Contemporary debates over categorical properties – mention of such properties, even – very often involves people talking past one another.
This did not used to be the case. It used to be that everyone knew what a categorical property was supposed to be. As it happens, they also used to know at least enough about what a dispositional property is supposed to be to know that they denied that such properties exist — which is to say that they denied that the world includes things (“things” as loosely construed as you like) that are inherently active in different kinds of ways. (And everyone understood what it meant to deny that.)
But time marches on.
Nowadays, there are multiple definitions of “categorical property” in play, and it is worth being sure about which one any given interlocutor is using. Here is a short list:
(1) a categorical property is one the having of which does not pertain to what its bearer can or can potentially do;
(2) a categorical property is a property that cannot itself do anything;
(3) a categorical property is a property the identity of which is not the same in all possible worlds, &/or is not essential to that property;
(4) a categorical property is a property that has no identity other than that of formal self-identity.
These definitions get mixed and matched and deployed in all manner of unfortunate discussion. Thinker A rejects pandispositionalism on the basis of (1). Thinker B endorses it on the basis of (4). Etc. (It should be easy to see, too, how confusion at this level winds up figuring in what has come to be called the debate over so-called “pure powers.”)
I expect to post more on this topic – and please, please feel free to weigh in – but for now let me just say that the categorical properties that everyone used to believe in when they didn’t believe in powers — those were categorical properties as per (1). That this is so, or was so, only makes sense, given that passivists deny that things have real powers to do, and passivism was then an unquestioned orthodoxy in the relevant circles. (No. It is not permissible to re-define “do” here. Not unless you are a sophist. :-) )
Powers, Capacities & Dispositions 
Nice post, as always. Let me ask: I see your point in defining (1) and (2) in their own respective way, but I’m not sure what different implications bear on the definition of categorical properties. If I don’t get that clear I don’t understand the relation you see between those who adopt categorical property on the basis of the refusal of powers.
Hi Andrea, I’m not completely sure what you are asking but it seems to be “Why is it important to be clear that these are completely different definitions?” Is that right? I’ll answer that question, but tell me if I’ve misunderstood. It’s important, I think, because 1-4 imply entirely different things about the world.
1 and 2 negatively posit real, live activity; their sense comes from denying that activity, either in the case of categorical properties only, or generally. If one thinks that there is no such thing as real live activity – i.e., activity that doesn’t reduce in the end to non-activity – then one will, on that basis, have to affirm both 1 and 2. By contrast, belief in activity has no bearing on 3 or 4.
This said, 2 implicitly commits one to the idea that it is a thing’s properties themselves that do the doing, rather than the (propertied) thing. That is a completely different view than is implicitly affirmed by #1. (I have a little paper against the implicit ontology of 2, if you’d ever like to see it. Against the idea that it’s the properties themselves that do the doing, I mean. It’s supposed to come out in a collection sometime, that paper.)
3 negatively posits not real, live activity, but essentialism, either with respect to the identity of properties or generally. Here what is distinctive about categorical properties is that they supposedly have no fixed identity. But there is nothing about a categorical property as per #1, say, that implies that its identity can’t be essential to it. Ellis, for example, believes in categorical properties with fixed identities. Apart from this being very clear in Scientific Essentialism, he told me so! So if you were rejecting the existence of categorical properties because you’d stipulated that they are “properties” with no fixed identities, you’d be rejecting them for a very different reason than does someone like Stephen, who would be rejecting them on the basis of denying that there are any properties that are like #1 or #2. And you’d be talking right past someone (such as Brian) who thinks that in addition to having powers with fixed identities, things also have categorical properties (as per #1) with fixed identities.
Finally, 4 – so far as I can tell – is just the (or at least one possible) end-game of 3 (i.e., you might think that a supposed property with no fixed identity is the same as a property with no identity at all, other than that of formal self-identity [whether that “other than …” is as simple or successful a thought as *it* seems is another matter all together]). 4 (like 3) is not what non-pandispositionalist powers theorists (i.e. powers theorists who think that not all properties are powers) are defending the existence of, when they defend them.
Does this help?
I agree with your point that there is much confusion about powers and categorical properties.
But it seems to me that each of your suggested definitions is itself extremely unclear and confusing.
As to (1), what does it mean for the having of a property to fail to “pertain” to what a thing can do? Does it mean entailment? Or perhaps metaphysical dependence/ground? I.e. do you intend one of the following, or something else?
(1*) a categorical property is one the having of which does not entail anything about what its bearer can or can potentially do.
(1**) a categorical property is one the having of which is not grounded in what its bearer can or can potentially do;
As to (2), Humeans certainly think properties “do things”, in the sense that they enter into causal relations. So on the most natural reading (2) is no help to define categorical properties. Perhaps you have something more substantive in mind by “do things”; but this isn’t expressed in this definition.
I don’t understand (3). Does “the identity of which is not the same in all possible worlds’ mean “is not identical to the same property in all possible worlds? But everything is identical to itself and no other thing. Any one should agree that there is no world in which mass fails to be identical to mass.
Similarly for (4). What could it mean for something to have any other kind of identity than self-identity? (What is ‘formal’ doing here?)
Hi Matthew! How exciting to have conversation!
1. The key point is not the meaning of “pertain,” but rather the aspect of activity. I would imagine that people who think of categorical properties as “pertaining” to do-ing, could endorse either 1* or 1**.
2.Yes, the sense of “doing” is more robust than that of “one impression, then another impression” (or a contemporary Humean equivalent thereof). I stipulated at the end of the original post that for the purposes of rigor with respect to thinking about the morass at hand, no re-definitions of activity terms such that they amount to what one might think of as not-activity would come into the discussion.
3. I’m not sure that I’m seeing what you’re confused about with respect to 3. The idea, for those who think that way, is that the identity of a purported property of the type in question is contingent upon external, possibly local conditions; the identity of such a property is therefore not essential to it.
4. Alexander Bird is someone who thinks that categorical properties don’t exist because if they did they would have self-identity as their only essential characteristic. He appropriates the term “quiddity” to refer to such a (purported) property. The point is that such a property could not be farther from the categorical properties defended by someone like Brian Ellis.
I hope this helps to clarify! Welcome!