From: Masoud Nikravesh (nikravesh@eecs.berkeley.edu)
Date: Sun Sep 30 2001 - 11:52:06 MET DST

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Causality is Undefinable - Toward a Theory of Hierarchical Definability

Attempts to formulate mathematically precise definitions of basic concepts such
as causality, randomness, and probability have a long history. The concept of
generalized definability that is outlined in the following suggests that such
definitions may not exist. Furthermore, it suggests that existing definitions of
many basic concepts, among them those of stability, statistical independence and
Pareto-optimality, may need to be redefined.
In essence, definability is concerned with whether and how a concept, X, can be
defined in a way that lends itself to mathematical analysis and computation. In
mathematics, definability of mathematical concepts is taken for granted. But as
we move farther into the age of machine intelligence and automated reasoning,
the issue of definability is certain to grow in importance and visibility,
raising basic questions that are not easy to resolve.
To be more specific, let X be the concept of, say, a summary, and assume that I
am instructing a machine to generate a summary of a given article or a book. To
execute my instruction, the machine must be provided with a definition of what
is meant by a summary. It is somewhat paradoxical that we have summarization
programs that can summarize, albeit in a narrowly prescribed sense, without
being able to formulate a general definition of summarization. The same applies
to the concepts of causality, randomness, and probability. Indeed, it may be
argued that these and many other basic concepts cannot be defined within the
conceptual framework of classical logic and set theory.
The point of departure in our approach to definability is the assumption that
definability has a hierarchical structure. Furthermore, it is understood that a
definition must be unambiguous, precise, operational, general, and co-extensive
with the concept it defines.
The hierarchy involves five different types of definability. The lowest level is
that of c-definability, with c standing for crisp. Thus, informally, a concept,
X, is c-definable if it is a crisp concept, e.g., a prime number, a linear
system, or a Gaussian distribution. The domain of X is the space of instances to
which X applies.
The next level is that of f-definability, with f standing for fuzzy. Thus, X is
a fuzzy concept if its denotation, F, is a fuzzy set in its universe of
discourse. A fuzzy concept is associated with a membership function that assigns
to each point, u, in the universe of discourse of X, the degree to which u is a
member of F. Alternatively, it may be defined algorithmically in terms of other
fuzzy concepts. Examples of fuzzy concepts are small number, strong evidence,
and similarity. It should be noted that many concepts associated with fuzzy sets
are crisp concepts. An example is the concept of a convex fuzzy set. Most fuzzy
concepts are context-dependent.
The next level is that of f.g-definability, with g standing for granular, and
f.g denoting the conjunction of fuzzy and granular. Informally, in the case of a
concept which is f.g-granular, the values of attributes are granulated, with a
granule being a clump of values that are drawn together by indistinguishability,
similarity, proximity, or functionality. f.g-granularity reflects the bounded
ability of the human mind to resolve detail and store information. An example of
an f.g-granular concept that is traditionally defined as a crisp concept, is
that of statistical independence. This is a case of misdefinition--a definition
that is applied to instances for which the concept is not defined, e.g., fuzzy
events. In particular, a common misdefinition is to treat a concept as if it
were c-definable, whereas in fact it is not.
The next level is that of PNL-definability, where PNL stands for Precisiated
Natural Language. Basically, PNL consists of propositions drawn from a natural
language that can be precisiated through translation into what is called
precisiation language. An example of a proposition in PNL is: It is very
unlikely that there will be a significant increase in the price of oil in the
near future.
In the case of PNL, the precisiation language is the Generalized Constraint
Language (GCL). A generic generalized constraint is represented by Z isr R,
where Z is the constrained variable, R is the constraining relation and r is a
discrete-valued indexing variable whose values define the ways in which R
constrains Z. The principal types of constraints are: possibilistic (r = blank);
veristic (r = v); probabilistic (r = p); random set (r = rs); usuality (r = u);
fuzzy graph (r = fg); and Pawlak set (r = ps). The rationale for constructing a
large variety of constraints is that conventional crisp constraints are
incapable of representing the meaning of propositions expressed in a natural
language--most of which are intrinsically imprecise--in a form that lends itself
to computation.
The elements of GCL are composite generalized constraints that are formed from
generic generalized constraints by combination, modification, and qualification.
An example of a generalized constraint in GCL is ((Z isp R) and (Z,Y) is S) is
unlikely.
By construction, the Generalized Constraint Language is maximally expressive.
What this implies is that PNL is the largest subset of a natural language that
admits precisiation. Informally, this implication serves as a basis for the
conclusion that if a concept, X, cannot be defined in terms of PNL, then, in
effect, it is undefinable or, synonymously, amorphic.
In this perspective, the highest level of definability hierarchy, which is the
level above PNL-definability, is that of undefinability or amorphicity. A
canonical example of an amorphic concept is that of causality. More
specifically, is it not possible to construct a general definition of causality
such that given any two events A and B and the question, "Did A cause B?" the
question could be answered based on the definition. Equivalently, given any
definition of causality, it will always be possible to construct examples to
which the definition would not apply or yield counterintuitive results.
In dealing with an amorphic concept, X, what is possible--and what we generally
do--is to restrict the domain of applicability of X to instances for which X is
definable. For example, in the case of the concept of a summary, which is an
amorphic concept, we could restrict the length, type, and other attributes of
what we want to summarize. In this sense, an amorphic concept may be partially
definable or, p-definable, for short. The concept of p-definability applies to
all levels of the definability hierarchy.
The theory of hierarchical definability is not a theory in the traditional
spirit. The definitions are informal and conclusions are not theorems.
Nonetheless, it serves a significant purpose by raising significant questions
about a basic issue--the issue of definability of concepts that lie at the
center of scientific theories.
Lotfi A. Zadeh is Professor in the Graduate School and director, Berkeley
initiative in Soft Computing (BISC), Computer Science Division and the
Electronics Research Laboratory, Department of EECs, Univeristy of California,
Berkeley, CA 94720-1776; Telephone: 510-642-4959; Fax: 510-642-1712;E-Mail:
zadeh@cs.berkeley.edu. Research supported in part by ONR Contract
N00014-99-C-0298, NASAContract NCC2-1006, NASA Grant NAC2-117, ONR Grant
N00014-96-1-0556, ONR Grant FDN0014991035, ARO Grant DAAH 04-961-0341 and the
BISC Program of UC Berkeley

```--
Dr. Masoud Nikravesh
Chairs: BISC-SIG-FLINT,ES, RT
Berkeley Initiative in Soft Computing (BISC)
Computer Science Division- Department of EECS
University of California, Berkeley, CA 94720
Phone: (510) 643-4522; Fax: (510) 642-5775
Email: Nikravesh@cs.berkeley.edu
URL: http://www.cs.berkeley.edu/~nikraves/
Staff Scientist
Lawrence Berkeley National Lab,
Imaging and Collaborative Computing Group
Masoud@media.lbl
URL: http://vision.lbl.gov/
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