Defining and measuring mathematically the level of knowledge, ignorance and uncertainty

Fujun Hou, Evangelos Triantaphyllou, Juri Yanase, Knowledge, ignorance, and uncertainty: An investigation from the perspective of some differential equations, Expert Systems with Applications, Volume 191, 2022 DOI: 10.1016/j.eswa.2021.116325.

People use knowledge on several cognitive tasks such as when they recognize objects, rank entities such as the alternatives in multi-criteria decision making, or for classification tasks of decision making / expert / intelligent systems. When people have sufficient relevant knowledge, they can make well-distinctive assessments among entities. Otherwise, they may exhibit some uncertainty. This paper establishes two differential equations, of which one is for the interaction between the knowledge level and the uncertainty level, and the other is for the interaction between the ignorance level and the uncertainty level. By solving these two differential equations under certain boundary conditions, one can derive that the proposed knowledge level indicator is equivalent to Wierman’s knowledge granularity measure up to a constant (exactly, ln2). Moreover, the knowledge level indicator and the ignorance level indicator are found to be in a complementary relationship with each other. That is, more knowledge implies less ignorance, and vice-versa. The results of this study bridge a critical gap that exists in the understanding of the concepts of knowledge and ignorance.

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