Thursday, May 10, 2007

The Impossibility Principle

Lotfi Zadeh, Extended Abstract, From Fuzzy Logic to Extended Fuzzy Logic—The Concept of F-validity and the Impossibility Principle, Lecture given at FUZZ-IEEE 2007, Imperial College, London, UK:
In an environment of imprecision, uncertainty, incompleteness of information, conflicting goals and partiality of truth, p-validity [provable validity; a provably valid solution] is not, in general, an achievable objective.
It does not follow, of course, Zadeh asserts, that logic dissolves into incoherence. Instead, logic becomes fuzzy -- in the extended sense that Zadeh describes in this lecture and on other occasions and in other publications.

The conception is bold. I am not a logician. But I would hesitate to dismiss Zadeh's (attempt at) radical (re)conceptualization of logic. Zadeh has grounds for making the following assertion (id.):

Fuzzy logic has come of age.
He adds:
During much of its early history, fuzzy logic has been an object of skepticism and derision, in part because fuzzy is a word which is usually used in a pejorative sense. Today, fuzzy logic is used in a wide variety of products and systems ranging from cameras, home appliances, medical instrumentation and automobiles to elevators, industrial control, subways, fraud detection and traffic control systems.
Whether or not fuzzy logic makes sense and whether or not it "works," Zadeh is plainly right in saying and complaining:
[T]here are still many misconceptions about fuzzy logic. To begin with, fuzzy logic is not fuzzy. Basically, fuzzy logic is a precise logic of imprecision.
Furthermore, there is more to fuzzy logic, he rightly adds, than the concept of a fuzzy set. However, for want of technical proficiency, I will not even begin to try to recount or summarize Zadeh's account of the four principal facets of fuzzy logic. Instead, I limit myself to quoting this statement:
More specifically, in fuzzy logic everything is or is allowed to be graduated, that is, be a matter of degree or, equivalently, fuzzy. Furthermore, in fuzzy logic everything is or is allowed to be granulated, with a granule being a clump of attribute values drawn together by indistinguishability, equivalence, similarity, proximity or functionality.
What does this mean? The answer cannot be simple. My intuitions are too poor to help me out here.

Zadeh proceeds to talk about natural language and he asserts (as he has done before) that "a natural language is viewed as a system for describing perceptions." He then proceeds to describe a program for the development of a logic or -- more precisely stated -- "a maximally expressive constraint definition language" that can "serve as a meaning representation/precisiation language for natural languages."

It is intriguing and revealing that Zadeh views the ability to use computations to mimic or manipulate (natural) words as almost equivalent to, or very closely related to, the ability to interpret perceptions:

Since a natural language is a system for describing perceptions, NL-Computation is closely related to computation with perception-based information. NL-capability is the capability of a theory to operate on information described in natural language or, equivalently, to operate on perception-based information. The importance of NL-capability derives from the fact that much of human knowledge is expressed in natural language.
The last sentence in the above quotation bears emphasis: Zadeh asserts (correctly, I think) that much genuine human knowledge is embedded in, or carried by, ordinary words [natural language].

Zadeh, as before, does not hesitate to embrace inference rules that look very different from the sorts of inference rules we are accustomed to seeing in traditional deductive and traditional if-then logic -- but, note, Zadeh refuses to cede any ground to traditional bivalent logic and insists on calling his new inference rules rules for drawing deductions:

Deduction in fuzzy logic is governed by a collection of rules of deduction which, in the main, are rules that govern propagation and counterpropagation of generalized constraints. The principal rule is the extension principle. Extension principle has many versions. The simplest version (Zadeh 1965) is the following. Let f be a function from reals to reals, Y=f(X). What we know is that X is A, where A is a fuzzy subset of the real line. Equivalently, what we know about X is its granular value, that is, its possibility distribution, A. What can be said about Y, that is, what is its granular value or, equivalently, its possibility distribution? In a more general form, (Zadeh 1975) X is A is replaced by f(X) is A. It is this form that is used in most practical applications. In a form that is used in fuzzy control, what is granulated is f, resulting in a granular function, f*, which is defined by a collection of fuzzy-if-then rules. More generally, the extension principle may be viewed as follows. Let Z =f(X), where X is a real-valued variable. Assume that we can compute Z for singular values of f and X. Basically, the extension principle serves to extend the definition of Z to granular values of f and X.
What does this all mean -- precisely? I am the wrong person to ask.

But to the eyes of this amateur, this ingenue, and this reckless autodidact, Zadeh's theory strikes me as one that must be taken very, very seriously.

In any event(!): There are delicious observations in Zadeh's abstract. For example:

Turning to Case 2, we observe that, in general, precision carries a cost.
This point is -- in some sense -- indubitably correct. In what sense? Well then, read Zadeh and then think about the question. And then -- and only then -- render your opinion.

Zadeh does not claim that he has already developed the broad sort of fuzzy logic that he thinks is required. He writes:

The concepts of extended fuzzy logic, FL+, and f-validity which are sketched in the following represent a more radical development. In essence, extended fuzzy logic may be viewed as an attempt at legitimizing the concept of fuzzy theorem (Zadeh 1975) and fuzzy validity. In large measure, the move from fuzzy logic, FL, to extended fuzzy logic, FL+, is a move into as yet uncharted territory.
Zadeh does not hesitate to stare directly at the seemingly anomalous, or paradoxical, character of the sort of logic he yearns to develop and justify:
A conclusion which is of key importance is that there are no crisp theorems in f-geometry.
There are no crisp arguments! What a baffling, strange, and intriguing proposition!
  • But, of course, lawyers are thoroughly familiar with this strange proposition: none of their arguments are "crisp." But they are full of arguments. And many lawyers even think that their arguments are arguments. They should therefore -- by all rights -- read Zadeh. There they will find a stout defender of their craft and of law's peculiar logic.
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