(Yes or No) but tolerates varying degrees of membership.
Using these criteria, an element is assigned a grade of membership in a parent set. The domain of this attribute is the closed interval [0, 1]. If the grade of membership values is restricted to the two extrema, then fuzzy logic reduces to two‐valued or crisp logic.
In his work, Zadeh proposed that people often base their thinking and decisions on imprecise or nonnumerical information. He further believed that the membership of an element in a set need not be restricted to the values 0 and 1 (corresponding to FALSE and TRUE) but could easily be extended to include all real numbers in the interval 0.0–1.0 including the endpoints. He further felt that such a concept should not be considered in isolation but rather viewed as a methodology that moves from a discrete world to a continuous one. To augment such thinking, he proposed a collection of operations supporting his new logic.
Zadeh introduced his ideas as a new way of representing the vagueness common in everyday thinking and language. His fuzzy sets are a natural generalization or superset of classical sets or Boolean logic that are one of the basic structures underlying contemporary mathematics. Under Boolean algebra, a proposition takes a narrow view that a value is either completely true or completely false. In contrast, fuzzy logic introduces the concept of partial truth under which values are expressed anywhere within, and including, the two extremes of TRUE and FALSE.
Based on the idea of the fuzzy variable, Zadeh (1979) further proposed a theory of approximate reasoning. This theory postulates the notion of a possibility distribution on a linguistic variable. Using this concept, he was able to reason using vague concepts such as young, old, tall, or short. Zadeh also introduced the ideas of semantic equivalence and semantic entailment on the possibility distributions of linguistic propositions. Using these concepts, he was able to determine that a statement and its double negative are equivalent and that very small is more restrictive than small. Such conclusions derive from either the equality or containment of corresponding distributions.
Zadeh's theory is generally effective in reconciling ambiguous natural language expressions. The scope of the work was initially limited to laboratory sentences comparing hair color, age, or height between various people. Zadeh's work provided a good tool for future efforts, particularly in combination with or to enhance other forms of reasoning.
As often follows the introduction of a new concept or idea, questions arise: Why does that thing do this? Why doesn't it do that? Can you make it do another thing? An early criticism of Zadeh's fuzzy sets was: “Why can't your fuzzy set members have an uncertainty associated with them?” Zadeh eventually dealt with the issue by proposing more sophisticated kinds of fuzzy sets. New criteria evolved the original concept into numbered types of fuzzy subsets. His initial work became type‐1 fuzzy sets. Additional concepts grew from type‐2 fuzzy sets to ultimately type‐n in a 1976 paper to incorporate greater uncertainty into set membership. Naturally, if a type‐2 or higher set has no uncertainty in its members, it reduces to a type‐1. In this text, we will work primarily with type‐1 fuzzy subsets.
1.7.3 Non‐monotonic Reasoning
Non‐monotonic reasoning is an attempt to duplicate the human ability to reason with incomplete knowledge and to make default assumptions when insufficient evidence exists to empirically support a hypothesis. This proposed method of reasoning may be contrasted with monotonic reasoning in the following way.
A monotonic logic states that if a conclusion can be derived from a set of premises X, and if X is a subset of some larger set of premises Y, then x, a member of X, may also be derived from Y. This does not hold true for non‐monotonic logic since Y may contain statements that may prevent the earlier conclusion from being derived.
Consider the following example scenario: The objective is to cross a river, and at the edge of the river there is a row boat and a set of oars. Using monotonic logic, one can conclude that it is possible to cross the river by rowing the boat across. If the new information that the boat is painted red is added, this will not alter the conclusion. On the other hand, using non‐monotonic logic, the same initial conclusion may be drawn; however, if the new information that the boat has a hole in it arises, the original conclusion can no longer be drawn.
An English philosopher, William of Ockham (or Occam) (1280–1348) held a number of beliefs that foreshadowed the development of non‐monotonic logic. In particular, there is an element of his philosophy called Occam's Razor that provides a succinct description of this form of logic. Occam stated “…that for the purposes of explaining, things not known to exist should not, unless it is absolutely necessary, be postulated as existing” (1280–1348); this has also been called the Law of Parsimony.
This belief may be reformulated slightly as “in the absence of any information to the contrary, assume….” This kind of reasoning may be defined as a plausible inference and is applied when conclusions must be drawn despite the absence of total knowledge about a world. These consequences then become a belief that might be modified with subsequent evidence. In a closed world, what is not known to be true must be false. Therefore, one can infer negation if proving the affirmative is not possible. Inferring negation becomes more difficult, of course, in an open world.
A first‐order theory implies a monotonic logic; however, a real‐world situation is non‐monotonic because of gaps or incompleteness in the knowledge base. The default inference can then be used to fill in these gaps, which is very similar to some of Piaget's arguments.
McCarthy (1980) presents an idea that he calls circumscription. Circumscription is a rule of conjecture that argues when deriving a conclusion, that the only relevant entities are the facts on hand, and those whose existence follows from these facts. The correctness of the conclusion depends upon all of the relevant facts having been taken into account. Rephrased, if A is a collection of facts, conclusions derived from circumscription are conjectures that A includes all the relevant facts and that the objects whose existence follows from A are all relevant objects.
Reiter (1980), on the other hand, argues for default inferences from a closed‐world perspective. Under such an assumption, he asserts that if R is some relation, then one can assume not R (the opposite of R or R does not exist) if assuming not R is consistent to do so. This consistence is based on not being able to prove R from the information on hand. If such a proof cannot be done, then the proof must not be true, or, similarly, if an object cannot be proven to exist in the current world, then the object does not exist.
Looking at the relationship between fuzzy logic and Reiter's form of non‐monotonic logic, Reiter asserts that a default inference provides a representation for (almost all) the fuzzy subsets (and with most in terms of defaults). Reiter's assertion is not strictly correct because the inference is either true or not true, whereas a fuzzy grade of membership expresses a degree of belief in the entity.
A fundamental difference between these two theories is that Reiter's theory appears to require a global domain, whereas McCarthy's theory does not. McDermott and Doyle (1980) argue that this may not be a weakness in Reiter's approach. In either case, the intention is to extend a given set of facts (beliefs) by inferring new beliefs from the existing ones. These new beliefs are held until the evidence is introduced to contradict them. When such counterevidence occurs, a reorganization of the belief system is required.
In the discussion of his TMS (Truth Maintenance System) system, Doyle (1979) suggests that such a reorganization may take either of the two forms: world model reorganization or routine revision. Routine revision requires maintaining a body of facts that are expressed as universally true but may have some exceptions. Such a need usually occurs as a result of inferences, default assumptions, or observations. World model reorganization involves more wholesale restructuring of the model when something goes wrong. The aforementioned world model reorganization is usually quite complex and is typically the result of induction hypothesis, testimony, analogy, and intuition.
Note that these two (monotonic and non‐monotonic reasoning) are very similar to Piaget's concepts