Evidential arguments from evil

Current evidential arguments from evil are of at least two main types. First, there are arguments inspired by the following passage in David Hume's Dialogues Concerning Natural Religion:

The key idea in Humean-inspired arguments from evil is that one can show that theism is unlikely to be true by arguing that some alternative hypothesis, logically incompatible with theism, is more probable than theism.

The first such Humean-inspired argument was developed by Paul Draper (Reference Draper1989), and involved comparing theism with the following ‘Hypothesis of Indifference’ (Draper Reference Draper1989, 332/Reference Draper1996, 13):

Draper subsequently developed and defended this type of argument further in later articles (Reference Draper, Copan and Meister2008, Reference Draper, Flint and Rea2009, Reference Draper, McBrayer and Howard-Snyder2013, Reference Draper, Dougherty and McBrayer2014a, and Reference Draper, Dougherty and McBrayer2014b). Then, more recently, Wes Morriston (Reference Morriston, Dougherty and McBrayer2014) constructed an alternative Humean-inspired argument in which he considered three hypotheses, all of which postulate the existence of an omnipotent and omniscient being, who is either (1) perfectly benevolent (theism), or (2) perfectly malevolent (demonism), or else (3) perfectly indifferent to good and evil, and Morriston asks, ‘How do these hypotheses fare when we consider the ‘mixed phenomena’ of good and ill in our world?’ (Reference Morriston, Dougherty and McBrayer2014, 225) Morriston then argues, first, that his indifference hypothesis is more probable than both theism and demonism, and second, that skeptical demonism fails to undermine his Humean argument against demonism, and that, for similar reasons, sceptical theism fails to undermine his Humean argument against theism.

A different kind of evidential argument from evil, which I have set out on three occasions (Tooley and Plantinga Reference Draper, Copan and Meister2008, Tooley Reference Tooley, Chandler and Harrison2012, and Tooley Reference Tooley2019), uses a substantive theory of inductive logic or logical probability to formulate an argument that, unlike Humean-inspired arguments from evil, involves no comparison with any specific hypotheses, neither ones involving some personal cause of the universe, nor purely naturalistic ones. Instead, equiprobability principles are used to derive an upper bound on the probability that God exists given the evils found in the world – a probability that depends on how many evils there are, but which is extremely low indeed.

Equiprobability-based evidential argument from evil

This second type of argument has the following structure. First of all, let E be a single event that, given only its known rightmaking and wrongmaking properties, would be morally wrong to allow. It follows that the probability, given only that information, that it would be morally permissible to allow E to occur, given the totality of its rightmaking and wrongmaking properties, both known and unknown, must be less than ½. Secondly, given N events, each of which, judged only by its known rightmaking and wrongmaking properties, would be morally wrong to allow, the probability, given only that information, that none of those N events would be morally wrong to allow, given the totality of its rightmaking and wrongmaking properties, both known and unknown, is of the order of 1/(N + 1)

There is, then, an upper bound on the probability that God exists, given the evils found in the world. That upper bound, moreover, is extremely low, since of the billions of persons and sentient non-persons that have existed, the proportion that have never suffered seriously in ways where the known wrongmaking properties of allowing that suffering outweigh the known rightmaking properties is surely very small. Accordingly, N must be extremely large, and thus the probability 1/(N + 1) of there being an omnipotent, omniscient, and perfectly good person must be very low indeed, relative to the evidence in question.

Calum Miller's first two objections: an overview

In his ‘Introduction’, Miller briefly describes the first two objections that he will advance against that type of argument:

Miller's first objection is thus that I have failed to recognize that ‘the fact that an event has happened can, at least in principle, be good evidence that the permission of that event by God is morally permissible’. His second objection is that I have also failed to consider ‘the probability of theism given all the other evidence we have’. (ibid., 209–210)

Neither objection, I shall argue, is sound. As regards the first, I shall show that the logical form of the propositions involved in my argument – namely, that they involve either names or definite descriptions of actual events – itself entails that the occurrence of an event cannot, on its own, alter the probability that allowing the event in question is morally permissible.

I shall also argue, however, that the latter is the case independently of my argument, and that the contrary view arises because one is implicitly expanding the relevant evidence to include positive arguments for the existence of God.

The upshot is that the mere ‘fact that an event has happened’ cannot alter the probability that God's permitting the event in question is morally permissible: that requires, instead, the conjunction of the proposition that the event has happened together with a proposition asserting that there are good arguments for the existence of God. But I would certainly not question the claim that that conjunction can alter the relevant probability. Miller's first objection has, therefore, no force against my argument.

Miller's second objection is that ‘the extent to which the occurrence of a prima facie evil state of affairs confirms or disconfirms the thesis that the permission of that state of affairs would be morally permissible by God depends crucially on the probability of theism given all the other evidence we have’. (ibid., 210) In response, I shall show that it is not true that, in my argument from evil as a whole, I have failed to consider relevant evidence bearing upon the probability that God exists.

The unsoundness of Miller's first objection

Let us now consider how Miller develops his first objection. First of all, he refers in his argument to the following proposition:

– where ‘WMP’ is short for ‘wrongmaking property’, and ‘RMP’ is short for ‘rightmaking property’.

Miller's discussion begins as follows:

What is one to say about this? First of all, the argument from evil in question focuses upon an actual event: the Lisbon earthquake – call it L – and it involves two propositions, which can be put as follows:

p: The known rightmaking and wrongmaking properties of allowing L are such that those known wrongmaking properties outweigh the known rightmaking properties.

q: The weight of all the wrongmaking properties of allowing L, both known and unknown, taken together, is greater than that of all of the rightmaking properties of allowing L, both known and unknown, taken together.

I then offered a proof that the probability of q given p is greater than 0.5, and, in response, not only does Miller not object to any of the calculations involved in the proof, he also says that ‘Tooley's suggestions pertaining to the equiprobability of similarly weighted unknown RMPs and WMPs are not particularly objectionable’ (ibid.), where it is precisely the equiprobability principles in question that are at the heart of the proof, and that lead to the conclusion stated in proposition (2) above.

Second, given that the term ‘L’ in proposition p is a definite description referring to an actual event – the Lisbon earthquake – proposition p trivially entails the following proposition:

r: Event L is an event that actually occurred.

But if p entails r, then p is logically equivalent to (p & r), and this entails:

s: The probability of q given (p and r) = the probability of q given p.

So adding the proposition that the Lisbon earthquake actually occurred cannot, contrary to what Miller claims, affect the probability in question.

The upshot is that the logical form of proposition p itself – the fact that it contains a definite description referring to an actual event – shows that Miller's claim, namely, that adding to proposition p the proposition that the event in question actually occurred can change the probability that q is the case, cannot possibly be true.

What is the source of Miller's mistaken view?

What is one to make, then, of Miller's contention that learning that a certain type of event is actual – such as a ‘Lisbon earthquake’ type of event – can make a difference to the probability that the weight of all of the wrongmaking properties of allowing such an event, both known and unknown, taken together, is greater than the weight of all of the rightmaking properties of allowing such an event, both known or unknown, taken together? Clearly, Miller must be thinking in terms of a quite different type of argument from my evidential argument – apparently, some sort of argument in which one focuses upon the mere possibility of a certain type of event, so that it is left open whether that type event has ever occurred or ever will occur, and what Miller is claiming is that the probability that God would permit an event of that kind – for example, a ‘Lisbon earthquake’ type of event – depends on whether there has been an actual event of that kind.

Let us explore, then, that issue, starting with the following probabilistic statement of the relevant sort, and one that seems promising, since it is a statement that comes out true given my equiprobability-principle-based approach:

1. (1) P(It would not be wrong for God to allow an event of the ‘Lisbon earthquake’ sort) is less than 0.5.

Consider now whether things might change if one learned that there is an actual event of the ‘Lisbon earthquake’ sort. To do that, one needs to consider the corresponding posterior probability statement, which is also true given my equiprobability-principle-based argument from evil:

1. (2) P(It would not be wrong for God to allow an event of the ‘Lisbon earthquake’ sort/There is an actual event of the ‘Lisbon earthquake’ sort) is less than 0.5.

But then, given that Miller holds that ‘the fact that an event has happened can, at least in principle, be good evidence that the permission of that event by God is morally permissible’ (Miller Reference Miller2021, 210), and given that God would allow an event of the ‘Lisbon earthquake’ sort only if doing so were morally permissible, it would seem that Miller must hold that (2) does not follow from (1), and thus that (2) might very well be false even though (1) is true.

Why might someone think this? Consider a person who thinks the ontological argument is sound. Then that person, in thinking that they are considering (2), might unconsciously be importing the proposition that the ontological argument is sound, and thus really be considering instead:

1. (3) P(It would not be wrong for God to allow an event of the ‘Lisbon earthquake’ sort/(There is an actual event of the ‘Lisbon earthquake’ sort and the ontological argument is sound)) is less than 0.5.

Proposition (3), of course, is clearly false, since the conjunction (There is an actual event of the ‘Lisbon earthquake’ sort and the ontological argument is sound) entails that God allowed an event of the ‘Lisbon earthquake’ sort, so that rather than (3) being true, one has instead

1. (4) P(It would not be wrong for God to allow an event of the ‘Lisbon earthquake’ sort/(There is an actual event of the ‘Lisbon earthquake’ sort and the ontological argument is sound)) is equal to 1.

Notice, in addition, that if one shifts from the a priori probability statement (1) to the following, related, a posteriori probability statement (5) involving the proposition that the ontological argument is sound:

1. (5) P(It would not be wrong for God to allow an event of the ‘Lisbon earthquake’ sort/The ontological argument is sound) is less than 0.5.

one has a proposition that, like proposition (1), is true.

The upshot is that someone who thought that the ontological argument was sound might, in thinking that they were considering propositions (1) and (2), both of which are true, actually be considering propositions (5) and (3) respectively, and then, since (5) is true, while (3) is false, would mistakenly conclude that while proposition (1) is true, proposition (2) is false.

I have put things in terms of someone who thinks that the ontological argument is sound to simplify the discussion, but the point is a perfectly general one. As long as one thinks that there is some argument or other for the existence of God that raises the probability that God exists above its a priori probability, the acceptance of that view may very well result in one's wrongly concluding, as Miller does, that the mere occurrence, on its own, and not conjoined with any argument in support of the existence of God, of an event of the type in question – such as a Lisbon earthquake type of event – makes a difference to the probability of its being wrong for God to allow such an event. It is thus only when one unconsciously adds on a proposition to the effect that there is some sound argument in support of the existence of God that the probability that it was wrong for God to allow an event of the type in question is altered, with how great a difference it makes depending upon the strength of the support that the positive argument in question provides for the proposition that God exists.

Miller's second objection

Let us turn, then, to Miller's second objection, which he describes at the very beginning of his article:

Let p now be the proposition that there are N events, each of which, judged simply by its known rightmaking and wrongmaking properties, it would be morally wrong to allow. It follows from my argument that the probability that God exists, given only proposition p, is of the order of 1/(N + 1), whereas given some sound theistic argument A, the probability that God exists – call it k – might instead be very high indeed, and, in any case, certainly much greater than 1/(N + 1). But then, surely it is reasonable to believe that the probability that God exists, given both proposition p and argument A, will presumably lie between k and 1/(N + 1), and could very well be greater than one half.

Moreover, as Alvin Plantinga and other philosophers have maintained (Walls & Dougherty Reference Walls and Dougherty2018), there are many arguments for the existence of God, and as more and more of these are taken into account, surely the probability that God exists will increase, moving further and further away from the value 1/(N + 1), thereby making it ever more plausible that the probability that God exists is ultimately greater than one half, thereby entailing that atheism is not an acceptable view.

Before addressing this argument, however, one should also note another, very closely related point that Miller makes, one that arises in connection with the following, initial premise in my argument:

In discussing this premise later, Miller says,

Miller's second objection is thus that there are two types of relevant considerations that I have either ignored or failed to address in anything like a satisfactory way: first of all, positive arguments in support of the existence of God, and second, possible theodicies that, if sound, show that what initially appear to be prima facie evils turn out to have counterbalancing rightmaking properties.

As we shall see shortly, neither charge is justified, and Miller's second objection is therefore unsatisfactory. Before turning to that, however, let us consider what conclusion Miller thinks one should draw, and the scope of that conclusion.

The conclusion Miller draws from his second objection

What conclusion should be drawn given Miller's contention that I fail to consider either positive arguments for the existence of God or the probabilities of the many theodicies that have been advanced? If one considers Miller's extended formal discussion on pages 216–221, it is evident that his view is that a much more complicated probabilistic argument is needed than what I offered.

Thus, for example, in formulating ‘a simple Bayesian argument’ (Miller Reference Miller2021, 221) that someone might propose in attempting to rescue my argument, Miller discusses the following equation, where ‘T’ is short for ‘theism’, ‘L’ for ‘The Lisbon earthquake occurred’, and ‘W’ for ‘It would be wrong for God to permit a Lisbon earthquake’ (ibid., 217):

$$\displaystyle{{{\rm P}( {{\rm T\vert L}} ) } \over {{\rm P}( {\sim {\rm T\vert L}} ) }} = \displaystyle{{{\rm P}( {{\rm L\vert T}} ) } \over {{\rm P}( {{\rm L\vert }\sim {\rm T}} ) }} \times \displaystyle{{{\rm P}( {\rm T} ) } \over {{\rm P}( {\sim {\rm T}} ) }}$$

As this equation illustrates, Miller thinks that one needs a probabilistic argument incorporating such things as the a priori probability P(T) that theism is true – something that my probabilistic argument certainly fails to provide. Moreover, as the following passage makes clear, Miller thinks that doing that alone will not be sufficient, since one will also need to consider ‘independent evidence’ for the existence of God:

Miller's contention, then, is that at the very least, my type of probabilistic argument will need to incorporate values for the posterior probabilities that theism has given each of the many arguments for the existence of God, and then, first of all, one will have to find some way of arriving at the a posteriori probability that God exists given the totality of those arguments, and second, one will also need to find some way of arriving at the overall probability that God exists, given the combination of the probability that the existence of God has given those arguments and the probability that the non-existence of God has relative to my equiprobability-based argument.

A daunting task indeed! So, if Miller's objection is right, anyone who tries to resurrect the type of argument from evil that I set out is embarking on an extremely arduous intellectual journey.

The scope of Miller's second objection

The only evidential argument from evil that Miller considers is my equiprobability-principle-based argument. But if the conclusion just described is correct, could not Miller equally well argue that Humean-style evidential arguments – advanced by Paul Draper (Reference Draper1989, Reference Draper, Copan and Meister2008, Reference Draper, Flint and Rea2009, Reference Draper, McBrayer and Howard-Snyder2013), and by Wes Morriston (Reference Morriston, Dougherty and McBrayer2014) – suffer from the same weakness? For do they not also fail to assign any evidential weight to arguments for the existence of God, or to theodicies that many believe suffice to show that all the evils in our world are logically necessary, either for goods we are aware of, or to avoid possible evils we can foresee? Consequently, in failing to take these things into account, are not those Humean-inspired arguments also unsound, violating as they do the requirement that the probability ultimately assigned to any proposition be based upon the total evidence available?

Response to Miller's second objection: a three-part structure for evidential arguments from evil

What, if anything, is wrong with Miller's second objection to my argument from evil? The answer is that Miller fails to discuss what I have said, first of all, about the overall structure of a complete and satisfactory argument from evil, and then, second, both about possible theodicies and about arguments in support of the existence of God.

Thus, as regards the former, in Tooley (Reference Tooley, Chandler and Harrison2012), I said that three tasks are involved:

In short, I certainly hold that in formulating an evidential argument from evil, one must determine whether either possible theodicies or arguments in support of the rationality of believing in the existence of God undermine the evidential argument from evil in question. Miller needed to consider, then, what I had said on those two matters, but failed to do so.

In the case of the Knowledge of God debate volume – which contains the only discussion of mine to which Miller refers – I did say relatively little as regards either the first or third tasks. This was especially so in the case of theodicies – but not surprisingly, since although Plantinga, who had previously been deeply sceptical of theodicies, had come to accept a religiously based, supralapsarian (‘O Felix Culpa’) theodicy at that time (Plantinga Reference Plantinga and van Inwagen2004), he made no appeal to that theodicy in our debate.

Then, in the case of arguments in support of the existence of God, I focused mainly on offering arguments against Plantinga's view that belief in the existence of God is non-inferentially justified (‘properly basic’) (Tooley and Plantinga Reference Draper, Copan and Meister2008, 241–246), although I also discussed arguments claiming to show that belief in the existence of God was inferentially justified. As regards the latter, I started by noting that ‘most arguments for the existence of a deity do not provide grounds for holding that the deity in question is even morally good, let alone morally perfect, and so they provide no counter at all to the argument from evil’ (ibid., 246).

That observation, moreover, is not an unfamiliar one, since other proponents of evidential arguments from evil have made the same point. Thus, as regards pro-theistic arguments, Paul Draper points out that

I then briefly addressed the ontological argument – which Plantinga had previously extensively defended – and there I raised two objections, including that the ontological argument can be exactly paralleled to show that there is an omnipotent, omniscient, and perfectly evil being, thereby generating a contradiction (Tooley and Plantinga Reference Draper, Copan and Meister2008, 246–247).

Finally, I also commented briefly on arguments from religious experiences and from miracles. As regards the former, I considered (ibid., 244–246) the idea that religious experiences can provide the basis for one's being non-inferentially justified in believing in the existence of God, and here my basic point was that equally thoughtful and well-informed people who have grown up in very different religious cultures – for example, Hindu and Christian – have religious experiences that support incompatible religious beliefs, and thus that there is good reason for concluding that such experiences are not veridical. I did note, however, that it might be claimed that mystical experiences differ in this regard, and thus that further discussion would be needed of an appeal to such experiences (ibid., 246).

Finally, there is the crucial case of miracle claims, where I referred to several careful studies that have been carried out by writers such as A. D. White (Reference White1896, chapter 13, part 2), Louis Rose (Reference Rose1971), William Nolen (Reference Nolen1974), and others, which I claimed provide excellent reasons for concluding that it is extremely unlikely that miracles actually occur.

Much more recently, in my book The Problem of Evil – whose publication in 2019 may well have been after Miller finished writing his article – I discussed arguments from miracles at greater length, citing many more relevant studies, including scientific experiments, as well as arguments from religious experiences – especially mystical experiences – and offered reasons for holding that neither provide support for the existence of God (Tooley Reference Tooley2019, 34–35).

My view, then, is that there are no arguments that raise the probability of the existence of God above its a priori probability. But what is its a priori probability? In the Knowledge of God book, I began by focusing on just three possibilities concerning the moral character of an omnipotent and omniscient deity – namely, perfect goodness, perfectly malevolence, and complete indifference to good and evil – and I argued that those properties should be treated as a priori equally probable, which entails, even leaving aside the a priori improbability of any kind of omnipotent being, that the a priori probability that God exists cannot be greater than 1/3. I then went on to say,

Now I did not assert at that point that I believed that that argument is correct, but in my later book The Problem of Evil I advanced, as my first equiprobability principle, the following principle:

This principle implies that the a priori probability that God exists is infinitesimal, so if I am right in claiming both that no arguments for the existence of God raise the probability of God's existence above its a priori probability, and that the existence of God cannot be non-inferentially justified to any degree, then it follows that the a posteriori probability that God exists is also infinitesimal.

Turning next to theodicies, and their relevance to the probability that God exists, in the online Stanford Encyclopedia of Philosophy article ‘The Problem of Evil’, posted in Reference Tooley2015, I discussed at length five of the most common types of theodicies. Then, more recently, in The Problem of Evil (Reference Tooley2019), I offered detailed objections to several theodicy themes, including (1) claims that no non-human animals are conscious (10), (2) appeals to the doctrine of karma and reincarnation (11–15), (3) John Hick's ‘soul-making’ theodicy (15–16), (4) contentions that both natural disasters and suffering by non-human animals are caused by supernatural beings exercising free will (16), and (5) claims that undeserved suffering resulting from natural evils are justified by the need for laws of nature (18–23)

In short, my view is that if one can show, first, that the probability of any theodicy ever proposed is virtually zero, and then secondly, that no argument raises the probability that God exists above its a priori probability – which the Families of Properties Equiprobability Principle entails is infinitesimally close to zero (Tooley and Plantinga Reference Draper, Copan and Meister2008, 92; Tooley Reference Tooley2019, 52) – then there is no need to complicate any argument from evil in a massive fashion by introducing, as Miller would have one do, probabilities concerning theodicies and pro-theistic arguments.

The upshot is that as regards, for example, the Lisbon earthquake, I hold that the following claims are justified:

1. (1) The logical probability that God exists, given only the following propositions, is less than one-half:

   1. (a) Not preventing the Lisbon earthquake has a known wrongmaking property, and no known counterbalancing rightmaking properties.

   2. (b) No argument for God's existence significantly raises the probability of God's existence above its very low a priori probability.

   3. (c) Belief in the existence of God is not non-inferentially justified.

I then implicitly assumed – mistakenly as we shall see – that aside from propositions concerning known evils other than the Lisbon earthquake that have no known counterbalancing rightmaking properties, (a), (b), and (c) cover everything that is relevant to the probability that God exists, and thus I concluded:

1. (2) The logical probability that God exists, given all of the relevant evidence, other than propositions about states of affairs other than the Lisbon earthquake whose known wrongmaking properties outweigh their known rightmaking properties, is less than one-half.

Finally, I generalized this conclusion by developing probabilistic arguments for the case of N evils, where the known wrongmaking properties of allowing each of those evils outweigh the known rightmaking properties of doing so. The quantitative results then depended upon the choice between a state-description equiprobability principle and a structure-description equiprobability principle, with the probability of God's existence being higher given the second of those principles, where it is of the order of 1/(N + 1). Thus my more general conclusion was this:

1. (3) The logical probability that God exists, given only the following propositions, and given a structure-description equiprobability principle, is of the order of 1/(N + 1), and it is even lower given a state-description equiprobability principle:

   1. (a) There are N events in the world, the allowing of any one of which possesses a known wrongmaking property and no known rightmaking properties that are counterbalancing.

   2. (b) No argument for God's existence significantly raises the probability of God's existence above its very low a priori probability.

   3. (c) Belief in the existence of God is not non-inferentially justified.

I then implicitly assumed, once again, that aside from propositions concerning evils other than the N evils in question that also have no counterbalancing rightmaking properties, (a), (b), and (c) cover everything that is relevant to the probability that God exists, and accordingly I concluded:

1. (4) The logical probability that God exists, given all of the relevant evidence, and given a structure-description equiprobability principle, is of the order of 1/(N + 1), and it is even lower given a state-description equiprobability principle.

Miller's third objection: the unsoundness of a structure-description approach to logical probability

In his third objection, Miller does not challenge the mathematical correctness of the calculations that I set out either in the case of a single evil or in that of multiple evils. What Miller does is instead offer an argument to show that an approach to logical probability based on a structure-description equiprobability principle is unsound, and his objection focuses upon 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:

$$\displaystyle{{( {1, \;000 + 63\;-\;1} ) !} \over {( {1, \;000) !\;( 63\;-\;1} ) !}} = \displaystyle{{( {1, \;063} ) \ldots ( {1, \;001} ) } \over {( {63} ) \ldots ( 1 ) }}$$

Miller then sets out a calculation showing that the proportion of the total structure-descriptions that contain only preserver Q-predicates is about 1 in 16. This implies that in the world described, the probability that God exists is very low, contrary to the claim that it is extremely clear that people in the world in question ‘should not be able to generate an inductive argument from evil against the existence of God’.

The unsoundness of Miller's argument against a structure-description approach to logical probability

Miller's scenario is an unusual one, in that it presupposes that one has quite detailed knowledge about the number of unknown rightmaking and wrongmaking properties and their relative weights. Still, it is logically possible that one might, through communication with an omniscient being, for example, have such knowledge.

What, then, if anything, is wrong with Miller's argument? The answer to this question requires, perhaps surprisingly, but as we shall now see, consideration of the metaphysics, and the resultant epistemology of, laws of nature.

The place to begin is with the fact that the world Miller describes is ‘populated only with prima facie good states of affairs’, of which there are 1,000 (Miller Reference Miller2021, 225). Given all the types of evils one can imagine, and of the unlimited magnitude that it is logically possible for familiar wrongmaking properties to have, surely it is a remarkable fact that all 1,000 of the independent states of affairs in the imagined world are prima facie good. Is it not plausible that there should be some explanation of this otherwise very improbable fact?

What form might such an explanation take? One possibility would be the existence of an omnipotent and omniscient being who was morally perfect, or at least a close approximation thereto. Alternatively, there could be a law of nature, somewhat akin to the Hindu and Buddhist doctrine of karma, which ensures that the good-making properties of every state of affairs outweigh its bad-making properties, if any.

Must there then not be a legitimate inductive inference from the fact that the good-making properties of each of the 1,000 independent states of affairs outweigh its bad-making properties to the likely existence of a law of nature that this is so for all states of affairs? But if there is such an inference, the resulting law is incompatible with Miller's quantitative conclusion, so must not Miller's calculation be unsound?

It is at this point that the metaphysics and the resulting epistemology of laws of nature become relevant. The great philosophical divide concerning laws of nature is between, on the one hand, reductionist views according to which non-probabilistic laws of nature are either simply cosmic regularities, or else cosmic regularities that satisfy some further constraint, such as the ‘best theory’ constraint proposed by David Lewis (Reference Lewis1973, 72–77), and, on the other hand, the non-reductionist view defended by Fred Dretske (Reference Dretske1977), David Armstrong (Reference Armstrong1983), and Tooley (Reference Tooley1977), according to which laws of nature involve second-order relations between first-order properties and relations.

But how is this metaphysical divide relevant to Miller's third objection? The answer lies in the dramatically contrasting epistemological implications of the two different approaches to laws of nature. As regards the reductionist approach, according to which laws of nature are basically cosmic regularities, possibly satisfying some further condition, Rudolf Carnap proved in the appendix of his Logical Foundations of Probability (Reference Carnap1950 and 1962, 570–571) that given a structure-description approach to logical probability, one cannot confirm the existence of any non-probabilistic laws of nature that have an unlimited number of instances.

Carnap's argument is quite complex, but here is a simple way of arriving at that result. First of all, the assumption that all structure-descriptions are a priori equally probable entails every Laplacean-style rule of succession, as Carnap demonstrated (Reference Carnap1950 and 1962, 565–571). The simplest such rule is a restricted form of Laplace's first rule of succession, which, properly stated, says that if n events of type C have all been followed by events having property P, where P belongs to a family of properties containing only two members, then the probability that the next event of type C will be followed by an event with property P is equal to ${{n + 1} \over {n + 2}}$. It then follows that the probability that the next m events of type C will all have property P is equal to ${{n + 1} \over {n + 2}}$ × ${{n + 2} \over {n + 3}}$ × ${{n + 3} \over {n + 4}}\;\times \;\ldots \;\;\times \;{{n + m-1} \over {n + m}} \times {{n + m} \over {n + m + 1}}$. The denominator of each fraction then gets cancelled along with the numerator of the next fraction, giving one that the probability in question is ${{n + 1} \over {n + m + 1}}$ , which, once m is greater than (n + 1), will be less than one-half, and which will also approach zero as m increases without limit.

In contrast, given a non-reductionist view of laws of nature, according to which laws are atomic states of affairs involving second-order relations between first-order properties, rather than an infinite conjunction of atomic states of affairs, it can be shown – though the arguments in question are far too lengthy to be given here – both that confirmation of laws of nature is possible, and that, in particular, the probability, in a world containing 1,000 prima facie good states of affairs that even one of them would have the single unknown bad-making property in question is extremely low. As a consequence, there is no incompatibility with the intuition, concerning that world, that ‘people in this world should not be able to generate an inductive argument from evil against the existence of God’ (Miller Reference Miller2021, 225–226).

The upshot is that Miller's third objection does not have any force against a structure-description equiprobability approach if one adopts the non-reductionist view that laws of nature are atomic states of affairs involving second-order relations between first-order properties.

What Miller's argument does show

The failure of Miller's argument notwithstanding, there is something very valuable that emerges from it, namely, that equiprobability-based arguments from evil cannot be based simply on the known evils in the world: the known goods that exist must also be considered, as is shown clearly by the following scenario, which, unlike Miller's, does not presuppose that humans have any knowledge at all about unknown rightmaking and wrongmaking properties, let alone that those properties are ‘balanced in favour of good’ (Miller Reference Miller2021, 226).

Consider a world with 1,000,000 known, intrinsically evil states of affairs E₁, E₂, . . . E_1,000,000, along with a million known, intrinsically good states of affairs G₁, G₂, . . . G_1,000,000, such that for every i from i = 1 to i = 1,000,000, E_i is logically necessary for G_i, and each combination of G_i and E_i is better than the absence of both. Suppose, also, that there is one more, and only one more, serious evil, E_1,000,001, where there is no known good G_1,000,001 for which E_1,000,001 is logically necessary, and that outweighs E_1,000,001. What is the probability, if there is an omnipotent and omniscient being, that that being is perfectly good?

Evils E₁, E₂, . . . E_1,000,000 do not themselves provide evidence for the conclusion that if there is an omnipotent and omniscient being, then that being is evil, since each is logically necessary for a known good that outweighs it. Consequently, the only negative evidence is that provided by evil E_1,000,001. Then, however, if my argument in the case of a single evil were sound, E_1,000,001 on its own would suffice to make the probability that God exists less than one-half. But that conclusion, surely, is not at all plausible. For setting E_1,000,001 aside, and considering only the million good states of affairs G₁, G₂, . . . G_1,000,000 and the million bad states of affairs E₁, E₂, . . . E_1,000,000, and the logical relations between the corresponding states of affairs, if there were an omnipotent and omniscient deity, what moral dispositions would it be reasonable to assign to that deity? Would they not include a disposition M to allow the existence of a seriously bad state of affairs only when there will be some good state of affairs that outweighs it, and for which the bad state of affairs is logically necessary? But if it is highly likely, given that evidence, that any omnipotent and omniscient being who exists has disposition M, surely it is eminently reasonable to believe that the bad state of affairs E_1,000,001 would not exist unless there were some counterbalancing good G_1,000,001, for which E_1,000,001 was logically necessary, and thus that, since, by hypothesis, we humans know of no such good, it must therefore lie outside our ken. Consequently, the probability that God exists, rather than being less than one-half, is presumably very high indeed. Thus my argument for the initial conclusion that, given a single evil, the probability that God exists is less than one-half, must be not only unsound but enormously so.

This alternative scenario therefore shows that my original three-part argument from evil cannot be a complete, evidential argument from evil against the existence of God, since it does not enable one to arrive, via an inductively sound inference, at a sound conclusion about the probability that God exists given all the relevant evidence.

An equiprobability-principles-based evidential argument from evil must involve four parts

Can my argument be expanded to deal with this problem? The answer is that it can be, and in a way paralleling that used to deal with theodicies and arguments for the existence of God. Thus, just as one can examine all known theodicies and pro-theistic arguments, and argue that none of them is satisfactory, so one can construct a list of the circumstances that make the move from the existence of a single evil in the world to the conclusion that it is unlikely that God exists unsound, and then see whether any such circumstances obtain.

Consider, then, the case described above. What makes the relevant inference unsound? The answer is that the existence of the 1,000,000 known, intrinsically evil states of affairs E₁, E₂, . . . E_1,000,000, along with 1,000,000 known, intrinsically good states of affairs G₁, G₂, . . . G_1,000,000, such that for every i from i = 1 to i = 1,000,000, E_i is logically necessary for G_i, and the combination of G_i and E_i is better than the absence of both, supports a strong abductive inference to the conclusion that any omnipotent and omniscient being that exists must be very strongly disposed not to allow the existence of an intrinsically bad state of affairs unless the latter is logically necessary for a good state of affairs that outweighs it. This then enables one to conclude that the bad state of affairs E_1,000,001 is very probably accompanied by some good state of affairs that outweighs it, and for which it is logically necessary, and thus that, since we humans do not know of any such good state of affairs, it must be one of which we have no knowledge.

The reasoning in this case could also be viewed, however, as based on a rule of succession stating that if in all n cases where a state of affairs of type C is known to have existed, it was accompanied by a state of affairs of type D, then in the next case where a state of affairs of type C is known to exist, there is a probability p – which depends upon n – that that state of affairs will be accompanied by a state of affairs of type D. In the case being considered, however, one has 1,000,000 instances where a bad state of affairs was accompanied by a good state of affairs that outweighed it, and for which the bad state of affairs was logically necessary. The rule of succession then allows one to conclude that it is very likely indeed that there is a good state of affairs G_1,000,001 that outweighs the bad state of affairs E_1,000,001, and for which the latter is logically necessary.

The conclusion, accordingly, is that inferences involved in my type of evidential argument from evil can turn out to be inductively unsound in at least two ways. One is that the known good states of affairs in the world, together with any known bad states of affairs that are logically necessary for those good states of affairs, may fall under a rule of succession according to which it is likely that any known bad state of affairs is logically necessary for some good state of affairs, either known or unknown, that outweighs the bad state of affairs in question.

The other way that such inferences may be inductively unsound is if the known good states of affairs in the world, and any known bad states of affairs that are logically necessary for those good states of affairs but outweighed by the latter, may provide the basis for an abductive inference to the conclusion that any omnipotent and omniscient being that exists will be morally perfectly good, or at least very nearly so.

Are there types of inferences other than the two just described that would make the inferences in question unsound? Perhaps, but if so, they would merely need to be added to the list of possibilities that must be considered before any move to a conclusion concerning the probability that God exists given all the relevant information is justified.

The actual world, and part 4 of an equiprobability-principles-based argument from evil

How are things in the actual world? Some evils in our world, such as undeserved, but mild insults, or failures to express gratitude, are relatively trivial and are also such that it is far from clear that intervening to prevent such occurrences would make the world a better place, given that it would thereby deprive people of important knowledge about the moral character of other people, unless humans were supplied with some non-sensory way of acquiring such knowledge, and the goodness or badness of such a combination of changes is surely at best very murky. Let us use the expression ‘serious evils’ to cover, then, cases where intervention to prevent those evils would clearly, other things being equal, make the world a better place.

Consider, then, the serious evils one finds in the world – including, first of all, natural evils such as pandemics causing grave illnesses and many deaths, hurricanes, volcanic eruptions, starvation due to famines, the painful deaths of animals due to predators, and so on, and, second, serious moral evils, including acts of torture, murders, rapes, brutal beatings, slavery, etc. In cases of such serious evils, how often is there some known good state of affairs that outweighs the evil, and for which the evil was logically necessary? Unless certain religious theodicies can be shown to be justified, the number would seem vanishingly small, if there are any at all.

Consequently, the two types of inductive inferences that were justified in the imaginary case described above – namely, an abductive inference to an omnipotent and omniscient person who was perfectly good, or at least very nearly so, and an inference via a rule of succession to the conclusion that a serious known evil that is not logically necessary for any known good is likely to be accompanied by an unknown good that outweighs the evil, and for which the latter is logically necessary – are not available in the actual world.