A framework for analysis of extended fuzzy logic

We address a framework for the analysis of extended fuzzy logic (FLe) and elaborate mainly the key characteristics of FLe by proving several qualification theorems and proposing a new mathematical tool named the A-granule. Specifically, we reveal that within FLe a solution in the presence of incomplete information approaches the one gained by complete information. It is also proved that the answers and their validities have a structural isomorphism within the same context. This relationship is then used to prove the representation theorem that addresses the rationality of FLe-based reasoning. As a consequence of the developed theoretical description of FLe, we assert that in order to solve a problem, having complete information is not a critical need; however, with more information, the answers achieved become more specific. Furthermore, reasoning based on FLe has the advantage of being computationally less expensive in the analysis of a given problem and is faster.