The success of set theory in mathematics owes to the fact that all compound entities and the relations between their parts can be represented in terms of sets. We claim that this also applies to commonsense set theory.
If we want to design artificial systems which will work in the real world, they must have a good knowledge of that world and be able to make inferences out of their knowledge. The common knowledge which is possessed by any child and the methods of making inferences from this knowledge are known as common sense. Common sense covers the fields of experience in which we all reason the same way and to the same effect. Any intelligent task requires it to some degree and designing programs with common sense is one of the most important problems in AI. McCarthy (1969) claims that the first task in the construction of a general intelligent program is to define a naive commonsense view of the world precisely enough, but also adds that this is a very difficult thing. He states that ``a program has common sense if it automatically deduces for itself a sufficiently wide class of immediate consequences of anything it is told and what it already knows,'' and proposes a program, the Advice Taker (McCarthy 1959).
It appears that in commonsense reasoning, a concept can be considered as an indivisible unit, or as composed of other parts, as in mathematics. Relationships, again as in mathematics, can also be represented with sets. For example, the notion of ``society'' can be considered to be a relationship between a set of people, rules, customs, traditions, etc. What is problematic here is that commonsense ideas do not have very precise definitions since the real world is too imprecise. We can face commonsense ideas in a variety of ways: by example, counter-example, analogy, or partial description (Perlis 1988). Even then we may not consider them in terms of indivisibles but in somehow composed ways. For example, consider the following definition of ``society'' (adapted from the Webster's Ninth New Collegiate Dictionary with some modifications):
``Society gives people having common traditions, institutions, and collective activities and interests a choice to come together to give support to and be supported by each other and continue their existence.''
It should be noted that the notions ``tradition,'' ``institution,'' and ``existence'' also appear to be as complex as the definition itself. So this definition should probably better be left to the experience of the reader with all these complex entities.
Nevertheless, sets may still be useful in commonsense reasoning. Whether or not a set theoretical definition is given, sets are useful for conceptualizing commonsense terms. For example, we may want to consider the set of ``traditions'' disjoint from the set of ``laws'' (one can quickly imagine two separate circles of a Venn diagram). We may not have a well-formed formula which defines either of these sets. Such a formation process of collecting entities for further thought is still important and simply corresponds to the set formation process of formal set theories, i.e., the comprehension principle. It helps us name the unities we have formed out of entities and use those names for further reference to those unities.
Having decided to investigate the use of sets in commonsense reasoning, we have to concentrate on the properties of such a theory. Instead of directly checking if certain set-theoretic technicalities have a place in our theory, we first look from the commonsense reasoning point of view and examine the set-theoretic principles which cannot be excluded from such reasoning.