The Failure of the Argument from Contingency

An Argument from Contingency is an argument for the existence of God which employs a broad explanatory principle asserting, for every contingent fact,  the existence of an explanation, reason or cause of some sort. It proceeds from the existence of contingency via the principle to an explanation of the contingency, whereupon it is inferred that this explanation must be necessary, and that this necessary being must be God. Here’s a basic version of the argument, which I intend to show unsound:

(1) Every contingent fact has an explanation. (The Principle of Sufficient Reason, or PSR)
(2) There is a contingent fact that includes all other contingent facts.
(3) Therefore, there is an explanation of this fact.
(4) This explanation must involve a necessary being.
(5) This necessary being is God.

Before I get to my criticism of this kind of argument, I must mention that ‘explanation’ here is always explanation of a certain sort. Put vaguely, an explanation of some fact serves to make that fact comprehensible, to make it less mysterious and surprising. But two ways to accomplish this. The first way is via descriptive explanation, which tells us in detail what the explanandum1 is, and in doing so makes comprehensible the truth of what is explained – to believe that some fact is true requires a clear conception of the nature of that fact, and descriptive explanations facilitate this semantic prerequisite of belief. The second way is via causal explanation, which tells us why the explanandum is. Unlike descriptive explanations, causal explanations involve the postulation of entities, those to which the explanans and explanandum refer. And unlike descriptive explanations, in which the explanans is identical to the explanandum, a causal explanans is always external to the explanandum. It cites a cause, or set of causes, which make the truth of the explanandum comprehensible by showing us how it is the necessary or likely product of some state of affairs which does not include it2.
Here is an example of what I mean: suppose I request an explanation of Republicanism in the United States from you. Your answer might involve a description of the political system, in terms of its three tiers of representation, the separation of its legislative, judicial and executive branches, the Constitution, and so on. Such an explanation is a descriptive explanation, for it tells me in what Republicanism in the United States consists. But you might instead give me an account of early American history, including the social influences that lead to the American Revolutionary War, and onwards to the drafting of the Constitution. This second explanation would be a causal explanation, for you presume me to know what Republicanism here is, but interpret me as asking after the conditions from which Republicanism could be seen to follow. Both answers count as explanations, as both serve to make comprehensible Republicanism in the United States, but they are obviously explanations of different kinds3.
With the distinction in hand, we see immediately that the sense of explanation relevant to the Argument from Contingency is that of causal explanation. What the PSR is intended to assert is not that every contingent fact has a description which would make it comprehensible, since one could not infer the existence of anything from this, but that for every contingent fact there is something external to it from which the truth of the fact can be seen to follow. So, in what follows I’ll be discussing causal explanations, and I’ll add the qualification to quotations. We proceed to a criticism of the PSR.

Van Inwagen’s Modal Fatalism Argument

Peter Van Inwagen, a theist himself, offers a reductio of the PSR:

(11) No necessary proposition [causally] explains a contingent proposition. (Premise.)
(12) No contingent proposition [causally] explains itself. (Premise.)
(13) If a proposition [causally] explains a conjunction, it [causally] explains every conjunct. (Premise.)
(14) A proposition q only [causally] explains a proposition p if q is true. (Premise.)
(15) There is a Big Conjunctive Contingent Fact (BCCF) which is the conjunction of all true contingent propositions, perhaps with logical redundancies removed, and the BCCF is contingent. (Premise.)
(16) Suppose the PSR holds. (For reductio.)
(17) Then, the BCCF has an [causal] explanation, q. (By (15) and (16).)
(18) The proposition q is not necessary. (By (11) and (15) and as the conjunction of true contingent propositions is contingent.)
(19) Therefore, q is a contingent true proposition. (By (14) and (18).)
(20) Thus, q is a conjunct in the BCCF. (By (15) and (19).)
(21) Thus, q [causally] explains itself. (By (13), (15), (17) and (19).)
(22) But q does not [causally] explain itself. (By (12) and (19).)
(23) Thus, q does and does not [causally] explain itself, which is absurd. Hence, the PSR is false.

Both this argument and the basic argument we met earlier have been lifted from Alexander Pruss’s extensive chapter on Leibnizian Cosmological arguments, from the Blackwell Companion to Natural Theology. And as we would expect, Pruss has something to say about Van Inwagen’s reductio. He thinks that the theist should not accept (11). He tells us:

The main reason to accept (11) is the idea that if a necessary proposition q [causally] explained a contingent proposition p, then there would be worlds where q is true but p is false, and so q cannot give the reason why p is true. This sketch of the argument can be formalized as follows:

(24) If it is possible for q to be true with p false, then q does not [causally] explain p. (Premise)
(25) If q is necessary and p is contingent, then it is possible for q to be true with p false. (A theorem in any plausible modal logic)
(26) Therefore, if q is necessary and p is contingent, then q does not [causally] explain p.

Instead of attacking (11) directly, I shall focus my attack on (24).. [which] seems to capture just about all the intuition behind (11). By contraposition (24) is equivalent to:

(27) If q [causally] explains p, then q entails p.

But Pruss thinks (27) is false. Briefly, it is false because a causal explanation need not entail that which it explains. In fact, a causal explanation need not even make probable what it explains, as in the explanation of syphilis by paresis: whereas the cause of paresis is known to be untreated syphilis, it is only in a small percentage of cases of untreated syphilis that paresis does result. But untreated syphilis is a causal explanation of paresis, even if not a sufficient (entailing) one, and so it follows that causal explanations need neither entail nor make probable their explanandum.
Pruss is right, of course, to say that (27) is false, and so that (24) is false. But showing that (24)-(26) is an unsound argument does not show that (11) is false; at most, it shows that one way of attempting to justifying (11) is a failure. So let’s try another way.

(A) Suppose that some necessary proposition q causally explains a contingent proposition p. (Premise for reductio)
(4) A proposition q only causally explains a proposition p if q is true.(Premise)
(B) So, q is a necessary truth. (From A and 4)
(C) The probability of a necessary truth is 1. (Premise)
(D) So, P(q) is equal to 1. (From B and C)
(E) Then, P(p|q) is equal to P(p). (From D4)
(F) But, if q causally explains p, then P(p|q) > P(p). (Premise)
(G) So, q does not causally explain p. (From E and F)
(H) Then, q both does and does not causally explain p, which is absurd. (From A and G)
(11) (Therefore) No necessary proposition causally explains a contingent proposition.

The thought behind (24)-(27) was that causal explanations entailed what they explained. The thought behind the argument I propose is different, and is captured by (F): though causal explanations may neither entail nor make probable what they explain, it is at least true of causal explanations that their being true increases the probability of that which they explain5. Consider again the syphilis-paresis case: though the presence of untreated syphilis does not make paresis probable, as it only rarely leads to paresis, still someone’s having syphilis makes paresis more likely than it otherwise would be, and plausibly this is required in order for it to be explanatory. By contrast, were it not the case that syphilis raised the probability of paresis, then it would be difficult to see in what sense syphilis would make the existence of paresis more comprehensible, less mysterious or surprising. It would be difficult to see what causal relevance syphilis had to paresis at all. We can bring this thought out with the following argument:

(I) To causally explain some fact X, one must cite some fact Y other than X which is yet relevant to X’s obtaining.
(II) To causally explain X is to simultaneously causally explain X’s being true.
(III) So, to causally explain X is to cite some Y other than X’s being true, which is relevant to X’s being true.
(IV) Some Y is relevant to X’s being true only if it either conduces to the truth of X, making it more probable, or conduces to the falsity of X, making it less probable.
(V) Whatever conduces to the falsity of X does not causally explain X.
(VI) Therefore, to causally explain X is to cite some Y other than X’s being true, which conduces to the the truth of X, making X more probable.

So, that a causal explanation increases the probability of its explanandum simply falls out of its being an explanation of some fact’s being true. And now we can see why a necessary fact fails to explain a contingent fact. If a necessary fact did explain a contingent fact, then it would explain why that contingent fact was true. So a necessary fact would have to offer something relevant to the truth of the contingent fact. And whatever is relevant to the truth of the contingent fact, which we could describe as causally explanatory, would have to conduce to the truth of that contingent fact. But a necessary fact is not conducive to the truth of any contingent fact for, what is equivalent, the truth of a necessary fact does not increase the probability of that contingent fact’s obtaining – a necessary fact will obtain whether or not the contingent fact also obtains, and so it tells us nothing about whether the contingent fact also obtains. Hence, Van Inwagen’s modal fatalism argument is sound: the Principle of Sufficient Reason is false, and all arguments which assume its truth are fatally flawed.

Conclusion

If the above argument is sound, then the PSR is false, and so one more argument for the existence of God is unsound. But though that is of interest, there is more we can conclude from the defense of Van Inwagen’s argument: it follows from the fact that no necessary proposition causally explains a contingent proposition that, if God is necessary, then God does not causally explain the universe. And given that part of what we understand by the term ‘God’ is ‘the cause of the universe’, it follows that if God is necessary, God does not exist. In that case I recommend that theists give up the notion that God is necessary, as well as the principle of sufficient reason6.

Addenum

There are, it is true, potential counterexamples to the principle that “if q causally explains p, then P(p|q)>P(p)“. I’ll address two, taken from the SEP’s article on Probabilistic Causation. The first:

(i) Probability-lowering Causes. Consider the following example, due to Deborah Rosen (reported in Suppes (1970)). A golfer badly slices a golf ball, which heads toward the rough, but then bounces off a tree and into the cup for a hole-in-one. The golfer’s slice lowered the probability that the ball would wind up in the cup, yet nonetheless caused this result.

In response, I deny that it is true that the golfer’s slice lowers the probability of a hole-in-one. That the golfer’s slicing the ball as opposed to hitting the ball cleanly lowers the probability of a hole-in-one is undoubtedly true, and in that case it is true that the golfer’s slicing the ball as opposed to hitting it cleanly does not causally explain the hole-in-one. On the other hand, slicing the ball simpliciter does raise the probability of a hole-in-one, since a hole-in-one is more likely on a slice than on no information regarding the initial conditions at all. Hence slicing the ball simpliciter is causally explanatory of the hole-in-one. The second case:

(ii) Preemption. A different sort of counterexample involves causal preemption. Suppose that an assassin puts a weak poison in the king’s drink, resulting in a 30% chance of death. The king drinks the poison and dies. If the assassin had not poisoned the drink, her associate would have spiked the drink with an even deadlier elixir (70% chance of death). In the example, the assassin caused the king to die by poisoning his drink, even though she lowered his chance of death (from 70% to 30%). Here the cause lowered the probability of death, because it preempted an even stronger cause.

My response here is of the same sort. It is not true that the assassin’s poisoning the king’s drink lowers the probability of the king’s death. What is true is that the assassin’s poisoning the drink as opposed to leaving the poisoning to their associate lowers the probability of the king’s death, and so, it is true that the assassin’s poisoning the drink as opposed to leaving the poisoning to an associate is not causally explanatory. However, as the assassin’s poisoning the king’s drink simpliciter does raise the probability of the king’s death, we can say that the assassin’s poisoning the king’s drink causally explains the king’s death.
As the SEP article notes, both of these are cases of singular causation (they make reference to particular individuals, places, and times). I suggest that this is not coincidental: To judge whether or not some X counts as a cause, we need to come to a conclusion about what causal powers X has, and this can only be done if we abstract away from the particular situation and advert to some comparison class of similar situations including X. But to ask after a cause in a case of singular causation leaves open just which details we are to hold constant in determining our comparison class, and so makes possible the kind of inconsistent treatment above – the probabilistic evaluation of a purported cause erroneously based on the irrelevant features of a singular case.

Notes

1 For convenience, I’ll be using the Latin terms ‘explanandum’ and ‘explanans’. An explanandum is what is explained, whereas an explanans is what does the explaining.

2 One response to the Argument from Contingency, and indeed all cosmological arguments, is to observe that they would appear to require a cause of the Big Bang. But, so the objection goes, it is part of the definition of a cause that it precedes its effect in time, and as nothing can precede the Big Bang in time, nothing can be its cause. For this post I’ll assume that the notion of cause is flexible enough to accommodate a timeless cause, which does not precede its effect.

3 There are two other classes of explanation sometimes added to my two. The first is intentional explanation – explanations which cite the beliefs and desires of agents as producing some effect. For my part, I see no good reason to assume that intentional explanations are not a species of causal explanation: though some philosophers may believe intentional explanations do not fit the scientific criteria of a cause, I take the lesson here to be not that they are non-causal explanations, but that there is a broader notion of cause upon which the scientific model is a restriction.
The second potential class is that of justification – explanation in terms of reasons for belief in what is explained. This too, I believe to be a species of causal explanation, for in giving reasons for belief one is often citing such reasons as causally efficacious in producing one’s belief. Cases where this does not happen, say, when reasons for belief are given merely to recommend a belief to others, or where they are cited in support of one’s belief without being productive of that belief, are not cases of genuine explanation – the former because explanation is factive, whereas one may recommend a belief to others without their being some belief which would be explained; the latter because it is an instance of rationalization, but rationalization merely masquerades as explanation.
In any case, it does not appear that insisting on either class in addition to our two would favor the Argument from Contingency: if the PSR is to be interpreted as demanding an intentional explanation for every contingent fact, then the argument from contingency would seem to beg the question; alternatively, if the PSR is interpreted as demanding a reason why one should believe any given contingent fact, then a logical demonstration would suffice to explain the contingent fact which includes all other contingent facts.

4 This follows from Bayes theorem. According to (D), P(q) is 1. Then, P(q|p) is also 1, and we can deduce as follows:
P(q|p) = P(p|q) x P(q) / P(p) [Bayes Theorem]
1 = P(p|q) x 1 / P(p) [Substitution]
1 = P(p|q) / P(p) [Elimination]
1 x P(p) = P(p|q) / P(p) x P(p) [Multiplication of both sides by common factor]
P(p) = P(p|q) [Result]

5 I stress that this is but necessary condition of a causal explanation, and is not intended as an analysis of what causal explanations are.

6 I feel I should add that this concession may not be theologically significant: often, the terms ‘necessary’ and ‘contingent’ are taken to be synonyms of ontological independence and dependence respectively. But as Ex-Apologist has argued, these concepts are not synonymous, and since it seems to me that it is ontological independence which is part of the core conception of God as opposed to necessity, I believe that this is an adjustment the orthodox theist can accept.

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17 thoughts on “The Failure of the Argument from Contingency

  1. Reidish says:

    Good work here, TaiChi. Lots to chew on.

    One consequence of your argument is that necessary facts cannot possibly explain contingent facts. How comfortable are you with this position? Personally I have a strong intuition against it.

  2. TaiChi says:

    Hi Reidish, how’s it going? Well, I hope.

    One consequence of your argument is that necessary facts cannot possibly explain contingent facts. How comfortable are you with this position? Personally I have a strong intuition against it.

    Pretty comfortable, so far as it goes. Remember, I’m only arguing that a necessary fact cannot be a causal explanation for a contingent fact, and that still leaves for necessary facts a role in descriptive explanation. Pruss gives two examples..

    For instance, in Metaphysics Z, Aristotle suggests explaining an eclipse of the sun by noting that an eclipse of the sun is identical with the earth’s entry into the moon’s shadow. Likewise, one might explain a knife’s being hot by noting that its being hot consists in, or maybe is constituted by, its molecules having high kinetic energy.

    .. which appeal to necessary facts but are clearly descriptive explanations (Pruss calls these ‘conceptual explanations’ instead). I think we can also add to list of necessary fact which explain, but do so descriptively, the proofs of mathematics and logic – these too are expansions of our original phenomena to be explained, rather than appeals to something which transcends them. In sum, the clarification that necessary facts explain only descriptively seems just about right to me.
    Does that quell your intuition somewhat? Can you think of any examples where necessary facts play something other than a descriptive/conceptual role?

  3. Reidish says:

    Ah, right. You’re only arguing against causal explanations. A proof of logic was indeed the counterexample I was going to provide.

    Does that quell your intuition somewhat? Can you think of any examples where necessary facts play something other than a descriptive/conceptual role?

    You can consider me quelled on that point for the time-being. Cheers, hope all is well.

  4. John D says:

    Very good.

    In my more metaphysically indulgent moments I consider myself some sort of Spinozist. So I wonder how a necessary being style pantheism would fair after this argument. I’m guessing it qualifies as a descriptive explanation, not a causal explanation. Would that be your intuition too?

  5. Reidish says:

    So, a key premise is (F):

    (F) But, if q causally explains p, then P(p|q) > P(p). (Premise)

    In other words: Given q, and given that q is a causal explanation of p, the probability of p on q is higher than the probability of p without q.

    But why couldn’t one prefer:

    (F)’If q causally explains p, then P(p|q) >= P(p). (Premise)

    That is, q does not decrease the probability of p. This is a weaker claim than (F). Also, using (F)’ instead, one could still deny (11).

    The question is, could we still, then, consider q a causal explanation of p? I don’t see why not.

  6. TaiChi says:

    @John D

    Thanks, John. I think there are two questions here. The first is whether or not one could be orthodox Spinozan necessary being pantheist and endorse my argument. The answer to that is no, from what I know of him: Spinoza takes God/Nature to necessary, and also to be his/its self-sustaining cause, which conflicts with conclusion of my argument.
    The second question is whether or not one can be a Spinozan-influenced necessary being pantheist and endorse my argument, to which I believe the answer is yes.

    Take the following argument, which is supposed to represent Spinoza:

    35. The Argument from the Intelligibility of the Universe
    (Spinoza’s God)
    1. All facts must have explanations.
    2. The fact that there is a universe at all — and that it is this universe, with just these laws of nature — has an explanation (from 1).
    3. There must, in principle, be a Theory of Everything that explains why just this universe, with these laws of nature, exists. (From 2. Note that this should not be interpreted as requiring that we have the capacity to come up with a Theory of Everything; it may elude the cognitive abilities we have.)
    4. If the Theory of Everything explains everything, it explains why it is the Theory of Everything.
    5. The only way that the Theory of Everything could explain why it is the Theory of Everything is if it is itself necessarily true (i.e., true in all possible worlds).
    6. The Theory of Everything is necessarily true (from 4 and 5).
    7. The universe, understood in terms of the Theory of Everything, exists necessarily and explains itself (from 6).
    8. That which exists necessarily and explains itself is God (a definition of “God”).
    9. The universe is God (from 7 and 8).
    10. God exists.

    Now, does this sort of argument require a causal explanation? Or does it call for a descriptive explanation? I think it pretty clear that it’s only the latter: the theory of everything is being identified with the universe itself, and in that case the explanans and the explanandum refer to one and the same entity, just under different descriptions (‘the universe’ being minimally specific, and ‘the theory of everything’ being maximally specific).

    @Reidish
    But why couldn’t one prefer:

    (F)’If q causally explains p, then P(p|q) >= P(p). (Premise)

    ..The question is, could we still, then, consider q a causal explanation of p? I don’t see why not.

    Okay, let’s try to put that into an example. Suppose that you are scientist, and you undertake an investigation of whether magnets are able to bring about pain relief. You institute a large double-blind study, in which there are two kinds of mattress which 1000 subjects are to sleep on (magnetized and non-magnetized), and neither you nor the subjects know during the study which they have been assigned to. The results come in: there is no difference between the control and experimental groups with regard to the relief of pain. You conclude, tentatively, that magnets do not cause pain relief.
    Well, what licenses you to draw this conclusion? Presumably, some principle that takes you from a sample to the whole. And, I say, the principle..

    (F) [I]f q causally explains p, then P(p|q) > P(p).

    .. since even if we assume that the sample represents the whole, we can only infer that magnets do not cause pain relief if we also affirm that causation requires that p(pain relief|magnets) > p(pain relief). If we instead adopt your principle (F)’, then the study shows nothing, for that there is no probabilistic difference in pain relief between those who sleep on magnetized mattresses and those who do not is a proposition is yet consistent with magnets causing pain relief on (F)’. But (I say) the study does show something on the assumption that the sample is representative: it shows that the magnets do not cause pain relief. And so (F) is true whereas (F)’ is false.

  7. tarrobread says:

    Hi, I’m new to blogs, but I thought I would share where I’ve always thought this argument goes wrong. I take PSR to be formulated as: Every contingent fact has a Sufficient explanation (hence the sufficient part). Now, in trying to explain BCCF, the only thing that the theist can point to as an explanation is God, since nothing can be the cause of itself (so no contingent being can be pointed to), and God is the only non-contingent being (leaving aside the possibility of mathematical Platonism or other metaphysical constructs which the theist would not want to resort to anyway.) Therefore, God must be a sufficient explanation for BCCF. However, since God exists in all possible worlds, and he provides a sufficient explanation for BCCF, then BCCF exists in all possible worlds. Here is the flaw. To formulate the argument from contingency, one must accept that contingent things exist, but once one accepts the argument, it turns out that no contingent things exist since BCCF is necessary. It literally contradicts itself. At least, that’s what I’ve always thought of the argument, so I don’t particularly see the need to invoke conditional probabilities or anything like that. I provide a more detailed account of this argument (as well as the KCA and fine-tuning) in my own blog. So, did I go wrong anywhere?

  8. TaiChi says:

    Hi tarrobread, welcome.

    The reason why I offer an argument from causes as probability raisers rather than offer your argument is that I’m trying to provide an objection against both strong and weak forms of the PSR. You state a strong form..

    Every contingent fact has a Sufficient explanation

    ..and I think your objection works against this. However, Pruss’s chapter urges the use of weaker forms of the PSR, like..

    (1) Every contingent fact has an explanation. (The Principle of Sufficient Reason, or PSR)

    ..from the argument I give at the beginning of the post, which your objection doesn’t rebut. Or, at least, not without a supplementary argument that the weak PSR entails the strong PSR. So, no, I don’t think you go wrong anywhere, but neither does your criticism cover every argument from contingency.

  9. tarrobread says:

    Thanks for your response TaiChi,

    Now I get why the theist would definitely want to make a weak/strong distinction of PSR seeing how the strong version is so easily defeated. Before I accept this particular argument against the weak version though, I still have some reservations about premise (F). In particular, I don’t see a necessary connection between x being a cause of y and P(y given x)>P(y). After all, to use this generalization as a premise one would have to establish this connection as a necessary one. In particular, cases of singular causation cannot be left out; they just have to be treated with care. Here is another counterexample which I just made up (it could fail as an example, I don’t know).

    Suppose there is a world where every person followed one of two diets. One diet consisted of purely fatty red meat while one consists of eating McDonalds every day. As a result, the vast majority of the population would die of heart disease pretty early on. So the probability of any person p dying of heart disease can be easily calculated as the # of people who die of heart disease divided by the # of total people. This probability would be very high. Now what is the probability of any person p dying of heart disease given that they follow the McDonald’s diet? (calculated, again, pretty easily by the # of people who eat McDonald’s diets and die of a heart disease divided by the number of people who eat a McDonald’s diet) Although it would be high, it would be less than the original probability. Here is the counter example. Clearly, the McDonalds diet caused that person to die of heart disease. Furthermore, P(any person p dying of heart disease) is more than P(any person p dying of heart disease given that they follow the McDonald’s diet). The probabilities were all calculated in a very straightforward manner. I fail to see what went wrong. It seems to me that premise (F) is contingent on the context of the situation. For example, in this world, the probability that someone dies of heart disease given that they eat McDonald’s every day is much more than just the regular probability of someone dying of heart disease, but that is not the case in this other world. Maybe one could object at the way I calculated the probabilities, but I simply used a frequency method. Perhaps your idea was to calculate P(any person p dies of heart disease) in this imagined world by not using frequencies? If it was, an argument would have to be made as to why calculating them the way I did is fundamentally incorrect.

    You say, “it is true that the assassin’s poisoning the drink as opposed to leaving the poisoning to an associate is not causally explanatory.” On what grounds do you believe this? In my opinion, adding an “as opposed to” clause at the end of the cause of something does not negate its causal powers. The grounds cannot simply be because if you add the opposed to clause it would actually lower the probability, because that would be question-begging. Anyway, those are my qualms about F. Still, this is a very thoughtful, well-argued post, it definitely taught me some things!

  10. TaiChi says:

    Perhaps your idea was to calculate P(any person p dies of heart disease) in this imagined world by not using frequencies? If it was, an argument would have to be made as to why calculating them the way I did is fundamentally incorrect.

    Yes, that would be my objection: though actual frequencies are evidence for objective probabilities, I don’t think one should equate one to the other. And isn’t it fairly easy to see why one shouldn’t? Suppose a die is manufactured, rolled once (it shows a six), and then destroyed. Is it true to say that the objective probability of the die’s landing on six after being rolled is 1, whereas the for all other numbers the probability is 0? If not, then the probability is not a function of the actual frequency.
    Well, then what alternative account should we adhere to? I’m not sure, and this is something I need to look into further. As it stands though, I imagine probability in terms of possible worlds. For example, though one might explain A’s entailing B in terms of B’s being true in every possible world in which A is also true, one could equally say that this explains p(B|A)=1, or what is the same, p(~B|A)=0. But if that’s so, then it ought to be the case that p(D|C)=.5 can be explained by D’s being true in half the worlds where C is true, and more generally that objective probabilities map onto proportions of possible worlds. Prior objective probabilities would just be equivalent to the proportion of possible worlds containing a given fact.
    I suppose that we explain objective probabilities in terms of possible worlds, but I should say that the space of possible worlds is itself going to be constructed from the knowledge we have of causal powers, which is where I think actual frequency comes in: it is on the basis of observed frequencies that we ascribe causal powers, and since the causal powers of various objects will determine the space of possible worlds, so, in the end, do frequencies come to determine what we judge objective probabilities to be. But we can be wrong about those objective probabilities, for the frequencies we observe may lead us to ascribe causal powers where none exist, to misjudge their potency, or even to fail to notice such causal powers.
    In fact, I think this last describes your calorific world: that everyone eats unhealthily will lead the people there to overlook (on the evidence of actual frequencies) the causal powers that the McDonald’s diet has in causing heart disease. Since they have a distorted understanding of the causal powers of McDonalds, their understanding of the space of possible worlds will too be affected, in particular, they will not assume a higher incidence of heart disease in worlds where people eat McDonalds, and so will not assume that p(Heart Disease|McDonalds) > p(Heart Disease). But they would be wrong in this, just as they would be wrong in their modal assumptions, just as they would be wrong in their understanding of causal powers.

    You say, “it is true that the assassin’s poisoning the drink as opposed to leaving the poisoning to an associate is not causally explanatory.” On what grounds do you believe this? In my opinion, adding an “as opposed to” clause at the end of the cause of something does not negate its causal powers.

    What I’m saying is not causally explanatory is not: poisoning-the-drink as opposed to letting-the associate-poison-the drink, but instead the unity: poisoning-the-drink-as-opposed-to-letting-the associate-poison-the drink. I take it that the unity does not have the causal powers that poisoning-the-drink alone does, since whatever causal powers a thing has depend not just on having some part with causal powers, but on being, as a whole, causally efficacious. (Trivial example: a wooden building contains a nail as a part, but the building does not have the causal power of penetrating wood with the application of relatively little force).

    Anyway, those are my qualms about F. Still, this is a very thoughtful, well-argued post, it definitely taught me some things!

    Cool, thanks for your comments, tarrobread.

  11. Nice argument.

    There are (at least) two ways to understand “q conduces to the truth of p”.
    (1) P(p|q) is greater than P(p)
    (2) P(p|q) is greater than P(p|~q)

    The two are equivalent when P(q) is strictly between 0 and 1. But when P(q) is either 0 or 1, they are no longer equivalent. (Of course, defining the conditional probabilities in (2) will then have to be done in some way other than by a ratio formula. But we already should have known that (see http://alexanderpruss.blogspot.com/2011/06/unconditional-probabilities.html and the Hajek paper linked therein).)

    Which should we take as the definition of “conduces to the truth of p”, then? I think (2) is superior. Here’s one reason. Suppose you randomly and uniformly choose a real number between 0 and 1. Let p = you didn’t choose 1/3. Let q = you chose neither 2/3 nor 1/3. Then P(p|q) = 1 and P(p) = 1, so if (1) is the criterion, we don’t have have truth-conducingness. But surely we do. q clearly conduces to the truth of p. And (2) concurs. P(p|q) = 1 and P(p | ~q) = 1/2 (if you chose either 2/3 or 1/3, the chance that it was 1/3 is 1/2).

    But if we accept (2) as the definition, then as long as we’re willing to countenance per impossibile conditional probabilities, it might very well be that P(p|q) is greater than P(p|~q) where q is a necessary truth. Let q be the proposition that 2+2=4. Let p be the proposition that if tomorrow I put two oranges in an empty box and then put two more in, there will be four oranges in that box. q doesn’t entail p (one orange might miraculously disappear), but q does explain p. And we do have truth-conduciveness by (2), though not by (1). For P(p|q) is close to 1, and P(p|~q) is much less than 1, but P(p|q)=P(p).

    • TaiChi says:

      Thanks for your interest, Professor. I’ll try and defend myself, though I don’t have anything decisive to say.

      You say that ‘P(p|q) > P(p|~q)’ is a way to understand “q conduces to the truth of p”. I disagree. It seems that the above statement says instead that q contributes more to the probability of p than does ~q, just as ‘P(p|q) > P(p|~r)’ says that q contributes more to the probability of p than does ~r. Whatever plausibility ‘P(p|q) > P(p|~q)’ has as an interpretation of “q conduces to the truth of p” seems to result from the fact that it and ‘P(p|q) > P(p)’ correlate.

      Suppose you randomly and uniformly choose a real number between 0 and 1. Let p = you didn’t choose 1/3. Let q = you chose neither 2/3 nor 1/3. Then P(p|q) = 1 and P(p) = 1, so if (1) is the criterion, we don’t have have truth-conducingness. But surely we do. q clearly conduces to the truth of p. And (2) concurs. P(p|q) = 1 and P(p | ~q) = 1/2 (if you chose either 2/3 or 1/3, the chance that it was 1/3 is 1/2).

      I’m not sure: it seems to me that q’s not conducing to the truth of p is in line with the family of oddities that the concept of infinity generates, and so I don’t find this further oddity a compelling reason to adopt (2). After all, we would anyway have to accept that my not choosing 1/3 and my choosing neither 1/3 nor 2/3 are equiprobable, both having a probability of 1, so isn’t it appropriate that my choosing neither 1/3 nor 2/3 fails to conduce to truth of my not choosing 1/3? In the very least, our intuitions around infinity don’t seem nearly reliable enough to adopt your interpretation over mine.

      But if we accept (2) as the definition, then as long as we’re willing to countenance per impossibile conditional probabilities, it might very well be that P(p|q) is greater than P(p|~q) where q is a necessary truth.

      I guess so, but then we need a good independent reason to bring these sort of probabilities in. It seems a strange idea that our assessments of probability should range wider than the space of possibilities.

  12. Rethinking stuff says:

    It seems odd to say that “2+2=4” causally explains how many oranges will be in the box.

  13. Rethinking stuff says:

    Well, if one orange miraculously disappears (and only 3 oranges remain), then I don’t see how 2+2=4 explains why 3 oranges remain. 2+2=4 explains that there are 4 oranges only if 4 oranges remain (and no orange miraculously disappears). So q explains p only if p. So ~p–>~(q explains p). So (q explains p)–>p. So your example only works if the explanation entails the explanandum.
    Anyway, Taichi’s post was about causal explanations (see premise F), and God’s existence plus his intentions is supposedly a causal explanation of why contingent things exist.

    TaiChi, I accept that a necessary proposition cannot explain a contingent one, but I don’t see how you concluded that the PSR is false. There is another proposition which Dr.Pruss objected to, which is: no contingent proposition can be self-explanatory. Dr.Pruss provided the following proposition to counter Inwagen’s premise, which is roughly: (God necessarily exists, and contingently chooses to instantiate the BCCF for reason R) is a contingent proposition and yet is self-explanatory since it includes a free choice. Assuming agent causation, free choices require no further explanation if they’re done for a certain reason.

    • Rethinking stuff says:

      So here’s how the reasoning goes, I think:

      The existence of contingent “objects” or “beings” requires an explanation in terms of an external “object” or “being”. This explanation cannot include a contingent being, so it must include a necessary being. The explanation is therefore: (God necessarily exists, and contingently creates contingent beings for reason R). As mentioned before, this explanation does not require a further explanation since it is self-explanatory (free choice).

      The way I presented the reasoning is sloppy, but I think the point is clear. (Stephen Davis provides this sort of argument as well, but he presents it better)

  14. Rethinking stuff says:

    Let me tighten that up:

    q: 2+2=4.
    p: if tomorrow I put two oranges in an empty box and then put two more in, there will be four oranges in that box.
    r: tomorrow I put two oranges in an empty box and then put two more in.
    s: there will be four oranges in that box

    if ~s, then ~(q explains p).
    So, if r&~s, then ~(q explains p).
    So if (q explains p), then (r–>s).
    But p is equivalent to (r–>s).
    So if (q explains p), then p.

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