Three Kinds of Negation in Knowledge and Their Mathematical Foundations

In the field of artificial intelligence, understanding, distinguishing, expressing, and computing the negation in knowledge is a fundamental issue in knowledge processing and research. In this paper, we examine and analyze the understanding and characteristics of negation in various fields such as philosophy, logic, and linguistics etc. Based on the distinction between the concepts of contradiction and opposition, we propose that there are three different types of negation in knowledge from a conceptual perspective: contradictory negation, opposite negation, and intermediary negation. To establish a mathematical foundation that fully reflects the intrinsic connections, properties, and laws of these different forms of negation, we introduce SCOI: sets with contradictory negation, opposite negation and intermediary negation, and LCOI: logic with contradictory negation, opposite negation and intermediary negation, and we proved the main operational properties of SCOI as well as the formal inference relations in LCOI.