Information Technology Reference
In-Depth Information
As we can see, CP is a numerical inference principle devoid of any causal
intuition. Here is the reconstruction of type I version of SP:
(1) Female and male populations are mutually exclusive and jointly exhaustive;
one can't be a student of both departments and satisfy the two conditions of SP.
(2) The acceptance rate of females is higher than that of males in Department
1. (observed from data)
(3) The acceptance rate of females is higher than that of males in Department
2. (observed from data)
(4) If (2) and (3) are true, then the acceptance rate for females is higher than
that of males overall. (from CP1)
(5) Hence the acceptance rate for females is higher than that of males overall.
(from (2), (3) and (4))
(6) However, fewer females are admitted overall. (observed from data)
(7) Overall acceptance rate for females is both higher and lower than that of
males. (from (5) and (6))
In our derivation of the paradox, premise (4) plays a crucial role. In type I
version of SP, as given in Table 1, CP1 does not hold (
A
1
>B
1
and
A
2
>B
2
,
but
ʱ<ʲ
). That CP1 is not generally true is shown by our derivation of a
contradiction. The same result can be obtained for Type II version of SP in
Table 2 where CP2 has to be given up if the paradox is to be avoided.
Our answer to the first question, (i), then, is simply that humans tend to
invoke CP uncritically, as a rule of thumb, and thereby make mistakes in certain
cases about proportions and ratios; they find it paradoxical when their usual
expectation that CP is applicable across the board, turns out to be incorrect.
And the reason we think people invoke CP uncritically, is its remarkable (formal)
resemblance with the two inference rules given below.
7
1. In elementary algebra, the following truth holds for real numbers:
x
1
>y
1
x
2
>y
2
∴
(
x
1
+
x
2
)
>
(
y
1
+
y
2
)
While it is correct to substitute
A
1
(
f
1
/F
1
)for
x
1
,
B
1
(
m
1
/M
1
)for
y
1
,
A
2
(
f
2
/F
2
)
for
x
2
and
B
2
(
m
2
/M
2
)for
y
2
, people might confuse (
x
1
+
x
2
)and(
y
1
+
y
2
)for
ʱ
((
f
1
+
f
2
)
/
(
F
1
+
F
2
)) and
ʲ
((
m
1
+
m
2
)
/
(
M
1
+
M
2
)) respectively, leading them
to think that CP is also a mathematical truth. Thus, mistakes about proportions
and ratios could lead the average person to see a superficial resemblance between
CP and the above mathematical truth.
2. In propositional logic, the following rule is valid:
P
1
→
Q
(A)
P
2
→
Q
(B)
∴
(
P
1
∨
P
2)
→
Q
(C)
7
We are thankful to Joseph Hanna and John G. Bennett for helpful emails on this
point.
