Theories of Set and Logic with Crisp or Fuzzy Information Granules

Four theories may be generated with the combination of information granules for sets and logics. First a descriptive set membership value may be assigned to be {0,1} or [0,1] in an information granule that describes a set with a relational constraint; and secondly, a logical assertion set membership value may be assigned to be {0,1} or [0,1] in an information granule that states a veristic constraint associated with the given set description stated in the first case. These two types of assignments must be first distinguished from each other and then combined for a new construction of the combined theories of sets and logics. In this discourse, therefore, two types of information granules are considered: (i) one for descriptive set assignments and (ii) the other for truthood assignments. The combination of these two distinct assignments let us generate four set and logic theories. This then leads to the concern of canonical forms and their derivation from truth tables for each of these theories. In this regard, some of the fundamental issues arising in this context are discussed and certain preliminary answers are provided in order to highlight the consequences of these theories.