Kirby asked:
What is wrong with this statement:
PII (standard definition):
If X and Y share ALL their properties (indiscernible), they are identical.
It is generally held that this definition is trivially true, so PII is redefined as:
If X and Y share all QUALITATIVE properties, they are identical.
I do not understand why the original definition is trivially true rather than just true, and I do not see any justification for the redefinition. This seems to be a case of taking a perfectly good principle, unjustifiably redefining, then arguing that the principle, as redefined, is false.
When PII is re-defined, a major argument for its falseness is that we can conceive of there being two, therefore nonidentical, objects that are qualitatively indiscernible. So spatial and/or temporal dispersal becomes a good argument against redefined PII. But, if we stay with the original definition of PII, the spatial and/or temporal dispersal argument is a major argument in favor of PII. Under the original PII, perceiving two objects in different regions of space is prima facie evidence of nonidentity; if they are in different regions of space at a time, they are not identical.
Answer by Geoffrey Klempner
The principle in question, for those who are not familiar with this issue is known as the Identity of Indiscernibles. It was one of the major components of Leibniz’s theory of monads, as elaborated in Monadology and other works.
The story goes that Leibniz amused the courtiers at the House of Hanover by challenging them to find two leaves that were identical, thus disproving the principle. You will not be surprised to learn that no-one ever did. The chances of discovering two leaves that appear the same even to the closest examination are very tiny. And yet, provided the courtiers weren’t allowed to use microscopes, just the naked eye, it is perfectly possible that two such leaves could have been found.
Would it have disproved Leibniz’s theory? No. Because difference in spatial position — for example, being held in your left hand and your right hand — would according to Leibniz suffice to establish non-identity. But here’s the finesse; according to Leibniz, there is no such thing as ‘space’ as we understand it. All that exists, in ultimate reality, are ‘subjects’ of varying degrees of consciousness, ‘monads’, each representing a world from a unique point of view. All spatial relations, according to Leibniz reduce to non-spatial properties.
Your intuition, which to many people seems plain common sense, is that space is something real. Two objects, say, two leaves, which are physically identical down to the atomic level can occupy different positions in space. That’s what makes them two and not one. Spatial ‘dispersal’, as you call it, of two otherwise indistinguishable objects is both necessary and sufficient for their non-identity.
However, one consequence of this intuitively attractive view, which perhaps you hadn’t thought of, is that it becomes logically true that two objects, say, two leaves or two pennies, cannot occupy the same space. Well, we know that that’s contrary to the laws of physics. But that wasn’t the question. The Identity of Indiscernibles isn’t a law of physics. It’s meant to be a law of logic, true even in possible worlds where physics is different from what it is in the actual world.
Well, here’s another experiment. I first heard about this in a lecture given many years ago by David Wiggins (author of Identity and Spatio-Temporal Continuity, later expanded to Sameness and Substance). I have two identical pennies, one in each hand. I move them together until they touch, then I press hard and, to my amazement, the pennies start to merge into one another. I pull the pennies apart again. Then I push them into one another, further this time, and pull them apart. After several goes, I have forced the pennies to occupy the same space. I now hold a single ‘penny’ which weighs twice as much as a normal penny. Or do I? When I give the ‘heavy penny’ a sharp tap, in separates into two pennies again. Why isn’t this a case of two identical objects occupying the same space? Remember, that this is a question about logic, not physics!
If you feel the slightest temptation to say that the law that two objects cannot occupy the same space is only a law of physics, not a law of logic, then you have to question the validity of the Principle of the Identity of Indiscernibles, re-defined, as suggested, to include spatio-temporal properties.
As a footnote, Wiggins’ own theory of identity as ‘spatio-temporal continuity under a covering sortal concept’ required, he claimed, that we reject the conclusion of the penny thought experiment. As a matter of logic, according to Wiggins, two objects cannot occupy the same space at the same time. There are good reasons for a philosopher wanting to hold this, and I appreciate those reasons. But I am not fully convinced.
Ask a Philosopher 
It is not at all a law of physics that objects cannot occupy the same physical location. Quite the opposite in fact, the laws of physics are known to require that some physical objects occupy the same physical location. Objects that have this property are known as bosons.
this was not my question at all!