Tooley on Laws of Nature and Counterfactual Support

So I was looking over Dan’s reconstruction of Tooley’s argument below, and I’m still somewhat worried about its validity.

Dan mentioned that I thought there might be some funny business with premises (3) and (10) (on his first reconstruction):

(3) It is a nomological truth that all salt, when in water, dissolves.

(10) It is true that if this piece of salt were in water and were not dissolving, it would not be in the vicinity of a piece of gold.

Dan’s certainly right that there’s no straight-up contradiction here (like there would be if (10) was stating a material conditional).  However, something still feels very odd about the line of argumentation being taken.  What originally bothered me was the fact that, in (10), we are using a putative law to support a counterfactual about what would happen in a situation in which that law (or, rather, the law from which it is derived) is violated.  And I’m not convinced that any law can support a counterfactual like this.

This is a problem because Tooley is trying to draw a distinction between two classes of nomological truths (the laws proper and the logical consequences of laws).  He is arguing that the second class cannot support certain counterfactuals which the first class can – and that, therefore, they must be treated differently.  However, if he demonstrates this by pointing to a certain class of counterfactuals which are not only problematic for the second class, but for the first class as well, then he’s failed to draw the distinction.

So, I’ve been convinced by Dan that Tooley’s argument demonstrates that nomological truths like (L) all salt, when in water and near gold, dissolves in water. have difficulty supporting certain counterfactuals.  However, I haven’t been convinced that laws proper don’t face the very same difficulty.  On my understanding of things, if I can show that a plain jane law faces similar difficulties with the same kind of counterfactual, then I will have undermined Tooley’s distinction.

I take it that some laws can be derived from more fundamental laws.  I think a good example of this is Snell’s law, which governs the angle of refraction light makes when moving from one medium to another.  It states that  n1(sinθ1)=n2(sinθ2), where n = the refraction index of the medium and θ = the measure of the interior angle the light makes with the line orthogonal to the border between the two media – as shown in the diagram.

from scienceword.wolfram.com

from scienceword.wolfram.com

We could represent this law logically in the following manner:

(x)(y)(z)[(N1x&N2y&θ1zxy) → θ2zxy]

where N1Φ = ‘Φ is a medium with refractive index n1.’

N2Φ = ‘Φ is a medium with refractive index n2.’

θ1ΦΧΨ = ‘Φ is light which makes angle θ1 with the line orthogonal to the border between medium Χ and Ψ while in medium Χ.’

and θ2ΦΧΨ = ‘Φ is light which makes angle arcsin((n1/n2)sinθ1) with the line orthogonal to the border between medium Χ and Ψ while in medium Ψ.’

Now, like I said before, Snell’s Law can be derived from more basic laws.  I’ve been led to believe that it can be derived from the wave-nature of light along with the law of conservation of momentum (but I don’t think it really matters for our purposes whether this is true).  So, it would be a mathematical truth that

(A) (W&C)→S

where W = ‘light is suitably wave-like,’ C = ‘total momentum is conserved,’ and S = ‘Snell’s Law is true.’*

Given this, it seems plausible to say that we could conjoin the proposition C = ‘total momentum is conserved.’ to the antecedent of Snell’s Law, and still be left with a ‘plain jane’ law:

(x)(y)(z){[(N1x&N2y&θ1zxy)&C] → θ2zxy}

However, from here, we can just follow the procedure outlined in Dan’s post to get the following counterfactual to come out true:

(B)  For media a and b, and light c, if it were the case that N1a and N2b and θ1cab, yet not the case that θ2cab, then it would not be the case that C.

However, (B) looks false.  Given (A), I feel inclined to say that, in the counterfactual scenario in which Snell’s Law is violated, it is either the case that not-C or it is the case that not-W.  To further illustrate the point, we could perform the exact same reasoning again and conclude that, because

(x)(y)(z){[(N1x&N2y&θ1zxy)&W] → θ2zxy}

is a ‘plain jane’ law, this counterfactual will come out as true:

(C) For media a and b, and light c, if it were the case that N1a and N2b and θ1cab, yet not the case that θ2cab, then it would not be the case that W.

On the standard picture of counterfactuals, this would allow us to conclude that, if Snell’s Law were violated, then it would be the case both that momentum is not conserved and that light is not suitably wave-like.  For the closest possible world in which the antecedent is satisfied in (B) will also be the closest possible world in which the antecedent is satisfied in (C) – after all, they are the same antecedent.

In conclusion, it looks to me like Tooley has discovered a problem with counterfactually speculating about situations in which laws are violated, and not that he has discovered any interesting difference between laws and the deductive consequences of laws.

[Admittedly, I should also give an example in which not only the derived law, but the law from which it was derived is violated – since this is more analogous to the example Tooley himself gave, but I’m not up to the task right now.  Maybe I’ll update later with an example like that.]

* You would undoubtedly have to stick some ancillary assumptions in here also.

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2 Responses to Tooley on Laws of Nature and Counterfactual Support

  1. Shen-yi Liao says:

    The controversial premise is in the sentence “I take it that some laws can be derived from more fundamental laws.” Being a complete amateur on this, is there some reason to be committed to there being non-fundamental laws? More importantly, does Tooley commit himself to this somewhere? As you recognize in the last parenthetical, if the answer to these questions cannot be answered affirmatively, it seems that your argument would lose some force.

  2. Hey there Dmitri,

    Good response. I think that you’re at least partially right. Tooley does need 6 and 13 (of my reconstruction) to ascribe incompatible properties to laws and other nomological truths, respectively. What I take it that you’ve attempted to show is that Tooley only succeeds in ascribing a property to mere nomological truths that is compatible with the properties of laws. You attempted to show this by showing that the property he ascribes to the mere nomonlogical truths (this property of supporting certain kinds of counterfactuals) also holds of the laws.

    I think that you’ve misunderstood the type of property that Tooley is attempting to ascribe to laws that the merely nomological truths do not have. I thought about making this more clear in my original post, but I decided that it was a detail that could be glossed. The issue is this:

    Strictly speaking, laws do not support counterfactuals; that is, for any law, there are some counterfactuals that the law does not support. For instance, the law that all salt dissolves in water does not support the counterfactual if Nixon would have pushed the button, etc. Rather, the idea is that there are a certain class of counterfactuals, that are in the relevant sense `about the law,’ that the law supports.

    The reason why I didn’t spell this out more clearly in the original post is that it’s hard to specify exactly what that class is for each law. But, I can provide an alternative solution here:

    Assume that all nomological truths (including laws) are of the form (x)(Fx -> Gx). (I realize that this is a big and probably false assumption.) Then, given some law L
    ( = (x)(Fx -> Gx) ), define the `canonical counterfactual’ of that law as if some object a where F then it would be G. Say that a nomological truth `canonically supports counterfactuals’ if it’s canonical counterfactual is true in virtue of it being true.

    Now, it should be clear that proper laws canonically support counterfactuals: if it’s a law that if something is salt and in water, then it dissolves, then the canonical counterfactual, if something were salt and were in water, then it would be dissolving, is true.

    On the other hand, mere nomological truths do not canonically support counterfactuals. We can see this using the example from the reconstruction: it’s a mere nomological truth that all salt that is in water and doesn’t dissolve isn’t near gold, but it’s not true that if something were salt and in water and doesn’t dissolve it wouldn’t be near gold (the canonical counterfactual of this nomological truth).

    I take this to be Tooley’s claim.

    Now, you may object to this property as being essential to laws, and thereby claim that Tooley as done little more than he would have if he had said that laws and mere truths are distinct in virtue of us calling them “laws” and “mere truths”. But, I don’t think this line is right; it is essential to being a law that it support it’s canonical counterfactual, and mere nomological truths do not have this property.

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