One further aspect to be considered is ``possible'' membership which might have many applications, mainly in language oriented problems. This concept can be handled by introducing partial functions---functions which might not have corresponding values for some of their arguments. A commonsense set theory may be helpful in providing representations for dynamic aspects of language by making use of partiality. For example, partiality has applications in modality (the part of linguistics which deals with modal sentences, i.e., sentences of necessity and possibility), dynamic processing of syntactic information, and situation semantics (Mislove et al. 1990).
We had mentioned above that situations can be modeled by sets. Consider a situation s in which you have to guess the name of a boy, viz.,
The boy's name is Jon or the boy's name is John.
This situation can be modeled by a set of two states of affairs. The
problem here is that neither assertion about the name of the boy can
be assured on the basis of s (because of the disjunction). A
solution to this problem is to represent this situation as a
partial set, one with two ``possible'' members. In this case s
still supports the disjunction above but does not have to support
either specific assertion. There is another notion called
clarification, which is a kind of general information-theoretic
ordering that helps determine the real members among possible ones.
If there exists another situation 
, where 
The
boy's name is Jon, then 
is called a clarification of s.