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In-Depth Information
The sentence
3
ʲ
1
(
a
1
)
∧
ʲ
1
(
a
2
)
∧
ʲ
1
(
a
3
)
∧
ʲ
4
(
a
4
)
∧
ʳ
[1
,
1
,
1
,
1]
(
a
j
,a
l
)
∧
ʳ
[1
,
4
,
1
,
2]
(
a
j
,a
4
)(7)
1
≤j<l≤
3
j
=1
is a state description for
a
1
,a
2
,a
3
,a
4
and the sentence
ʲ
1
(
a
1
)
∧
ʲ
1
(
a
2
)
∧
ʲ
1
(
a
3
)
∧
ʲ
3
(
a
4
)
∧
ʳ
[1
,
1
,
1
,
1]
(
a
1
,a
2
)
∧
ʳ
[1
,
3
,
1
,
2]
(
a
1
,a
4
)
( )
is a partial state description for
a
1
,a
2
,a
3
,a
4
.
Imagine a simple situation where individuals
a
1
,a
2
,...
could do just two
things: think other individuals (also themselves) to be good cooks or not, and
like to eat fish or not. If we interpret each
a
j
as
a
j
,
R
(
x
)as'
x
likes to eat fish'
and
Q
(
x, y
)as'
x
thinks
y
is a good cook', then (7) says that
a
1
,a
2
,a
3
all like
to eat fish and think everybody including themselves to be good cooks and that
a
4
does not like to eat fish and does not think anybody including him/herself
to be a good cook; (8) says that
a
1
,a
2
,a
3
all like to eat fish and each thinks
him/herself to be a good cook,
a
4
does not like to eat fish but thinks him/herself
to be a good cook,
a
1
and
a
2
think each other to be good cooks and
a
1
also
thinks
a
4
to be a good cook but
a
4
does not think
a
1
to be a good cook.
Now consider in light of this example what a binary variant of Johnson's
Sucientness Postulate might be. It appears reasonable that for a partial state
description
ʘ
(
a
1
,...,a
m
) as given by (2), the probability of an extension of it by
some
ʲ
k
(
a
m
+1
) (how a new individual behaves in isolation) would depend only
on the
ʲ
v
j
(how other individuals behave in isolation) rather than the
ʳ
h
j,l
,and
an extension of it by some
ʳ
[
k,c,h,d
]
(
a
s
,a
t
)for1
A
(how
a
s
and
a
t
relate to each other given how each of them behaves in isolation) would
depend only on those
ʳ
h
j,l
where
a
j
and
a
l
behave in isolation just as
a
s
and
a
t
do.
Taking this a step further along the lines of the unary Johnson's Sucient-
ness Postulate and using the notation from page 210, this appears to lead to
the requirement that the conditional probability of
ʲ
k
(
a
m
+1
)given(2)should
depend only on
m
k
and
m
,andfor
k
=
v
s
,
c
=
v
t
the conditional probability of
ʳ
[
k,c,h,d
]
(
a
s
,a
t
) given (2) should depend only on
n
[
k,c,h,d
]
and
n
k,c
.
However, there is a little catch when
k
=
c
and
h
≤
s<t
≤
m
,
s, t
∈
/
=
d
for the following reason.
Assuming again that
k
=
v
s
,
c
=
v
t
,for
k
=
c
the conditional probability
of
ʳ
[
k,c,h,d
]
(
a
s
,a
t
) given (2) equals the probability of increasing
n
[
k,c,h,d
]
by 1
using
a
s
,a
t
since only
ʳ
[
k,c,h,d
]
(
a
s
,a
t
), not
ʳ
[
c,k,d,h
]
(
a
s
,a
t
), is consistent with
(2). Similarly when
k
=
c
and
h
=
d
because the two possibilities become the
same. But when
h
=
d
,both
ʳ
[
k,k,h,d
]
(
a
s
,a
t
),
ʳ
[
k,k,d,h
]
(
a
s
,a
t
) can extend (2)
and hence increasing
n
[
k,k,h,d
]
by 1 using
a
s
,a
t
canbedoneintwoways;the
conditional probability of each should therefore arguably depend on
n
[
k,k,h,d
]
and
n
k,k
differently than when
k
=
c
or
k
=
c, h
=
d
.
Binary Carnap Continuum
Motivated by the above, we will say that the atoms
ʳ
[
k,k,h,d
]
where
h
=
d double
,
and we shall consider the following principle.
