A Definition of Entropy based on Qualitative Descriptions

A new concept of generalized absolute orders of magnitude qualitative spaces is introduced in this paper. The new structure makes it possible to define sets of qualitative labels of any cardinality, and is consistent with the classical structure of qualitative spaces of absolute orders of magnitude and with the classical interval algebra. In addition, the algebraic structure of these spaces ensures initial conditions for adapting measure theory to a qualitative environment. This theory provides the appropriate framework in which to introduce the concept of entropy and, consequently, the opportunity to measure the gain or loss of information when working within qualitative spaces. The results obtained are significant in terms of situations which arise naturally in many real applications when dealing with different levels of precision.