatives in the
non-exclusive fashion, (2) the attempt to interpret our terms
intensively instead of extensively, and (3) the attempt to
extend our rules so as to embrace what is called the Logic of
Relatives; that is, of relative terms as distinguished from
mere class terms.

I. As regards the first of these departures from our sys-
tem something was said in the second chapter. So to clear

CHAP. XIX.] Variations in symbolic procedure. 3SX

the ground from confusion I will merely remind the reader
briefly that we have nothing whatever now to do with

Online LibraryJohn VennSymbolic logic → online text (page 31 of 37)