An object is an instance-of a class if it is a member of the set denoted by that class. One would normally state the fact that individual i is an instance of class C with with the relational form (C i), but it is equivalent to say (INSTANCE-OF i C). Instance-of is useful for defining the second-order relations and classes that are about class/instance networks.
An individual may be an instance of many classes, some of which may be subclasses of others. Thus, there is no assumption in the meaning of instance-of about specificity or uniqueness. See DIRECT-INSTANCE-OF.
(<=> (Instance-Of ?Individual ?Class) (And (Class ?Class) (Holds ?Class ?Individual)))
Because instance-of is in common usage, and member-of can get confused with the set and list operators.
Because these words are used to mean many different things in different contexts.
In KEE, instance-of is called member.of.
In Loom, instance-of is implicit in the syntax (unary predicates).
In Epikit, there is no notion of instances, although by convention people use unary relations to denote instance-of relationships.