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. `

(1) If x=

(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 K

(5) If x∈K

(6) If x, y∈K, and x=

(7) If K

(8) If K

(9) If x∈K

(10) For all x, (Nx∈K or Nx∉K)

(11) There are no two natural kinds, K

(12) The class of things defined as the intersection of the extensions of two distinct natural kinds K

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=

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

Reading A. If x, y∈K, and x=

Reading B. If x, y∈K, and x=

But let's ingore this.

Second, both readings seem false. Just take y to be x, and take K

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 K

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

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'
- 'K
_{1}⊂K_{2}' reads 'K_{1}is a species of K_{2}' - 'x=
_{e}y' reads: 'x is essentially the same as y' - 'x=
_{i}y' reads: 'x and y are intrinsically identical in their causal powers, capacities and propensities.'

(1) If x=

_{i}y, then x=_{e}y(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 K

_{1}⊂K_{2}, then PeK_{1}(5) If x∈K

_{1}and K_{1}⊂K_{2}, then x∈K_{2}(6) If x, y∈K, and x=

_{i}y, and there is a K_{1}and K_{2}such that x∈K_{1},y∈K_{2}, K_{1}, K_{2}⊂K and K_{1}≠K_{2}(7) If K

_{1}≠K_{2}, then there is a property P such that it is not the case that PeK_{1}≡PeK_{2}(8) If K

_{1}⊂K_{2}, and K_{2}⊂K_{3}, then K_{1}⊂K_{3}(9) If x∈K

_{1}, K_{2}, and K_{1}≠K_{2}, then either K_{1}⊂K_{2}or K_{2}⊂K_{1}, or there is a K such that K_{1},K_{2}⊂K(10) For all x, (Nx∈K or Nx∉K)

(11) There are no two natural kinds, K

_{1}and K_{2}, such that necessarily for all x, x∈ K_{1}or x∈K_{2}(12) The class of things defined as the intersection of the extensions of two distinct natural kinds K

_{1}and K_{2}is not necessarily the extension of a natural kind, unless K_{1}⊂K_{2}or K_{2}⊂K_{1}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=

_{i}y, and there is a K_{1}and K_{2}such that x∈K_{1},y∈K_{2}, K_{1}, K_{2}⊂K and K_{1}≠K_{2}The first observation is that this is ambiguous between two readings:

Reading A. If x, y∈K, and x=

_{i}y, and there is a K_{1}and K_{2}such that x∈K_{1},y∈K_{2}, K_{1}, then K_{2}⊂K and K_{1}≠K_{2}Reading B. If x, y∈K, and x=

_{i}y, and there is a K_{1}and K_{2}such that x∈K_{1},y∈K_{2}, then K_{1}, K_{2}⊂K and K_{1}≠K_{2}But let's ingore this.

Second, both readings seem false. Just take y to be x, and take K

_{1}and K_{2}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 K

_{1}be a superspecies of K, and let K_{2}be a superspecies of K_{1}. 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??

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