Yesterday, I discussed Thomas Flint’s response to the grounding objection in chapter 5 of Divine Providence: The Molinist Account. Today, I want to discuss his response to Robert Adams in chapter 7.
Adams’ objection turns on a notion of explanatory priority which, Flint complains, is not adequately defined. Flint argues that there is an equivocation in the argument, and that Adams relies on a transitivity assumption which is not plausible when applied across the different sorts of priority involved. I think, however, that Flint is mistaken on both counts: first, the notion in question is not equivocal. Rather, it is a genus containing several species. Second, transitivity is not actually required. What’s required is just an anti-circularity principle. The anti-circularity principle is abundantly well-justified across the entire genus.
The notion of priority here corresponds to the notion of objective explanation. That is, A is prior to B iff B because A. That’s simple enough. Of course, there are many different uses of ‘because’ and I’m inclined to agree that the anti-circularity principle won’t apply to all of them. That’s why we require that the because or priority here track objective explanation, i.e., that A really be a reason why B is true, and not merely a fact that helps make B intelligible to some particular mind. It is extremely plausible to suppose that there can be no cycles in chains of objective explanation.
The types of priority/explanation at issue include these:
- The priority of reasons (and, more generally, considerations) to actions (whether divine or creaturely).
- The priority of God’s creative act to all creaturely activity.
- The priority of causes to effects.
- The priority of free choices to free actions.
Now, it is, as I said, extremely plausible that an anti-circularity constraint applies here. For instance, it is incoherent to suppose that I should choose to act in a certain way because I am going to act in that way. Similarly, if my action causes it to be the case that P, then P can’t be among the reasons for my action, since (barring overdetermination, etc.) P won’t be true unless I take the action. (Of course, I might take the action because taking the action will cause it to be the case that P. That’s different.)
Now, let C be a proposition describing a total circumstance and let A be a proposition stating that a creature takes some free action in that circumstance. The Molinist is clearly committed to:
(1) C -> A is prior to God’s decision to weakly actualize C.
(2) God’s decision to weakly actualize C is prior to the agent’s having the reasons, considerations, etc., which lead her to choose A.
(3) The agent’s reasons, considerations, etc., are prior to her choice that A.
(4) The agent’s choice that A is prior to A.
By the anti-circularity constraint, this implies that neither the agent’s choice that A, nor A itself, is prior to C -> A.
But then why is C -> A true? If the Molinist says, for no reason at all, she runs into the randomness objection. The anti-circularity constraint prevents the Molinist from saying it’s because of the agent’s choice or the agent’s action. The Molinist obviously can’t say it’s due to God. If it’s due to the agent’s essence, nature, character, etc., then we’re presupposing a compatibilist theory of freedom and don’t need to bother with all the complexities of Molinism. There’s a serious problem here, and Flint hasn’t defused it.
(Cross-posted at blog.kennypearce.net.)
When: December 4th‐5th 2015
Where: VU University Amsterdam, The Netherlands
Organizers: Hans van Eyghen, Rik Peels, and Gijsbert van den Brink
Although Cognitive Science of Religion (CSR) is still a rather young discipline, its main theories have been the subject of considerable debate. One main point of discussion is whether cognitive theories explain religion. The title of Pascal Boyer’s book Religion Explained (2002) signals that at least one goal of CSR is to explain religion. Many authors have interpreted ‘explaining’ as explaining away and have argued that CSR‐theories have not explained religion away because the truth of religion is compatible with the main theories in CSR.
Trent’s interesting post about evil and hiddenness has reminded me of the following draft that I wrote some time ago:
The problem of evil challenges theism by raising the following question: if God is omnipotent and omnibenevolent, why is there evil in the actual world? Theists have proposed many responses to the problem, such as the free will response, the soul-making response, the greater good response, and so on. Whether any succeeds has been debated for hundreds of years.
Suppose now, for the sake of argument, that there is a successful theistic response to the problem of evil explaining the reason, call it X, that God has to allow evil. Unfortunately, this does not end the story because the existence of X raises a new question: If God is omnipotent and omnibenevolent, why does He not tell us that X is the reason that He has to allow evil? A state of affairs in which we remain puzzled by not being told by God that X is the reason that He has to allow evil seems to undermine the existence of an omnipotent and omnibenevolent God. Let us call this the ‘second-order problem of evil’.
Suppose, for the sake of argument, that there is a successful theistic response to the second-order problem of evil explaining the reason, call it Y, that God cannot tell us that X is the reason that He has to allow evil. Unfortunately, this does not the end the story because the existence of Y raises a new question: If God is omnipotent and omnibenevolent, why does He not tell us that Y is the reason that He cannot tell us that X is the reason that He has to allow evil? A state of affairs in which we remain puzzled by not being told by God that Y is the reason that He cannot tell us that X is the reason that He has to allow evil seems to undermine the existence of an omnipotent and omnibenevolent God. Let us call this the ‘third-order problem of evil’.
And so on, ad infinitum.
What does this observation teach us? First, it teaches us that theists who think that they have found a successful response to the problem of evil should beware of overconfidence; such a response raises new challenges for them. Second, it encourages theists to investigate a link between evil and God’s hiddenness. The only plausible explanation, if there is any, that God does not prevent evil, does not tell us X is the reason that He has to allow evil, does not tell us Y is the reason that He cannot tell us that X is the reason that He has to allow evil, and so on, appears to be that God has to remain hidden from us; that is, God has to avoid any form of interaction with us which suggests His existence. We can see this clearly by showing that the above infinite regress does not arise for the problem of divine hiddenness, despite the fact that the problem of divine hiddenness is structurally parallel to the problem of evil. Suppose that there is a successful theistic response to the problem of divine hiddenness explaining the reason, call it Z, that God has to remain hidden from us. Unlike the case of the problem of evil, the existence of Z does not raise the following second-order question: If God is omnipotent and omnibenevolent, why does He not tell us that Z is the reason that He has to hide Himself? If there is any valid reason that God has to hide himself then He cannot tell us that that is the reason because by telling it to us God would fail to hide Himself from us. This seems to indicate that there is a link between the problem of evil and the problem of divine hiddenness and that theists might be able to stop the infinite regress of the higher-order problems of evil by appealing to God’s hiddenness. Conversely, it might be that the higher-order problems of evil cannot be resolved without first resolving the problem of divine hiddenness.
Suppose that we’ve observed a dozen randomly chosen ravens and they’re all black. We (cautiously) make the obvious inference that all ravens are black. But then we find out that regardless of parental color, newly conceived raven embryos have a 50% chance of being black and a 50% chance of being white, and that they have equal life expectancy in the two cases. When we find this out, we thereby also find out that it was just a fluke that our dozen ravens were all black. Thus, finding out that it’s random with probability 1/2 that a given raven will be black defeats the obvious inference that all ravens are black, and even defeats the inference that the next raven we will see will be black. The probability that the next raven we observe will be black is 1/2.
Next, suppose that instead of finding out about probabilities, we find out that there is no propensity either way of a conception resulting in a black raven or its resulting in a white raven. Perhaps an alien uniformly randomly tosses a perfectly sharp dart at a target, and makes a new raven be black whenever the dart lands in a maximally nonmeasurable subset S of the target and makes the raven be white if it lands outside S. (A subset S of a probability space Ω is maximally nonmeasurable provided that every measurable subset of S has probability zero and every measurable superset of S has probability one.) This is just as much a defeater as finding out that the event was random with probability 1/2. (The results of this paper are driving my intuitions here.) It’s still just a fluke that the dozen ravens we observed were all black. We still have a defeater for the claim that all ravens are black, or even that the next raven is black.
Finally, suppose instead that we find out that ravens come into existence with no cause, for no reason, stochastic or otherwise, and their colors are likewise brute and unexplained. This surely is just as good a defeater for inferences about the colors of ravens. It’s just a fluke that all the ones we saw so far were black.
Now suppose that the initial state of the universe is a brute fact, something with no explanation, stochastic or otherwise. We have (indirect) observations of a portion of that initial state: for instance, we find the portion of the state that has evolved into the observed parts of the universe to have had very low entropy. And science appropriately makes inferences from the portions of the initial state that have been observed by us to the portions that have not been observed, and even to the portions that are not observable. Thus, it is widely accepted that the whole of the initial state had very low entropy, not just the portion of it that has formed the basis of our observations. But if the initial state and all of its features are brute facts, then this bruteness is a defeater for inductive inferences from the observed to the unobserved portions of the initial state.
So some cosmological inductive inferences require that the initial state of the universe not be entirely brute. I don’t know just how much cosmology depends on the initial state not being entirely brute, but I suspect quite a bit.
What if there is no initial state? What if instead there is an infinite regress? Here I am more tentative, but I suspect that the same problem comes back when one considers the boundary conditions, say at time negative infinity. If these boundary conditions are brute, then we’ve got the same problem as with a brute initial state. Likewise, a contingent first cause will not help, either, since the argument can be applied to its state.
It seems that the only way out of scepticism about cosmology is if there is a necessary first cause. And I also suspect that the impact of the argument may go beyond cosmology. Presumably, we continue to come into causal contact with portions of the initial state that we have previously not been in contact with, and couldn’t that affect us in all sorts of ways that undermine more ordinary inductive inferences (e.g., a burst of radiation might kill us all tomorrow, and no probabilities can be assigned to the burst, and hence no probabilities can be assigned to any positive facts about what we will do tomorrow)? If so, then we lose quite a bit of our predictive ability about the future if we hold the initial state to be brute.
I have now completed my series of posts on The Puzzle of Existence. I’ll conclude by saying that I enjoyed most of the essays in this book quite a lot, and found them interesting food for thought. Further reflection on the points raised by the various authors stands to enrich metaphysics, philosophy of religion, and the theory of explanation. Additionally, most of the essays are quite accessible for non-specialists, including advanced undergraduate students. Assuming that a less expensive paperback version becomes available, this book would be a great choice for graduate or advanced undergraduate courses covering explanation in metaphysics, the argument from contingency from the existence of God, or the theory of modality.
Here is a list of my 16 blog posts, corresponding to the 16 chapters of the book:
- Introducing The Puzzle of Existence (Goldschmidt)
- O’Connor on Explaining Everything
- Oppy on Theism, Naturalism, and Explanation
- Kleinschmidt on the Principle of Sufficient Reason
- Jacob Ross on the PSR
- Christopher Hughes on Contingency and Plurality
- Conee on the Ontological Argument
- John Leslie’s Axiarchism
- How to Determine Whether there Might have Been Nothing (Efird and Stoneham)
- Why do We Ask Why? (Heil)
- Lowe on Metaphysical Nihilism
- Rodriguez-Pereyra on Ontological Subtraction
- Kotzen on the Improbability of Nothing
- Lange on the Natural Necessity of Something
- Maitzen on the Explanatory Power of Penguins
- McDaniel’s Ontological Pluralism
(Cross-posted at blog.kennypearce.net.)
In his contribution to The Puzzle of Existence, Stephen Maitzen defends the surprising claim that penguins hold the answer to the deep mysteries of the universe.
Well, that’s not exactly what he says. Maitzen’s position is that the only interpretation of ‘why is there something rather than nothing?’ on which that sentence expresses a legitimate, well-formed question is one on which it is not a deep mystery at all, but a trivial empirical question to which ‘because there are penguins’ is a perfectly adequate answer.
It is interesting to note that Maitzen’s article is, in a way, just the reverse of Lange’s. Lange thinks causal explanations are paradigmatic examples of good scientific explanations. However, Lange’s distinctness principle eliminates the possibility of a causal explanation of why there is something rather than nothing. Furthermore, the distinctness principle rules out a large class of explanations we might call ‘constitutive explanations.’ These are cases in which we explain why a state of affairs obtains by indicating some lower-level state of affairs that constitutes it. However, in my comments on Lange, I indicated that constitutive explanations are indeed part of scientific practice, and hence that Lange was wrong to rule them out.
Maitzen’s article reverses Lange’s in the sense that Maitzen holds that causal explanations should be regarded with suspicion in the absence of an agreed-upon metaphysical analysis of causation, but that constitutive explanations are good (265-266). Thus Maitzen’s view is, essentially, that the existence of something (Maitzen is specifically concerned with concrete, contingent things) is (partly) constituted by the existence of penguins and the fact that there are penguins can thus explain the fact that there is something.
Maitzen considers a number of objections to this explanation, but I think the most important is the one he labels ‘Objection D.’ (I suspect that Maitzen agrees, since this is the objection he spends the most time on). Here is how he states the objection:
Because it invokes CCTs [i.e., concrete contingent things] that already exist, [Maitzen’s] naturalistic method of explanation has no chance of explaining why there are any CCTs in the first place, any CCTs to begin with, any CCTs at all (259).
Maitzen’s strategy for dealing with this objection is to concede that an explanation of why there are penguins that appealed to more penguins would not explain why there are any penguins at all, but argue that this is because penguin is a substantial kind. CCT, Maitzen thinks, is not a substantial kind.
This is effectively a version of the ‘no sufficiently comprehensive beings’ objection to the argument from contingency, endorsed earlier in the volume by Ross. That is, Maitzen admits that the fact that a particular kind has instances is the sort of thing that needs to be explained without appeal to instances of the kind, but denies that there is any kind comprehensive enough to encompass all of the contingent, concrete beings and so force us to posit anything beyond them.
According to Maitzen, the question ‘why are there penguins?’ is not well-answered with ‘because there are emperor penguins’ (what I have been calling a constitutive explanation) but ‘why are there red things?’ is well-answered with ‘because there are things that reflect light of wavelengths roughly in the range 630-740 nm’ (263-264).
In fact, something Maitzen says later strongly suggests that he has misunderstood the contrast here. Maitzen admits that ‘because there are emperor penguins’ may be “a sufficient answer to the question ‘Why are there any penguins at all left on earth?’ in circumstances in which emperor penguins are the only penguins left on earth” (265). But in the appropriate context, the question ‘why are there penguins?’ could mean just that. For instance, consider the following dialog, which takes place in a dystopian future, c. 2200AD:
A: From the 19th century through the middle of the 21st century, humans relied on fossil fuels as their primary source of power. The resulting climate change was especially damaging to arctic and antarctic wildlife. Almost nothing survives, but there are still some penguins in Antarctica.
B: Why are there penguins?
A: Because there are emperor penguins. They were hardy enough to survive the changes to their ecosystem.
The conversation would sound quite unnatural to me without that last sentence, but this is only because B’s next question (“yes, but why are there emperor penguins?”) is so obvious that A is expected to anticipate it, as she does in that last sentence. Maitzen is in no position to object to this, since he argues explicitly that explanations are not to be considered inadequate just because they raise further ‘why’ questions (255-257).
On the other hand, the following dialog also seems perfectly natural:
A: Why are there red things?
B: Because there are things that reflect light of wavelengths roughly in the range 630-740 nm.
A: No, no, that’s not what I mean. I know perfectly well that red things are red in virtue of their reflective profile. What I want to know is, why are there any such things? I mean, couldn’t it have been the case that there was just nothing in the universe with that reflective profile? Why isn’t the universe like that instead of like this?
Of course, A’s response is, mutatis mutandis, just the response the arguer from contingency will want to give to Maitzen’s penguin explanation.
What all of this suggests is that, in fact, a question of the form ‘why are there xs?’ admits of (at least) two very different sorts of answers, one or the other of which may be desired on a particular occasion. The distinction between substantial kind terms and other sortals does not track the distinction between the circumstances in which the two types of explanations are desired. Furthermore, as my little dialog shows, even once we’ve got the constitutive explanation, we may still legitimately ask for the other kind. Thus Maitzen hasn’t shown that non-trivial interpretations of the question ‘why is there something rather than nothing?’ are illegitimate.
(Cross-posted at blog.kennypearce.net.)
Marc Lange’s contribution to The Puzzle of Existence, begins with this remark:
I read recently about a baby who was trapped during the night of February 26, 2011, in a locked bank vault in Conyers, Georgia. Naturally, I wondered why that had happened (235).
In the article which follows this fantastic opening, Lange appeals to the theory of necessity and laws of nature from his 2009 book, Laws and Lawmakers, to argue that one can explain why there is something rather than nothing only by showing that something exists as a matter of natural necessity (or, in a qualification he makes at 246n11, showing that it is naturally necessary that something has a nonzero probability of existing). Lange begins, therefore, with a destructive line of argument, designed to show that the only candidate answers to the question why there is something rather than nothing are non-causal scientific explanations, then proceeds with the constructive project of showing how, on his theory, such an explanation can be given. It is, I think, to Lange’s credit that the constructive portion of his essay is stronger than the destructive portion; the reverse is (and always has been) more often the case in philosophy.
Lange’s destructive argument can be reconstructed as follows:
- Every candidate answer to the question, ‘why is there something rather than nothing?’, must be a scientific explanation (238).
- Scientific explanations obey the distinctness principle (236-237).
- Any causal explanation of why there is something rather than nothing would violate the distinctness principle (239-240).
- Every candidate answer to the question, ‘why is there something rather than nothing?’, must be a non-causal scientific explanation.
Every premise of this argument is false.
To Lange’s credit, he does recognize that premise 1 is a substantive premise – that is, that not all (good) answers to ‘why?’ questions are scientific explanations. Nevertheless, all he says in defense of premise 1 is this:
I have taken for granted that in asking why there is something rather than nothing, we are demanding a scientific explanation. If an answer to this question does not have to satisfy the usual criteria of adequacy for a scientific explanation … then I do not know what it must do. Of course, not all explanations are scientific explanations; there are explanations in mathematics, moral explanations, legal explanations, and even baseball explanations (e.g., for why a given baserunner is entitled to third base). But none of these kinds of explanations is demanded by the riddle of existence (238).
However, Lange goes on, immediately thereafter, to observe that “Some philosophers who claim to regard the riddle of existence as demanding a scientific explanation may not actually so regard it.” There follows a brief discussion of attempts at axiological explanations (explanations that say that the world exists because it is good for it to exist). Similarly, one might appeal to other kinds of teleological explanations, or the ‘personal explanations’ in which some philosophers believe. Furthermore, at 242n4, Lange discusses David Lewis’s view that the existence of something is metaphysically necessary, and notes that on Lewis’s view of explanation this does not actually explain why there is something rather than nothing. However, Lange rejects Lewis’s view of explanation, and so holds that if Lewis were right about worlds, the existence of something rather than nothing would thereby be explained. Lange seems to think that this would be a scientific explanation, but it sure looks to me like a distinctively metaphysical explanation, different from anything found in natural science. So Lange does not give adequate reason for thinking that answers must take the form of scientific explanations and, indeed, there seems to be reason to suppose just the opposite. (Perhaps, though, an argument could be produced to show that, among the many candidate answers, the scientific explanations are, for whatever reason, more likely to succeed. This kind of argument would not rule the alternative answers out of court as Lange seems to want to do.)
Lange defines the distinctness principle, to which he appeals in premise 2, as follows:
If F suffices (or even helps) to constitute G‘s truth, then F is too close to G to help scientifically explain why G obtains (236).
The explanation of the laws of thermodynamics by statistical mechanics is a counterexample to this principle: the obtaining of the microphysical laws, together with the statistical facts about the microstates, constitute the obtaining of the thermodynamic laws and also explain their obtaining.
It seems plausible to me that Lange’s distinctness principle holds for explanations of particular facts, although not for general facts like special science laws. Thus, for instance, plausibly the position and momentum of the various gas particles in the room does not explain why the air temperature and pressure are as they are. It is unclear, though, on which side of this contrast the fact that there is something rather than nothing belongs.
Premise 3 is false because Lange takes the question to be about “why there exists some contingent thing rather than no such thing” (239). But some necessary thing or things could have caused the existence of contingent things in a non-necessitating manner, such as indeterministic physical causation or libertarian free choice. To cite such a cause would be to give a causal explanation of the existence of something rather than nothing without violating the distinctness principle.
So Lange’s argument that his sort of explanation is the only candidate explanation fails. But, as I said, in this piece Lange does a better job building up than tearing down, so let’s turn to Lange’s positive proposal.
The general idea of Lange’s view is that subjunctive conditionals are to be taken as primitive and the different species of necessity are to be defined in terms of them. Possibility and contingency then get defined in terms of necessity in the usual way, and all naturally (i.e., physically or nomologically) necessary propositions count as laws of nature. What Lange argues is that it may well be the case that it is a law of nature (in his sense) that some particular entity or entities exist, and that if this were the case it would amount to a non-causal scientific explanation of why there is something rather than nothing.
The analysis of necessity in terms of counterfactuals, as it is explained in the essay, goes like this:
Take a set of truths that is “logically closed” (i.e., that includes every logical consequence of its members) and is neither the empty set nor the set of all truths. Call such a set stable exactly when every member p of the set would still have been true had q been the case, for each of the counterfactual suppositions q that is logically consistent with every member of the set. I suggest that p is a natural necessity exactly when p belongs to a “stable” set (245).
As Lange indicates in a footnote, there are some further complications discussed in his book, but the general idea is that for any species of necessity, in order to get a necessarily false consequent on a true counterfactual, you have to start with a necessarily false antecedent. Natural necessity is a species of necessity which is weaker than logical necessity (hence the logical consistency requirement).
From here, the idea is very simple: Newton thought that if absolute space did not exist, the Newtonian laws of motion would not hold. On Lange’s view of laws, if one adds to this the two claims that (a) the Newtonian laws of motion are laws of nature, and (b) the existence of absolute space is logically contingent, then one gets the conclusion that it is a law of nature that absolute space exists. (Newton would not, of course, have called this a law of nature, and it is unclear – to me at least – whether Newton thought absolute space was logically contingent, but this is beside the point.) Lange thinks that, if Newtonian physics were true, then this would constitute a non-causal scientific explanation of why there is something than nothing. In fact, Newtonian physics is not true but, Lange thinks, it is nevertheless plausible, perhaps even likely, that an explanation of this general form is the correct explanation of why there is something rather than nothing.
(Cross-posted at blog.kennypearce.net.)
When someone asks ‘why p rather than q?’, it is sometimes a good answer to say, ‘p is far more probable than q.’ When someone asks, ‘why is p more probable than q?’, it is sometimes a good answer to say, ‘there are many more ways for p to be true than for q to be true.’ According to a well-known paper by Peter Van Inwagen, the question ‘why is there something rather than nothing?’ can be answered in just this fashion: something is far more probable than nothing, because there are infinitely many ways for there to be something, but there is only one way for there to be nothing. In his contribution to The Puzzle of Existence, Matthew Kotzen argues that, this sort of answer is only sometimes a good one, and that we cannot know a priori whether it is a good answer to the question of something rather than nothing.
Kotzen’s general line of response is a standard one: he argues that there are many possible measures, and not all of them assign probability 0 to the empty world. Van Inwagen is perfectly aware of this problem, but argues that a priori considerations allow us to select a natural measure. Kotzen’s strategy is to identify some everyday examples where this pattern of explanation looks good, and some where it looks bad, and show that van Inwagen’s a priori considerations don’t draw the line between good and bad in the right place. Furthermore, he argues (p. 228) that van Inwagen’s considerations may not actually be sufficient to assign unique probabilities in the relevant cases, since it is not always clear what space the measure should be assigned over.
I think Kotzen’s argument against van Inwagen is quite compelling. The best thing about Kotzen’s article, though, is that it does a great job explaining these complex issues at a moderate level of rigor and detail without assuming hardly any background. This would be a great article to assign to undergraduate students.
In the rest of this post, I’m going to do two things. First, I’m going to explain the issue about measures in a much lesser level of rigor and detail than Kotzen does, just to make sure we are all up to speed. Second, I am going to raise the question of whether van Inwagen’s argument might have an even bigger problem: whether, instead of too many equally eligible measures, there might be none.
The simplest, most familiar, cases where the probabilistic pattern of explanation with which we are concerned works are finite and discrete. This is the case, for instance, with dice rolls or coin flips. The coin either comes up heads or tails; each die shows one of its six faces. So then, as one learns in one’s very first introduction to probabilities, in the case of the dice roll, the probability of any particular proposition about that dice roll is the number of cases in which the proposition is true divided by the total number of possible cases (for two six-sided dice, 36). In real life, by dividing the outcomes into discrete cases like this, we care about certain factors (which face is up) and not about others (e.g., where on the table the dice land). This division into discrete cases is called a partition. The reason the probabilities are so simple in the dice case, with each case in the partition being equally likely, is because we chose a good partition. (Well, actually, it’s because a fair die is defined as one that makes each of those outcomes equally probable, but let’s ignore that for now and imagine that fair dice just occur in nature rather than being made by humans on purpose.) Suppose that, on one of our dice, the face with six dots is painted red rather than white and, for some reason, what we really care about is whether the red face is up. Well then we might partition the outcomes accordingly, into the red outcome and the non-red outcomes. But these two cases (red and non-red) are not equally probable.
Sometimes the thing we care about is not a discrete case like this, but a fundamentally continuous case like (in a standard example) where on a dartboard a perfectly thin dart lands. A measure is basically the equivalent, in this continuous case, of the partition in the discrete case. For the dart board, there is a natural measure, one that ‘just makes sense’, and this is provided by our ordinary spatial concepts. So if, for instance, the bullseye takes up 1/10 of the area of the dartboard then, if the dart is thrown randomly, it will have a 1/10 chance of landing there. (Again, this is really just what it means for the dart to be thrown randomly.) This isn’t the only possible measure, but it’s the one that, in some sense, ‘just makes sense.’ But the question is, is there a natural measure on the space of possible worlds? That is, is there some ‘correct’ or ‘sensible’ or ‘natural’ way of saying how ‘far apart’ two possible worlds are? This is far from clear. The Lewis-Stalnaker semantics for counterfactuals supposes that we can talk about some worlds being ‘closer together’ than others, but this is not enough to define a measure. Furthermore, Lewis, at least, thinks that the closeness of worlds might change based on contextual factors (which respects of similarity we most care about), so it seems like there’s a plurality of measures there. Perhaps one could claim that all of these reasonably natural measures agree in assigning nothing probability 0, but that’s not clear either. For instance, Leibniz seems to think that one reason why the existence of something cries out for explanation is that “a nothing is simpler and easier than a something” (“Principles of Nature and Grace,” tr. Woolhouse and Francks, sect. 7). So maybe we should adopt a measure in which worlds get lower probability the more complicated they are. (I think Swinburne might also have a view like this.) On this kind of view, the empty world (if there is such a world) will be the most probable world. So the plurality of measures seems like a problem.
It’s not the only problem, though. Kotzen notes that “the Lebesque measure can be defined only in spaces that can be represented as Euclidean n-dimensional real-valued spaces” (222). (The Lebesgue measure is the standard measure used, for instance, in the dart board case: the bigger space it takes up the bigger its measure.) But the space of possible worlds is not like this! David Lewis has argued that the cardinality of the space of possible worlds must be greater than the cardinality of the continuum (Plurality of Worlds, 118). The reason is relatively simple: suppose that it is possible that there should be a two-dimensional Euclidean space in which every point is either occupied or unoccupied. The set of possible patterns of occupied and unoccupied points in such a space (each representing a distinct possibility) will be larger than the continuum. But if this is right, then there can be no Lebesgue measure on the possible worlds because there are too many worlds. Even if this exact class of worlds is not really possible (for reasons such as the considerations about space in modern physics I raised last time) it seems likely that there are too many worlds for the space of possible worlds to have a Lebesgue measure. Yet Kotzen attributes to van Inwagen that view “that we ought to associate a proposition’s probability with its Lebesgue measure in the relevant space” (227).
Maybe van Inwagen is not in quite this much trouble. He doesn’t actually seem to say anything about a Lebesgue measure in the paper, so I’m not sure exactly why Kotzen thinks van Inwagen is committed to this. In fact, in the paper Kotzen is discussing, van Inwagen cites his earlier discussion in Daniel Howard-Snyder’s collection, The Evidential Argument from Evil. In endnote 3 (pp. 239-240) of that article, van Inwagen says “the notion of the measure of a set of worlds gets most of such content as it has from the intuitive notion of the proportion of logical psace that a set of worlds occupies.” I find it a little bit ironic that van Inwagen says this, because he’s always denying that he has intuitions about things! I don’t have intuitions about proportions of logical space. In any event, it seems to me that van Inwagen is here disavowing the project of giving a well-defined measure in the mathematician’s sense.
Suppose one did want to identify a natural measure that was well-defined in the mathematician’s sense. I’m not sure about all the technicalities of trying to do this for sets of larger-than-continuum cardinality, and whether it can be done at all. Even if it can, thought, it’s going to be hard to say that one measure is more intuitive or natural than another in such an exotic realm. Things might be even worse: Pruss thinks (PSR, p. 100) that, for any cardinality k, it is possible that there be k many photons. If this is true, then there is a proper class of possible worlds, and one certainly can’t define a measure on a proper class. (This is another thing I don’t think I have intuitions about.)
All this to say: anyone who wants to assign a priori probabilities to all propositions (as van Inwagen does) is fighting an uphill battle, but if such probabilities cannot be assigned, then it does not seem that the probabilistic pattern of explanation can be used to tell us why there is something rather than nothing.
(Cross-posted at blog.kennypearce.net)