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 of assimilation and accommodation. From Doyle's point of view, non‐monotonic logic is reasoning with revision and that if a default election is made from a number of possible alternatives based on the alternatives not being believed, then the concept or argument under debate or consideration is not extensible.
1.8 Sets and Logic
1.8.1 Classical Sets
Classical sets are considered crisp because their members satisfy precise properties. For example, for illustration, let H be the set of integer real numbers from 6 to 8. Using set notation, one can express H as:
(1.1)
One can also define a function μH(r) called a membership function to specify the membership of r in the set H,
(1.2)
The expression states:
r is a member of the set H (membership in H = 1) if its value is 6, 7, or 8. Otherwise, it is not a member of the set (membership in H = 0).
One can also present the same information graphically as in Figure 1.2.
Whichever representation is chosen, it remains clear that every real number, r, is surrounded by crisp boundaries and is either in the set H or not in the set H.
Moreover, because the membership function μ maps the associated universe of discourse of every classical set onto the set {0, 1}, it should be evident that crisp sets correspond to a two‐valued logic. An element is either in the set or it is not in the set, and it is either TRUE or it is FALSE.
Figure 1.2 Membership in subset H.
1.8.2 Fuzzy Subsets
In relation to crisp sets, as we noted, fuzzy sets are supersets (of crisp sets) whose members are composed of collections of objects that satisfy imprecise properties to varying degrees. As an example, we can write the statement that X is a real number close to 7 as:
F = set of real numbers close to 7
But what do “set” and “close to” mean and how do we represent such a statement in mathematically correct terms?
Zadeh suggests that F is a fuzzy subset of the set of real numbers and proposes that it can be represented by its membership function, mF. The value of mF is the extent or grade of membership of each real number r in the subset of numbers close to 7. With such a construct, it is evident that fuzzy subsets correspond to a continuously valued logic and that any element can have various degrees of membership in the subset.
Let's look at another example. Consider that a car might be traveling on a freeway at a velocity between 20 and 90 mph. In the fuzzy world, we identify or define such a range as the universe of discourse. Within that range, we might also say that the range of 50–60 is the average velocity.
In
