Wolfgang Schwarz

Property Subtraction

Sometimes, a property A entails a property B while B does not entail A, and yet there seems to be no interesting property C that is the remainder of A minus B. For instance, being red entails being coloured, but there is no interesting property C such that being red could be analysed as: being coloured & being C. In particular, there seems to be no such property C that doesn't itself entail being coloured.

This fact has occasionally been used to justify the claim that various other properties A entail a property B without being decomposable into B and something else. I will try to raise doubts about a certain class of such cases.

First, a fact about intrinsic and extrinsic properties:

Subtraction: Every normal extrinsic property is the conjunction of an intrinsic property and a purely extrinsic property.

What does this mean? By an intrinsic property, I mean a property that never differs between perfect duplicates of objects. If you were to create an atom-by-atom replica of the Mona Lisa, the two would share all their intrinsic properties. They would still differ in how they were created, how far away they are from a large tower, and how much they are worth: those are not intrinsic properties. Intuitively, they also concern what happens elsewhere in the world. They are extrinsic. A normal extrinsic property is one that is such that whenever something x has it, there is a possible perfect duplicate of x that doesn't have it. All the extrinsic properties above are normal. E.g. for any x that is 200m from a large tower, there is a possible perfect duplicate of x that is not 200m away from a large tower. An example of a non-normal extrinsic property is: being either spherical or 200m away from a large tower. Finally, a purely extrinsic property is one that is independent of intrinsic character; that is, F is purely extrinsic iff for any possible object x, there are possible objects y, z that are perfect duplicates of x such that y is F and z is not F. Perhaps the extrinsic properties mentioned above are purely extrinsic, but perhaps not. A clear example of a normal extrinsic property that is not purely extrinsic is: being both spherical and 200m away from a large tower.

For convenience, I here take a property to be a class of possible individuals (or tuples of individuals for polyadic properties). So I think in terms of counterparts and timeslices. Nothing much would change if instead we took properties to be functions from worlds and times and places to classes of individuals.

Here is a proof of Subtraction: Assume F is a normal extrinsic property. Let I be the class of all perfect duplicates of all things in F. By definition, I is intrinsic. Let E be an arbitrary property that contains a) all Fs, b) no non-F duplicates of F-things, and c) some, but not all duplicates of any object x that is not a duplicate of an F-thing. E is purely extrinsic. For take any object x. If x is not a duplicate of an F-thing, then by (c), there are duplicates y,z of x such that E contains y but not z. If x is a duplicate of an F-thing, then by the normality of F, x has at least one duplicate, y, that is F, and another, z, that is not F; by (a), E contains y, and by (b), E doesn't contain z.

If a property F is the conjunction of properties I and E, does that mean that F-ness can be analysed as I&E-ness? Often, yes. But it depends on what is meant by "analysed". Let's take "F-ness can be analysed as I&E-ness" to mean: "having F is analytically equivalent to having I&E". The remaining question is whether our words, or concepts, "F", "I" and "E" transparently express the relevant properties in the way in which "water" does not transparently express the property of being H2O.

With the help of 2D semantics, we can say that each property expression actually expresses two properties: the property primarily expressed by "F" is (roughly) the class of all possible things x we would count as satisfying "F" if we turned out to be where they are. For instance, the property primarily expressed by "water" is the property of being watery stuff: if it turned out that we live on a planet where XYZ is the watery stuff, we would count XYZ as water. The property secondarily expressed by "F" is (roughly) the class of all possible things x we count as satisfying "F" holding fixed our own position in (centered) logical space. The property secondarily expressed by "water" is being H2O.

Suitably defined, the properties primarily expressed by our terms will be transparent. That is, if F, I and E are primarily expressed by "F", "I, and "E", and F = I&E, then "F = I&E" is a priori and analytic. From now on, I will mean "primarily expressed" by "expressed". Hence Subtraction gives us:

Analytic Subtraction: Every predicate that expresses a normal extrinsic property is analysable as the conjunction of a predicate that expresses an intrinsic property and one that expresses a purely extrinsic property.

Tentative applications:

1. Knowledge. The property expressed by "knowing that it rains" is a normal extrinsic property of people: for any person that knows that it rains, there is a possible perfect duplicate of her located in different weather conditions who doesn't known that it rains. On the other hand, knowing that it rains is not purely extrinsic. Unlike the property of being such that it rains, knowing that it rains requires something about internal representations of your surroundings, and those are not completely independent of your intrinsic structure. By Subtraction, knowing that it rains can be analysed as the conjunction of a purely intrinsic property capturing the relevant intrinsic requirements and a purely extrinsic property. (I don't say there is no unique extrinsic property of this kind.)

2. Phenomenal states. Suppose being a pain state analytically entails satisfying a certain causal role, but not vice versa. That is, there is more to being pain than satisfying this role. Plausibly, this additional requirement is intrinsic: it is not a matter of how the pain state relates to its surroundings and interacts with it, but with what it is like by itself. By Subtraction, we can isolate this intrinsic aspect: being a pain state can be analysed as the conjunction of a purely intrinsic property and a purely extrinsic property.

3. Necessitation. Suppose it is a law of nature that all Fs are Gs iff a certain relation N obtains between the universals corresponding to F and G. Arguably, N is a normal extrinsic relation. It is extrinsic because it being a law that all Fs are Gs entails that there is no F that is not a G, and this is something external to the two universals for F and G. And N is normal because in possible worlds where the laws are different, the two universals (or their counterparts) can be intrinsically exactly as they are here without standing in N to one another. Now consider the relation between these two facts: a) F-ness and G-ness are N-related and b) F-ness and G-ness are such that all Fs are Gs. The latter describes a purely extrinsic relation between the universals. The former presumably goes beyond this by adding an intrinsic link between the two (intrinsic to the pair, not to the relata, in Lewis's terminology): the fact that it is a law that all Fs are Gs partly concerns the regularities in the world, but partly something that is just about the two universals. By Subtraction, we can isolate this second part: N is analysable as the conjunction of an intrinsic relation and a purely extrinsic relation. (And then it seems that whatever N adds to all Fs being Gs is something that is logically independent of all Fs being Gs, and hence something of which it is unclear why it should have anything to do with laws of nature.)

This is all very sloppy, and I need to think more about how these applications are supposed to work in detail. There are lots of pitfalls here. For instance, one might think that by Subtraction, every normal extrinsic property that entails an intrinsic property is the conjunction of that intrinsic property with a purely extrinsic property. But this is not so. Take being round and 200m away from a tower: it entails being either round or cubical, but we can't recover roundness by adding a purely extrinsic constraint on being round or cubical. Likewise it is not true that every normal extrinsic property that entails a purely extrinsic property is the conjunction of that purely extrinsic property with an intrinsic property. (Take being either 200m away or 300m away from a tower.)