Property inheritance and generics

Some properties are inherited from wholes to their parts: if x is (completely) made of steel, then its parts are also (completely) made of steel; if x is in the top drawer, then its parts are also in the top drawer. Other properties are upwards inherited from parts to wholes: if a part of x contains steel, then x contains steel; if a part of x touches the ground, then x touches the ground. Yet other properties are not inherited either way: if x is a hand, then x usually has non-hands as parts and is part of non-hands.

Plural properties, too, come in these forms. If the birds have died, then all of the birds have died (downwards inheritance); if a few of the birds have reached the house, then the birds have reached the house (upwards inheritance); but if the birds form the shape of a hand, it's usually not true that every subgroup of the birds forms that shape, nor is the latter sufficient for the former (no inheritance).

Many other properties are only weakly downwards-heritable: when x is red (and composite), then at least one proper part of x must be red. It need not be the case, however, that all its parts are red: a red object may well have a few non-red parts. Same with plurals: when the birds in this forest are aggressive, at least some of the birds must be aggressive, but it's okay if a few of them are not. (Note that weakly downwards-heritable plural properties need not be distributive: if the birds ate the cake or surrounded the building, at least some of them must have eaten the cake or surrounded the building, but it's okay if others of them did not. Strong downwards-inheritable properties, by contrast, must be distributive.)

In fact, having a single red part is usually not enough to qualify as red, and having one aggressive member is usually not enough for a group to qualify as aggressive. These properties, unlike typical upwards-heritable properties, are had by a whole x iff a sufficient fraction of x has it.

What is a sufficient fraction? That depends on context. In some contexts, x may count as red if only half of its surface is red, or only the part of its surface facing us. In other contexts, x has to be red all over. When I say "be careful, the poker is hot!", what I say might be true even if only a small part of the poker is hot. Likewise for "be careful, the birds carry plasmodia!"

Of course, it's not really the properties that are context-dependent. Rather, our words are indeterminate between many candidate properties that differ in their inheritance requirements, and context partly resolves the indeterminacy. (In this respect, even color words for determinate shades of red would be indeterminate.)

We can narrow down the indeterminacy by modifying the predicate to make the inheritance requirements more explicit: "x is wholly/completely red", "x is partly red", "x is mostly red", "x is red at y"; "the xx are all aggressive", "the xx are partly aggressive", "the xx are often aggressive".

Or we can modify the subject: "all of x is red", "part of x is red", "the y-part of x is red"; "all the xx carry plasmodia", "some of the xx carry plasmodia", "all the xx that are y carry plasmodia".

Perhaps "the xx" itself is a modification of this kind, modifying "xx": "birds" plurally refers to all birds, and "birds are aggressive" claims that a sufficient fraction of all birds in the domain are aggressive; but "the birds" refers only to the salient ones among all birds in the domain -- like "the bird" refers to a salient bird among those in the domain, as in "the bird is larger than the other birds".

If the generic "birds" plurally refers to all birds, it is obvious why generic predication easily mixes with essentially plural predication: "birds have wings and are common on all continents"; "some animals have wings and are common on all continents". That would make little sense if "birds are F" meant something like "each normal bird is F". (This problem was mentioned by Bernhard Nickels in a talk here last week.)

"All birds" also plurally refers to all birds (as witness, "all birds have a common ancestor", which doesn't mean that each bird has a common ancestor; this example is from JC Bjerring). But "birds" and "all birds" don't behave the same. The difference seems to be that "all birds" fixes the inheritance requirement, whereas "birds" leaves it open: "birds carry plasmodia" can be true if only some birds carry plasmodia (like "Dutchmen are good sailors"); but "all birds carry plasmodia" is false under this condition.

I don't have a good explanation why this happens. The same phenomenon occurs with singulars: "Fred" refers to the whole Fred, as does "the whole Fred" (and "all of Fred"). But "Fred was burned" leaves it open how much of Fred was burned, while "the whole Fred was burned" and "all of Fred was burned" do not.

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