Fuzzy Logic

Welcome to another issue of “Intractable Problems”.  In this issue I want to stimulate your thinking skills by discussing fuzzy logic, a logic system that evaluates truth in shades of gray.

Ah, logic, the weapon of reason and the nemesis of emotion! Logic, epitomized by Spock, Aristotle, and the occasional departmental reorganization.

We all know that classical logic deals with propositions that are required to be either true or false. However, this assumption has been questioned since the time of Aristotle. In his writings Aristotle argued that the truth status of matters cannot be established for future events. Aristotle maintained that propositions about future events are neither true or false, but potentially either; hence their truth value is undetermined.

This leads me to wonder: If a politician speaks on future events, then what can we say about the truth of the politician’s statements? Will taxes really go down?  Will the new road get built this year?

Anyway, all kidding aside, how do we solve problems in the absence of complete truth? Well, if we know that something is absolutely true then we can assign it a truth value of 1. Similarly, if we know it to be absolutely false, then it has a value of 0. If we know nothing about its truth or falsehood, then we can give it a value of ½. Our degree of uncertainty, our fuzziness about a proposition, can be used throughout our logic as a confidence level in the validity of our statements.

So, if we can now ascertain the fuzziness of truth, then how do we propagate this fuzziness, particularly when we conjoin truth clauses such as (A and B) or think about the disjunction such as (A or B).  Zadeh[1], the founder of the theory, discussed methods to do this where conjunction was related to the product of the fuzziness and disjunction was related to the sum of the fuzziness.  With these methods we can now rework all our famous theorems in logic to discover their fuzzy complements.  We can apply fuzzy logic in new ways, because it is exciting to think of our world in a non-crisp way, as most things in our world appear to be.

To explain the significance of fuzzy logic, it is appropriate to quote Zadeh:

“Essentially, our contention is that the conventional quantitative techniques of system analysis are intrinsically unsuited for dealing with humanistic systems or, for that matter, any system whose complexity is comparable to that of a humanistic system.

An alternative approach is based on the premise that the key elements in human thinking are not numbers but labels of fuzzy sets. … Indeed, the pervasiveness of fuzziness in human thought processes suggests that much of the logic behind human reasoning is not the traditional two-valued or even multivalued logic, but a logic with fuzzy truths, fuzzy connectives, and fuzzy rules of inference. In our view, it is this fuzzy, and not well understood, logic that plays a basic role in what may well be one of the most important facets of human thinking, namely, the ability to summarize information.”

So, with our new-found knowledge about fuzzy logic, we can now better understand some of the difficulties with some people’s thought processes. Clearly, thinking can be fuzzy!

Fuzzy logic and other technological advances have already helped us to build systems that were previously intractable by their nature and complexity. We can use fuzzy relationships to simplify control processes for complex systems. Did you know that your washing machine may incorporate a fuzzy controller to help determine when the clothes are clean? Or are they perhaps only fuzzily clean? In general, a good problem simplification minimizes the loss of information relevant to the problem of concern. Information and complexity are thus (fuzzily) related.

To better understand human information systems, in our next issue I will discuss the nature and characteristics of information. You will be surprised to find that information is not what you think it is!





Footnotes

1. Zadeh, L.A. “Outline of a new approach to the analysis of complex systems and decision processes.” IEEE Trans. On Systems, Man, and Cybernetics, SMC-1, pp. 28-44.