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 Jürgen Lawrenz

I disagree with Geoffrey Klempner’s claim in his answer to this question that the PII of Leibniz is a purely logical principle, not to be misunderstood as pertaining to the physical universe. The very example quoted at the beginning (the leaves in the garden) disproves that supposition, by showing that Leibniz was (without room for hedging about) concerned with things that exist.

I also don’t consider the example of the pennies to be valid. It is an intuition pump deliberately engineered to produce a false conclusion. Where do these ‘logical’ pennies come from? If they’re not ‘in space’, then they don’t exist and the issue of PII never enters. In fact it would have been better to use two drops of water, which is at least an experience we are all familiar with. Then the PII has something of merit to contribute, namely confronting us with our notion of ‘1’.

The point with regard to Leibniz is precisely that the purely nominal status of space requires the concept of a volume to be replaced with the relational configuration of existents. These existents create what we call ‘space’ in their mutual relations, as something addressed to our perceptions. Accordingly the whole issue revolves around actualised monads, i.e. existents. The theoretical matrix for this is the multiple worlds in God’s mind prior to his choice of implementing ‘the best’ of these possible worlds.

It is easy to fall into the trap set for us by the likes of Everett or Wheeler that the unactualised world(s) in the collapse of a quantum wave train may be actual as simultaneous but unperceived worlds. This is another ‘logical’ game, conveniently forgetting (or disregarding) that the whole wave train is actual and terminated by the experimenter, who thus gains sight of only the instantaneous remnant of the wave, the rest being fatally disrupted and dispersed. Next time it rains, I invite you to make the analogous macroscopic experiment of inserting a piece of cardboard in the rain. Except in this case you get the whole picture, not a fragment per analogiam!

This comparison should alert us to the problem of dimensions, which as far as I can see is persistently kept out of sight. Leibniz arrived at his monads and his spatial doctrine from the recognition that all monads are a species of force (be careful not to read ‘energy’ here!): accordingly they are ‘simple, having no parts’ — in our language zero dimensional. Thus two or two trillion monads would occupy the same space, i.e. no space, to a Newtonian observer. Which brings us to the essence of the doctrine, namely the perceptions of the monads, each perceiving the others as ‘others’ and adjudging those which can only partially be perceived (because they are obscured by the nearer ones) as spatially more distant. But always ‘in its perceptions’. This is the basis for the PII thesis. As monads collectivise, each such collective comprising an existent would be influenced by its dominant monad to perceive other existents as spatially separate.

So in Leibniz’s canon, the PII is a physical, not a logical argument. It is physical not primarily because the term ‘matter’ is especially meaningful in the context, but because the species of force that binds collectives of monadic force together produce loci of force that are perceived by monads as coherent. If it is not coherent then there is no existent. Then the PII does not apply, nor does it apply to the unactualised monads that didn’t make it into God’s chosen world.

Obviously there is more to it than I’ve depicted here. Not least, perhaps, whether Leibniz’s God is dispensable to the argument or not. I believe it makes no difference if God is left out or replaced by a residual electric potential. ‘Look you, the situation is the same as in Macedonia’. Whatever exists, is subject to the Identity of Indiscernibles. And Leibniz’s monadology is about existence, about ‘why is there something rather than nothing?’ The world created by his God is such that no two identical objects can ‘occupy’ the same space because there is no space. But the moment we speak of relations, we need relata; and even in logic it is impossible to marry one relatum to itself!