Speak with the vulgar.

Think with me.

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 $X$ that contains all the sets. But in that case $X$ 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 $X$ is hypergunk iff it satisfies the following condition:

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

From this condition it follows that “part of $X$” is an indefinitely extensible property: $X$ 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.

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

Written by Jeff

November 16, 2008 at 12:00 am

26 Responses

1. At the original posting, Anonymous said this:

I’m not sure why you think hypergunk is any less spooky than sets. As you construe it, it seems like it would be a merely possible substance. Wouldn’t hypergunk be just as inadmissible for the ontology of the nominalists as sets? Wouldn’t it be no less abstract? I could be missing something, and this all seems interesting as hell no matter what. Thanks for posting it!

I’m glad you brought this out—I didn’t make this point clear enough in the post. I’m not saying, “There is possible supergunk, so absolutism is false.” That would only be convincing to someone who thought that merely possible objects really exist. Rather, I’m saying “There could be supergunk, so absolutism could be false. Furthermore, if absolutism could be false, then it is false (since it isn’t a contingent matter).” Even someone who doesn’t believe in possibilia should still believe that my description of supergunk is possible—and that’s all I’m relying on.

On the other hand, I’m less sure now that there’s no plausible contingent absolutist story. More on that later.

Jeff

December 6, 2008 at 6:47 pm

2. Fair enough. But then what makes the claim There is possible supergunk’ true? Is the semantics for such a claim nominalistically acceptable? I’ll admit I am not familiar enough with their position to have any idea, but I’m not entirely sure your argument would be persuasive to the absolutists you mention. If it is, it would be pretty slick.

Anonymous

December 6, 2008 at 8:03 pm

3. Yeah, I’m assuming my target doesn’t have a general problem with modal claims like “There could be a talking donkey”. I wouldn’t think that restriction rules out too many people in this day and age (or anyway, too many people interested in ontology). At any rate, it’s a separate issue from nominalism.

Of course, like everyone else, the nominalist needs an answer to the question, what are possibility claims are about?—and her answer can’t be “possibilia” (unless, like David Lewis, she thinks possibilia are concrete). Presumably she’ll have to go instead for some story like Prior’s or Kit Fine’s or Ted Sider’s, where operators—ways for sentences to hold—are at the bottom, not things.

Jeff

December 6, 2008 at 8:44 pm

4. It’s interesting that you mention Prior. Don’t you lose S5 if you use Prior’s modal logic? Granted, I don’t really think Prior is on the right track when he denies S5, but my point is that I’m not sure that your key premise premise would be acceptable to some people who have non-realist semantics. Now, I’m not saying that Prior is a non-realist; in fact I have no idea. I’m just pointing out that the conceptual possibility of hypergunk might not worry the absolutist. There could be supergunk’ and There could be a talking donkey’ have very different things going on in them, and while I think most absolutists might be able to make sense of the latter statement, I’m not sure they can make sense of the former. Hearkening back to Prior, he had a paper in the Review of Metaphysics that presents an example of a situation that seems like it’s possible, but he claims really wouldn’t be. The example is much like the grandfather paradox, where you think you should be able to kill your grandfather but really can’t, except in his case you think you should be able to think something but really can’t. It’s a pretty interesting case, and it leads me to believe that it would be perfectly reasonable to deny that hypergunk really even possibly exists.

Here’s another say of thinking about my concern. The absolutist says it’s necessary that talking about everything is intelligible, it’s just that you can’t talk about thinks like sets, cardinals, and properties when you’re doing so and still be speaking intelligibly because such things are indefinitely extensible. So they deny the existence of such things. Well now if hypergunk really does exist (or possibly exist), and it is indefinitely extensible, then it still doesn’t make sense to talk about everything if hypergunk is among the things you’re talking about. What stops the absolutist from making the same move here and denying the existence of hypergunk? More precisely, the absolutist need not necessarilly deny the existence of sets, cardinals, properties, or hypergunk, but merely their intelligibility. Such a weaker absolutist can say Sure, sets, cardinals, properties and hypergunk are possible, and even possibly exist. My denial of their existence does not amount to saying that they don’t even possibly exist, but merely that they do not possibly exist in intelligible discourse. Thus, even in worlds where hypergunk genuinely exists, the people residing in such a world cannot have anything like hypergunk in the scope of everything’. The existence of hypergunk at such worlds doesn’t make everything’ any less intelligible because everything’ is never concerned with hypergunk. The world could behave oddly, but intelligible people cannot. I hope this is making sense.

Anonymous

December 7, 2008 at 9:22 am

5. A few things:

1. Can you point me to the Prior reference you mention? I don’t actually know much about Prior’s view beyond the fact that he’s broadly a “modalist” (i.e. operators first), which was all I was calling attention to. I didn’t catch what the specific problem you were raising there, or how limitations on possible thoughts would make a difference for supergunk—but maybe you can fill me in?

2. Certainly one possible response to the argument, for the nominalist who really wants to hang onto absolutism, is to deny the possibility in question. All I can say is that’s a cost. It means denying an apparent possibility to save your theory, and that’s the sort of thing metaphysicians aren’t generally happy about.

3. But I think you had something different in mind, and I’m having a hard time making sense of it. What’s the difference between “existing” and “existing in intelligible discourse”? I would have thought that the F’s exist in intelligible discourse just in case one can intelligibly and truly say, “The F’s exist”, and one can intelligibly and truly say “The F’s exist” just in case the F’s exist—so there’s no difference between the two. But it looks like you have something else in mind.

Jeff

December 7, 2008 at 9:22 pm

6. Responses, in order:

The Prior reference I mention isn’t as relevant for his denial of S5. That should be in his Time and Modality where he presents his quantified modal logic Q. I don’t know it in detail, I just think that your argument might not work in his logic.

Here (and for what you ask at the end of the first item) is where the Prior paper I specifically referenced is probably more relevant. That paper is Identifiable Individuals, Review of Metaphysics, Vol 13, June 1960. It might not be super-duper relevant, but he talks about logical illusions, i.e. things that seem logically possible but really aren’t. He gives the well-known example of the barber who shaves all those people and only those people who do not shave themselves, but he also gives a novel example concerning two men in two rooms that would be too long to explain here. Another obvious example that would be along the lines of Prior’s account would be Nicholas Smith’s response to the grandfather paradox, which is just to say that, while it seems like you should be able to kill your own grandfather if you had a time machine, that you actually can’t; it’s ultimately an absurd claim. It would seem the absolutist could, along similar lines, say that hypergunk isn’t really possible. It seems like it’s possible, but indefinitely extensible things really cannot possibly exist, so the concept of hypergunk rests on a confusion of what is possible. I don’t necessarily agree with that, I’m just curious about what your response would be for the absolutist who claims it.

I would think the difference between existing’ and existing in intelligible discourse’ would just amount to the difference between the way the world is and the way we can speak consistently. If the world is inherently inconsistent, or at least “funny” in some way, then we would never be able to consistently talk about it and speak accurately. So, I think the absolutist might be able to deny that One can intelligibly and truly say Fs exist just in case the Fs exist’ by simply denying that one can intelligibly and truly talk about something that is internally inconsistent or based on contradictory ontological commitments. If one can truly say such things exist, then one is being unintelligible, even though the things exist.

I hope I’m making more sense now! Your comments really helped me to organize my thoughts much better. :)

Anonymous

December 8, 2008 at 3:59 am

7. Hi Jeff,

Thanks, that was an interesting post. I was a bit confused about your definition of supergunk. Was the condition on the x’s supposed to be that they were linearly ordered by parthood?

It seems to me like supergunk is impossible: take the linearly ordered (by parthood) subsets of the the set of parts of X. Then by Zorns lemma you have a maximal element, the y’s, under subsethood, which can’t satisfy your condition.

Maybe a better case for indefinite extensibility is the existence of antigunk: a world such that everything is a proper part of something else. In order to be consistent with unrestricted composition, it must be the case that your quantifiers can always be extended. That is, to copy your argument against a universal set: “say you have all of the concrete objects. Then there ought to be a fusion of just those things— an object X that contains all the other objects as parts. But in that case it is a proper part of another object X+, by antigunkiness. Since X is the fusion of everything, X+<X, and of course X<X+. Paradox.”

Nice blog by the way!

Andrew

Andrew

December 8, 2008 at 5:24 am

8. Anonymous:

Yes, for all I said in the post there might be some hidden incoherence in the description of supergunk, and that would be a reason to reject its possibility (as in the grandfather case, for instance). As it happens, though, I’m pretty confident there isn’t any incoherence in this case; at least, I can show that supergunk is consistent relative to set theory, by giving a set-theoretic model for supergunk (well, not quite a model, since its domain has proper-class-many elements, but good enough). I’m sure there are simpler models, but one I particularly like is the class of fusions of intervals of John Conway’s “surreal numbers”. I’ll describe that in a little more detail in another post, if there’s interest.

You’re still losing me on your second point. What would it be for the world to be inconsistent? The best I can come up with is if there were some true sentences that were inconsistent. But in that case, presumably one could intelligibly and truly assert those sentences.

Getting at it another way: it sounds to me like you want it to be possible to say things like “‘$\phi$‘ is unintelligible, though in fact $\phi$.” But wouldn’t that utterance be, by its own lights, unintelligible?

Jeff

December 8, 2008 at 1:52 pm

9. Jeff:

I’d definitely be interested in that post!

Perhaps I’m using the word intelligible’ too literally, where intelligible’ just means able to be understood’. It certainly seems that the world could be a certain way but that we couldn’t actually understand anything we said about it. And if the world there were some true inconsistent sentences, my sense is we would not be able to intelligibly assert them, i.e. understand what we are saying. As Aristotle would say, the mere assertion of a contradiction would betray our unintelligibility.

Anonymous

December 8, 2008 at 2:11 pm

10. Hi Andrew,

Glad to see you stopped by.

1. Yes, that is what I meant by the condition.

The problem with applying Zorn’s lemma is that supergunk has too many parts to form a set. We can also see that the argument breaks down by analogy with the ordinals: the upshot of the indefinite extensibility argument in that case is that there is no maximal chain of ordinals. As with the ordinals, so with the parts of supergunk. So there shouldn’t be a problem with Zorn’s lemma.

2. That’s a nice point about antigunk. I’d thought there would be an analogous case there, but I hadn’t followed up on the idea. It is a bit more striking in that case, since the principle of unrestricted fusion has a lot more traction than its counterpart, unrestricted overlap (which is tricky without a null object).

Of course, you can make the same Zorn’s lemma objection to antigunk: again Zorn’s lemma implies that there is a maximal chain of parts; but it has a fusion, which is by hypothesis a proper part of some object $X$. But then $X$ can’t be a member of the chain, and so the chain isn’t maximal. The objection fails in this case for precisely the same reason.

Jeff

December 8, 2008 at 2:12 pm

11. I’m sceptical there’s a model for supergunk. At least, one in which you have full interpretations of the plural quantification (and assuming global choice.)

I believe there are models for Nolan’s hypergunk though (e.g. the regular open sets in 2^k for inaccessible k, I think would do it, although I haven’t checked it thoroughly.)

Andrew

December 8, 2008 at 2:14 pm

12. Andrew:

I’m not sure that Zorn’s lemma would be a problem here. The maximal element of X would be X itself, I suppose an improper part of X. But I don’t see how the definition of hypergunk forbids the existence of such an improper part for it.

Anonymous

December 8, 2008 at 2:15 pm

13. […] 3 comments In my last post I argued that the set-theoretic problems with “absolutely everything” carry over even […]

14. That looks like I just ignored your post, but I think we must have posted simultaneously.

Yeah, the Zorns lemma was supposed to be an argument against you being able to construct a (set theoretic) model. Assuming global second order reflection, that should mean there’s no proper class sized model either.

As for the ordinals, naturally there is no maximal ordinal. But there is a maximal *class* (or plurality, or concept, or whatever your second order variables range over. That is, assuming we’re always taking full models.) Thus when you say “for any parts of X, the x’s,…”, there will be those parts corresponding to the maximal linear order, even if it’s proper class sized.

Yes, that’s true what you say about antigunk. I guess that is inevitable if you’re going to take the line that we can’t quantify over everything. It’s an interesting disanalogy with the Nolan hypergunk, since that is consistent in the ordinary sense (that it has a model.)

Anonynmous: X wasn’t even in the partial order, so I’m not sure what you mean!

Andrew

December 8, 2008 at 2:36 pm

15. Andrew:

My point was that I don’t see why X wouldn’t be in a partial order of all parts of X since, it seems to me, for all y, y is a part of y. That may be an idiosyncratic view. My familiarity with the philosophy of parts is quite weak (not to mention that you each clearly know much more set theory than I do).

Also, I’m not completely clear on what condition of hypergunk the maximal elements of X (which I’m guessing you mean is a token of hypergunk) don’t satisfy. It’s probably just me, but I’m not sure where your original point was going. If you could explain it to me I’d be most appreciative.

Anonymous

December 8, 2008 at 2:55 pm

16. The parts of X, call it P, wasn’t the partial order in question. The partial order I was concerned with was the set of linearly ordered subsets of P.

My point was, supergunk doesn’t seem to be consistent in the ordinary model theoretic sense. It is at best consistent in the way that the set comprehension schema is: if you’re willing to let the interpretation of the quantifiers vary within a sentence.

One worry with this is that we might end up with no (quantificational) contradictions at all. After, the derivation of Russell’s paradox doesn’t use any non-logical principles.

Andrew

December 8, 2008 at 3:20 pm

17. Andrew,

Now I’m having difficulty reconciling these two sections:

“It seems to me like supergunk is impossible: take the linearly ordered (by parthood) subsets of the the set of parts of X. Then by Zorns lemma you have a maximal element, the y’s, under subsethood, which can’t satisfy your condition.”

“The parts of X, call it P, wasn’t the partial order in question. The partial order I was concerned with was the set of linearly ordered subsets of P.”

So I take it you mean P is the set of parts of X?

But then I don’t fully understand what condition the set of linearly ordered subsets of P would fail to satisfy. The conditions for being supergunk? I’m not sure how, if this set has a maximal element, it would fail to satisfy any of the conditions for being supergunk. And, if in fact it does, why should the set of partially ordered subsets of supergunk’s not being supergunk demonstrate that supergunk is not possible? Is it that supergunk must be made of supergunk?

Please don’t take my questioning to be of the rhetorical, obnoxious sort. The questions come from genuine curiosity and confusion. I’m beginning to realize that there is a much larger literature I’m missing out on here than I thought when I first posted. I haven’t been thinking about this particularly set theoretically at all, and it appears that that’s what the two of you have been doing this whole time. I hope I’m not pestering you. This has all been very fascinating for me. I would welcome suggestions for getting up on some of the literature of parts and doing all of this in a mathematically rigorous fashion.

Anonymous

December 9, 2008 at 4:50 am

18. Ok, to spell out completely, any (full) model of supergunk will be of the form $M := \langle P, \leq \rangle$, where, intuitively, P is the set of X’s parts, and $\leq$ the parthood relation. M will be a partial order (in fact, a complete Boolean algebra). That’s just from the fact it satisfies the mereological axioms.

If M also satisfies the condition for supergunk then for any linearly ordered subset of P, L say, there will be an element, x, in P which is strictly less than every element in L: $\forall y \in L, x < y$. Suppose now that L is a maximal linearly ordered subset of P. By the supergunk condition there will be an x which is less than every element of L. So $L\cup \{x\}$ is a linear order in P which properly includes L. That’s a contradiction because L was assumed to be maximal (not a proper subset of any other linear order in P.)

Given a suitable reflection principle, the fact that there is no set theoretic model for supergunk implies that there is no proper class sized model either.

A couple of notes. This argument does not preclude the existence of a Henkin model: one where the second order quantifiers range over less than all the subsets. Secondly, I expect you could argue for its inconsistency directly in the object language using global choice.

As for literature, mabye a good place to start would be the SEP entry on mereology. Shapiro’s book on second order logic would be useful.

Andrew

December 9, 2008 at 9:39 am

19. That’s completely right, Andrew. I was a bit loose with my original formulation of the supergunk condition—the reason being that it’s very hard to consistently describe extensibility.

To be clear, certainly the supergunk argument is no more compelling than the set-theoretic versions. The only difference is who it should compel, namely the nominalist. But besides that, all of the problems with the set version still apply to the mereological version.

In particular, this means my claim to have a “model” was grossly imprecise. Supergunk has a model only in the same sense that the indefinitely-extensible universe of sets has a model. Still, I think that’s pretty good.

Jeff

December 9, 2008 at 12:43 pm

20. Andrew,

Thank you so much! That made things significantly clearer for me. I was getting confused and thinking of being <-maximal as being the largest part, one which is not a part of anything else that is a part of X, as opposed to being a smallest part, i.e. a part of X which has no parts. Foolish mistake on my part, but now your argument makes perfect sense to me.

Also, thank you for the suggestions on the literature. I had only vaguely heard of mereology as a research subject, and my familiarity with set theory is still only at an undergraduate level. So, I look forward to getting acquainted with the enormous amount of stuff that is going on here.

This has really been instructive for me and a lot of fun. Thank you for the initial post, Jeff. I doubt I was able to help you as much as Andrew was, but if your blog is meant to educate as much as elucidate, then you have certainly already reached your goal with me!

Anonymous

December 9, 2008 at 1:47 pm

21. I’m glad that helped :-D.

Yeah, I’m usually quite lazy about spelling things out in full (esp in blog comments), so it’s good you pressed me on it!

Good luck with the literature.

Andrew Bacon

December 9, 2008 at 2:18 pm

22. “I was getting confused and thinking of being <-maximal as being the largest part, one which is not a part of anything else that is a part of X, as opposed to being a smallest part, i.e. a part of X which has no parts.”

Wait, that’s not quite right. When I was talking about maximal elements, I was talking about about maximal elements *in the set of linearly ordered subset of P, under the subset ordering*. Nothing to do with parthood. What you described would be a <-minimal object surely?

Andrew Bacon

December 9, 2008 at 2:47 pm

23. Thanks, Anonymous. I appreciate the feedback a lot.

Jeff

December 9, 2008 at 3:16 pm

24. Ah now I think I get it. Zorn’s lemma entails that every linear order has a maximal linearly ordered subset, and the condition on supergunk makes such a subset impossible. Is that finally right?

You’re right, I thought that you were saying Zorn’s lemma entailed that there was a <-minimal element in the set of parts, and if that was the case then there couldn’t be supergunk.

Before I thought that, I think I was thinking you meant Zorn’s lemma entailed that there was a <-maximal element in the set of parts of X, which I figured wouldn’t be problematic for supergunk because that part would just be X.

My goodness do I ever need to brush up!

Anonymous

December 10, 2008 at 9:00 am

25. Sorry, not every linear order’ but every set that is partially ordered’. That should make it right.

Anonymous

December 11, 2008 at 5:41 pm

26. […] » 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 […]