Oops

I’ve been cleaning up my “Indefinite Divisibility” paper from last year. One of my arguments in it concerned supergunk: X is supergunk iff for every chain of parts of X, there is some y which is a proper part of each member of the chain. I claimed that supergunk was possible, and argued on that basis against absolutely unrestricted quantification. I even thought I had a kind of consistency proof for supergunk: in particular, a (proper class) model that satisfied the supergunk condition as long as the plural quantifier was restricted to set-sized collections. Call something like this set-supergunk.

Well, I was wrong. I’ve been suspicious for a while, and I finally proved it today: set-supergunk is impossible. So I thought I’d share my failure. In fact, an even stronger claim holds:

Theorem. If is atomless, then has a countable chain of parts such that nothing is a part of each of them.

Proof. Since is atomless, there is a (countable) sequence . For each positive integer , let be . Then let be the sum of . Note that the ’s are a countable chain. Note also that each is part of .

Now suppose that is a part of each . In that case, is part of each . But since is disjoint from , this means that is disjoint from each , and so by the definition of a mereological sum, is disjoint from each . This is a contradiction.

Indefinite divisibility

If you’re interested, I’ve written a short paper on my nominalistic indefinite extensibility arguments. (This is also my way of making good on my offer in the comments to discuss a sort of consistency result for supergunk—it’s in the appendix.)

The “strict philosophical sense”

Here’s an inconsistent triad:

• The question of the ontological status of ordinary material objects is a serious question: its answer isn’t obvious.
• Obviously there is a chair I’m sitting on.
• Ontology is about what there is. (So, specifically, the question of the ontological status of ordinary material objects is just the question of whether there are such objects (chairs being among them).)

All three principles are pretty compelling. How can we resolve their inconsistency?

I suggest that there is an equivocation on “there is”. When we say ontology is about what there is, we are using “there is” in a different way than when we say there is a chair I’m sitting on. It is responsive to different constraints.

This is Quine’s picture: to find out what there is, we look at what we quantify over in our simplest theory of the world. The quantifiers are the symbols that appear in certain inferences: If is a , then there is a ; If we can infer that isn’t a from premises not involving , then we can infer from the same premises that there isn’t a . These rules, or something like them, constrain what we mean by “there is”, when we are doing our philosophical theory-building.

But the natural meaning of “there is” is constrained by the facts of English usage (perhaps together with some facts about the natural properties out there for us to talk about). There’s no reason to think beforehand that the constraints of theory-building are going to coincide with the constraints of ordinary usage. Clearly there’s an etymological relationship between the “strict philosophical” sense of “there is” and the ordinary English sense, but it looks plausible to me that they aren’t quite the same thing.

An analogy. We have an ordinary use of “animal” that excludes human beings. But biologists have discovered that there is a more useful category for systematic theory-building, one which mostly coincides with ordinary “animal”, but which includes human beings. This “strict biological sense” of the word “animal” doesn’t mean that a sign that says “No animals are allowed in the bus” is (strictly speaking) wrong. It’s just employing a different sense of “animal”.

I think a lot of philosophers think that when they say “strictly speaking”, they are manipulating the pragmatics of the discourse: the “strict philosophical sense” is the most literal sense. If what I’m saying is right, then this is a mistake. The strict philosophical sense isn’t any more literal than the ordinary sense; it is simply a sense that belongs to a different, philosophical register.

Getting in touch with the universe

In my last post I argued that the set-theoretic problems with “absolutely everything” carry over even for those who don’t believe in sets, by appealing to the possibility of “supergunk”. There’s another route to the same conclusion by way of some principles about contact. I think it’s kind of neat.

Let’s take contact to be a two-place relation between objects; it is reflexive (we count overlap as contact), symmetric, and monotonic: if touches a part of , then touches . These are all standard so far.

The following additional principles seem jointly possible:

1. A pretty weak separation principle: if and don’t touch, then there is some further that doesn’t touch either of them. (Think of as being located between and , keeping them apart.)

2. A very strong distribution principle: if touches the fusion of the ’s, then touches some . (Since the last post, I’ve switched from plural quantification to schemes, since I think it helps avoid some issues.) We might call this contact supervenience: what touches the whole touches some part.

The finite version of distribution is completely tame and standard: if touches , then touches or touches . It’s very hard to imagine the finite version failing. It turns out that the general version can fail, though. For instance, none of the intervals touches the interval ; but their fusion does (under ordinary topology). But this is pretty counterintuitive (John Hawthorne has written a whole paper about the principle’s failure). And so, even if it turns out that actually contact doesn’t supervene on parts, it still strikes me as a way things could have been.

But these two principles together give rise to another extensibility argument. Suppose that something doesn’t touch . Given any ’s that don’t touch , their fusion doesn’t touch by (2), and so by (1) there is some further thing that doesn’t touch . So the ’s, whatever they may be, don’t exhaust the things that don’t touch : the non--touchers are indefinitely extensible. Thus, in a world where (1) and (2) hold, it doesn’t make sense to talk about absolutely everything there is.

To sum up: (1) and (2) are jointly possible; therefore, generality absolutism is possibly false. Since generality absolutism isn’t contingent, generality absolutism is actually false.

All things great and small

This is a blog-sized summary of a paper I’m working on.

For more than a century now, there’s been a problem with “everything”. Here’s a simple version: say you have all of the sets. Then there ought to be a set of just those things—a set that contains all the sets. But in that case is a member of itself, which no set can be. Paradox!

In 1906 Bertrand Russell writes,

[T]he contradiction results from the fact that…there are what we may call self-reproducing processes and classes. That is, there are some properties such that, given any class of terms all having such a property, we can always define a new term also having the property in question. Hence we can never collect all of the terms having the said property into a whole; because, whenever we hope we have them all, the collection which we have immediately proceeds to generate a new term also having the said property.

Michael Dummett (1993) calls properties like this indefinitely extensible—the main example is “set”, but related paradoxes also show up for “cardinal number”, “order-type”, “property”, and “proposition”. Because of this a lot of philosophers are driven to conclude that we can’t speak intelligibly of all the sets (cardinals, properties, etc.). Whenever we think we’ve caught them all, another pops up to defy us. And if we can’t talk about every set, then we also can’t talk about plain everything—since that would have to include all the sets.

This kind of argument leaves open an escape to somebody with enough nerve: one way out is to deny outright that there are any sets (cardinals, properties, etc.). This is kind of an attractive view anyway, since sets are a lot spookier than, say, tables and chairs and galaxies and electrons—even without the paradoxes. The strong-nerved people who deny the existence of such things are called nominalists (contrasted with platonists or realists).

I have a way to close of the nominalists’ escape route. What we need is a new indefinitely extensible property that isn’t “abstract” (like “set”, etc.): instead, it applies to concrete, material objects. (Even nominalists don’t want to deny those!) I don’t claim that there actually are any such things, though: instead I claim that there could be. This is enough, because it would be very odd if it turned out that “absolutely everything”-talk was intelligible just by luck. The people who think it makes sense to talk that way think that it necessarily makes sense to talk that way. If they’re right, then it shouldn’t even be possible for something to be the way I suggest.

Here’s the idea. Material things could be made of atoms: they might have smallest parts that cannot be divided any further. Alternatively, they could be made of “atomless gunk” (David Lewis’s term (1991)): any piece of it contains ever-smaller bits. Inside our “atoms” we find protons, in the protons we find quarks, and it never stops. Gunk has a long pedigree as a theory of how the world is—and even if it happens to be false about our world, it sure seems like a way a world could possibly be.

But gunk doesn’t by itself give us what we need: it could be that the parts of a gunky material object eventually run out. If you follow finite chains of decreasing objects, there is always something further down—but if you follow infinite chains, you may succeed in getting all the way to the bottom, with nothing smaller below. But also, (it seems) that might not happen. As you go further and further down to smaller and smaller parts, there are always smaller parts further on. An object with parts like this I’ll call supergunk.1

More precisely, an object is hypergunk iff it satisfies the following condition:

• For any parts of , the ’s, such that each is a part of or has as a part each of the ’s, there is something that is a proper part of each of the ’s.

From this condition it follows that “part of ” is an indefinitely extensible property: is indefinitely divisible. So if there’s trouble for the sets, there is just as much trouble for supergunk. And it sure seems like there could be supergunk (even if there isn’t any in the actual world). So the nominalist has a problem with “everything”, too.

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1. Daniel Nolan (2004) describes something he calls “hypergunk”, but unfortunately that’s a bit different. ↩

Fatalism and fundamentality

Here’s another argument for fatalism (from a conversation with Dean Zimmerman):

1. If P is true, then P is true in virtue of some Q which is fundamentally true.
2. If P is true in virtue of Q, and Q is necessarily true, then P is necessarily true.
3. Whatever is fundamentally true is necessarily true.
4. Therefore, if P is true then P is necessarily true.

Understand “P is necessarily true” as “P cannot be changed”. The conclusion is that whatever is a fact cannot be changed. Thus if there are facts about the future, then the future is fixed, so that no one can do anything about it.

The most suspicious premise of the three is the third—and indeed, I think it is false. But it does have some tug. I think the tug comes from a principle of sufficient reason (PSR):

1. If P is contingently true, then there is some further reason for why P is true.
2. If P is fundamentally true, then there is no further reason for why P is true.
3. So if P is fundamentally true, then P is necessarily true.

The full argument is more or less Leibniz’s. It is unsound, since this version of the PSR is false (though I think there is a good methodological principle in the neighborhood). But I won’t defend this claim right now.

For now I just want offer a sociological speculation: I suspect that something like this kind of reasoning is what drives people to views like presentism in order to rescue our freedom. Suppose that there are future things; why would their existence threaten our power to make it such that there be different things instead? Existing future things would threaten this freedom, if tenseless existence facts are fundamental (at least for fundamental sorts of things), and the fundamental facts could not be changed. The right thing to say to this is that (some) fundamental facts, including tenseless existence facts, can be changed.

(I heard a good joke today—Adam Elga attributed it to Steve Yablo: “Everyone talks about how people could have done otherwise. But why doesn’t anyone?”)