A New Contingency Axiomatic System for Rough Sets

Contingency logic is well-known to study the principles of reasoning involving necessity, possibility, contingency and non-contingency. However, there are some defects in existing contingency axiomatic systems. For instances, (NCR)\(_i\) is an infinite inference rule. The definition of accessibility relations and the corresponding axiom schema are very complex. To tackle these issues, a new contingency axiomatic system is proposed in this paper. Firstly, a new concise accessibility relation is defined for the axiomatic system; Then, two simpler axiom schemas of the axiomatic system are designed to replace the axiom schema K. This is helpful to prove the soundness and completeness theorems for the axiomatic system. Finally, rough sets can be perfectly formalized by our proposed axiomatic system. Theoretical analysis proves that a complete formal system is achieved. In addition, the concepts of “precise” or “rough” of rough sets can be described without the help of semantics functions of metalanguage.