### Brian Ellis on the logic of natural kinds

I'm reading Brian Ellis' Natural Kinds and Natural Kind Reasoning (in Natural Kinds, Laws of Nature and Scientific Methodology). I'm looking especially at section 6, The Logic of Natural Kinds. There, he introduces some notation and puts forward a bunch of principles that are meant to be necessarily true about natural kinds (it's almost like reading an early piece presenting axiomatic approach to modal logic: here's the language, here's the intuitive reading, and here are the principles).

Anyway, one of the principles doesn't seem quite right, and honestly, I don't know what Ellis wanted to say there. Here's a brief description of the "logic".

Abbreviations. `
• 'x∈K' reads: 'x is a member of the natural kind K'
• 'PeK' reads 'P is an essential property of K'
• 'K1⊂K2' reads 'K1 is a species of K2'
• 'x=ey' reads: 'x is essentially the same as y'
• 'x=iy' reads: 'x and y are intrinsically identical in their causal powers, capacities and propensities.'
Now for the principles, which Ellis says, are necessarily true (pp. 23-24). Here they are, pretty much as they stand in the paper (I only replaced the box with 'N'):

(1) If x=iy, then x=ey
(2) For every K, there is an intrinsic property P such that PeK
(3) If x∈K and PeK, then NPx
(4) If PeK, and K1⊂K2, then PeK1
(5) If x∈K1 and K1⊂K2, then x∈K2
(6) If x, y∈K, and x=iy, and there is a K1 and K2 such that x∈K1,y∈K2, K1, K2⊂K and K1≠K2
(7) If K1≠K2, then there is a property P such that it is not the case that PeK1≡PeK2
(8) If K1⊂K2, and K2⊂K3, then K1⊂K3
(9) If x∈K1, K2, and K1≠K2, then either K1⊂K2 or K2⊂K1, or there is a K such that K1,K2⊂K
(10) For all x, (Nx∈K or Nx∉K)
(11) There are no two natural kinds, K1 and K2, such that necessarily for all x, x∈ K1 or x∈K2
(12) The class of things defined as the intersection of the extensions of two distinct natural kinds K1 and K2 is not necessarily the extension of a natural kind, unless K1⊂K2 or K2⊂K1

Now, apart from the disadvantages of the attempt to determine a logic in a purely axiomatic manner, I'm worried especially about principle (6). Let's take a look at it again:

If x, y∈K, and x=iy, and there is a K1 and K2 such that x∈K1,y∈K2, K1, K2⊂K and K1≠K2

The first observation is that this is ambiguous between two readings:

Reading A. If x, y∈K, and x=iy, and there is a K1 and K2 such that x∈K1,y∈K2, K1, then K2⊂K and K1≠K2

Reading B. If x, y∈K, and x=iy, and there is a K1 and K2 such that x∈K1,y∈K2, then K1, K2⊂K and K1≠K2

But let's ingore this.

Second, both readings seem false. Just take y to be x, and take K1 and K2 to be K. Clearly, x belongs to K and is intrinsically identical to itself. Yet, neither K is a subspecies of K, nor is it different from K. The same substitution falsifies reading B.

So, perhaps, Ellis was assuming that different variables are referring to different objects/kinds (I'm trying to be charitable here, and try this reading, event though this reading is unlikely in the light of the fact that Ellis wants to have non-identity of kinds in the consequent)??

Well, again, this won't fly. For take x and y to be two distinct, and yet intrinsically identical objects belonging to one species K. Let K1 be a superspecies of K, and let K2 be a superspecies of K1. This interpretation falsifies both readings.

Am I missing something? Is there another natural principle that Ellis might've had in mind, but failed to state??