Leonard & Goodman’s Axioms: Antisymmetry = Extensionality

May 27, 2010

In my PQ paper, I made the claim that the antisymmetry of parthood is intimately tied up with extensionality principles in mereology. I made some informal arguments (that were perhaps not as strong as I would have liked). I basically said that antisymmetry is like an extensionality principle for improper parthood. I also argued that rejecting antisymmetry causes failures of all extensionality principles, and that’s, in effect, all that one loses since we can recapture classical mereology given certain axiom sets.

Some axiomatizations of classical mereology are much more explicit about this than others. In fact, the standard axiomatization (Partial Order + Strong Supplementation + Unrestricted Fusions) actually hides this fact quite well, so much so that people have tended to blame Strong Supplementation for extensionality. But in fact, even Weak Supplementation generates extensionality principles in the presence of Tarski/Lewis style fusions and antisymmetry.

Leonard and Goodman’s axiomatization is actually quite elegant, rivaled perhaps only by Tarski’s axiomatization (Transitivity + Uniqueness of Fusions). And moreover, it is extraordinarily explicit about the role antisymmetry is playing in extensionality principles.

Leonard and Goodman take disjointness ( $\wr$ ) as their mereological primitive. Let’s start with some definitions:

Parthood: $x \leq y := \forall z ( z \wr y \rightarrow w \wr x)$
Proper Parthood: $x < y := x \leq y \wedge x \neq y$
Overlap: $x \circ y := \exists z (z \leq x \wedge z \leq y)$
Fusion: $\mathsf{Fu}(t,\varphi(x)) := \forall y (y \wr t \leftrightarrow \forall x (\varphi(x) \rightarrow y \wr x)))$

Here, x is a part of y whenever anything disjoint from y is also disjoint from x. One thing to not is that there is quantification built into the definition of parthood here. Proper parthood is normal (although all the same issues will come up when rejecting antisymmetry). Overlap is normal as well. L&G fusions are a lot like standard fusions, except it’s done in terms of disjointness instead of overlap (in fact, they’re equivalent given Axiom 2 below).

Now for the axioms:

Antisymmetry: $(x \leq y \wedge y \leq x) \rightarrow x = y$
Duality: $x \circ y \leftrightarrow \neg x \wr y$
Unrestricted Fusion: $\exists x \varphi(x) \rightarrow \exists z \mathsf{Fu}(z, \varphi(x))$

(1)-(3) guarantee classical mereology. But look carefully at the first axiom. A little unpacking shows that it is equivalent to:

$\forall z ( z \wr x \leftrightarrow z \wr y) \rightarrow x = y$

But that is just explicitly an extensionality principle for disjointness. Indeed, in the presence of the second axiom, we can contrapose the extensionality of disjointness to get the extensionality of overlap:

$\forall z (z \circ x \leftrightarrow z \circ y) \rightarrow x = y$

And again this is pretty uncontroversially problematic for the antiextensionalist. Just look at the LRD of the extensionality of overlap:

$\forall z (z \circ x \rightarrow z \circ y) \rightarrow x \leq y$

Weakening gives us:

$\forall z (z \leq x \rightarrow z \circ y) \rightarrow x \leq y$

Contraposing again:

$x \nleq y \rightarrow \exists z (z \leq x \wedge \neg z \circ y)$

But that’s just the Strong Supplementation axiom, the axiom that has gotten so much bad press for extensionality. On L&G’s definitions and axiom 2, antisymmetry implies strong supplementation. None of this is terribly deep. And if you work much in this area, you know that you require antisymmetry to prove extensionality principles from supplementation principles. But this fact can be hidden — and L&G’s axiomatization is a pretty clear way to draw this out.

From → Mereology, Metaphysics

12 Comments

Andrew Bacon permalink

Hi Aaron,

I’m having difficulty seeing why this dispute isn’t just verbal. For example, suppose I’m an anti-extensionality guy (because, e.g., I don’t want unique fusions) who thinks that it’s supplementation, weak and strong, that’s responsible for my troubles. (Also suppose I’m taking parthood as primitive.) I’ll certainly grant you that the relation “everything that overlaps x overlaps y” isn’t antisymmetric – I *have* to say that – but I still think parthood is antisymmetric. Instead I reject the LG equivalence of parthood in terms of overlap/disjointness. Furthermore, I’ll interpret your antisymmetric mereology as being about the relation “everything that overlaps x overlaps y” and not about parthood. I’ll just view you as speaking an expressively handicapped language, because I have, in addition to all your notions, a notion of parthood you can’t define. Is there anything you can say that will convince me we’re really disagreeing?

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Andrew Bacon permalink

Oh, by the way (and I know this is a bit late) but congrats on getting the paper published!

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Jeff permalink

There are (meta-)metaphysical views floating around these days on which, when you have two equivalent axiomatizations of a theory, there’s still a serious question as to which axioms are more fundamental, and which ideology is more primitive. On that kind of view, then it’s a substantive issue whether to take overlap or parthood or disjointness as primitive. And then the disagreement between Andrew’s anti-extensionalist and the L&G-type theorist (or anyway, the theorist who takes L&G’s formulation as carving at the joints of reality) is not a verbal disagreement. It isn’t a first-order mereological dispute, either; it’s a disagreement about metaphysical primitiveness.

I’m not sure I like those kinds of (meta-)metaphysical views, at least as they apply to this kind of case. But it seems like this is the rough sort of background picture you need to sustain questions like “Where does extensionality really come from?”

Another thought. In general, extensionality principles are like special forms of the principle of identity of indiscernibles (PII): they say no distinction (between objects of this kind) without a difference (of this sort). That is: in second-order logic, PII says,

$\displaystyle \forall x \forall y \, (\forall X \, (Xx \equiv Xy) \to x = y)$ .

An extensionality principle says exactly the same thing, but with all of the quantifiers restricted: the $\forall x$ and $\forall y$ get restricted to some category of objects, and $\forall X$ gets restricted to some special class of properties. For example: the extensionality principle from set theory says no distinction between sets without a difference in members (i.e., a difference with respect to properties of the form $\lambda x \, a \in x$ , for arbitrary $a$ ). Similarly, mereological extensionality says: no distinction between objects without a difference in overlappers ( $\lambda x \, a \circ x$ ).

Anti-symmetry is really close to a principle of this form, namely: No distinction between objects without a difference with respect to parthood (that is, a difference with respect to properties of the form $\lambda x \, x \leq a$ or $\lambda x \, a \leq x$ ). “Really close” in that it’s logically equivalent in the presence of transitivity and reflexivity. So it’s natural to think of anti-symmetry as a kind of shorthand for this extensionality principle. Maybe this point is what you were referring to in the first paragraph. (I’m afraid I haven’t read the paper.)

I think the relationship between this principle of part-extensionality and the other principles of overlap-extensionality or disjointness-extensionality is going to come down to your views on the relationship between part, overlap, and disjointness. But regardless of whether anti-symmetry is really the ringleader, it’s nice to see it clearly as part of the gang.

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acotnoir permalink

Right, Andrew. Thanks for the comment. One option: I could suggest that there are reasons for thinking that if everything that overlaps x overlaps y, then x is part of y. Maybe they aren’t equivalent. Maybe “everything that overlaps x overlaps y” isn’t the parthood relation. But perhaps we could come up with extensionality-independent reasons for thinking that it implies parthood.

Another option: I could suggest that there is something conceptually important to the “remainder” intuition captured by strong supp. or complementation that one ought to preserve if possible. But if one rejects that “Everything that overlaps x overlaps y” implies that “x is part of y”, then you must reject the remainder intuition.

Hi Jeff,

These would be meta-arguments about which axioms are more “natural” or whatever. I’m inclined to think that such disputes usually are merely terminological when disagreeing over different equivalent axiomatizations. But the axiomatizations we’re talking about aren’t equivalent. And so we do have to ask ourselves, “which axiomatization expresses parthood?”.

I’m also not really interested in defending the claim that antisymmetry is responsible for all extensionality principles no matter what the axiomatization. I don’t think I need to say that antisymmetry is the “ring-leader”, as you put it. I just want to say something like “if you don’t like extensionality principles in mereology, then there are plausible reasons for not liking antisymmetry.”

I did make the point about antisymmetry being analogous to an extensionality principle for parthood. I didn’t, however, put the point as clearly as you did. Thanks.

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Andrew Bacon permalink

“I could suggest that there is something conceptually important to the “remainder” intuition captured by strong supp. or complementation that one ought to preserve if possible. But if one rejects that “Everything that overlaps x overlaps y” implies that “x is part of y”, then you must reject the remainder intuition.”

Well if I’m being a charitable interpreter I’m also going to interpret your intuitions about supplementation to be about sublapping, rather than parthood, as well. (I’ll use ‘sublap’ to mean ‘everything that overlaps x, overlaps y’ and superlaps for the converse.)

“But the axiomatizations we’re talking about aren’t equivalent. ”

If I’m reading your use of “parthood” as “sublaps” then I see no reason why they can’t be equivalent (although you haven’t yet said which axiomatization of nonwellfounded parthood your talking about.)

@Jeff. Yes, that’s one way to go, although I think that thought has led some people, e.g. Sider, to asking some pretty silly questions such as whether $\forall$ or $\exists$ is fundamental or to be defined out of the other. I get the feeling that resting the distinction on the debate between whether overlap or parthood is more fundamental is getting that way too.

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- Andrew permalink
  
  I reckon that given a rich enough non-well-founded mereology you’ll be able to prove some kind of theorem that says that for any model of it, you can always define an antisymmetric parthood relation on the same set whose sublap relation is the original non-antisymmetric parthood relation.
  
  Here’s an example: take the non-antisymmetric mereology $\mathcal{P}(\mathbb{R}^n)\times 2$ , where parthood is being a subset in the left coordinate (so, e.g., $\langle x, 0\rangle$ and $\langle x, 1\rangle$ will always be distinct mutual parts for any x.)
  
  Now let my new parthood relation be $\langle x, a\rangle \sqsubseteq \langle y, b\rangle \leftrightarrow x \subseteq y \wedge a\leq b$ . The sublap relation of this relation will be our original parthood relation.
  
  Reply
  - Andrew permalink
    
    (The general idea for the theorem being to well order each equivalence class under mutual parthood, to get the new parthood relation.)
    
    Reply
acotnoir permalink

Okay, so this is getting interesting.

There’s still a general line of response, which takes mereology to be about formalizing people’s folk intuitions about parthood and composition (or ordinary uses of the terms “part” and “composes” if you prefer.) So one could, for example, argue that the remainder intuition is an intuition about parthood, and so you shouldn’t call your relation “parthood”. (Of course, you might also argue that antisymmetry is an intuition about parthood, and hence I shouldn’t call my relation “parthood”. Etc, etc. I’m not sure ordinary people have intuitions about sublapping — unless sublapping is parthood. There are also going to be differences regarding the defintion of fusion. (Standard type fusions behave very differently when antisymmetry is present than when it isn’t.) So we should also check folk intuitions about fusions against the rival definitions. [NB: I'm not sure I like going this route. Intuitions probably underdetermine this stuff anyway. Plus, If our mereology is driven primarily by folk intuitions, there's reason to think we won't have unrestricted fusions anyway (of any sort).]

There’s still also the other line of response, mentioned by Jeff: talking about fundamentality of certain relations over others. I’m not sure I can really get a grip on those kinds of debates. I’m likely to agree with you that such questions can get kind of silly.

So let’s set those aside. I’m mostly wondering whether the two rival axiomatizations (one antisymmetric, one not) turn out to be equivalent.

I was thinking that the axiomatizations wouldn’t be equivalent with respect to fusions either. You can’t use the same definition of fusion in both cases because sublapping and parthood aren’t equivalent. But aren’t there going to be translations from one into the other?

In order to figure this out, we need to be clear about the axiomatizations involved. Maybe I should put this up in a separate post.

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- acotnoir permalink
  
  Here’s a question: what antisymmetric non-extensional mereology are you thinking of?
  
  Option 1: the partial order axioms, plus Weak Sup, plus unrestricted fusions, where fusion is defined as:
  $\mathsf{Fu}_1(t,\varphi(x)) := \forall y (y \circ t \leftrightarrow \forall x (\varphi(x) \rightarrow y \circ x)))$
  Of course, with this axiomatization, strengthening Weak Supp to Strong Supp gives you classical (and so all kinds of extensionality).
  
  Option 2: the partial order axioms, plus unrestricted fusions, where fusion is defined as:
  $\mathsf{Fu}_2(t, \varphi(x)) := \forall x ( \varphi(x) \rightarrow x \leq t ) \wedge \forall y ( y \leq t \rightarrow \exists x (\varphi(x) \wedge y \circ x))$
  Adding Weak Supp to this gives you classical mereology.
  
  It’s not even clear to me whether these two axiomatizations are equivalent.
  
  The non-antisymmetric axiomatizations I’m thinking of are these:
  
  Option 3: pre-order axioms, plus Strong Supp., plus unrestricted fusions, where fusion is defined via $\mathsf{Fu}_1$ .
  
  Option 4: pre-order axioms, plus Complementation given by:
  $y \nleq x \rightarrow \exists z \forall w( w \leq z \leftrightarrow (w \leq y \wedge \neg w \circ x))$
  plus unrestricted fusions, where fusion is defined as minimal upper bounds:
  $\mathsf{Fu}_3(t, \varphi(x)) := \forall x ( \varphi(x) \rightarrow x \leq t ) \wedge \forall y( \forall x (\varphi(x) \rightarrow x \leq y) \rightarrow t \leq y)$
  
  I know that Option 3 and Option 4 are equivalent axiomatizations (thanks to Paul Hovda). If Options 1 and 2 are equivalent, then I suppose it doesn’t matter which we pick. In any case, the easiest thing to check is whether Option 1 and Option 3 are equivalent with parthood in Option 3 being Sublapping in Option 1. But it’s not obvious how this would go, given the Fusion axioms are going to be expressing very different things.
  
  Reply
acotnoir permalink

It took me a bit to convince myself that the sublap relation on your $\sqsubseteq$ is non-antisymmetric. But it is. The 1-variant of x and the 0-variant of x will overlap with exactly the same zs; even if the common parts between z and the 1-variant of x are different from the common parts between z and the 0-variant of x.

This sort of model is nice because it’s kind of natural. Antiextensionalists who accept antisymmetry think the statue/clay example is such that the clay is part of the statue (but not vice versa). I’ve always wondered why think that’s the case. It has always seemed to me that whatever reasons you might give for the clay being part of the statue would also apply to the converse. That is, unless you thought that there is something *more* to the statue than there is to the clay (the statue has some additional properties, and the like). But then we can ask: are those additional features *parts* of the statue, or not? If not, then why is there a *mereological* difference between them — why is one part of the other but not vice versa. If these features are parts of the statue, then of course we no longer have a counterexample to extensionality. It just seems to me that the additional properties of the statue trigger some kind of remainder intuition, which leads to people rejecting mutual parthood. But that remainder should either be mereological (in which case we still have extensionality) or non-mereological (in which case we don’t get asymmetric parthood).

In any case, your model makes this explicit, especially if you think of the second coordinates as non-mereological (perhaps modal) properties. Anyway, sorry for the side-track.

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- Andrew permalink
  
  Hi Aaron,
  
  Just to quickly answer some of the questions you asked:
  
  “You can’t use the same definition of fusion in both cases because sublapping and parthood aren’t equivalent. But aren’t there going to be translations from one into the other?”
  
  I’m not seeing this yet, at least, not if the thing I claimed was true is true. (Of course, once you’ve interpreted the parthood symbol as sublapping, the definition of fusion will be translated non-standardly too…)
  
  Regarding the different axiomatizations I guess I want the constraint that D/~ will be a standard mereology (at least, that’s what my example relied on, and the generalization I was suggesting.) Given all that, one would think that by taking each equivalence class under ~ and carefully choosing some kind of complete partial order on it you could get a partial order mereology without extensionality.
  
  (Note also in the example I could have let and be maximal. This would have resulted in a partial order mereology without a preference for 1 or 0 and non-unique fusions. Maybe this is the preferable strategy: take each equivalence class under ~ and make each member maximal by deleting the parthood relations between each other.)
  
  Reply
Andrew permalink

Sorry – the HTML ate part of the bit in parentheses. It should have been “let $\langle x, 0\rangle$ and $\langle x, 1\rangle$ be maximal.” In other words “latex \langle x, a\rangle \sqsubseteq \langle y, b\rangle \equiv (x \subset y \vee (x=y \wedge a=b))$.

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