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a situation is a case of SP if and only if:
C
≡
(
C
1
&
C
2
&
C
3
)
(a)
ʱ
)
>
0
4
(b)
Both (a) and (b) are necessary conditions, but they jointly constitute sucient
conditions for generating SP [1].
5
Both conditions for the paradox generate two
key theorems which specify the relationship between the two acceptance rates
in both sub-populations. These are: 1.
A
1
C
4
≡
ʸ
=(
A
1
−
B
1
)+(
A
2
−
B
2
)+(
ʲ
−
=
B
2
.Table3shows
why the condition for Theorem 1 needs to hold. Since
A
1
=
A
2
, i.e., 25% = 25%,
no paradox results. Similarly, in Table 4, since
B
1
=
B
2
, i.e., 25% = 25%, the
paradox does not occur. Proofs of these theorems are provided in the appendix.
=
A
2
,and2.
B
1
Tabl e 3.
No SP (
A
1
=
A
2
)
Acceptance
Rates
Dept 1
Dept 2
Overall
Acceptance Rates
Two
Groups
Accept Reject Accept Reject
Dept 1 Dept 2
Females
75
225
75
225
25%
25%
25%
Males
10
90
20
80
10%
20%
15%
Tabl e 4.
No SP (
B
1
=
B
2
)
Acceptance
Rates
Dept 1
Dept 2
Overall
Acceptance Rates
Two
Groups
Accept Reject Accept Reject
Dept 1 Dept 2
Females
10
90
20
80
10%
20%
15%
Males
75
225
75
225
25%
25%
25%
There are four points worth mentioning. First, Clark Glymour [5] would call our
account an application of the “Socratic method” in which we provide necessary
and sucient conditions for the analysis of a concept.
6
Second, the character-
ization of the puzzle in terms of our two conditions captures the paradoxical
4
See Blyth [3] for similar conditions. However, our conditions and notations are
slightly different from his.
5
See [6], [16]. The latter paper shows that SP reversal involves Boolean disjunction
of events in an algebra rather than being restricted to cells of a partition.
6
Glymour contrasts this method with what he calls the “Euclidean”-method based
theories where one could derive interesting consequences from them although
Euclidean-method based theories, according to him, are invariably incomplete. It
is interesting to note two very different points. First, although Glymour is not fond
of the Socratic-method on which, however, a large part of the western philosophical
tradition rests, our Socratic-method based logical account at the same time is also
able to generate some interesting logical consequences (See [1,2]). Second, it is not
only the Greeks who applied this method. In classical Indian philosophical tradi-
tion, the Socratic method is also very much prevalent where a definition of a term
is evaluated in terms of whether it is able to escape from being both “too narrow”
and “too wide.”
