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In the formalization of ontology, the notion of identity is especially important, because only it can be explained by the set-theoretical model, in which each property is expressed as a set of individuals. Primarily, an identity condition (IC) is used to judge if two individuals are identical or not. In this paper, we consider multiple identity conditions for a sortal property in order to solve subsumption inconsistency, that is, lack of relevant ICs between two sortal properties. We employ modal logic and provide functions for identity conditions. Our theory of multiple ICs would contribute to the consistent updating of ontological knowledge not only by removing inadequate links on incompatible ICs but also by verifying the IC set of each sortal property on change of subsumption. We present a practical example to illustrate knowledge updating mechanism concerning with rigid properties and their ICs.