Causation and contingency

A correspondent yesterday reminded me of a classic objection to the “inductive” approach to the causal principle that all contingent things have causes in the context of cosmological arguments. As I understand the objection, it goes like this:

1. Granted, we have good reason to think that all the contingent things we observe do have causes. However, all these causes are contingent causes, and so we have equally good inductive support to think that all contingent things have contingent causes. Thus, to extend this reasoning to conclude that the cosmos—the sum total of all contingent things—has a cause is illegitimate, since the cosmos cannot have a contingent cause on pain of circularity.

An initial response is that (1) as it stands appears to rely on a false principle of inductive reasoning:

2. Suppose that all observed Fs are Gs, and that all observed Fs are also Hs. Then we have equally good inductive support for the hypothesis that all Fs are Hs as that all Fs are Gs.

But (2) is false. All observed emeralds are green and all observed emeralds are grue, where an emerald is grue if it is green and observed before 2100 or it is blue and not observed before 2100. It is reasonable to conclude that all emeralds are green but not that they are all grue. Or even more simply, from the facts that all observed electrons are charged and all observed electrons are observed, it is reasonable to conclude that all electrons are charged but not that all electrons are observed.

Nonetheless, this response to (1) does not seem entirely satisfying. The predicate “has a contingent cause” seems to be projectible, i.e., friendly to induction, in a way in which “is grue” or “is observed” are not.

Still, I think there is something more to be said for this response to (1). While “has a contingent cause” is not as obviously non-projectible as “is observed”, it has something in common with it. We are more suspicious of inductive inferences from all observed Fs being Gs to all Fs being Gs when being G includes features that are known prior to these observations to be concommitants of observation. For instance, consider the following variant of the germ theory of disease:

3. All infectious diseases are caused by germs that are at least 500 nm in size.

Until the advent of electron microscopy, all the infectious diseases whose causes were known were indeed caused by germs at least 500 nm in size, as that is the lower limit of what can be seen with visible light. But it would not be very reasonable to have concluded at the time that 500 nm is the lower limit on the size of a disease-causing germ. Now, something similar is happening in the contingent cause case. All observable things are physical. All physical things are contingent. So being contingent is a concommitant of being observed.

Finally, there is another epistemological problem with (1). The fact that some evidence gives as good support for q as for p does not mean that q is as likely to be true as p given the evidence. For the prior probability of q might be lower than that of p. And indeed that is the case in the reasoning in (1). The prior probability that everything contingent has a contingent cause is zero, precisely for the reason stated in (1): it is impossible that everything contingent have a contingent cause! But the prior probability that everything contingent has a cause is not zero.

Divine simplicity and knowledge of contingent truth

I think the hardest problem for divine simplicity is the problem of God’s contingent beliefs. In our world, God believes there are horses. In a horseless world, God doesn’t believe there are horses. Yet according to divine simplicity, God has the same intrinsic features in both the horsey and the horseless worlds.

There is only one thing the defender of simplicity can say: God’s contingent beliefs are not intrinsic features of God. The difficult task is to make this claim easier to believe.

It’s worth noting that our beliefs are partly extrinsic. Consider a world just like ours, but where a mischievous alien did some genetic modification work to make cows that look and behave just like horses to the eyes of humans before modern science, and where humans thought and talked about them just as in our world they talked about horses. If a 14th century Englishman in the fake-horse world sincerely said he believed he owned a “horse”, he would be expressing a different belief from a 14th century Englishman in our world who uttered the same sounds, since “horse” in the fake-horse world doesn’t refer to horses but to genetically modified cows. Their beliefs about rideable animals would be different, but inside their minds, intrinsically, there need be no difference between their thought processes.

But it is difficult to stretch this story to the case of God, since it relies on observational limitations. Moreover, it is hard to extend the story to more major differences. If instead of fake horses, the alien produced tauntauns, no doubt the minds of the people in that world would be intrinsically different in thinking about riding tauntauns from our minds when think about riding horses (even if accidentally their English speakers used “horse” to denote a tauntaun).

While our beliefs are partly extrinsic, God’s contingent beliefs are radically extrinsic according to divine simplicity. There are no intrinsic differences in God no matter how radical the differences in belief are.

This feels hard to accept. Still, once we have accepted that beliefs can be partly extrinsic, it is difficult to mount a principled argument against radical extrinsicness of divine belief. All we really have is that this extrinsicness is counterintuitive—but given God’s radical difference from creatures, we should expect God to be counterintuitive in many (infinitely many!) ways.

But I want to share a thought that has helped me be more accepting of the radical extrinsicness thesis about divine belief. There is something awkward in talking of God’s having beliefs. The much more natural way to talk is of God’s having knowledge. But knowledge is way more extrinsic in us than belief is. For you to know something, that something has to be true. So what you know depends very heavily on the external world. You know that your car is in the garage in part precisely in virtue of the fact that your car is in the garage. If your car weren’t in the garage, you wouldn’t have this knowledge.

In us, belief and knowledge are separable. Belief is much more of an intrinsic state, while knowledge is much more of an extrinsic one. When we know something outside ourselves, what makes it be the case that we know it is both a state of belief and a state of the external world. This separation makes error possible: it is possible to have the belief without the external world matching up.

But in a being that is epistemically perfect, there is no possibility of belief without knowledge. I want to suggest the plausibility of this thesis: in a being that epistemically perfect, there is not even a metaphysical separation between knowledge and belief. For such a being, to believe is to know. But knowledge of contingent external states of affairs is significantly extrinsic. So if to believe for such a being is to know, then we would expect beliefs about contingent external states of affairs to be significantly extrinsic as well.

In other words, the extrinsicness of belief that divine simplicity requires matches up with an extrinsicness that is quite plausible given considerations of the perfection of divine epistemology.

A necessary truth that explains a contingent one

Van Inwagen’s famous argument against the Principle of Sufficient Reason rests on the principle:

1. A necessary truth cannot explain a contingent one.

For a discussion of the argument, see here.

I just found a nice little counterexample to (1).

Consider the contingent proposition, p, that it is not the case that my next ten tosses of a fair coin will be all heads, and suppose that p is true (if it is false, replace “heads” with “tails”). The explanation of this contingent truth can be given entirely in terms of necessary truths:

2. Either it is or is not the case that I will ever engage in ten tosses of a fair coin.

3. If it is not the case that I will, then p is true.

4. If I will, then by the laws of probability, the probability of my next ten tosses of a fair coin being all heads is 1/2¹⁰ = 1/1024, which is pretty small.

My explanation here used only necessary truths, namely the law of excluded middle, and the laws of probability as applied to a fair coin, and so if we conjoin the explanatory claims, we get a counterexample to 1.

It is, of course, a contingent question whether I will ever engage in ten tosses of a fair coin. I have never, after all, done so in the past (no real-life coin is literally fair). But my explanation does not require that contingent question to be decided.

This counterexample reminds me of Hawthorne’s work on a priori probabilistic knowledge of contingent truths.

Arbitrariness and contingency

I’ve come to be impressed by the idea that where there is apparent arbitrariness, there is probably contingency in the vicinity.

The earth and the moon on average are 384400 km apart. This looks arbitrary. And here the fact itself is contingent.

Humans have two arms and two legs. This looks arbitrary. But it is actually a necessary truth. However there is contingency in the vicinity: it is a contingent fact that humans, rather than eight-armed intelligent animals, exist on earth.

Ethical obligations have apparent arbitrariness, too. For instance, we should prefer mercy to retribution. Here, there are two possibilities. First, perhaps it is contingent that we should prefer mercy to just retribution. The best story I know which makes that work out is Divine Command Theory: God commands us to prefer mercy to just retribution but could have commanded the opposite. Second, perhaps it is necessary that we should prefer mercy to retribution, because our nature requires it, but it is contingent that we rather than beings whose nature carries the opposite obligation exist.

Now here is where I start to get uncomfortable: mathematics. When I think about the vast number of possible combinations of axioms of set theory, far beyond where any intuitions apply, axioms that cannot be proved from the standard ZFC axioms (unless these are inconsistent), it’s all starting to look very arbitrary. This pushes me to one of three uncomfortable positions:

• anti-realism about set theory

• Hamkins’ set-theoretic multiverse

• contingent mathematical truth.

A theory of contingency and an argument for a causal Principle of Sufficient Reason

Consider this theory, a modification of my causal power account of possibility:

• A proposition p is contingent provided that something has a power for p and something has a power for not-p.

Here, I say that x has a power for p if and only if x has the power to bring p about, or x has the power to bring it about that something has the power to bring p about, or ....

It follows from this theory that every contingent true propositions has a causal explanation.

For suppose for reductio ad absurdum that p is contingently true and has no causal explanation. Let q be the conjunction of p with the claim that p has no causal explanation. Then q is true, and it is not necessarily true since p is not necessarily true, so q is contingent. It follows from our account of contingency that something has the power to ... bring q about (where the "..." is a possible chain of causal power claims). But that's absurd, since something that brings q about thereby also brings p about, and then p isn't bereft of causal explanation!

An Aristotelian argument for a necessary concrete being

All of the quantifications in the following are to be understood tenselessly. Consider these premises:

1. If y is an entity grounded solely in the xs and maybe their token relationships, then it is impossible that y exist while none of the xs exist.
2. All y is an abstract being, then there are concrete xs such that y is grounded solely in the xs and maybe their token relationships.
3. There is a possible world in which none of the actual world's concrete contingent beings exist.
4. There is a necessarily existing abstract being.

Alright, then:

5. Suppose there are no necessary concrete beings. (For reductio)
6. Let y be a necessarily existing abstract being. (4)
7. Let the xs be concrete entities such that y is grouned solely in the xs and maybe their relationships. (2 and 6)
8. The x are contingent. (5 and 7)
9. Possibly none of the xs exist. (3 and 8)
10. Possibly y does not exist. (1,7 and 9)
11. y does and does not necessarily exist. (5 and 10). Which is a contradiction.
12. So, by reductio, there is a necessary concrete being.

Premise 2 is a basic assumption of Aristotelianism. Premise 1 is more problematic. Note, however, that it is very plausible that this computer could not have existed had none of its discrete parts (CPU, screen, etc.) existed (i.e., ever existed, since the quantifications are tenseless). An object can have its parts get gradually replaced, but by essentiality of origins it must at least start off out of some of the stuff it started out of. And so it must have at least some of its constituents (at some time) in any world where it exists.

Further, premise 1 follows from the thought that when y is grounded solely in the xs and maybe their token relationships, then there is nothing more to the being of y than the being of the xs and maybe their relationships. But the token relationships of the xs couldn't exist if the xs never existed.

Premise 3 is very plausible. It must, of course, be distinguished from the much more controversial claim that there could be no contingent beings. Premise 3 is, on its own, compatible with the thesis that necessarily something contingent or other exists, as long as there aren't any contingent things that necessarily exist.

If premise 3 is the sticking point, but S5 is granted, an alternate argument can be given. Very plausibly, there is a possible world w containing a concrete being c with the property that all the concrete beings of w modally depend on w, i.e., they couldn't exist without c. (For instance, maybe they are solely grounded in c and its properties, or maybe c is a common part of them all, or maybe there is nothing but c.) Then running our argument in that world we conclude that c is a necessary being in w, and, by S5, actually.

A general form of philosophical argument

This is a bit cynical, but while reading Spinoza I was really struck by the prevalence of the following implicit line of philosophical argument, not just in Spinoza:

1. My theory cannot handle Xs.
2. So, there are no Xs.

It seemed obvious to me that the thing for Spinoza to do was not to conclude that there is no contingency, but to conclude that his theory was inadequate to handle contingency.

I use this form of argument myself.  Perhaps too much.  It takes wisdom to know when the thing to say is that the theory is inadequate to handle Xs and when to conclude that there are no Xs.

Could it contingently be the case that the laws of nature hold necessarily?

Here is a nice cautionary tale about being careful with scope and modality. A friend asked me whether one could argue from the possibility of the laws of nature being necessary to laws of nature being actually necessary. I answered in the affirmative, thinking about S5, and imagining the following argument:

1. Possibly, the laws of nature hold necessarily. (Premise)
2. If possibly necessarily p holds, then necessarily p holds. (Premise: S5)
3. Therefore, the laws of nature hold necessarily.

But the argument is badly fallacious. One way to see the fallacy is to note that in (2) the proposition p is referred to rigidly, while in (3) "the laws of nature" are a definite description. Compare the fallacious argument:

4. Possibly, what I will write on the board holds necessarily. (Premise)
5. If possibly necessarily p holds, then necessarily p holds. (Premise: S5)
6. Therefore, what I will write on the board holds necessarily.

Claim (4) is true: it is possible that I will write on the board something that holds necessarily (e.g., "5+7=12"). But one cannot conclude to (6), since after all I might also write something that holds merely contingently or not at all.

Another way to see the fallacy is to see it as a scope ambiguity in (4). For (6) to follow, (4) must be read as saying that possibly p holds necessarily, where p is what I will write on the board. But that claim is unjustified. All I am justified in saying is that possibly I will write something that holds necessarily.

And in fact we can see that it is prima facie possible that it a merely contingent fact that the laws of nature hold necessarily. Suppose that the laws of nature come in two classes. Some laws of nature are metaphysically necessary and some are not. For instance, prima facie, it might be a metaphysically necessary law that electrons have electric charge, but a metaphysically contingent law that opposite charges attract. Then it might turn out that there is a world w where there are no metaphysically contingent laws. It would then be true at w that all the laws of nature hold necessarily. But this would be only contingently the case, because there are worlds that have some of the contingent laws as well.

Interestingly, we can make the argument from (1) and (2) to (3) work if we add the following premise:

7. Necessarily: (For all p, either it is a law that p is a law, or it is a law that p is not a law).

For then, by (1), suppose that w₁ is a world where the laws hold necessarily. Suppose now that w₂ is any world and p is any law in w₂. Then, p either is or is not a law in w₁. If p is a law in w₁, then it is a law in w₁ that p is a law. But the laws in w₁ hold necessarily, so it follows that p holds necessarily. And if p is not a law in w₁, then it is a law in w₁ that p is not a law. Since the laws in w₁ hold necessarily, p is not a law in w₂, which is absurd. Hence, every law of w₂ holds necessarily.

Note: The above argument assumed that "p is a law" entails p, and that the claim "law p holds necessarily" entails that p holds necessarily (I did not additionally assume that "law p holds necessarily" entails that necessarily p is a law, though that does follow from (7)).

[I fixed some typos, and more importantly edited (2) and (4) in response to Comment #1 below. The original version of the post had "p" instead of "necessarily p" in the consequents of (2) and (4), which made the arguments not work. This emphasizes again the first sentence of the post.]