Dear all:
Permit me to add a comment on some of the issues which
emerged in the course of our recent discussions about causality. The
issues in question are of basic importance, especially in the realm of law.
First, it is almost always the case that an event, B, is
caused not by a single event, A0, but by a multiplicity of collateral
events, A1, A2, ..., some obvious and some not. For example, what caused
yesterdays decline in the stock market? What caused John to leave Mary?
Clearly, the collateral events have different degrees of
materiality as possible causes of B. For example, in the raincoats
problem, rainy weather may have a higher degree of materiality than a
minor change in styling. Thus, if you accept that an event may have a
multiplicity of causes, then you must accept that causality is a matter
of degree.
What should be underscored is that the degree to which A
caused B is not the same as the probability that A caused B. More
specifically, the degree to which A caused B should be interpreted as
the truth value of the proposition "A caused B;" rather than as the
probability that A caused B. In fact, a basic shortcoming of
bivalent-logic-based probability theory is that it does address
partiality of certainty but does not address partiality of truth.
Second, the collateral events could be structured as a
hierarchy or, more generally, a network, with events at the top having a
higher degree of materiality than those at the bottom. The problem is
that different judges, observers and analysts may construct different
hierarchies. To qualify as a theory of causality, a theory should
provide an objective method of constructing a hierarchy of causes, with
each cause associated with a degree of causal strength, dependence or
materiality. No existing theory has this capability. A concept which may
be of relevance is that of a fuzzy cognitive map due to Bart Kosko.
Third, there is a linkage between causality and
responsibility but the two concepts are not coextensive. For example, if
a child sets a house on fire by playing with matches which were left in
the house by parents, then it would be correct to say that the child
caused the fire, but it would be the parents who would be held
responsible. Thus, a hierarchy of causes would not be the same as a
hierarchy of responsibilities. Clearly, the problem of how to associate
a hierarchy of responsibilities with a hierarchy of causes should have a
position of centrality in the realm of law.
What is the rationale for granulation of degrees of causal
dependence? Basically, granulation is a form of fuzzy quantization. For
example, coarse granular values of age are young, middle-aged and old;
coarse granular values of height are short, medium and tall; and coarse
granular values of probability are low, medium and high. Such granular
values may be viewed as labels of fuzzy subsets of the real line.
Granulation is a necessity when the available information
about a variable is not sufficient to assign a numerical value to it;
and granulation is an option when there is a tolerance for imprecision
which may be exploited to achieve tractability, robustness and lower
cost. When we deal with degrees of causality, granulation is a necessity
because of the intrinsic imprecision of world knowledge and case-based
knowledge--the bodies of knowledge which we rely upon to estimate the
degrees of causal dependence. It is my belief--as stated in an earlier
comment--that, in realistic settings, the degree of causal dependence
should be represented as a bimodal distribution. In such distribution,
an ordered pair of the form (low; high) means that the probability that
the degree of causal dependence is high, is low. In effect, a bimodal
distribution associates a granular value of probability with a granular
value of the degree of causal dependence. Thus, a bimodal distribution
generalizes the artifice of association of probability with causal
dependence, in the spirit of the elegant theories of Joe Halpern et al.
I have been told repeatedly that existing theories can deal
with the raincoats example. But so far I have not seen a demonstration
of how this can be done. Will I see such a demonstration? For reasons
articulated in my comments, I doubt that this will happen.
Regards to all
Lotfi
--
Professor in the Graduate School, Computer Science Division
Department of Electrical Engineering and Computer Sciences
University of California
Berkeley, CA 94720 -1776
Director, Berkeley Initiative in Soft Computing (BISC)
