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the present paper is then given.]



90 ON THE TYPES OF COMPOUND STATEMENT

which is the same thing, all hard, wet, black things are
nasty). The statement ABC=0 (no hard, wet, black
things exist, or all hard, black things are dry) is to be
regarded as made of these two, ABCD=0, ABCd=0 (no
hard, wet, black, nice things exist, and no hard, wet,
black, nasty things exist) and so is called a compound,
(in this case a twofold) statement. The notion of types
is defined in art. 1.

1. Four classes or terms A, B, C, D, give rise to
sixteen cross divisions or marks such as AbCd. A
denial of the existence of one of these cross divisions, or
of anything having its mark (such as A&Cc/=0), is called
a simple statement. A denial of two or more cross
divisions is called a compound statement, and more-
over, twofold, threefold, etc., according to the number
denied.

When two compound statements can be converted
into one another by interchange of the classes A, B, C,
D, with each other or with their complementary classes
a, by c> d, they are called similar ; and all similar
statements are said to belong to the same type. The
problem before us is to enumerate all the types of com-
pound statement that can be made with four terms.

2. Two statements are called complementary when
they deny between them all the sixteen marks without
both denying any mark, or, which is the same thing,
when each denies just those marks which the other
permits to exist. It is obvious that when two statements
are similar, the complementary statements will also be
similar ; and, consequently, for every type of 7i-fold
statement there is a complementary type of 16-^-fold
statement. It follows that we need only enumerate the




INVOLVING FOUR CLASSES. 91

types as far as the eighth order ; for the types of more
than eightfold statement will already have been given as
complementary to types of lower orders. Every eight-
fold statement is complementary to an eightfold state-
ment ; but these are not necessarily of the same type.

3. One mark ABCD may be converted into another

A^bCd by interchanging one or more of the classes A, B,

C, D with its complementary class. The number of such

changes is called the distances of the two marks. Thus

in the example given the distance is 2. In two similar

compound statements the distances of the marks denied

must be the same ; but it does not follow that when all

the distances are the same the two statements are similar.

There is, however, as we shall see, only one example of

two dissimilar statements having the same distances.

When the distance is 4, the two marks are said to be

obverse to one another, and the statements denying them

are called obverse statements as ABCD, abed, or, again,

A.bCd, or, again, AbCd, aBcD. When any one mark is

given (called the origin), all the others may be grouped

in respect of their relations to it as follows : Four are

at distance one from it, and may be called proximates ;

six at distance two, and may be called mediates ; four at

distance three, and may be called ultimates. Finally, the

obverse is at distance four.

aBCD abCD Abed



ABCrf ABCD AiCD /K abcV a b c d a B cd




ABoD ABcrf abGd

Origin and 4 proximates. 6 mediates. Obverse and 4 ultimates.

It will be seen from the above table that the four proxi-



92 ON THE TYPES OF COMPOUND STATEMENT

mates are respectively obverse to the four ultimates, and
that the mediates form three pairs of obverses. Every
proximate or ultimate is distant 1 and 3 respectively
from such a pair of mediates. Thus each proximate or
ultimate divides the mediates into two classes ; three of
them are at distance 1 from it, and three at distance 3.
Two mediates which are not obverse are at distance 2.
Two proximates or two ultimates, or an ultimate and a
proximate which are not obverse, are also at distance 2.
This view of the mutual relations of the marks is
the basis of the following enumeration of types.

4. There is clearly only one type of simple state-
ment. But of twofold statements there are four types ;
viz. the distance may be 1,2, 3, or 4 ; and so, in general,
with n classes there are n types of twofold statement.

5. A compound statement containing no pair of
obverses is called pure. In a threefold statement there
are three distances ; one of these must be not less than
either of the others. If this be 2, the remaining mark
must be at odd distance from both of these or at even
distance from both ; thus we get the types 1, 1, 2, and
2, 2, 2. If the not-less distance be 3, the remaining
distances must be one even and the other odd ; the
even distance must be 2, the odd one either 1 or 3 ;
and the types are 1, 2, 3 ; 2, 3, 3. Thus there are 4
pure threefold types. With a pair of obverses, the re-
maining mark must be at odd or even distance from
them; 1, 3, 4 ; 2, 2, 4. In all six threefold types
observe that there is necessarily one even distance.

6. A fortiori, in a fourfold statement there must be
one even distance. In a pure fourfold statement this
distance is 2. From this pair of marks let both the



INVOLVING FOUR CLASSES. 93

others be oddly distant ; then they must be evenly
distant from one another i.e. at distance 2, obverses
being excluded. The odd distances are 1 or 3 ; and it
will be easily seen that the following are all the possible
cases :

LL 1 LU l 1 1 l |3 l 1 3 3J3

*1 | l' *1 | 3* *3 [3* *3 j l' *3 | 3* *3 | 3'

In these figures the dots indicate the four marks, the
cross lines indicate distance 2, and the other figures the
distances between the marks on either side of them.
Next, from the pairs of marks at distance 2 let one of
the others at least be evenly distant, i.e. at distance 2.
Then we have three marks which are all at distance 2
from one another ; and it is easy to show that they are
all proximates of a certain other mark. For, select one
of them as origin ; then the other two are mediates
which are not obverse, and which consequently are at
distance 1, from some one proximate. With this proxi-
mate as origin, therefore, all three are proximates. We
have therefore only to inquire what different relations
the fourth mark can bear to these three. It may be
the origin, its obverse* the remaining proximate, its
obverse, or one of two kinds of mediates, viz. at distance
1 or 3 from the remaining proximate. Thus we have 6
types, in which the distances of the fourth mark from the
triad are respectively 1 1 1, 3 3 3, 2 2 2, 2 2 2, 1 3 3, 1 1 3.
The third and fourth of these are especially interesting, as
being distinct types with the same set of distances ; I call
them proper and improper groups respectively : viz., a
proper group is the four proximates of any origin ; an
improper group is three proximates with the obverse of



94 ON THE TYPES OF COMPOUND STATEMENT

the fourth. On the whole we get 12 types of pure
fourfold statement.

7. In a fourfold statement with one pair of obverses,
take one of them for origin ; the remaining two marks
must then be either a pair of proximates or ultimates, a
proximate and an ultimate, a pair of mediates, or a
proximate or ultimate, with one of two kinds of mediate
in all, 5 types, with the distances 1 3 2 , 1 3 ; 1 3 2 , 3 1 ;
2 2 2 , 2 2 ; 1 3 l , 2 2 ; 1 3 3 , 2 2. With two pairs of ob-
verses they must be either at odd or even distances from
one another; two types. Altogether 12 + 5 + 2 = 19
fourfold types.

8. In a pure fivefold statement there is always a
triad of marks at distance 2 from one another. For
there is a pair evenly distant ; if there is not another
mark evenly distant from these, the remaining three are
all oddly distant, and therefore evenly distant from one
another. First, then, let the remaining two marks be
both oddly distant from the triad. In regard to the
origin of which these are proximates, the two to be
added must be either two mediates, like (of two kinds)
or unlike, or a mediate of either kind with the origin or
the obverse ; 7 types. Next, if one of the two marks
be evenly distant from the triad, it must form with the
triad either a proper or an improper group of four. To
a proper group we may add the origin, the obverse, or
a mediate ; to an improper group, the origin or the
obverse (the mediates give no new type), 5 types ; or,
in all, 12 pure fivefold types.

9. In a fivefold statement with one pair of obverses
there must be another pair of marks at distance 2. We
have therefore to add one mark to each of the following



INVOLVING FOUR CLASSES. m 95

three types of fourfold statement, a pair of obverses
together with (1) two proximates, (2) a proximate and
an ultimate, (3) two mediates. To the first we may
add another proximate, an ultimate or a mediate of
three kinds, viz. at distances 1 1, 1 3, 3 3 from the two
proximates ; 5 types. To the second if we add a proxi-
mate or an ultimate, we fall back on one of the pre-
vious cases ; but there are again three kinds of mediates,
at distances 1 1, 3 3, 1 3 from the proximate and ultimate ;
3 types. To the third we may add another mediate,
whereby the type becomes a proper group together
with the obverse of one of its marks, which is the same
thing as an improper group together with the obverse
of one of its marks or a proximate or ultimate which
are of three kinds, at distances 11, 13, 33 from
the two mediates ; 4 types. Thus there are 12 five-
fold types with one pair of obverses. With two pair
of obverses at odd distances, there is only one type,
all the remaining marks being similarly related to
them ; at even distance the remaining mark may be
evenly or oddly distant from them ; 2 types. On
the whole we have 12 + 12 + 3 = 27 types of fivefold
statement.

It is to be remarked that there is no pure fivefold
statement in which all the distances are even, and that,
if there is only one pair of obverses with all the distances
even, the type is a proper group together with the
obverse of one of its marks.

10. We may now prove, as a consequence of the
last remark, that a pure sixfold statement either
contains a group of four with a pair oddly distant from
it, or consists of two triads oddly distant from one another.



96 ON THE TYPES OF COMPOUND STATEMENT

For there must be a pair at distance 2 : if the other
four are all oddly distant from these, they form a
group ; if one is evenly distant, and three oddly distant,
we have the case of the two triads ; if two are evenly
distant, we again have a group. We must add, then,
first to a proper group , and then to an improper group,
a pair oddly distant from it. To a proper group con-
sisting of the proximates to a certain origin we may
add the origin or its obverse with a mediate, or two
mediates ; 3 types. An improper group is symmetrical ;
that is to say, if we substitute for any one of its marks
the obverse of that mark, we shall obtain a proper
group. In this way we shall get four origins distant
1113 from the group, and four obverses distant
1333; if we add to these the obverses of the marks
in the group itself, we have described the relation of
the twelve remaining marks to the group. To form,
therefore, a pure sixfold statement we may add either
two origins or two obverses or an origin and an obverse ;
3 types.

In the case of the two triads, since they are oddly
distant from one another their origins must be oddly
distant ; that is, they must be distant either 1 or 3. If
they are distant 1, neither, both, or one of the origins
may appear in the statement ; if they are distant 3,
neither, both, or one of the obverses : 6 types. Thus
we obtain 12 types of purely sixfold statement.

11. If a sixfold statement contains one pair of
obverses, the remaining four marks cannot all be evenly
distant from this pair. For in that case they would
constitute a group ; and it is easy to see that the marks
evenly distant from a group, whether proper or improper,



INVOLVING FOUR CLASSES. 97

do not contain a pair of obverses. We have therefore
only these four cases to consider :

(1) The four marks are all oddly distant from the
obverses.

(2) One is evenly distant and three oddly distant.

(3) Two are evenly distant and two oddly.

(4) Three are evenly distant and one oddly.

In the first case the four marks form a group. If
this is a proper group, the pair of obverses must be
either the origin and obverse of the group, or a pair of
mediates ; 2 types. If the group is improper, the pair
must be an origin and an obverse ; 1 type. In the
second case we have an origin, an obverse, and a
mediate, to which we must add 3 marks taken out of
the proximates and ultimates. We may add 3 proxi-
mates distant respectively 113 or 133 from the
mediates (2 types), or 2 proximates distant respec-
tively 11, 13, 33 from the mediate, and with each of
these combinations an ultimate distant either 1 or 3 (6
types). To interchange proximates with ultimates clearly
makes no difference ; so that in reckoning the cases of
1 proximate and 2 ultimates or 3 ultimates, we should
find no new types. In the third case we have an origin,
an obverse, and two mediates distant 2 from each other ;
and to these we have to add either two proximates or a
proximate and an ultimate. The two proximates may
be distant from the two mediates 11, 13, or 11, 33, or
1 3, 1 3, or 1 3, 3 3 ; 4 types. The proximate or ultimate
must not be respectively distant 11, 33, or 33, 11;
for then they would form a pair of obverses ; there
remain the cases 1 1 with 11 or 1 3, 1 3 with 1 3, and
3 3 with 1 3 or 3 3 ; 5 types. In the fourth case we have

VOL. II. H



98 ON THE TYPES OF COMPOUND STATEMENT

an origin, obverse, and three mediates distant 2 from
another ; the remaining mark must be distant either
1 or 3 from these mediates ; 2 types. This makes
twenty-two types of sixfold statement with one pair
of obverses.

12. If a sixfold statement contains two pairs of
obverses, these must be either evenly or oddly distant.
If they are evenly distant we have an origin, obverse,
and two obverse mediates, to which two other marks
are to be added. These may be both evenly distant ;
taking one of them as origin, it is associated with 5
mediates, so that there is 1 type only. Or both oddly
distant ; here there are two cases, according as the dis-
tances are 11, 33, or 1 3, 1 3. Or one oddly and one
evenly distant ; the latter is any one of the four remain-
ing mediates, and then the former is distant 1 or 3 from
it ; 2 types. If the two pairs of obverses be oddly dis-
tant they form an aggregate which is related in the
same way to all the remaining twelve marks ; viz. any
one of these being taken as origin, we have a pair of
mediates and a proximate with its obverse ultimate.
The thing to be considered, therefore, is the distance
between the two marks to be added, which may be 1,
2, or 3, and each in two ways ; 6 types.

A sixfold statement with three pairs of obverses is
one of two types only ; viz. these are all evenly distant
when they are the mediates to one origin, or two
evenly distant and one oddly distant from both of
them.

13. A pure sevenfold statement must consist of a
group and a triad ; for it must contain a triad, by the
same reasoning by which this was proved for a fivefold



INVOLVING FOUR CLASSES.



statement ; and then either all the other four marks are
oddly distant from this, and so form a group by them-
selves, or else one of them is evenly distant from the
triad and so forms a group with it. If the group is
proper, being the proximates to a certain origin, the
triad must consist of two mediates and either the origin,
the obverse, or another mediate ; and in the latter case
the three mediates are distant 1 1 1 or 3 3 3 from some
proximate ; 4 types. If the group is improper, the
triad is either all origins or all obverses, or two origins
and an obverse, or an origin and two obverses ; 4 types.
In all, 8 types of pure sevenfold statement.

14. A sevenfold statement with one pair of obverses
must consist either of four marks evenly distant from
one another and three oddly distant from them ; or of
five marks evenly distant from one another and two
oddly distant from them. In the former case the pair
of obverses may be in the four or in the three. If they
are in the four, the three form a triad which are proxi-
mates to one origin ; and then the pair may be the origin
and obverse or a pair of mediates. If the pair are
origin and obverse, the other two (at distance 2) are
mediates, distance 11, 13 or 33 from the proximate
which is not in the triad ; if the pair are mediates, the
two may be the origin or obverse with a mediate distant
1 or 3 from that proximate (4 types), or two mediate
distant 11, 13, 33 from it (3 types). If the pair of
obverses are in the set of three marks, the four form a
group, which may be proper or improper. If proper,
the three may be origin and obverse with a mediate, or
a pair of mediates with origin, obverse, or another
mediate ; 4 types. If improper, the three must be two

H 2



100 ON THE TYPES OF COMPOUND STATEMENT

origins and an obverse, or an origin and two obverses ;
3 types.

Five marks evenly distant containing only one pair
of obverses, must be a proper group with the obverse
of one of its marks ; see end of art 9. To these we
may add the origin or obverse of the proper group with
a mediate distant 1 or 3 from the extra mark, or else
two mediates distant 1 1, 1 3, or 3 3 from that mark ; 7
types.

15. A sevenfold statement with two pairs of ob-
verses may have six marks evenly distant from one
another and one oddly distant from them ; in this case
the six are an origin and five mediates in two different
ways, or say two pairs and a two ; the remaining mark
may be distant 11, 13, or 33 from the two, which
gives 3 types.

Otherwise the sevenfold statement must subdivide
(as in the last case) into five and two or into four and
three. If it subdivide into five and two, the two may
be a pair or not. In the first case we have a proper
group and the obverse of one of its marks, together
with the origin and obverse of the group or a pair of
mediates ; 2 types. In the second case we have five
mediates of an origin or its obverse, to which we may
add two proximates distant 1 1, 1 3 or 3 3 from the old
mediate, or a proximate and an ultimate distant 1 1,
1 3 or 3 3 respectively from the odd mediate ; 6 types.

If the sevenfold statement subdivide into four and
three, the two pairs may be both in the four, or one in the
four and one in the three. In the former case we have
a triad, to which may be added the origin and obverse
and a pair of mediates or two pairs of mediates ; 2



INVOLVING FOUR CLASSES. 101

types. In the latter case the four consists of an origin
and obverse and two mediates ; we must add a pair
consisting of a proximate and an ultimate, which may
be distant 11, 33 or 13, 13 from the two mediates,
and then another proximate or ultimate which may be
distant 1 1, 1 3, or 3 3 from the two mediates ; 6 types.

16. Three pairs of obverses in a sevenfold statement
may be all evenly distant, or two evenly and the other
pair oddly distant from each. If they are all evenly
distant they are the mediates to a certain origin or its
obverse, and the seventh mark may be the origin or a
proximate, 2 types. In the other case we have an
origin, obverse, and pair of mediates together with a
proximate and its obverse ultimate ; we may add
a proximate or a mediate, 2 types.

17. A pure eightfold statement must consist of two
groups, either both proper or both improper, or one
of each. Two proper groups may have their origins
distant 1 or 3 ; 2 types. To an improper group we
may add a proper group made of one origin and three
obverses or of three origins and one obverse, or an
improper group made of four origins or four obverses
or two origins and two obverses ; 5 types. Altogether
there are 7 types of pure eightfold statement.

18. An eightfold statement with one pair of obverses
must subdivide into four and four, or into five and
three. In the former case we have a pair of obverses,
viz. an origin and its obverse, and two mediates ; to
which we must add a group formed out of the proxi-
mates and ultimates. This group may be proper, (1
type), or improper, the mediates being in regard

two origins, two obverses, or an origin and an




102 ON THE TYPES OF COMPOUND STATEMENT

3 types. In the latter case the five marks must be a
proper group with the obverse of one mark, to which
we must add a triad made out of the origin, obverse,
and mediates of the group. This triad may be the
origin or obverse together with two mediates distant 1 1,
1 3, 3 3 from the ultimate, 6 types ; or else it may be
three mediates distant 111, 113, 133, 333 from the
ultimate, 4 types.

19. An eightfold statement- with two pairs of
obverses must subdivide into four and four, or into five
and three, or into six and two. In the first case the
two pairs of obverses may be evenly distant, when the
remaining marks form a group either proper, with its
origin, obverse, and pair of mediates, or two pairs of
mediates, or else improper, 3 types ; or oddly distant,
when the remainder form one of the six pure fourfold
statements enumerated art. 6. Two marks distant 2 from
each other may be distant 11, 33 or 13, 13 from the
pair of obverses which are oddly distant from them ;
thus each of the six fourfold statements gives 3 types of
eightfold statement, except the third, which gives 4 ;
in all 19. In the second case the three may be a triad
or may contain a pair of obverses. If it is a triad, the
five are mediates to one origin and its obverse, and we
add three proximates distant 113 or 133 or two
proximates distant 11, 13 or 33, with an ultimate dis-
tance respectively 11 or 33 from the old mediate ; 6
types. If the three contain a pair of obverses, the five
make a proper group with obverse of one mark ; to this
we may add the origin and obverse of the group with
mediate distant 1 or 3 from the ultimate, or a pair of






INVOLVING FOUR CLASSES. 103



obverse mediates with a mediate distant 1 or 3 as
before ; 4 types. In the third case the six must be an
origin and five mediates, and we may add two proxi-
mates distant 11, 13, 33 from the old mediate, or a
proximate and an ultimate, or two ultimates, distant as
before ; 9 types.

20. In an eightfold statement with three pairs of ob-
verses these may be either all evenly distant, or two of
them evenly distant and the other oddly distant from
both. In the first case they are mediates to a certain
origin and its obverse, and we may add the origin with a
proximate or ultimate, two proximates, or a proximate
and ultimate ; 4 types. In the second case take the
oddly distant pair for origin and obverse ; then these
are associated with two proximates and their obverse
ultimates, and we may add the two other proximates,
a proximate and an ultimate, a proximate and a mediate
(distant 11, 13, 31, 33 from this proximate and the
remaining one), or two mediates distant 1 1, 3 3 or 13,
13 from the two proximates ; 8 types.

Lastly, in an eightfold statement with four pairs of
obverses they may be all evenly distant, or the statement
may subdivide into six and two, or into four and four ;
in the latter case there are 2 types.

21. To obtain the whole number of types, we
observe that for every less-than-eightfold type there is
a complementary more-than-eightfold type (art. 2) ; so
that we must add the number of eightfold types (78) to
twice the number of less-than-eightfold types (159) ;
the result is 396.



104 ON THE TYPES OF COMPOUND STATEMENT



TABLE.
Art.

4. 1-fold 1

2-fold, distance 1, 2, 3, 4 4

5. 3-fold, pure, distance 112, 222, 123, 233 . . . 4

Ipair obv., dist. 134, 224 2

6

6. 4-fold, pure, two and two :

1 |l 1 [1 Ijl 1)3 1 |'S 3 J3
"l| l"l | 3' 'SJT '3TT '3J3' '



three and one 4

group, proper or improper . . .2

12 12

7. 1 pair obv 5

2 pair obv., dist. odd or even 2

8. 6-fold, pure, three and two .... 7

four and one .... 6

12

9. 1 pair obv. + two prox 6

+ prox. + ult 3

+ two med 4

32

12
2 pair obv., odd dist., 1 ; even, 2 . . . .3



27 27



10. 6-fold, pure, three and three .... 6

four and two . ... 6

12

12

11. 1 pair obv., two and four . . .3

three and three . . .8
four and two . . .0
five and one .... 2

22

22

12. 2 pair obv., odd dist., 6 ; even, 6 . . . .11

3 pair obv 2



47 47



INVOLVING FOUR CLASSES. 105

Art.

13. 7-fold, pure ; proper group, 4 ; improper, 4 ... 8

14. 1 pair obv., four and three . . .10

three and four . . .7


1 2 3 4 5 7 9 10 11 12 13 14 15 16 17 18 19 20 21 22

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