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3.4. OPEN WORLD REPRESENTATION

Relational representation of traditional logic is occluded by the Closed World Assumption, which includes only a limited number of concepts and relations, and supports the hypothesis that the whole world is explorable in a well-defined way. Uncertainty methods open some windows to the real world of unexplored, unexpected phenomena. This is especially true for the nontraditional uncertainty methods which ignore the hypothesis of excluded middle and of independence of basic events. The price of a more permissive method is the increased softness of their basic model and consequently the inefficiency of reasoning capabilities. From the point of view of logical and mathematical rigor, they are less and less welcome. This used to be the history of all methods beyond the classical probabilistic theory. One of the compromises is that all methods, the fuzzy representation included, can be attached at any instance of an acceptable hypothesis to the relational representation of classical logic. This is the intention of practically all methods concerning an open world: restricting the part of soft representation to the unavoidable limits.

The other way towards the open world of reality is the development of modern logic. It grew out of the modalities of classical logic, i.e. the limitations of the validity to certain subjects, interpretation fields. This is done by the different interpretations of the quantifiers (Figure 8). The next, far leading step is a deeper intrusion into the meaning. The quantifiers define an external frame for the interpretation, the meaning of the variables, analyzed and interpreted by intensional logic, opens every possible interpretation due to the specific usage of the words in different cultures, different ages, different disciplines, different mood, different social environment (Figure 9).

Some readings of Modal Operators
[open box] A [open diamond] A
It is necessarily true that A It is possibly true that A Alethic logic
It will always be true that A It will be sometimes that A Temporal logic
It ought to be that A It can be that A Deontic logic
It is known that A The opposite of A is not known Logics of knowledge
It is believed that that A The opposite of A is not believed Logics of belief
After every terminating execution of the program, A is true There is an execution of the program that terminates with A true Dynamics logic

FIGURE 8 Modal representation. (After A. Thayse, D. Snyers: Languages and logics, in Thayse (Ed.): From Modal Logic to Deductive Databases, Chichester: John Wiley, 1989.)

Peter looks for his car

-- In the world of a certain parking place where automobiles are parking
* type of parking automobiles
-- In the world of a railway station where railway cars are assigned to the ticket owners
* type of railway cars
-- In the world of desires where somebody has a lottery ticket, can win a car, and is just looking at the lottery list
* type of lottery prizes

FIGURE 9 Intentional representation.

The open world is full of contradictions. Because not all cases, relations, phenomena are known, one meets new facts in everyday practice, information which is not in strict logical relation with the previous knowledge, cannot be included into the framework of previous definitions. Three ways are used in common practice. The first preserves the earlier knowledge, its frames, structures and recognizes those which could not be fit in as exceptions. The second tries to detach this world of different phenomena by creating other new worlds. The third tries to create a unified new world of renewed concepts and relational hypotheses. The first can be illustrated by the favorite example of nonmonotonic logic, the exception of the birds (like the penguin or the oyster) which cannot fly, like the "regular" birds of the northern hemisphere. The second is the case of discovering the viral and bacterial variants of pneumonia or the difference between hepatitis A and B. The third is the case of the Kuhnian new paradigms in science, quantummechanics unifying the theories of the micro and the macroworld, or relativity theory, the Newtonian and the cosmological world. All these have representational methods in modern logic.

Nonmonotonic logic is the conceptual frame of logical relations where some new information is in discord with earlier information; the flow of information does not corroborate the consequence in a monotonic way. Two obvious procedures are at hand: narrowing down the conceptual frame to the extent where no contradiction exists, or the extension of the concept with new rules valid for the exceptions. The first is the circumscription of McCarthy, the second is the use of default rules of Reiter.

In a system of an open world, not all statements corroborate the previous ones, or those hypotheses or conclusions that are logical antecedents or consequences of these statements. Monotonic logic is created in a closed world where all further statements, data, and facts add to the validity of the previous ones.

Nonmonotonic logic looks at these contradictory sets of statements and attempts to find a consistent way of resolving the situation by adding further new conditions or canceling some old ones.

In classical logic, theorems are results of valid inferences; in nonmonotonic logic, one starts with all inferred statements, whether they are consistent or not. Then in most cases, one starts to find a fixed point, i.e., a minimal set where all statements get a consistent context. A typical paradigm is Tweety the bird, Tweety the penguin. Either we should exclude the penguin from the class of birds or flying should not be a characteristic attribute.


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