### You're correct that the

You're correct that the Identity of Indiscernibles says that *qualitative* identity (i.e., identity of properties) implies *numerical* identity (i.e., just one individual rather than more than one).

You then asked about the converse principle, which says that *numerical* identity implies *qualitative* identity: in other words, any individual has all and only the properties that it has. This converse principle, the Indiscernibility of Identicals, is even more secure than the Identity of Indiscernibles. Even those who challenge the Identity of Indiscernibles (such as Max Black, in his classic dialogue "The Identity of Indiscernibles") tend to accept the Indiscernibility of Identicals.

As for Bill with *n* hairs and Bill with *n*-1 hairs: The defender of the Indiscernibility of Identicals would probably insist on describing Bill's properties in a more fine-grained way. For example: Bill has the property of *having exactly n hairs at time t1* and the property of *having exactly n-1 hairs at time t2*. Because those properties are themselves time-indexed, Bill has both of those properties: not at one time but timelessly or tenselessly. If so, then it's not that Bill has one of the properties and lacks the other, in which case the Indiscernibility of Identicals looks safe.

There's much more to be said, of course, about this interesting topic. You might start with this SEP entry.

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