Show-Me the Argument

This is the paper I presented in the Phil. of Sci. seminar where we had no time for discussion. Would love to hear comments from you guys, both people in the seminar and those who are not. (This is a part of the practice for the seminar, and the paper is rather too long for a post. For those who are not in the seminar, if you could just read Part II, III, and VI, and offer some comments, that would be great.)
I. Introduction
For the past decade or so, John Carroll’s Mirror argument (John Carroll, 1994) has centered the debate over Humean supervenience (HS), which is, according to Carroll, the thesis that “no two possible worlds have propositions that agree on all their nonnomic features and also disagree on their status as laws.”[ There are different formulations of HS. For simplicity, I will simply reply on Carroll's formulation. For more recently characterization of HS, see Earman and Roberts (2005).] (Carroll, 1994: 58). In the Mirror argument, Carroll imagines a series of possible worlds that share all the nonnomic facts but nevertheless disagree on their laws, and Carroll claims that this disproves HS. Proponents of HS usually contend that Carroll presupposes in the argument a concept of law that is incompatible with the Humean concept of law, and thus simply begs the question against HS (Roberts, 1998; BeeBee, 2000).

Recently, Barry Ward proposes a new interpretation of the Mirror argument such that the argument need not presuppose a concept of law that is incompatible with Humean lawhood. I will argue that Ward’s new interpretation also presupposes a non-Humean concept of law , and thus it still faces the charge of begging the question. I will then suggest an alternative defence for the Mirror argument that does not beg the question against HS.

II. The Mirror Argument
Humeans, such as David Lewis, believe that what the world contains is simply particles, fields, and whatever entities discovered by our science, physics in particular. Laws of nature are simply generalizations of these nonnomic facts that appear “as a theorem (or axiom) in each of the true deductive system that achieves a best combination of simplicity and strength” (Lewis: 73). HS, thus, is usually thought to follow from this Humean characterization of laws of nature - worlds that share all the nonnomic facts share the same laws.

The Mirror argument, along with other earlier attempts[ See also Bas van Fraanssen (1989), chapter 3, Michael Tooley (1977), and John Carroll (1990).], is supposed to show that HS is false. In this argument, Carroll asks us to imagine two possible worlds U1 and U2, both of which contain only X-particles, Y-fields, and a mirror, which is posited in a certain angle such that it does not deflect any X-particle away from any Y-field. Let  be a law that all X-particles have spin up when they enter a Y-field. Also, suppose in U1, all X-particles do enter Y-fields and have spin up in Y-fields, and U2 is exactly the same as U1, except that there is one particular X-particle, b, which does not get spin up in the Y-field it enters. That is to say,  is a law in U1, but it is not a law in U2. Humeans will agree with Carroll on all of these so far.

Now, imagine two further possible worlds U1* and U2*, in which all the other things are exactly the same as in U1 and U2 respectively, except that the mirror is turned to a different angle such that it deflects the X-particle b away from Y-fields. Consequently, U1* and U2* now share all the nonnomic facts, since b never enters any Y-field. Thus, if HS were true,  should be a law in both U1* and U2*. But, as we see,  is not a law in U2, so  should not be a law in U2* as well, for the only difference, according to our assumption, between U2 and U2* is that the mirror is turned to a different angle. How could the issue of whether something is a law or not depend on such a contingent fact as the angle of a mirror? Therefore, it must be false that two possible worlds that share all the nonnomic facts must also agree on their laws.

The Mirror argument, as Carroll puts it, clearly relies on the following principle (Carroll, 1994: 59):
(SC) If, in a certain possible world, P is not true but is physically possible and Q is not a law, then Q is still not a law in the closest possible world where P is true.

It is because of this principle, we are able to say that since  is not a law in U2,   is also not a law in U2*, which is the closest possible world to U2, where the mirror is rotated to that particular angle. This very principle, however, has been called into question by proponents of HS.

III. Typical Humean Response
Helen BeeBee (BeeBee:588) contends that whether SC is true depends on whether U2* interpreted in Carroll’s way is indeed the closest possible world to U2, where the mirror is turned to that particular angle. There are two possible interpretations of U2*. One is Carroll’s interpretation, namely, the closest physically possible world, which shares the same set of laws with U2. However, given such an interpretation, U2* must differ from U2 in some respect in the history before b hits the mirror, since if it were identical to U2 with respect to that history, it would be identical to U2 for all future times, including the positioning of the mirror, given that the laws remain unchanged. So, if b does hit the mirror in U2*, U2* must be a pretty distant world from U2.

On the other hand, if U2* is interpreted as just the closest possible world to U2, then it will share all the history with U2 right before the deflection. But there is a small “miracle” just before the deflection so that the laws in U2* allow the mirror to rotate to the right angle - that is to say, U2* has different laws, which allow the “miracle” to occur. SC is thus false if U2* is interpreted in the second way.

There are two conclusions that Humeans can draw from the previous discussion. A stronger conclusion is that U2* interpreted in the second way, the way they prefer, is closer to U2, and thus SC is false. A weaker conclusion is that, as BeeBee puts it, to interpret U2* as the closest physically possible world where the laws remain intact is to presuppose a governing concept of law - laws are somehow out there, “prior to and watching over the particular matters of fact to make sure that they don’t step out of line” (BeeBee: 584), which is precisely the one that Humeans will reject. Thus, to interpret U2* in the first way is to beg the question against Humeans.

The stronger conclusion is of course very problematic. One worry is that it is not clear that there is a widely accepted way of determining the closest possible world. Why is U2* interpreted in a Humean way closer to U2 than U2* interpreted in Carroll’s way? Second, granted that there must be a change of laws in U2*, interpreted in a Humean way, such that the mirror’s rotation can occur, why does the change of those relevant laws necessarily belies  as a law in U2*? The law that allows the mirror to rotate need not be the same law as . Finally, why should we suppose that U2*, interpreted in Carroll’s way, must differ from U2 in regard to the history before the deflection? To suppose that is to presuppose determinism. But if indeterminacy is compatible with the existence of laws, as it is, why cannot Carroll say that whether the mirror rotates to that particular angle is as indeterminate an event as the movement of a quantum? So, there need not be a difference in the history before the deflection occurs; what differs in U2 and U2* may just be that the mirror indeterminately rotates to a particular angle. Consequently, U2*, interpreted in Carroll’s way need not be that distant from U2.

Therefore, I think Humeans cannot have the stronger conclusion. However, the weaker conclusion still stands. Humeans can contend that the second interpretation is at least as good as Carroll’s. Under the second interpretation, SC is not true. Thus, the assumption that the Mirror argument relies on, namely, whether there is a law does not depends on such contingent fact as the angle of the mirror “is clearly born of the view that particular matters of fact depend upon the laws and not vice versa … [i]f the only support to be had for SC is such an explicit avowal of the anti-Humean conception of laws, then the Mirror Argument is clearly question-begging.” (BeeBee: 589)

John Roberts has a similar line of objection. He argues that, according to Lewisian conception of law, among the deductive systems that are true on one set of nonnomic facts, “either there is, or there is not, a unique one that achieves the best balance of strength and simplicity.” (Roberts: 434) Moreover, for any given law claim, it either does or does not belong to that unique deductive system. Therefore, since U1* and U2* share all the nonnomic facts, and thus they must share the same set of deductive system that achieves the best balance of strength and simplicity. Consequently,  either is or is not a law in both U1* and U2* (ibid). So, to suppose that  can be a law in U1* but not in U2* is to reject the Lewisian conception of law right at the start.

The debate between people who believe HS and those who do not, then,  is really about which of the two is a more plausible intuition: HS or SC? (Roberts: 434) The main contention from Humeans has been that the Mirror argument shows a surprising consequence of HS, but the intuition that is offended is really “a mixed modal-nomological intuition” of laws (Roberts: 436), which is exactly what Humean rejects. If the Mirror argument is really to succeed, it has to show that there are strong intuitions other than such an already modally-loaded intuition that are offended by HS.

Barry Ward recently suggests a defence for the Mirror argument in roughly this direction.

IV. Ward’s New Proposal
Ward argues that one may believe something like SC not because of a metaphysical presupposition of governing laws, but because of a more commonly accepted idea that laws explain and predict. More specifically, laws explain and predict by generation - that is, “the law statements, along with statements of the relevant boundary or initial conditions, are (non-redundant) assumptions in a derivation that yields a specification or model of the behaviour of the system in question.”(Ward: 545)

According to Ward, it is characteristic of laws that they are rules that generate models describing the possibilia of a given system for the purpose of explanations and predictions, and that “the models generated by laws may be used contrastively to single out the factor crucial to the occurrence of the feature to be explained” (ibid). In this sense, laws are still constrained by nonnomic facts, but the constraint is much weaker than the constraint imposed by HS. On Ward’s view, the only constraint is that laws need to be consistent with the history of nonnomic facts (Ward: 549). And a law claim is consistent with such a history “if, and only if, the application of the specified MGR [Model Generating Rules] (or set of MGRs) to the boundary conditions for that history yields a model consistent with the history.” (ibid)

Therefore, a model generating rule can be a law as long as it is consistent with the history of the nonnomic facts in that world. For example, Ward claims that two model generating rules about electrostatic force, F=qE and F= -qE, can both be laws in a world where no positively charged particles enter an electronic field, because both of them generate models that are consistent with the history of the nonnomic fact, namely, the fact that all the particles, which are all negatively charged, are subject to a force in the opposite direction when entering an electronic field. Moreover, the history provides no basis for deciding between the two models generated by different laws. (Ward: 545-46) Likewise, since both the set of laws that contain  and the set of laws that do not generate models that are consistent with the nonnomic facts in U2*, the history of U2* simply cannot adjudicate between them. So, one can hold  as a law in U2* without presupposing a metaphysical notion of governing laws.

One more thing needs to be said about Ward’s view before we move on. Although Ward argues against HS, he is also not committed to any realism about laws. He denies, for example, that laws are metaphysical entities, which Carroll believes. Nevertheless, according to Ward, essential to the conception of law is the model-generating role, which is to explain and predict.

V. Problem for Ward
Although Ward’s proposal avoids presupposing a governing concept of law, I think he still faces the charge of begging the question against Humeans.

One of the key claims in a Lewisian conception of law, as Roberts emphasizes, is that for any set of nonnomic facts, there is a unique set of deductive system that achieves the best balance of simplicity and strength. Laws are not only constrained by the requirement that it should be consistent with the nonnomic facts, but also by the requirements of simplicity and strength. However, Ward claims that nonnomic facts cannot adjudicate which competing MGR is a law if they generate models that can explain and predict the facts equally well. Ward’s conception of law, thus, acknowledges only the consistency constraint, and simply rejects the constraints of simplicity and strength, which are clearly required by the Lewisian conception of law.

However, we may interpret Ward not as proposing an alternative conception of law that sraightforwardly rejects a Lewisian lawhood, but as pointing out a serious defect in that conception of law. The defect is that the Lewisian conception of law may fail to adjudicate between explanatorily and predictively equal MGRs, e.g. F=qE and F= -qE, on the basis of nonnomic facts in a given world. These two equations may be said to have an equal balance of simplicity and strength, and thus are equally qualified as laws in that world.

That said, the same argument cannot be applied to the justification of SC. Given the nonnomic facts in U2*, the set of laws that contain  and the set of laws that do not contain  are equally consistent with the facts. However, the latter set is clearly simpler than the former - it contains fewer axioms, whereas the former has no obvious advantage in its strength - it explains nothing in a world which contains no relevant instantiation. So, even if the Lewisian conception is inadequate for determining laws in Ward’s case, it clearly eliminates uninstantiated “laws” as in the Mirror argument. To claim that  is a law in U2* on the base of its generating role is to beg the question against Humeans.

VI. Two Offended Intuitions (Very Sketchy)
So, Ward fails to defend SC without begging the question against HS. To defend the Mirror argument, we need to appeal to other non-modal-nomologically loaded intuitions that are also offended by HS. In what follows, I will offer two such intuitions.

First, it is imaginable that we can have laws that can be deduced from whatever system of laws that Lewisian conception of law allows, but the deduced law need not be instantiated in this world. Think about Mendeleev’s Periodic Table of the chemical elements. There is at least one element, ununseptium (117), on that table that has not been discovered, and there is a possibility that it may not exist. However, we are not completely ignorant of its potential chemical/physical properties. For example, ununseptium is expected to decay by alpha decay to element 115. This property is presumably inferred from the chemical and physical laws that we already have. It is intuitive to say it is a law that ununseptium decay by alpha decay to element 115, though the corresponding events may not be instantiated at all in this world. But if HS is true, this intuition must be rejected.

I can imagine two responses. One may be that it is not possible that there exists no ununseptium. It is just that we have not found it yet. However, this amounts to saying that anything that can be deduced from our existing physical system must have instantiations. This is simply too bold a claim to accept. A unicorn may be perfectly deducible object from the current physical system, but it need not exist. Seond, one may deny that “ununseptium decay by alpha decay to element 115″ is a law. I find the intuition that this claim is a law is too strong to be simply denied without a good reason. Notice that there is no presupposition of any governing concept of law, the claim is deduced from our current system of laws, which Lewisian theory admits. So, to reject that claim as a law is not simply to offend a non-Humean modal-nomological intuition, but to offend an intuition that is modally innocent, and much stronger.

My second, and related, argument goes like this. It is usually admitted that we have negative laws, such as “no solid spheres of uranium-235 have a diameter larger than one mile”. However, this law is true precisely because of the following: if a sphere of uranium-235 would have a diameter larger than one mile, it would have a chain reaction that leads to its elimination. However, the latter could not be a law according to HS, because it is presumably not instantiated in the world at all. There does not exist a sphere of uranium-235 that has a diameter larger than one mile and goes out of existence. But it is bizarre to think that something could be a law though what entails it is not. So, HS offends another rather plausible intuition. One may want to deny that sentences, such as “no solid spheres of uranium-235 have a diameter larger than one mile”, are laws after all. But, again, this is to deny an intuition that seems to be widely held and has no explicit presupposition of a non-Humean modal-nomological implication.

Therefore, I conclude that there are strong intuitions other than a mere  non-Humean modal-nomological intuition that are offended by HS, and thus there are good reasons to prefer SC to HS.

This entry was posted on Tuesday, November 18th, 2008 at 11:46 pm and is filed under Ariew Seminar (Fall 08), Philosophy of Science. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.