Truths about Simpson’s Paradox: Saving the Paradox from Falsity - Logic and Its Applications - page 58

Truths about Simpson's Paradox:
Saving the Paradox from Falsity
Prasanta S. Bandyopadhyay 1 , R. Venkata Raghavan 2 , Don Wallace Dcruz 2
andGordonBrittanJr. 1
1 Department of Philosophy, Montana State University, Bozeman, USA
[e-mail], [e-mail]
2 Department of Philosophy, University of Hyderabad, Hyderabad, India
{ raghavan.rv,don.wallace } @uohyd.ac.in
Abstract. There are three questions associated with Simpson's paradox
(SP): (i) Why is SP paradoxical? (ii) What conditions generate SP? and
(iii) How to proceed when confronted with SP? An adequate analysis
of the paradox starts by distinguishing these three questions. Then, by
developing a formal account of SP, and substantiating it with a counter-
example to causal accounts, we argue that there are no causal factors at
play in answering questions (i) and (ii). Causality enters only in connec-
tion with action.
Keywords: Simpson's Paradox, Formal Analysis, Collapsibility Princi-
ple, Inference rules, Causal Accounts, Definition of Paradox, First-level-
truth, Second-level-truth.
1 Overview
In his recent topic, Saving Truth from Paradox , Hartry Field discusses the philo-
sophical significance of paradoxes. According to him, “[a]ny resolution of the
paradoxes will involve giving up (or at least restricting) some very firmly held
principles:... [and] [t]he principles to be given up, are the ones to which the aver-
age person simply can't conceive of alternatives. That's why the paradoxes are
paradoxes .” [4, p.17]. Their significance and the firmly held principles which we
have to give up in resolving them is a recurring theme in philosophical logic.
We will illustrate this in the case of Simpson's paradox (SP), which involves the
reversal of the direction of a comparison or the cessation of an association when
data from several groups are combined to form a single whole [17]. At least three
distinct questions are important in understanding the nature of the paradox: (i)
Why or in what sense, is SP a paradox? (ii) What are the conditions in which
the paradox arises? (iii) How should one proceed when confronted with a typi-
cal case of the paradox, hereafter to be called the “what-to-do” question? 1 The
three questions are distinct: answering one of them does not entail answers to the
1 Daniel Hausman was perhaps the first philosopher who drew our attention to the
significance of these three types of questions (in an email communication).