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In our case, let P1 = “A student applies to Department 1”, P2 = “A student
applies to Department 2” and Q = “The student has greater chance of being
accepted, if the gender of the student is female”. Now, (A) partially captures
the condition
A
1
>B
1
whereas (B) partially captures
A
2
>B
2
.(C),which
reads, “If a student applied to Department 1 or Department 2 then, the student
has greater chance of being accepted if the gender of the student is female” re-
sembles the condition
ʱ>ʲ
. We do not suggest that propositional logic can
capture the essence of the paradox. The reasoning leading to SP involves proba-
bilistic considerations which, unlike propositional logic, is not truth-functional.
For example, the probability of a disjunction is not a function of the probability
of its disjuncts. Likewise, SP is a weighted average of probabilities, or, in other
words, averages of averages. No such concept of weighted average exists in truth-
functional logic. The above comparison of CP with a valid propositional rule no
more than suggests why people tend to use CP even in cases where it leads to
contradiction.
3 Causal Accounts of SP
3.1 Pearl's Account
Pearl argues that the arithmetical inferences in SP seem counter-intuitive only
because we commonly make two incompatible assumptions, that causal relation-
ships are governed by the laws of probability and that causal relationships are
more
stable
than probabilistic relationships [9, pp. 180, 25]. Once we reject ei-
ther of these assumptions, and he opts for rejecting the first, the “paradox” is no
longer paradoxical. On the other hand, when we fail to distinguish causal from
statistical hypotheses, the paradox results.
Pearl makes two basic points. One, SP is to be understood in causal terms
for its correct diagnosis. In the type I version, for example, the effect on “accep-
tance” (A) of the explanatory variable, “gender” (G), is hopelessly mixed up (or
“confounded”) with the effects on A of the other variable, “department” (D).
We are interested in the direct effect of G on A and not an indirect effect by
way of another variable like D. His other point is that causal hypotheses, which
support counterfactuals, often cannot be analyzed in statistical terms. Suppose
we would like to know Bill Clinton's place in US history had he not met Monica
Lewinsky. The counterfactual for the causal hypothesis is “Clinton's status in
the US history would be different had he not met Monica Lewinsky” [9, p. 34].
However, there is no statistical model one could construct that would provide
the joint occurrence of 'Clinton' and 'no Lewinsky'. There simply are no appro-
priate data, as there are, for instance, in the fair coin-flipping experiments where
the model about flipping a coin and data about it are well known.
3.2 SGS Account
Spirtes, Glymour, and Scheines suggest a subject-matter-neutral automated
causal inference engine that provides causal relationships among variables from
