The structure of the argument

Preliminary problems

The potential success of theodicies

An important preliminary consideration in interpreting Tooley's argument charitably is that it is primarily an argument against sceptical theists who concede that theodicies are generally unsuccessful. In particular, there are sceptical theists who think that there is at least one state of affairs in the world such that, had it not occurred, we would expect God not to permit it (on the basis of its WMPs). But, say these sceptical theists, our cognitive and epistemic limitations imply that it is unreasonable to conclude from our knowledge of these states of affairs that the permission of these states of affairs is overall evil, and so that God does not exist. Tooley's argument is an attempt to show that this inference is indeed reasonable – that our knowledge regarding certain prima facie evil states of affairs is good reason to think that, probably, the permission of these states of affairs is overall evil. Since this conclusion entails atheism, it follows from the probability calculus that, probably, God does not exist.

Note, therefore, that Tooley does not give a detailed defence of premise (1) – conceded by this variety of sceptical theist – that any action of choosing not to prevent the Lisbon earthquake has a known WMP such that there are no RMPs known to be counterbalancing.Footnote ⁷ Of this premise, Tooley writes:

[This statement] makes a claim that would be challenged by philosophers who respond to the evidential argument from evil by offering a theodicy. Nevertheless, the claim seems very reasonable, given the relevant facts about the world, together with the moral knowledge that we possess. For what rightmaking properties can one point to that one has good reason to believe would be present in the case of an action of allowing the Lisbon earthquake, and that would be sufficiently serious to counterbalance the wrongmaking property of allowing more than 50,000 ordinary people to be killed? (Plantinga & Tooley (Reference Plantinga and Michael2008), 122)

This relatively cursory dismissal of theodicies will be seen as deeply unsatisfactory by many philosophers, and is a relatively conspicuous shortcoming of Tooley's argument. There are many intelligent, educated people who are sincerely persuaded that they are aware of some reason which justifies the suffering in the world. One might think that the goods of free will, forgiveness, or redemption are all such important goods that, were they to be instantiated, would outweigh the wrongmaking properties of the action of allowing the negative corollaries of such goods to obtain. It is not obvious that these goods are justifying, but Tooley has not demonstrated that they are not. Importantly, Tooley needs this premise to be certain for his argument to work. For Tooley's intermediate conclusion is of the form P(X|Y) > 0.5, where X =_df an action of choosing not to prevent the Lisbon earthquake is morally wrong, all things considered, and Y =_df choosing not to prevent the Lisbon earthquake has a WMP that we know of, and that there are no RMPs that are known to be counterbalancing. But we do not know Y for certain, so we have to factor in our uncertainty into our final evaluation of P(X). This is done by weighting P(X) by P(Y):

$${\rm P}\lpar {\rm X} \rpar = {\rm P}\lpar {{\rm X} \vert {\rm Y}} \rpar \times {\rm P}\lpar {\rm Y} \rpar + {\rm P}\lpar {{\rm X} \vert \neg {\rm Y}} \rpar \times {\rm P}\lpar {\neg {\rm Y}} \rpar $$

If P(Y) = 1, then P(X) = P(X|Y), and so P(X) > 0.5 (assuming the argument is otherwise correct). But suppose P(Y) is only 0.6. Then P(X) = P(X|Y) × 0.6 + P(X|¬Y) × 0.4. But then knowing that P(X|Y) > 0.5 does not show us that P(X) > 0.5. It shows us only that P(X) > 0.3. And this conclusion does not help Tooley much. So even if Y is more probable than not, it may not suffice for the argument in the presence of significant uncertainty about Y. So the various kinds of uncertainty about premise 1 – uncertainty about whether God's allowing the Lisbon earthquake has a WMP, or uncertainty about whether a known RMP is true of the Lisbon earthquake or whether a known RMP is justifying – can be appealed to here.

A more serious problem

This leads us to the most fatal problem of Tooley's argument from single cases. The issue is not, in fact, with his argument from premise (2). That is fine as far as it goes. Conceive of an action that it seems it would be wrong for God to permit – perhaps, permitting every created human to suffer eternal conscious torment. Most of us would confidently agree that it is likely that this is impermissible. And if someone objected that there might be unknown RMPs outweighing the prima facie wrongness of this action, we would respond that there might well be, but the likelihood is still that the action is impermissible. So to that extent Tooley's suggestions pertaining to the equiprobability of similarly weighted unknown RMPs and WMPS are not particularly objectionable.

The problem, however, is that Tooley does not just say that, in the absence of such an event occurring, we should judge that prima facie evils are ultima facie evils. His argument seems to require that we must say the same once such an event has actually occurred. And although this might seem innocuous initially, there is a compelling case that the occurrence of such an event does, in fact, change the probability that that event is permissible by God. Let me explain.

Armed with the premise that the probability that the action of God's permitting a Lisbon earthquake would probably be morally wrong given only the known RMPs and WMPs of permitting the Lisbon earthquake, with no knowledge of whether the earthquake occurred, is greater than 0.5, we can give a clearer formalization of the argument to highlight the more troublesome mistake. This premise can be formalized:

• $${\rm P}\lpar {{\rm W} \vert {\rm WMP}} \rpar \gt 0.5$$

• W = _df It would be wrong for God to permit a Lisbon earthquake

• WMP = _df Our knowledge regarding the RMPs and WMPs of God's permitting a Lisbon earthquake (most notably, that it has an epistemically weighty WMP)

My contention is that even this probabilistic claim does not do the required work. To see why, we can formalize the rest of the argument.

WMP is taken as data. We know that P(W|WMP) > 0.5. And since W and L jointly entail ~T (atheism), it can be shown from the Kolmogorov axioms that P(~T|WMP&L) > P(W|WMP&L). So as long as P(W|WMP&L) is also greater than 0.5 – that is, as long as knowing that the Lisbon earthquake occurred makes no difference to the probability that it would be morally permissible for God to allow it – it follows that P(~T|WMP&L) is also greater than 0.5. This is just to say that the probability of atheism given WMP and L is greater than 0.5. And WMP and L appear not to be in doubt. The argument succeeds. But it makes the assumption that L does not change the probability of W. Tooley assumes, that is, that P(W|WMP) = P(W|WMP&L). This implicit assumption is crucial to the argument, but is in fact probably false.

While Tooley briefly acknowledges that there might be ‘countervailing positive evidence in support of the existence of God’, his probabilistic treatment neglects the impact that the prior probability of theism and evidence for theism can, in principle, have on P(W). Interestingly, he also neglects the impact that the occurrence of the Lisbon earthquake can have on P(W). Indeed, I shall show that his formalization of the argument actually seems to preclude these considerations, and so is in tension with his claim that there might be countervailing positive evidence in support of theism.

This, it turns out, is a critical flaw in Tooley's argument. To see this, consider the following analogy: suppose you are hiding in a trench in a battle. To exit the trench and run out into the open seems, on the basis of your knowledge of military tactics, fatal, and unlikely to help your side at all. It seems reasonable to conclude on this basis that, probably, your good and loyal military commander would not command you to do such a thing, since the command would have a serious WMP with no known counterbalancing RMPs. Suppose you are then commanded to exit the trench and run out into the open. According to the known RMPs and WMPs properties of this command, the command is probably an evil one. Since a good and loyal military commander could not command something evil (provided we define ‘good’ in this context with sufficient care), this entails that the probability that your military commander is good is less than 0.5. But this clearly ignores any positive evidence you might have for thinking that your military commander is good, and the prior probability that your military commander is good. It is perfectly conceivable that the occurrence of the command itself ends up modifying the probability that such a command is morally permissible. But if Tooley's argument works, it is difficult to see how any evidence for the goodness of your commander could outweigh that, in theory, the command seemed wrong to you. Even if you had amazing evidence for your commander's goodness, the fact that, on balance, this command seemed wrong to you before it was given shows that your commander simply isn't good. This is an absurd result. So Tooley's approach seems to contradict the seeming importance of positive evidence or prior probability at this juncture.Footnote ¹¹

Another counter-intuitive consequence of Tooley's method is the following: suppose, for whatever reason, that we consider theism to have an overwhelming prior probability (whether because of a high intrinsic probability or because of strong evidence), but that we are aware of just one prima facie evil state of affairs. This might be a completely trivial evil: I drop an ice cream, and I am unaware of a counterbalancing good that comes from this event. This, according to Tooley, has so much weight that, regardless of any considerations of prior probability, the probability of theism drops to below 0.5. Again, this seems implausible. Not only that, but it is in tension with his claim that countervailing positive evidence might raise the probability of theism above 0.5. On Tooley's schema, it is not clear how this is possible.

What is going on here? The issue is that Tooley is not really looking for P(~T|WMP), as his prose suggests, but rather for P(~T|WMP&L). That is, he wants to know the probability of atheism given WMP and that the Lisbon earthquake actually occurred. Starting with P(W|WMP) > 0.5, Tooley seems to think that P(W|WMP) = P(W|WMP&L), and so P(W|WMP&L) is likewise greater than 0.5. That is, he assumes that L makes no difference to the probability of W. Then, since P(W|WMP&L) = P(W&L|WMP&L), it follows that P(W&L|WMP&L) > 0.5. And because W&L entails ~T, it follows that P(~T|WMP&L) > 0.5. This is just a formalization of Tooley's argument as described in section 2.1.

We noted earlier that it is not clear how evidence for God even could be accommodated in Tooley's framework: after all, in Tooley's framework, whether W is probably true surely just depends on the known WMP and the symmetry principle. And before we knew that the Lisbon earthquake occurred, we might well have agreed. Even with evidence of God's existence – perhaps very strong evidence – we might still judge that God would not have allowed the Lisbon earthquake, and so be surprised when the Lisbon earthquake occurs. That is the basic premise Tooley is appealing to. But then how does evidence for God relate to Tooley's argument? The answer is that the prior probability of theism (which will depend on the intrinsic probability of theism and evidence for or against theism) is channelled through the probability of the Lisbon earthquake occurring. This is to say that although a theist could reasonably agree that evidence for God makes no difference to W in itself (that is, unless one knows that the earthquake has occurred, a theist could think that W is probably true, and that the evidence for God makes no difference to this), the evidence for God does make a difference to the probability of W when one finds out that the earthquake has actually occurred – just as one can consider the military commander's command wrong in itself (and the evidence for the goodness of the commander is irrelevant until the command is given), but change one's opinion when one finds out that the command has actually been given. A consequence of this is that the occurrence of the Lisbon earthquake affects the probability of the permission of the Lisbon earthquake being morally permissible by God. Thus, it is false that P(W|WMP) = P(W|WMP&L). There is in fact a relatively simple mathematical proof available of this:

$$\eqalign{{\rm P}\lpar {{\rm W} \vert {\rm WMP}\& {\rm L}} \rpar = &{\rm P}\lpar {{\rm W} \vert {\rm WMP}\& {\rm L}\& {\rm T}} \rpar \times {\rm P}\lpar {{\rm T} \vert {\rm WMP}\& {\rm L}} \rpar \cr &+ {\rm P}\lpar {{\rm W} \vert {\rm WMP}\& {\rm L}\& \sim{\rm T}} \rpar \times {\rm P}\lpar {\sim{\rm T} \vert {\rm WMP}\& {\rm L}} \rpar }$$

P(W|WMP&L&T) is 0, since theism and the Lisbon earthquake's occurring entail that God's permitting the Lisbon earthquake is permissible (so W is false). Thus:

$${\rm P}\lpar {{\rm W} \vert {\rm WMP}\& {\rm L}} \rpar = {\rm P}\lpar {{\rm W} \vert {\rm WMP}\& {\rm L}\& \sim{\rm T}} \rpar \times {\rm P}\lpar {\sim{\rm T} \vert {\rm WMP}\& {\rm L}} \rpar $$

P(W|WMP&L&~T) is the probability that an action of God's permitting the Lisbon earthquake would be wrong, given its uncounterbalanced WMP, given that it has occurred, and given that God does not exist. But Tooley seems to think that L makes no difference to the probability of W. And it is not clear that learning ~T should increase one's confidence in W, at least on Tooley's account, since for Tooley, the probability of W is judged only by WMP. So, for Tooley, P(W|WMP) = P(W|WMP&L) = P(W|WMP&L&~T). But if so, then:

$${\rm P}\lpar {{\rm W} \vert {\rm WMP}\& {\rm L}} \rpar = {\rm P}\lpar {{\rm W} \vert {\rm WMP}\& {\rm L}} \rpar \times {\rm P}\lpar {\sim{\rm T} \vert {\rm WMP}\& {\rm L}} \rpar $$

This is consistent only if P(~T|WMP&L) = 1. But since P(~T|WMP&L) is not 1 (because the Lisbon earthquake's having an uncounterbalanced WMP and its occurrence does not entail atheism), we have a contradiction. Thus, it cannot be the case that L makes no difference to the probability of W.

On reflection, this is entirely intuitive. For in the military case, it is intuitively clear that learning that a captain has given a command does, in fact, affect the probability that that command is good (specifically, if we have good evidence that the captain is good, it increases it).

We can get more of a grip on how L might affect the probability of W by the odds form of Bayes’ Theorem. At this point, it will be helpful for ease of reading to exclude WMP from the conditional. We will take it as a given that WMP is part of the background knowledge.

$$\displaystyle{{{\rm P}\lpar {{\rm W} \vert {\rm L}} \rpar } \over {{\rm P}\lpar {\sim{\rm W} \vert {\rm L}} \rpar }} = \displaystyle{{{\rm P}\lpar {{\rm L} \vert {\rm W}} \rpar } \over {{\rm P}\lpar {{\rm L} \vert \sim{\rm W}} \rpar }} \times \displaystyle{{{\rm P}\lpar {\rm W} \rpar } \over {{\rm P}\lpar {\sim{\rm W}} \rpar }}$$

According to this, adding L into our conditional lowers the probability of W just if the first multiplicand is less than 1. So it is worth comparing P(L|W) and P(L|~W). These are the probabilities that the Lisbon earthquake would occur if it would be wrong for God to permit it and if it would not be wrong for God to permit it. Intuitively, these should be different: if it would be wrong, then there are fewer ways the earthquake could occur – it could not occur in any world where God exists, for example. But if it is not wrong, there are more ways the earthquake could occur: in a variety of worlds where God exists. This is borne out by looking more closely at the probabilities in question:

$${\rm P}\lpar {{\rm L} \vert {\rm W}} \rpar = {\rm P}\lpar {{\rm L} \vert {\rm W}\& {\rm T}} \rpar \times {\rm P}\lpar {{\rm T} \vert {\rm W}} \rpar + {\rm P}\lpar {{\rm L} \vert {\rm W}\& \sim{\rm T}} \rpar \times {\rm P}\lpar {\sim{\rm T} \vert {\rm W}} \rpar $$

$${\rm P}\lpar {{\rm L} \vert \sim{\rm W}} \rpar = {\rm P}\lpar {{\rm L} \vert \sim{\rm W}\& {\rm T}} \rpar \times {\rm P}\lpar {{\rm T} \vert \sim{\rm W}} \rpar + {\rm P}\lpar {{\rm L} \vert \sim{\rm W}\& \sim{\rm T}} \rpar \times {\rm P}\lpar {\sim{\rm T} \vert \sim{\rm W}} \rpar $$

Turning this into intuitive judgements is not easy. But it can be seen how P(L|~W) could exceed P(L|W): for the first conjunct in the first line is 0, whereas the corresponding first conjunct in the second line is more than 0. How much more than 0? That depends on the values of its constituents. But one of these constituents is P(T|~W), which is just the prior probability of theism given that permitting the Lisbon earthquake is morally acceptable. This could well be high if there is good evidence for theism. So P(L|~W) may be considerably higher than P(L|W) if theism is independently probable. The only reason P(L|~W) would not exceed P(L|W) is if the second conjunct in the first line exceeds the second conjunct in the second line by the same magnitude. I leave this consideration aside for reasons of space, only to note that I have thought carefully about whether it does so and can see no reason to think it should. And in any case, that is Tooley's burden. So, quite clearly, P(L|~W) and P(L|W) could be discrepant, and hence Tooley has to factor L in to his judgement on W. He has not done so. And if he did do so, he would have to provide a reason to think that P(T|~W) is not high.Footnote ¹²

Part of this complexity arises because Tooley has not framed his argument in a standard Bayesian format. So we can briefly consider whether there is a strictly Bayesian (i.e. in the sense of using Bayes’ Theorem) argument to be made from his premises. Tooley's argument is essentially intended to vindicate our intuitive judgement that we would not expect God to perform a certain action from a sceptical theist rebuttal. Even if theism is true, we wouldn't expect this of God. So his premise can be understood as claiming that P(W|T) (with WMP implicitly understood in the background knowledge) > 0.5.

1. (1) P(W|T) > 0.5 (premise)

2. (2) P(~W|T) < 0.5 (from 1)

3. (3) P(L|T) = P(L|T&W) × P(W|T) + P(L|T&~W) × P(~W|T) (theorem)

4. (4) P(L|T) = P(L|T&~W) × P(~W|T) (from 3, since P(L|T&W) = 0)

5. (5) P(L|T) < P(L|T&~W) × 0.5 (from 2 and 4)

What this generates is an argument that P(L|T) is less than 0.5 – perhaps considerably less than 0.5, depending on the value of P(L|T&~W). That is, on theism, we would not expect the Lisbon earthquake. This could form the basis of a simple Bayesian argument:

$$\displaystyle{{{\rm P}\lpar {{\rm T} \vert {\rm L}} \rpar } \over {{\rm P}\lpar {\sim{\rm T} \vert {\rm L}} \rpar }} = \displaystyle{{{\rm P}\lpar {{\rm L} \vert {\rm T}} \rpar } \over {{\rm P}\lpar {{\rm L} \vert \sim{\rm T}} \rpar }} \times \displaystyle{{{\rm P}\lpar {\rm T} \rpar } \over {{\rm P}\lpar {\sim{\rm T}} \rpar }}$$

Tooley could therefore argue that since P(L|T) < 0.5, and P(L|~T) is plausibly higher,Footnote ¹³ the final ratio exceeds the first ratio, and so L increases the probability of atheism. If Tooley has an argument that the final ratio is 1 or lower, then the final result will be that atheism is probable. But this is a weak result. For it depends crucially on the prior probabilities of T and ~T, which might be modified by other evidence. And even if Tooley can establish this prior probability, he still needs to show that P(L|~T) exceeds P(L|T). He has shown at most that P(L|T) is less than 0.5. But this shows nothing on its own. So the result is a relatively uninteresting one in the absence of these other premises. The best case scenario, in the absence of a discussion of prior probability and independent evidence, is that L supports atheism to some extent. But many theists already grant this premise, and it is not typically thought to deal theism a mortal wound. So a simple Bayesian reconstruction of Tooley's argument suggests that it is not powerful.

Structure-descriptions and the generalization to many prima facie evils

Tooley does offer an extension of the argument to encompass the number of events in the world which seem, given their RMPs and WMPs, impermissible by God. The aim is to make God's existence very improbable indeed, not just less probable than not. This argument suffers from all the same problems as the previous argument, but I will also show that it is vulnerable to at least two further criticisms, one of which is redeemable and the other of which is fatal. The argument is as follows:Footnote ¹⁴

• n = number of prima facie evil events in the world

• k = number of unknown WMPs and RMPs

Define a Q-predicate as a predicate which is maximal with respect to U, the set of unknown WMPs and RMPs. A predicate is maximal with respect to U if, when applied to an individual, it indicates, for every property in U, whether or not that individual has that property or not. So if there are three basic properties P, Q, and R in U, the following predicates would be maximal with respect to U:

• (P & Q & R)   (P & Q & ~R)

• (P & ~Q & R)   (P & ~Q & ~R)

• (~P & Q & R)   (~P & Q & ~R)

• (~P & ~Q & R)   (~P & ~Q & ~R)

As can be seen, maximal predicates offer a helpful way of enumerating all the possible combinations of predicates or properties applying to any given object.

In this case, there are k members of U.Footnote ¹⁵ U is the set of unknown morally relevant properties. Since y basic properties generate 2^y maximal predicates, it follows that there are 2^k Q-predicates.Footnote ¹⁶ And each of these Q-predicates will have its own moral weight. If the RMPs in a given Q-predicate outweigh the WMPs, then the Q-predicate is ‘positive’. But, as far as we know, there is no reason to think that there are more unknown RMPs than unknown WMPs or vice versa. And we have no reason to think that unknown RMPs are weightier on average than unknown WMPs. And so these symmetry considerationsFootnote ¹⁷ suggest that no more than half of all Q-predicates are positive (in fact, less than half will be positive, since some will be neutral). Assuming, charitably, that half of all Q-predicates are positive, the number of positive Q-predicates is 2^k−1.

Predicates can be applied to objects. And there is a simple formula governing how many possible combinations there are, for any given number of basic properties and any number of objects. So, if we have one basic property P and three objects a, b, and c, we have the following possibilities:

• Pa & Pb & Pc    Pa & Pb & ~Pc

• Pa & ~Pb & Pc    Pa & ~Pb & ~Pc

• ~Pa & Pb & Pc    ~Pa & Pb & ~Pc

• ~Pa & ~Pb & Pc    ~Pa & ~Pb & ~Pc

These are called state-descriptions, and the general rule is that the number of state descriptions is equal to mⁿ, where m is the number of maximal predicates and n is the number of objects (which in our case, are states of affairs). So in this case, where one basic property generates two maximal predicates, m = 2. And since n = 3, we have 2³ = 8 state-descriptions.

These state-descriptions can be organized into structure-descriptions, where state-descriptions within a structure-description differ only in the permutations of objects therein. For example:

• Structure-description 1: Pa & Pb & Pc

• Structure-description 2: Pa & Pb & ~Pc

• Pa & ~Pb & Pc

• ~Pa & Pb & Pc

• Structure-description 3: Pa & ~Pb & ~Pc

• ~Pa & Pb & ~Pc

• ~Pa & ~Pb & Pc

• Structure-description 4: ~Pa & ~Pb & ~Pc

Tooley, following Carnap, proposes that the correct prior probability distribution applies a principle of indifference not across state-descriptions but across structure-descriptions, so that the prior probability of Pa & Pb & Pc is not 1/8 but 1/4.

There is a helpful formula given by Carnap which calculates the number of structure-descriptions given by m maximal predicates and n objects. That formula is:

$$\displaystyle{{\lpar {{\rm n} + {\rm m}-1} \rpar !} \over {{\rm n}!\lpar {{\rm m}-1} \rpar !}}$$

Again, since in this case m = 2 and n = 3, the total amounts to 4, which fits with our enumeration of structure-descriptions. The formula generalizes for any integral number of maximal predicates and any integral number of objects.

We have said that there are 2^k maximal predicates, where k is the number of unknown, basic RMPs and WMPs. So the total number of structure-descriptions is:

$$\displaystyle{{\lpar {{\rm n} + 2^{\rm k}-1} \rpar !} \over {{\rm n}!\lpar {2^{\rm k}-1} \rpar !}}$$

Now, in order for the theist to say that no prima facie evil event in the world is ultima facie evil (by which I mean that it would be impermissible for God to permit the occurrence of that event), they will have to hold that for every such event, the unknown RMPs and WMPs outweigh the known WMPs of the event. And that can only be the case if a positive Q-predicate is predicated of every one of those events. So we will be looking for structure-descriptions which involve only positive Q-predicates. Since there are 2^k−1 positive Q-predicates, the number of structure-descriptions which attribute positive Q-predicates to every prima facie evil event is:

$$\displaystyle{{\lpar {{\rm n} + 2^{{\rm k}-1} - 1} \rpar !} \over {{\rm n}!\lpar {2^{{\rm k}-1} - 1} \rpar !}}$$

Now the crucial step. We know that there are a certain number of structure descriptions for all prima facie evils and a given number of unknown moral properties. And we know that there is a certain number of structure descriptions on which all these prima facie evils come out as justifiable. The probability that they do all come out as justifiable – call this P(k,n) – is, according to Tooley, simply the ratio of these:

$$\displaystyle{{\lpar {{\rm n} + 2^{{\rm k}-1}-1} \rpar !/{\rm n}!\lpar {2^{{\rm k}-1}-1} \rpar !} \over {\lpar {{\rm n} + 2^{\rm k}-1} \rpar !/{\rm n}!\lpar {2^{\rm k}-1} \rpar !}}$$

Tooley shows that this is a very small number as follows. It can then seen by comparing P(k,n), P(k,n + 1) and P(k + 1,n) that P(k,n) is a monotonically decreasing function of k and n, except where n = 1. That is, as k or n increases, P(k,n) decreases. This is because the ratios P(k,n)/P(k,n + 1) and P(k,n)/P(k + 1,n) are both greater than 1 (except where n = 1). Since P(k,n) is therefore at a maximum when k = 1, we can set an upper bound for P(k,n) as P(1,n):

$$\eqalign{& \displaystyle{{\lpar {{\rm n} + 2^{1-1}-1} \rpar !/{\rm n}!\lpar {2^{1-1}-1} \rpar !} \over {\lpar {{\rm n} + 2^1-1} \rpar !/{\rm n}!\lpar {2^1-1} \rpar !}} \cr & = \displaystyle{{{\rm n}!/{\rm n}!} \over {\lpar {{\rm n} + 1} \rpar !/{\rm n}!}} \cr & = \displaystyle{1 \over {{\rm n} + 1}}} $$

Since this is a maximum value for P(k,n), we can say that P(k,n) is at most 1/(n + 1), where n is the number of prima facie evil events in the world. Since there are many such events, P(k,n) is very low. Conversely, the probability that there is at least one ultima facie evil event in the world is very high. And since the existence of at least one ultima facie evil event in the world entails God's non-existence, the probability that God does not exist is even higher. Given that this argument involves some assumptions charitable to the theist, Tooley says, 1/(n + 1) is a significant overestimate of the probability that all prima facie evils are justifiable, and so an even more significant overestimate of the probability of theism.

As I previously mentioned, this argument succumbs to all my criticisms of Tooley's first argument. For example, Tooley again neglects the fact that whether or not an event has occurred may drastically change the probability that that event is permissible by God. But even setting aside those difficulties, the Carnapian method is demonstrably unreasonable. One problem is that it fails to adequately represent the extreme conditional dependence of the evilness of various n_i. That is, finding out that there are some prima facie evils should make it very likely that there are other prima facie evils of similar kinds. But Tooley's argument seems insensitive to this fact – a fact which severely limits the force of the argument from evil against theism.Footnote ¹⁸

The clearest problem, however, is that the improbability here is not generated by any enormous tension between theism and suffering in the world, but simply by the magnitude of n and k. And there are parodies which can show exactly why this is so problematic. I will give just one. Consider the following scenario:

Humans exist in a world populated only with prima facie good states of affairs. They manage to individuate 1,000 independent such states of affairs. Then n = 1,000. There are just 6 unknown morally relevant properties, and so there are 64 Q-predicates. Thus, the total number of structure descriptions is:

$$\displaystyle{{\lpar {1,000 + 64-1} \rpar !} \over {1,000!\lpar {64-1} \rpar !}} = \displaystyle{{\lpar {1,063} \rpar \ldots \lpar {1,001} \rpar } \over {\lpar {63} \rpar \ldots \lpar 1 \rpar }}$$

Suppose humans are in the possession of the following knowledge: the unknown RMPs and WMPs are heavily balanced in favour of RMPs, and RMPs are both weightier and more common than WMPs. RMPs and WMPs are balanced such that only one of the 64 Q-predicates is sufficient to render these prima facie good states of affairs evil. Call the 63 Q-predicates preserving goodness preserver Q-predicates, and call the other predicate the evil Q-predicate. The details of these RMPs and WMPs remain unknown, of course.

This case is obviously favourable to theism. The world contains nothing but prima facie good states of affairs, and humans even have the knowledge that the unknown RMPs and WMPs of their world are balanced in favour of good. Few things in philosophy are as clear as that people in this world should not be able to generate an inductive argument from evil against the existence of God.

If Tooley's approach is correct, however, these people can do exactly that. For, as Tooley does, they note that a single instantiation of the evil Q-predicate is sufficient for God's non-existence. And they point out to the theists of their world that God exists only if the correct structure-description of the world contains only preserver Q-predicates. But, they say, there are (1,063) . . . (1,001)/(63) . . . (1) structure-descriptions. And only a small proportion of them contain only preserver Q-predicates. To be precise, the number of structure-descriptions containing only preserver Q-predicates is:

$$\displaystyle{{\lpar {1,000 + 63-1} \rpar !} \over {1,000!\lpar {63-1} \rpar !}} = \displaystyle{{\lpar {1,062} \rpar \ldots \lpar {1,001} \rpar } \over {\lpar {62} \rpar \ldots \lpar 1 \rpar }}$$

The proportion of total structure-descriptions which only contain preserver Q-predicates is therefore equal to:

$$\displaystyle{{\lpar {1,062} \rpar \ldots \lpar {1,001} \rpar \times \lpar {63} \rpar \ldots \lpar 1 \rpar } \over {\lpar {62} \rpar \ldots \lpar 1 \rpar \times \lpar {1,063} \rpar \ldots \lpar {1,001} \rpar }} = \displaystyle{{63} \over {1,063}}$$

So it is very likely that at least one of these 1,000 prima facie good states of affairs actually exemplifies the evil Q-predicate. And so it is very likely that God does not exist.

According to this inductive method, the probability of all of these seemingly permissible states of affairs being permissible is still incredibly low even when there is no prima facie evil in the world, and even when almost all Q-predicates render these states of affairs permissible. This improbability is generated simply by the fact that there are a large number of states of affairs, and by the fact that there are a modest number of unknown RMPs and WMPs. But it is painstakingly obvious that there simply being a large number of states of affairs should not count as such compelling and apparently indefeasible evidence against theism. This parody therefore shows Tooley's Carnapian inductive method to be unreasonable.