Now eliminate from these equations, after reduction by
the methods described in a former chapter, every term except
t and those terms in which F(x, y, z) was to be determined.
Then develop t, by the well-known methods, and what was
required is done ; for t or F(x, y, z) will be described in terms
of those symbols, and those symbols only, which it was desired
to make use of in describing it.
If certainty and completeness of symbolic procedure were
all we had to look for, there oan be no doubt that Boole's
316 Completion of the Logical Problem. [CHAP.
method would be the best, as the whole answer is at once and
completely given by it; that is, we obtain at one and the
same time the converse propositions referred to a page or two
back, as well as the direct proposition describing F(x, y, z).
But it is a terribly long process ; a sort of machine meant to
be looked at and explained, rather than to be put in use.
Consequently if ever we do feel occasion to solve such a
problem there can be little doubt that the comparatively
empirical method above offered will answer our purpose best.
I call it empirical because it requires us to build up the
given function in detail, by exercise so to say of our own
observation and sagacity, instead of taking, and trusting to,
a precise rule for the purpose of effecting it.
How tedious Boole's method is in practice, in spite of its
theoretic perfection, may be seen by taking the very simple
example already referred to ; viz.
Given ^ \ find xz.
We should write them thus :
xy a = 01
xz ~t = O
Then proceed to eliminate a?, y, and z, from these equa-
tions and nothing will be left but a, c, and t. This will give
therefore an equation for solving t in terms of a and c, which
will involve the required answer. I should proceed to do
this by first resolving all these three equations into their
negative constituents, as follows ;
xya\ yzc\ xzi\
ax> cy\ >=0.
ay] cz\ tz)
XV.] Completion of the Logical Problem. 317
The full formula for eliminating x, y, and z from this
expression would be rather long : it would be as follows :
(a + c + i)(a + c + f)(a + c + t)(a + c + i)(a + c + t) 4 = 0,
this being the result of the requisite substitutions in the
the formula for the elimination of three terms fromf(x,y,z) = 0.
Boole has given various methods of reduction for the
simplification of such a formula, but its management will
always be tedious. After a page or two of work, -which I
need not repeat here, the student may find himself brought
to the result
(ct + a + ct)(a + cf) = 0,
, whence t
ac ac ac
From this, by the ordinary process of development, we have
t = ac + $ac.
That is, xz = ac + $ac.
The desired result is therefore obtained, and so far we have
an illustration of the power and certainty of the method ; but
it can hardly be claimed as showing also its ease and simpli-
city. (As regards these latter qualifications the reader may
contrast the processes in Chapter xil. p. 279). Of course what
can be done in this case can be done in any case whatever,
omitting considerations of mere tediousness ; for, since we
have no equations of anything higher than the first degree,
every solution is theoretically obtainable with equal certainty,
and by the employment of the same method.
We see therefore that this problem, which may be
described as the most general problem of Symbolic Logic, is
318 The Syllogism. [CHAP.
theoretically attainable in every case, and the method for so
obtaining it has now been fully exhibited.
The problem just discussed is closely connected with the
Syllogism of ordinary Logic, of which indeed it may (as will
presently be pointed out) be regarded as a generalized ren-
dering. We must frankly remark that from our point of
view we do not greatly care for this venerable structure,
highly useful though it be for purposes of elementary training
in thought and expression, and almost perfect as it technically
is when regarded from its own standing point. But its ways
of thinking are not ours, and it obeys rules to which we own
no allegiance. To it the distinction between subject and
predicate is essential, to us this is about as important as the
difference between the two ends of a ruler which one may
hold either way at will. To it the position of the middle
term is consequently worth founding a distinction upon, to us
this is as significant as is the order in which one adds up the
figures in an addition sum.
There are reasons nevertheless for taking some account
of the syllogism here ; partly because the contrast of treat-
ment will serve to emphasize this difference in the point of
view, partly because the omission of any such reference might
possibly be taken as a confession of failure on the part of the
Symbolic Logic. Since the Syllogism is a sound process it
must admit of some kind of treatment upon any scheme,
though we shall take the liberty of freeing it from what we
are bound to regard as certain unnecessary restrictions. We
cannot consistently recognize the differences upon which the
distinctions of Figure are founded, so we look only at the
first Figure. Even then we must simplify further. Firstly,
we refuse to see any difference of character between x and
xv.] The Syllogism. 319
not-x; whereupon vanishes the distinction between syllogisms
with affirmative and with negative major premises. Secondly,
our punctilious aversion to anything like enumeration or
quantification, in other words, to the mathematical conces-
sions of the common Logic, induces us to regard 'some X'
in the subject of the major, as a distributed class-term,
instead of regarding 'X' as an undistributed term. The
justification for doing this is that, when we look only to the
first Figure, precisely the same indefinite class 'some X'
occurs in the minor premise and in the conclusion, so that its
indefinite character has no bearing whatever on the actual
process of reasoning 1 . Accordingly our syllogistic type is
reduced to the one form,
XY=0, or X
When all the three letters here stand for whole terms,
these being positive, we have Barbara. When X stands for
1 The reader will remember that 'some-X' and refer it to a part of
this was not the form adopted in the class 'not-Z '. But it is a per-
Chap. vii. for the expression of par- fectly legitimate way of expressing it
ticular propositions. But the pro- when, as in the first Figure, this
blem before us in the two cases is same indefinite class 'some-^' occurs
not the same. What we were there on each occasion as one of three
considering was the best way of ex- terms. When a Symbolist is forced
pressing the real force of such propo- to syllogize, it is, I consider, a fair
sitions, in symbolic language ; taking rendering of Ferio to put it,
account, as far as we could, of the All Y is not-Z,
attendant difficulties of implication. Some-X is Y,
What is now before us is the best way .'. Some-X is not-Z.
of similarly expressing a certain pro- We do not syllogize willingly,
cess of reasoning. I think that it is nor profess to do it gracefully, but it
not by any means the best rendering seems to me that we do not do it in-
Of 'Some X is not Z' to say that accurately.
we thus contemplate the whole class
320 The Syllogism. [CHAP.
a part-class (i.e. for some-X), and Z is a negative term, we
have Ferio ; and so on with the two remaining forms. That
is, the above symbolic arrangement will, by suitable interpre-
tations of X, Y, and Z, cover all the Moods of the first figure,
and consequently all the Moods of the other figures.
The above rendering of the syllogism, it will be seen, is
really nothing but a symbolic translation of the Dictum of
Aristotle ; as any single comprehensive rendering of it ought,
I suppose, to be. Or rather it is a slight generalization of
that Dictum, for, since we recognize no difference of character
between x and x, we make but one dictum out of omne and
Supposing that we feel bound to treat the syllogism at
all, the above certainly seems to me the best way of doing so ;
indeed the only way, in strict consistency with our own prin-
ciples. One serious defect, as it seems to me, in the great
majority of the attempts to treat Logic symbolically has con-
sisted in the fact that the authors have not sufficiently shaken
themselves free from the old trammels. They have felt
bound to adhere as far as possible to all the old distinctions
in the form, order, and so forth, of the constituent proposi-
tions even of the syllogism. The majority of the older sym-
bolists (for instance, Maimon) have really done little more
than go in detail through the nineteen moods, clothing each
in a new symbolic dress. This is highly unsatisfactory, since
most sets of symbols require some violence to force them into
recognizing distinctions so utterly alien to their genius and
Boole's plan is very different from the one offered above.
He has certainly solved a very general problem, and one
which can be made to include, amongst other things, a num-
ber of the syllogistic moods. I cannot however regard it as
quite a fair generalization of that process; nor, for that
matter, did he seem to regard it so himself. His plan is as
follows. He considers the two prepositional forms, vx = v'y,
wz = w'y, as containing under them all the possible forms of
premise needed for a syllogism. Thus, put v = 1 in vx = v'y,
and we have 'All x is y'', leave v = 1, but regard y as nega-
tive, and we have ' No x is y'; make v and v' indefinite, and
we have ' Some x is y'; do the same, regarding y as negative,
a,nd we have ' Some x is not y '. Similarly with the inter-
pretation of the other premise. Now eliminate y from these
two equations and determine the relation of x to z, and a
part 1 of the syllogistic scheme will be completed.
The answer obtained by Boole from these premises, involv-
ing seven terms, is somewhat intricate. It is given in the
Laws of Thought (p. 232). I will not however pursue the sub-
ject further for two reasons. In the first place, the whole
enquiry seems to me to be carried on upon a wrong line of
1 Only a part, on two accounts.
Firstly, in the propositions as above
expressed, the same middle term y
occurs in both. But, in the syl-
logism, we require that 'all y' and
' no y ' shall enter as a middle term ;
that is, we must take account of the
pair of equations vx = v'y, wz = w'y,
or we shall omit some of the recog-
nized pairs of premises. Secondly,
syllogistic conclusion demands the
determination not only of the re-
lation of x, but also of that of x and
vx, to z, or we shall omit some of the
recognized conclusions. Consequent-
ly the form above worked out is only
one of six which demand exami-
nation, and which are duly discussed
It deserves notice that Lambert
attempted much the same problem.
Starting from two perfectly general
expressions for the premises, he ob-
tains a similar one for the conclu-
sion, pointing out that by due deter-
mination of the arbitrary letters in-
volved we may specialize for any
desired figure and mood. His forms
(of which some account is given in
Chap, xx.) are closely analogous to
those of Boole. Thus,
"Major = ,
Minor = ,
, un., mv .
Conclusio C = A.
In dieser allgemeinen Formel kann
man die Buchstaben nach Belieben
bestimmen, wenn man daraus beson-
dere Foimeln fiir die Schliisse her-
leiten will" (Log. Ab. i. 103).
322 The Syllogism. [CHAP.
attack, inasmuch as it involves a concession to a variety of
rules and assumptions which from our point of view must be
regarded as arbitrary and almost unmeaning. Moreover such
a form as vx = v'y cannot be regarded as a true representa-
tion of a particular proposition unless we reject the value
for v and v, a restriction which is not claimed for x and y.
Consequently we are mixing up in the same investigation
terms which are subject to ditferent laws of valuation.
If we simply regard v, v, w, w, as ordinary class terms,
like x, y, z, the problem in question acquires a very ready in-
terpretation, but one widely remote from anything contem-
plated in syllogistic Logic. It then becomes, "If every x
which is v is a, y which is v (and vice versd), and every z
which is w is a y which is w (and vice versd}, what is the de-
scription of x as given by the other terms, omitting y"1 A
concrete example of this is given by Boole himself in the
words : " Suppose a number of pieces of cloth striped with
different colours were submitted to inspection, and that the
two following observations were made upon them ;
(1) That every piece striped with white and green was
also striped with black and yellow, and vice versd.
(2) That every piece striped with red and orange was
also striped with blue and yellow, and vice versd.
Suppose it then required to determine how the pieces
marked with green stood affected with reference to the colours
white, black, red, orange, and blue ".
If we are to take this kind of view of the syllogism at all
there seems to me to be a simpler and more symmetrical
arrangement. Instead of starting with the premises employed
by Boole, let us make use of the following,
ocy = a
xv.] The Syllogism. 323
By suitable modifications, as explained in Chap, vil., we
can make these forms represent a number of the propositions
needed for the syllogism, as well as some others besides. We
have only to put a and c respectively equal to 1, v, 0. This
will furnish one type of pairs of propositions, from which we
can then determine xz, xz and xz. For the other type we
must take a pair of propositions in which the middle term
occurs in the forms y and y. Thus,
and then again determine xz, xz and xz from this pair. The
problem thus solved is essentially the same as that under-
taken by Boole, and can therefore hardly be considered to be
a true representation of the syllogism, but it seems to me
simpler and more symmetrical than his rendering. Of the
results, one or other of which represents every syllogistic mood
and which correspond to the six alluded to above, I give two :
From the premises
we deduce xz = ac +$ac ............ (a).
And from the premises
we deduce xz ^ac+^ac .............. (/5).
It would take too much space to work out all the results,
but two may be chosen as samples.
(1) In the premises scy = a,yz = c, put z = w, a = v, c = 0,
and we have
xy = v)
Then, employing formula (a), we have xw = v + or xw = v.
324 The Syllogism. [CHAP.
Put into words this stands, inverting the order of the
All y is w\
Some x is y\ Darii
:. Some x is w]
(2) In the second pair of premises xy = a, yz = c, put
a = 0, c = 0, and we have
Then, employing formula (), we have az = 0.
Put into words it stands
No y is x\
All is y> Celarent.
:. No s is x]
The syllogistic regulations are however so remote from the
ways of the symbolic system that it hardly seems worth the
trouble to follow out the enquiry any further. We can
syllogize, after a fashion, just as one could drive a stagecoach
from London to Birmingham along the railroad, but to do
this would be a needless deference to tradition now that we have
engines and carriages specially constructed for the new system.
I take this opportunity of cor- All Fs are A"s,
rccting an error, or rather a sug- No Fs are X'B,
gestion of error, in the note on p. 176. Some Fs are X's,
It slipped my memory at the time Some Fs are not-A"s,
(and indeed till that sheet was prin- All not-Fs are X's,
ted off) that Boole when discussing No not-Fs are X's,
the Syllogism in his later work, the Some not-Fs are Z's,
Laws of Thought, had not confined Some not-Fs are not-X's.
himself to the four old forms of pro- These forms, as he distinctly says,
position, but had added on four are not the same as those of 1 1 ami 1
more, thus adopting a scheme equi- ton. They are simply an enlarge-
\iik-nt to that of De Morgan. Thus,, ment of the old scheme by the intro-
duction of negative subjects, and
they leave the predicate, as regards
its quantity or distribution, in its
customary condition. There is no
attempt to distinguish whether we
mean 'some only' or 'all' the predi-
cate to be taken; and this I have
always considered to be the whole
point of the Quantification doctrine.
Moreover his whole treatment of these
forms is antagonistic to this doctrine.
Thus he here expresses (p. 229)
'All Ps are X'a', in his usual way,
y = vx ; but gives at once, as equi-
valent forms, yx 0, x = % y . These
forms, as I have said in the text of
the page referred to, seem to me
as directly hostile to all that I have
ever understood to be meant by Quan-
tification of the Predicate, as any
forms which I could invent for the
purpose of expressing such hostility.
If any one will point out to me a
passage in which Boole has admitted
the distinctive propositions ' All X is
some Y', 'Some Xis not some Y', I
shall admit, not that his system is
founded on the Quantification of pre-
dicates, but that he has there used
expressions inconsistent with his
system of symbols.
IN the course of the preceding chapters we have frequently
had to touch incidentally upon the treatment of Hypotheti-
cal propositions, both in the symbolic expression of our data
and in the interpretation of our results. The subject how-
ever is too large and intricate for merely incidental treatment'
so a separate chapter is here devoted to a more detailed
discussion of it.
The reader will have had repeated opportunities of in-
ferring that the only tenable symbolic view is that there is no
real distinction between the hypothetical and categorical forms
of statement, these distinct renderings being regarded as a
matter of private option, so that it is open to any one to read
off the symbols in whichever way he pleases. In the sym-
bolic statements themselves there is nothing to intimate in
which way the premises were worded when they were handed
over to us, nor is there anything to force us to translate them
back again into one form rather than the other.
Here, as at every other point, we have to consider the de-
mands of our own scheme and not those of other schemes.
Doubtless there are systems of Logic to which the distinction
XVL] Hypothetical. 327
in question is by no means insignificant. For instance, those
who adopt a more subjective view of the nature of the reason-
ing processes might fairly insist upon the distinction between
categorical and hypothetical judgments. On the other hand
there are ways of treating Logic objectively, for example by
the discussion of Induction, in which the distinctive charac-
teristic of the Hypothetical brings it (as will presently be
pointed out) into such close proximity with the estimation of
Probability as to entitle it to distinct consideration*
As just remarked, all that here directly concerns us is the
most appropriate symbolic account of the matter. In this we
must have regard not merely to the fact that we make very
large employment of symbols, but also to the character of the
propositions to which this employment introduces us. Now
the sort of proposition with which we constantly have to
deal, and which we class with others of a more simple and
familiar description, is one wherein the subject or predicate,
or both, are complex. Take, as a type of this character, the
statement AB = ABCD. This may be read off in a variety
of ways : All AB is CD ; If any A is B then it is CD ; If
any B is A then it is CD ; Every A which is B is a C which
is D ; Whenever A concurs with B then will C concur with
D ; and so forth. That these different renderings of the same
original, statement involve somewhat different judgments may
be freely conceded ; and therefore, as already remarked, on a
more subjective system of Logic we might have to distinguish
between them. But it has been abundantly illustrated in
the course of this work that what we look to are the class re-
lations involved, and therefore we are prepared to allow con-
siderable latitude in respect of the way of reading off our
statements, provided the relations themselves be left un-
It is quite true that logical equations which involve com-
328 Hypotheticals. [CHAP.
plex terms are more naturally interpreted in a hypothetical
form than are those of a simpler kind. Indeed, in a recent
treatise on Symbolic Logic 1 , the form xy = c is chosen as the
type of the Hypothetical, as a; = c is of the Categorical. It
will be instructive to enquire into the ground of this choice.
There is no doubt that each of these forms can, if we please,
be interpreted in either way. Thus xy = c may be read off
either 'All xy is c', or 'If any x is y then it is c'; whilst
x c may be read off either ' All x is c' or ' If anything is x
then it is c'. Why then is the former regarded as more
appropriately hypothetical ? Mainly, I apprehend, for the
following reason. It was shown in a previous chapter (Chap.
VI.) that every universal affirmative proposition must be in-
terpreted as involving something of a hypothesis ; 'All x is c'
having to be understood, if we wish to work with it success-
fully, as meaning 'All x, if there be any, is c'. But this
hypothetical element is generally so faint as scarcely to be
perceptible in common discourse, where indeed it is often
entirely rejected. Names are seldom employed except to
denote what we suppose to exist, so that we come to feel
much reluctance to assert that 'All a; is c' unless we are con-
vinced of the existence of x. Hence any doubt on this score
commonly drops out of sight, and the categorical is safely
assumed in most cases to carry with it an assurance of the
existence of the subject and predicate. But though such an
assurance may be justified in the case of x and y separately,
1 Macfarlane's A Igebra of Logic, folgenden verwechselt werden ; Alles
p. 81. Lambert, so far as I know, A so B ist, 1st C. Nun 1st, Alles
was the first to explicitly assign this A so B ist .-(7. 1 ; folglich, Alles . / /.'
notation for the expression of a ist C. Daher dieZeichnung: AB>C,
hypothetical: "Die allgemeinste AB=mC" (Log. Abhandlungen, i.
Formel der hypothetischen Sa'tze ist 128). Some explanation of the sym-
diese : Wenn A ein B ist, so ist CM C. hols thus employed will be found iu
Uif.se Formel kaun allezeit mit der the final chapter.
xvi.] Hypoiheticals. 329
it is quite another thing to justify it in the case of the com-
pound xy. These symbols, when occurring separately, stand
presumably for common terms which are familiar to every
one, but their combination into one may be something en-
tirely novel. Hence the enquiry whether there be any such
thing as the subject of the proposition, which would seem al-
most impertinent in the case of the simple propositions typified
by x = c, becomes quite pertinent in the case of the complex
ones typified by xy = c. The doubt thus suggested naturally
expresses itself by our adopting, as the verbal equivalent of
the latter form, such a conditional statement as, 'If x is y then
it is c', the rest of the complete statement, indicated by the
symbols, taking the form ' c is xy, if there be such'. Proposi-
tions therefore with complex subjects almost force upon us
that hypothetical interpretation which we have found it
advisable to extend to all propositions without exception.
There are, it must be admitted, certain grammatical diffi-
culties in this way of framing our hypothetical propositions,
but these seem really common to all systems of Logic. Take
as an example, ' If the harvest is bad in England, then corn
dealers in America will gain', which seems a fair concrete
example of ' If A is B then C is D'. We have said (Chap,
xu) that we should frame this in the symbolic expression
(1 - AF) CD = 0, or CD = GDAR This identifies ' G that
is not D' with some uncertain part of 'A that is not B'\ that
is, literally interpreted, we seem to be declaring that ' Ameri-
can corn dealers who do not gain' are ' harvests which are
not bad in England',
Such a difficulty as this ought hardly to trouble any
logician. It surely arises only out of the impossibility of in-
flecting our literal symbols, which compels us to make con-
siderable modifications in the structure of our sentences
before submitting them to symbolic treatment. If, in accord-