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# Symbolic logic

. (page 11 of 37)

constant. Interpretation is demanded at the primary step of
writing down our data, and at the final step of reading off
the answer, but in all the path between the start and the
finish we are not really obliged to think of interpreting our
formulae at all. Accordingly when the alternative is before
us of inventing new symbols, or only assigning some new

iv.] On the Choice of Symbolic Language. 93

meaning to old and familiar ones, experience and reason seem
decisively in favour of the latter plan. Certainly the experi-
ence of the mathematicians appears to tell in this direction,
which, ought to count for much. When they have to denote
a new conception or a new law of operation, of course they
may want a new symbol for it. But when the law of opera-
tion is the same, or even partially the same, they continue to
use the old symbol even though the signification may have
undergone a very considerable change. To take then one of
the simplest instances : which is easiest, to use the familiar
sign ( + ) as we have always been accustomed to use it,
bearing in mind, as we do so, that x + y does not mean
addition of a toy (an early prejudice which the mathematician
has long laid aside), or to invent a new symbol ( -I- ) where
we have to learn anew both the laws of operation and the
signification ?

Whilst then we shift the signification of the symbols
we retain their laws of operation as far as possible un-
changed. Indeed the only change we venture to make is
of the nature of special limitation rather than of actual
alteration. Thus to identify x z with as is admissible in cer-
tain cases even in mathematics, for instance when x = 1 ;
and to forbid division by x is also admitted, in case x = 0.
We can hardly be said therefore to transgress universally
binding usage. As an illustration of what must be called the
false use of such symbols a practice which has the sanction of
several good names must be noticed, that, namely, of em-
ploying + and to mark respectively affirmation and denial.
The analogy on which this usage is founded is very slight,
amounting indeed to little more than the fact that two
denials (in the case of contradictions) result in re-affirmation,
just as ( ) twice repeated yields ( + ). But the commutative
law is here rejected, for if X + Y means that 'All X is Y

94 On the Choice of Symbolic Language. '[CHAP.

we must clearly refuse to identify this with Y+ X. Again
if we express 'if S, then not P' by (+S P), we might be
tempted, following familiar usage, to regard this as equiva-
lent to ( P + 8) which would be, of course, to fall into a
familiar fallacy 1 . The fact is, as was abundantly illustrated
in the preceding chapters, that the contrast between affirma-
tion and negation, important as it is on a predicative view of
logic, becomes quite superficial when we adopt a thorough-
going class view. We have therefore no right here to use
such a pair of symbols as ( + ) and ( ) to indicate it.

Those who propose a new notation commonly, and not
unnaturally, assume that it is to supersede all others. But
those who approach it as strangers know that the odds are
decidedly that it will only prove one more of those many
attempts which perplex and annoy the lecturer, historian,
and critic. Hence we may fairly use the argument, dear to

1 As indicated above, the actual been puzzled by Hamilton's symbolic
usage here is various. Maimon equivalent for the Law of Contradic-
made + and - equivalent respective- tion. " This law is logically ex-
ly to ' is ' and ' is not ', so that ' a + 1 ' pressed in the formula, what is con-
meant *o is &', and '0-6' meant tradictory is unthinkable. A = not.
'a is not b'. As Darjes used them .4=0, or A-A = 0" [Logic, i. 81:
they might be best rendered by 'posit' Is this, by the way, an attempt at
and 'sublate', for he affixed them to rendering a passage in Bachmann
each term. Thus +S-P meant (Logik, p. 43), "Keine position und
'posit S and we sublate P'. Dro- negation, setzen und aufheben, ( + .4
bisch's use is sounder, as he seems -A), in einem Denkakte unmittelbar
to confine them to mark propositions verbunden, vernichten sich, weil sie
aa wholes: thus +, -u, +p, -p, einander rein entgegengesetzt sind
stand respectively for what are com- (.4-4=0)"?]. Mr Chase again ( First
monly indicated by A, E, I t O. But Logic Book) making + and - do duty
in all cases alike the usage seems to for affirmation and negation employs
me faulty and misleading. the negative particle as well, writing

For downright grotesque perver- e.g., Cesare, thus:
eion of mathematical terms some of No Z - Y,

the non- mathematical logicians are All X+Y,

unequalled. Many readers must have No X - Z.

On the Choice of Symbolic Language. 95

those in authority, that if we loosen the sanctions of orthodoxy
heresies will multiply. Only those whose professional employ-
ment compels them to study a number of different works
.have any idea of the bewildering variety of notation which is
already before the world. A new notation is not like a new
fact or theory from which, so to say, the passer-by may learn
something. It is meant for habitual use, and thus practically
aims at the exclusion of all rivals. No doubt it would be
rank intolerance to forbid such new attempts, but an attitude
of slight social repression towards them may serve to check
too luxuriant a growth of new proposals 1 .

There are two subordinate advantages in employing an
already widely-used and familiar set of symbols. One of
these is in their occasional suggestiveness. Take for instance
that inverse process to class-restriction which was explained
in the last chapter. It is not by any means an obvious
process, and though perfectly intelligible in itself it is not at
all likely that it would have suggested itself to the mind
except by way of the symbols. We are very familiar with

1 In order to gain some idea of a a x (Shroder),

what has been from time to time 1 a -a (Segner),

proposed in this way, the reader may tt na (Maass),

turn to the final chapter of this whilst the same symbols: capital

volume where he will find a detailed and small letters respectively : have

account of over twenty distinct nota- been made to do duty for the fol-

tions for that simplest of all proposi- lowing meanings :
tions : the Universal Negative^ The class A and its contradictory

The following illustrations may (De Morgan, Jevons),
be given here. The same meaning, The class A distributed and un-

the distinction between a term and distributed (Ploucquet),
its contradictory ; has been variously The concept and its extension

symbolized as follows: (Maass, Logik, p. 100),

A a (De Morgan, Jevons), The determined and the determin-

a a (Boole, E. Grassmann), ing concepts (Wundt, Logik, p. 223),
a a' (Delboeuf, Maccoll), Universal and particular proposi-

a 1-a (Boole), tions (Gergonne).

96 On the Choice of Symbolic Language. [CHAP.

the particular inverse process of Division in relation to
Multiplication, and when we use the latter sign to denote
class-restriction the enquiry seems forced upon us to deter-
mine what there is in Logic corresponding to the former.
The moment we write down xy = z, we can hardly refrain

ty

from writing also x = -, and then the interpretation of the

latter is forced upon us. Every mathematician knows what
a fertile source of new theorems is the attempt to ascertain
the analogues to such and such a familiar process in some
other branch of analysis. Of course we must not permit such
hints as these to be anything more than hints, for every
logical rule must be established on its own proper grounds,
but even hints may be of great value.

It may be remarked that the analogy just mentioned was
seized from the first by logical symbolists ; (as was shown, in
the last chapter, in the case of Lambert and Holland). As
they interpreted the step indeed, viz. as denoting Abstraction,
the logical process was one which was already quite familiar,
so that very likely the symbolic step was first suggested and
justified by the logical. As we feel bound to interpret it,
however, viz. in respect of extension or denotation, the case
is very different. As just remarked, the step is not an easy
one to grasp, and it is very doubtful if we should have been
able to see our way to it without the help of the slight
pressure in the right direction afforded by our wish to justify
and explain a familiar symbolic procedure.

Again it seems really important to impress upon the
mind of the student certain characteristics of symbolic
language. The distinction between the mere laws of opera-
tion, and the interpretation of them, is apt to be overlooked,
and this will very likely be still more the case if we insist
upon introducing a new notation for one special class of

iv.j On the Choice of Symbolic Language. 97

interpretations. We may thus lose our appreciation both of
the generalized extent over which the same laws of operation
can prevail, and the very various though connected signi-
fications which this extent of application will serve to
cover.

Various other ways have been adopted in order to prevent
any confusion between the special logical usage of symbols
and the ordinary mathematical usage of the same, and yet
not to lose sight of the common properties. Thus Mr C. S.
Peirce has proposed 1 to differentiate the logical use by the
insertion of a distinguishing mark (a comma underneath).
Instead of writing a ] a = a, with Professor Jevons, in order
to represent the fact that logical 'addition' does not double
the number of the common members, he writes it a + a = a.
This would be interpreted to mean that 'the (logical) ad-
dition, or aggregation, of a to a is (logically) equivalent
to the class taken simply.' So again, as a consequence, if
we know that a and b are mutually exclusive we have the
formula a + b = a + b ; for in this case the results of the logical

Or again, an entirely new set of symbols might be
introduced, intended to be applied not to logic separately in
contrast with mathematics, but made so general as designedly
to cover both. Among those who have actually offered
something in this way, of a logical kind, may be noticed
H. Grassmann. He proposes the symbol ^ to denote 'con-
nection' (verkniipfung) in general, and ^ to denote its
inverse, so that a x - x b denotes the form which when joined
by ^ to b will yield a. He does not indeed in his definition
refer to anything outside the domain of mathematics, but his
language seems intended to be perfectly general : "By a

1 (Proc. of American Acad. of Arts and Sc. 1867.)
V. L. 7

08 On the Choice of Symbolic Language. [CHAP.

general science of symbols [Formenlehre] we understand
that body of truths which apply alike to every branch of
mathematics, and which presuppose only the universal con-
cepts of similarity and difference, connection and disjunction"
(Ausdehnungslehre, p. 2). As a result of this generalized
use we shall have to notice, in another chapter, a curious
anticipation, in certain respects, of one detail in Boole's pro-
cedure.

It is to this generalized symbolic language, such as we
are here employing it, that some writers have applied, by a
revival of an old word, the term Algorithm. Thus, for
example, Delbceuf entitles his work, written on the same
kind of subject as this, "Logique algorithmique." There is
no objection whatever to the word, but I have preferred
to speak of "Symbolic Logic" as being more familiar in our
language : 'symbolic* as I understand it, being almost exactly
the equivalent of 'algorithmic.'

There is also another old term which will be familiar to
readers of Leibnitz and Wolf, characteristic, which seems
to me to cover much the same ground as Algorithmic and
Symbolic : the word is thus defined by Wolf (Psychologia
empirica, 294 seq.) "Ars characteristica appellatur ea quse
explicat signorum, in rebus aut earundem perceptionibus

denotandis, usum. Ars hoec adhuc in desideratis est." "In

Algebra istiusmodi signa habemus pro quantitatibus ; sed
desiderantur talia in philosophia pro rerum qualitatibus."...
"Ars ilia qusG docet signa ad inveniendum utilia et modum
eadem combinandi eorumdemque combinatlonem certa lege
variandi, dicitur Ars characteristica combinatoria. Vocattir

a Leibnitzio etiam speciosa generalis." "Si quis mente

perpendit qualis numerorum in Arithmetica, magnitudinum
in Algebra, syllogismorum in Logica, notio prsesupposita
fuerit antequam characteristica ad numeros magnitudines et

iv.] On the Choice of Symbolic Language. 99

syllogismos applicari potuerit, et quamdiu in Arithmetica
atque Algebra commodi desiderati fuerunt characteres; is
difficultatem artis characteristics combinatorics generalis haud
difficulter sestimabit."

These extracts seem to indicate a tolerably clear appreci-
ation of the end to be aimed at in constructing a generalized
symbolic logic but the discussions on this subject are much
mixed up with the wider question of a general philosophical
language. As the reader very likely knows, this problem
was keenly discussed in the seventeenth and eighteenth
centuries and occupied the attention of Leibnitz more or less
throughout his life. Speaking from a very slight acquaint-
ance 1 , I should say that what was mostly contemplated by
the writers in question was more what we should now call
either a universal language, or a general system of shorthand,
than a logic. I mean that they do not attempt any analysis
of the reasoning processes; and that the words or symbols
proposed by them do not stand perfectly generally for any
classes whatever, like our x and y, but specially for such and
such well-known classes as are already designated by general
names; they differ, in fact, as language does and should differ
from logic.

1 I refer here to such works as this subject, -has discussed many of

the Ars magna sciendi of Athan. the' special sets of symbols appro-

Kircher (1631) ; Bp. WiMns's often priate to particular arts and sciences,

mentioned Essay towards a real cha- Of course the growth of international

racter and a philosophical language telegraphy, and other causes, have

(1668) and Dalgarno's Ars signorum greatly varied the relative importance

(1661). A discussion of Leibnitz's of the schemes known to him.

speculations and attempts La this (Neues Organon. Semiotik. 1. Von

direction will be found in Trendelen- der symbolischen Kenntniss ueber-

burg's Historische Beitrage, in. 1 48. haupt.)
Lambert, who took much interest in

72

CHAPTER V.

DIAGRAMMATIC REPRESENTATION.

THE majority of modern logical treatises make at any
rate occasional appeal to diagrammatic aid, in order to
give sensible illustration of the relations of terms and pro-
positions to one another. With one such scheme, namely
that which is commonly known as the Eulerian, every
a decided majority of the familiar treatises make more or
less frequent use of it 1 . Such a prevalent use as this clearly
makes it desirable to understand what exactly this particular
scheme undertakes to do and whether or not it performs its
work satisfactorily.

As regards the inapplicability of this scheme for the
purposes of a really general Logic something was said
by implication in the first chapter, for it was there pointed

1 Until I came to look somewhat suited for this purpose: somewhat

closely into the matter I had no idea at random, as they happened to be

how prevalent such an appeal as this most accessible : it appeared that

had become. Thus of the first sixty thirty-four appealed to the aid of

logical treatises, published during the diagrams, nearly all of these making

last century or so, which were con- use of the Eulerian Scheme.

CHAP, v.] Diagrammatic Representation. 101

out how very special and remote from common usage is
the system of propositions for which alone it is an adequate
representation. To my thinking it fits in but badly even
with the four propositions of the common Logic to which it is
usually applied, but to see how very ineffective it is to meet
the requirements of a generalized or symbolic Logic it will be
well to spend a few minutes in calling the reader's attention
to what these requirements are.

At the basis of our Symbolic Logic, however represented,
whether by words by letters or by diagrams, we shall always
find the same state of things. What we ultimately have to
do is to break up the entire field into a definite number of
classes or compartments which are mutually exclusive and
collectively exhaustive. The nature of this process of sub-
division will have to be more fully explained in a future
chapter, so that it will suffice to remark here that nothing
more is demanded than a generalization of a very familiar
logical process, viz. that of dichotomy. But its results are
simple and intelligible enough. With two classes, X and F,
we have four subdivisions ; the X that is Y, the X that is
not Y, the Y that is not X, and that which is neither X nor
Y. And so with any larger number of classes. How then
are these ultimate class divisions to be described ?

For one thing, we can of course always represent the
products of such a subdivision in the language of common
Logic, or even in that of common life, if we choose to do so.
They do not readily offer themselves for this purpose, but
when pressed will consent, though failing sadly in the desired
symmetry and compactness. The relative cumbrousness of
such a mode of expression is obviously the real measure of
our need for a reformed or symbolic language. We must not
however forget that we are not dealing with mathematical
conceptions which common language will hardly avail to

102 Diagrammatic Representation. [CHAP.

describe, but only with logical classes -which can be com-
pletely and unambiguously determined by the traditional
modes of speech. However complicated the description of
any given class may be we could always build it up by means
of JTand not-.X, Y and not-F, and so forth; whether JTand
F remain as letters, or be replaced by concrete terms such as

But it need not be insisted on that we require something
far more manageable and concise than this, if we wish to deal
effectively with really complicated groups of propositions.
For this purpose nothing better can be employed than
letters such as we use in algebra. This is done of* course to
some extent in ordinary Logic, the only innovation upon
which we have to insist being that of introducing equally
concise symbols for negative terms. We could never work
with not-#, in that form, and must therefore look about for
some substitute. The full significant substitute is, as already
shown, 1 #, and this will often have to be employed. But
this is too cumbrous for actual calculation. Of the various
substitutes that have been proposed for not-# we shall make
a practice of employing x.

The reader will see at once how conveniently and briefly
we can thus indicate any desired combination of class terms,
and, by consequence, any desired proposition. Thus xyz
represents what is x, y, and z\ x~yz what is x, but neither
y nor z\ xw (yz + yz) stands for 'what is not x, but is w; and
is also either y but not z, or z but not y', and so forth. The
significance of such expressions, when built up into proposi-
tions, will be fully discussed in a future chapter.

That such a scheme is complete there can be no doubt.
But unfortunately, owing to this very completeness, it is apt
to prove terribly lengthy. The powers of 2 soon mount up ;
so that a pair of terms will yield 2 a combinations, three will

v.] Diagrammatic Representation. 103

yield 2 3 , or 8, and so on; the total number doubling every
time. Of course in any particular proposition or problem we
shall most likely not require to make appeal to more than
some of these constituents, perhaps only to a few of them.
But the existence of all has to be recognized, and a notation
provided for every one of them. Moreover it is always
possible that a problem may be so stated as to demand an
explicit reference to a great number, or even to all. Suppose
for instance a proposition were given involving five terms,
and we were told to enumerate all the ultimate combinations
denied by it, we should have a certain number of class terms
on our hands; and if to this were added the enumeration of
all the combinations which were not denied, we should have
all the rest of the total of thirty-two before us.

This then is the state of thing which a reformed scheme
of diagrammatic notation has to meet. It must correspond
in all essential respects to that regular system of class sub-
division which has just been referred to under its verbal and
its literal or symbolic aspects. Theoretically, as we shall see,
this is perfectly attainable. . Indeed up to four or five terms
inclusive it works very successfully in practice; where it
begins to fail is in the accidental circumstance that its farther
development soon becomes intricate and awkward, though
never ceasing to be feasible.

I. On the proposed scheme we have to make a broad
distinction, not recognized on the common scheme, between
the representation of terms and the representation of proposi-
tions. We begin with the former. What we propose to do
is to form a framework of figures which shall correspond
to the table of combinations of x, y, z, &c. All that is
necessary for this purpose is to describe a series of closed
figures, of any kind, so that each successive one shall in-
tersect all the compartments already produced, and thus

104 Diagrammatic Representation. [CHAP.

double their number. That this is what is done with the
letter symbols is readily seen. Thus with two terms, x and
y, we have four combinations; xy, xy, xy, xy. Introduce the
term z, and we at once split up each of these four into its z
and its not-z parts, and so double their number. Provided
our diagrams are so contrived as to indicate this, they will
precisely correspond, in every relevant respect, to the table of
combinations of letters.

The leading conception of such a scheme is simple
enough, but it demands some consideration in order to
decide upon the most effective and symmetrical plan of
carrying it out in detail 1 . Up to three terms inclusive,
indeed, there is but little opening for any variety; but as
the departure from the familiar Eulerian conception has to
be made from the very first, it will be well to examine the
simplest cases with some care. Our primary diagram for two
terms is thus sketched :

'QD

On the common plan this would represent a proposition,
and is indeed commonly regarded as stand ing for the proposi-
tion 'some x is y'; though (as was mentioned in the first
chapter) it equally involves in addition the two independent

1 A brief historic sketch is given in unsuitable for the purpose. Though

the concluding chapter of some pre- the method here described may be

vious attempts, before and after Euler, said to be founded on Boole's system

to carry out the geometric notation of Logic I may remark that it is not

of propositions. I tried at first, as in any way directly derived from him.

others have done, to illustrate the He does not make employment of

generalized processes of the Symbolic diagrams himself, nor does he give

Logic by aid of the familiar methods, any suggestions for their introduc-

Lut soon found that these were quite tiou.

v.] Diagrammatic Representation. 105

propositions 'some x is not y\ and 'some y is not x', if we
want to say all that it has to tell us. With us however it
does not as yet represent a proposition at all, but only the
framework into which propositions may be fitted; that is, it
indicates only the four combinations represented by the letter
compounds, xy, xy, xy, xy.

Now suppose that we have to reckon with the presence,
and consequently with the absence, of a term z. We just
draw a third circle intersecting the above two, thus :

Each circle is thus cut up into four parts, and each com-
mon part of two circles into two parts, so that, including what
lies outside them all three, there are eight compartments.
These of course correspond precisely to the eight combinations
given by the three literal symbols; viz. xyz, xyz, xyz, xyz,
xyz, xyz, xyz, xyz. Put a finger upon any compartment, and
we have a symbolic name ready provided for it; mention the
name, and there can be no doubt as to the compartment
thereby referred to.

Both schemes, that of letters and that of spaces, agree

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