Does Lewisian combinatorialism imply quidditism?

From Combinatorialism to Legal Contingency

In “On the Plurality of Worlds”, Lewis endorses the following principle:

Lewisian Combinatorialism: For any x1,x2,… (perhaps from different worlds) and any spatiotemporal arrangement R (except one that co-locates two or more of its relata), there is a possible world where there are perfectly-natural duplicates y1,y2… of x1,x2,… (respectively), such that Ry1,y2,…

Definition: x and y are “perfectly-natural duplicates” just in case they have exactly the same perfectly natural properties.

It is rather uncontroversial, or at least it should be, that Lewisian combinatorialism implies that the laws of nature are contingent. Consider a (hopefully uncontroversially) possible world w where there are two positively charged particles p1 and p2 that bear the following spatiotemporal relationship to one another: at t0 they are d0 apart, at t1 they are d1 apart, at t2 they are d2 apart, where d1 is greater than d0 and d2 is greater than d1. By Lewisian combinatorialism, there is a possible world w* where there are two particles p1* and p2* which are duplicates of p1 and p2 (respectively) and which bear the following spatiotemporal relationship to one another: at t0 they are d2 apart, at t1 they are d1 apart, and at t2 they are d0 apart. In short, w is a world where p1 and p2 move away from one another and w* is a world where there perfectly-natural duplicates, p1* and p2*, move toward each other.

Is the law that positively charged objects repel one another violated in w*? That depends on whether p1* and p2* are (both) positively charged: if so, then it is; if not, then it isn’t. Now we said that p1* and p2* are perfectly-natural duplicates of p1 and p2 (respectively) which are both positively charged. So our question is this: is positive charge a perfectly natural property? From the perspective of current science, positive charge is a perfectly natural property. Let us assume then that it is—if science later discovers that positive charge is a structural property composed of yet more natural properties, we will use one of those more natural properties in our example. In that case, Lewisian combinatorialism indeed implies that there is a possible world—namely, w*–where the law that positively charged particles repel is violated. By suitably complicating the example, we can show along the same lines that Lewisian combinatorialism implies that there is a possible world where that law is rampantly violated, and hence is a world where it is not a law at all that positively charged objects repel. In short, Lewisian combinatorialism implies that the laws of nature are contingent.[i]

Legal Contingency is not Quidditism

Despite some impressions to the contrary, legal contingency is not quidditism. Legal contingency is the claim that the laws of nature are contingent. All that is required for this claim to be true is that in some possible world some property, say, positive charge, does not realize the nomological role that it realizes in our world. (This can happen in one of two ways—by realizing a different role or no role.) Quidditism, however, is the claim that some nomological role that is in fact realized, is, in some other world, realized by a property distinct from the property that realizes it in our world.

From Combinatorialism to Quidditism?

Consider a (hopefully uncontroversially) possible world w where there are two particles, p1 and p2, with masses m1 and m2 and positive charges q1 and q2. Now assume that the particles move in accordance with classical dynamics, so that the acceleration of p1 is (Gm1m2/r2 + q1q2/4pε0r2)/m1, and similarly for p2. By Lewisian combinatorialism, there is a possible world w* where there are two particles p1* and p2* that are perfectly natural duplicates of p1 and p2, respectively, and which move as follows: the acceleration of p1 is (Gq1q2/r2 + m1m2/4pε0r2)/q1, and similarly for p2. It looks as if the nomological role that is realized by mass in w is realized by positive charge in w* and it looks as if the nomological role realized by positive charge in w is realized by mass in w*. If so, then we have here an argument from combinatorialism for quidditism.

But not so fast. Are we really justified in concluding that positive charge and mass have swapped nomological roles in w*? All we know about w* is that the particles exhibit a certain pattern of behavior—namely, that p1* and p2* accelerate at a certain rate. A regularity theorist about natural laws would thus conclude that the properties have swapped nomological roles, especially if the example were made such that the world in question contained many more particles and the pattern of behavior was much more widespread. But those who have a dislike for the regularity account of laws ought to hesitate before concluding that the two properties have swapped nomological roles in w*.

Concluding Remark

The reason that combinatorialism rather uncontroversially implies legal contingency but does not uncontroversially imply quidditism, concerns the relationship between laws and mere regularities: it is rather uncontroversial that certain laws imply certain regularities (and hence uncontroversial that violations of those regularities—at least widespread violations of those regularities—imply that those laws don’t hold) but it is very controversial whether certain regularities imply certain laws (and hence controversial whether certain regularities imply the existence of certain laws).


[i] This is not to say that defenders of the necessity of laws could not accept other forms of combinatorialism.

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5 Responses to Does Lewisian combinatorialism imply quidditism?

  1. Shen-yi Liao says:

    i didn’t quite follow the second case. is the reason why they may not have swapped nomological roles that there might be other laws in that w* where mass and positive charge play the same nomological roles as they do in w?

    also, let me know if the following assumptions are incorrect: 1. by possible you mean something wider than nomic, at least; 2. in the first case it is stipulated that in both worlds, time’s arrow goes from t0 to t2.

  2. dtlocke says:

    Hi Sam, thanks for the questions!

    “by possible do you mean something wider than nomic possibility?”

    What I mean is metaphysical possibility. I can’t say whether this is wider than nomic possibility without begging the question of whether the natural laws are necessary (i.e., metaphysically necessary).

    “in the first case it is stipulated that in both worlds, time’s arrow goes from t0 to t2?”

    Yes, and that is an excellent question indeed (as are the other two).

    “is the reason why they may not have swapped nomological roles that there might be other laws in that w* where mass and positive charge play the same nomological roles as they do in w?”

    No. The laws of w cannot hold in w* (i.e., the properties cannot realize the same nomological roles in w* that they do in w) because those laws are violated in w*. However, this does not imply that the properties have swapped nomological roles—it might be that (A) they realize some completely different nomological roles or (B) they realize no nomological roles at all. All we know about w* is that a certain pattern of behavior is exhibited. Unless we are regularity theorists, we cannot conclude from this what the laws are (i.e., what the roles are and which properties realize them). But, as I noted, we can conclude that they are *not* the laws of w. The general assumption here is that everyone agrees that certain patters are necessary for certain laws, but only the regularity theorists thinks that certain patterns are sufficient for certain laws.

    I hope my answers clears things up a bit!

  3. Thom Blake says:

    It seems that you’re leaving out the logical possibility that the ordinary (w) laws of gravity and magnetism are still in effect in w*, but happen to be canceled out in w* by even more powerful forces (the natural attraction and repulsion of, say, green and red objects). Of course, being able to even make that distinction probably begs the question.

  4. dtlocke says:

    Hi, Thom. Thanks for bringing this up! I realized after posting this that I wasn’t as explicit about when I described w and w* as I should have been. Both descriptions should have included a “and that’s all clause”–i.e., both w and w* should have been stipulated to contain no particles other than the ones mentioned and the particles should have been stipulated to have no properties other than the ones mentioned (and those that are entailed by the ones mentioned). Thanks for reminding me to be more explicit!

  5. Jon says:

    Big D,

    Two things. First, I don’t understand why you connect the legal contingency->quidditism move with Lewis’s being a regularity theorist. It seems to me that you could easily be a regularity theorist (and a believer in Lewisian combinatorialism) without being a quidditist, simply by identifying properties with their nomological roles (as indeed Ramsey sentences suggest one should do). I don’t have “Ramseyan Humility” at hand to reference, but from what I remember Lewis has some explanation of his belief in quiddities that doesn’t have to do with his neo-Humeanism about laws. (Perhaps it’s rooted in his commitments about counterparts? It’s been a long time since I’ve read the article so you’ll have to remind me of what he does here.) Seems to me that the definition of quidditism you’re using here obscures the independence of being committed to quidditism (qua metaphysical thesis about transworld, nomological-role-independent identity of properties) from whatever commitments one has about laws. Of course you’re more up on this stuff than I am so I’m sure you have your reasons, but I’d like to hear more about why quidditism gets glossed as a claim explicitly about nomological roles.

    Second, I agree that in your example it sounds weird to say that mass is playing the positive charge role, etc., because what we take mass to be is so far away from what it would have to be if it were to do the sorts of things that positive charges do in this world. My intuitions aren’t nearly so clear in the case Lewis actually gives, where negative and positive charge are to switch roles. I assume the point of using the properties you do is to clear up the intuitions, but I wonder what it is about your version of it that does the clearing up. I don’t have a good answer to that at the moment, but I wonder if you have similarly clear intuitions about the case as Lewis presents it. If so, I guess I”ll have to try to articulate why I get different intuitions in response to the two cases.

    Best,
    j

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