Arguing with Bayes's theorem

Swinburne's argument for the existence of God is inductive. It thus does not belong to logic in the strict (deductive) sense. Nonetheless, he places great emphasis on its rational, well-reasoned character and on following the rules of (valid) inductive inference. These rules stem from Bayesian confirmation theory. Principally, a Bayesian approach to confirmation is quantitative; this means that to confirm a hypothesis, its probability must be raised. Swinburne (Reference Swinburne1973) discusses this subject in greater detail. One might disapprove of the notion of quantitative confirmation as opposed to a rather qualitative approach – a position endorsed by Dawes (Reference Dawes2009), for instance. However, this fundamental issue is not my point here. For now, I take Bayesianism for granted and focus on what is required to provide a sound Bayesian-style argument.

Bayes's theorem is believed to describe epistemic relations between propositions (Prevost (Reference Prevost1985), 175), namely, between a particular hypothesis and propositions describing evidence in support of it. Basically, it tells us that the probability of a hypothesis h being true based on evidence e is equal to its prior probability (its unconditional probability before the evidence is taken into account), multiplied by the likelihood of the hypothesis in light of the evidenceFootnote ² divided by the prior probability of the evidence.Footnote ³ Stated formally:

$${\rm P(}h|e\& k) = \displaystyle{{P(e|h\& k)P(h\& k)} \over {P(e\& k)}} = \displaystyle{{P(e|h\& k)P(h|k)} \over {P(e|h\& k)P(h|k) + P(e| \sim h\& k)P( \sim h\& k)}}$$

where h represents the hypothesis in question, e is our evidence, and k refers to background knowledge. Bayes's Theorem is a consequence of our common definition of conditional probability and thus, as a mathematical theorem, it follows directly from the basic ideas of probability calculus. Probability calculus, like every modern mathematical theory, is formally described by a set of axioms which define the structural properties of what may be called ‘probability’. According to the Kolmogoroff Axioms, probability is a normalized and additive measure. That means that probabilities lie between 0 and 1 and the probabilities of disjoint (mutually exclusive) events or hypotheses may be summed up.Footnote ⁴ I stress this point because we use the terms ‘probable’ and ‘probability’ in different and multiple ways in ordinary language, and we are perfectly warranted in doing so. However, if we want to calculate with probabilities, we must employ the term in a strict mathematical and not a loose ordinary sense.

Considering the probability of theism

In The Existence of God, the hypothesis h in question is that God exists. Swinburne has worked out a more detailed but supposedly equivalent definition of h, as well; thus we must keep in mind two explications of h which I label ‘h _min’ and ‘h _max’:

Swinburne forms his single arguments by considering eleven domains of empirical evidence e, starting from very general features of the world and ending up with more specific ones. The first two arguments are actually reformulations of the classical cosmological and teleological arguments. This evidence is sequentially inserted into Bayes's theorem, thereby becoming background knowledge for subsequent examination of his more specific arguments. Swinburne's aim is to show that the sum of our cumulative evidence favours God's existence. Thus by going through all the single domains, we should arrive at P(h|e&k)>0.5. This is called by Swinburne a ‘P-inductive argument’, whereas the single arguments, taken by themselves, merely form ‘C-inductive arguments’.Footnote ⁶ As such, they simply raise the probability of h, regardless of its actual value. Stated formally it reads as P(h|e&k)>P(h|k). Following Swinburne's argumentation, a valid P-inductive argument is a sound criterion for accepting a hypothesis. If it is more probable for a hypothesis to be true than to be false, it is rational to believe it, since P(h|e&k)⩾0.5 implies that P(∼h|e&k)⩽0.5.

Epistemic probabilities: what are they all about?

Now let us turn to the core questions of how probabilities are interpreted by Swinburne and on what grounds he calculates their values. In the first edition of The Existence of God (1979), no concrete numerical values are given; rather we find qualitative and comparative pronouncements, such as ‘very low’, ‘not too low’, and ‘much lower than’. Swinburne (Reference Swinburne2004, 131, 339) gives at least some values, but hastily admits that they might be ‘artificially precise’. So, just what is meant by saying that hypotheses have prior probabilities or probabilities based on evidence? A few more comments on the concept of probability may be useful before going into more detail.

Basically, Swinburne is concerned with epistemic or logical probability, a notion which goes back to Carnap (Reference Carnap1950). It is meant to be a measure of ‘how probable q makes p’, that is, ‘[a way] to assess whether and (within rough limits) how much some evidence supports some hypothesis’ (Swinburne (Reference Swinburne2004), 15). Yet, Swinburne also tells us that ‘[p]robability on evidence, or epistemic probability, may be ascribed to propositions of all kinds – moral, aesthetic, and theological propositions, as well as historical and scientific ones’ (Swinburne (Reference Swinburne1973), 5). But surely this claim does not follow from mathematics; rather, it constitutes a philosophical doctrine which is not at all uncontroversial (Sober (Reference Sober and Swinburne2005a), 21).

Presumably, the practical consequence of postulating such a wide scope of epistemic probability statements should be to establish rules or criteria that tell us how to generate their values, but Swinburne is quite reluctant on this issue. One concrete example given by Swinburne (Reference Swinburne2004, 16) is actually based on statistical data: ‘It is an a posteriori matter whether, in 1,000 tosses, 505 have landed heads; but an a priori matter whether that evidence gives a[n] [epistemic] probability of 0.505 to the next toss landing heads.’ Although, by all means, it is perfectly sound that ‘statistical data are of crucial importance for calculating epistemic probabilities’ (Swinburne (Reference Swinburne1973), 13), this rule is not of much help when it comes to assigning probabilities to non-statistical hypotheses, such as the claim that there is a God. We need something more to bridge the gap between quantitative (statistical) and purely qualitative hypotheses. Since probability is a quantitative measure, you either have to feed in quantitative data, or you need to provide rules that define how to transform qualitative assignments into quantitative values.

Apparently, Swinburne seems to be aware of this issue; in a more recent publication, he very briefly suggests a somewhat statistical interpretation of epistemic probability based on a possible-world semantics. At this point, he states that the prior probability of some piece of evidence should be regarded as ‘the sum of the probabilities of its occurrence in the different possible world states of h being true and ∼h … being true’ (Swinburne (Reference Swinburne and Swinburne2005), 10).

What shall we make of this suggestion? To be sure, possible worlds are primarily thought of as a logical construct to provide semantics for the central concepts of modal logic – namely, logical possibility and logical necessity. However, it seems doubtful that these semantics can be transferred to probability calculus. In other words, do possible worlds constitute an adequate model for thinking about epistemic probabilities? Apparently, in order to base epistemic probabilities on possible-world states, precise knowledge about them is needed. More technically, we are looking for a mathematical definition of the space of possible-world states that meets the criteria of additivity and normalizability.

I envisage two interrelated problems here. First, when we operate with mathematical probabilities, we need a definition of basic or elementary events. By convention, elementary events constitute the ‘atoms’ of our probability space and are therefore mutually exclusive or disjoint. The condition of normalizability requires that the probabilities of all elementary events in sum add up to 1. However, Swinburne never provides an explicit definition of ‘elementary events’ within his possible-worlds framework. It would seem to me that we need a way of describing ‘bedrock properties’ of possible-world states, but it is hard to imagine on which grounds this could be feasible.

Second, there is probably no way to define a finite amount of possible-world states. By convention, logical possibility means lack of contradiction. Thus, the set of possible-world states is constrained by the single core requirement of consistency, which means that propositions describing matters of fact must not be self-contradictory. In addition, possible worlds may be constrained by language and imagination. But these latter criteria are quite subjective and pliable, and they fail to impose sufficient constraints. Certainly, if we use a language as complex as our ordinary one, and if we presume brains to be as complex as they are, the number of possibilities soon becomes tremendous. The vast amount of science fiction literature may provide evidence for this claim. From a strictly logical point of view, we may imagine a number of possible-world states that is not just tremendous but infinite. Let us assume that, from any given world-state, we could always create a new state by adding one more discrete element (be it some property or event). In this way, it would never be possible to reach an upper limit, which means that the number of possible-world states is infinite. However, infinity has precarious consequences: either, the probabilities of the single possible-world states decline to 0 (at least if we assume a normal distribution as the basis), or the sum of all probabilities is greater than 1 (it is actually infinite in this case).Footnote ⁷

Likelihoods and explanation

The crucial issue of this article concerns the precise definition of likelihoods. Generally, likelihoods tell us about the conditional probabilities of the evidence we actually observe to occur, given the hypothesis in question is true. Hence, the term P(e|h&k) expresses how likely is it that evidence e would exist, assuming that the hypothesis h (‘God exists’) is true. Thus, the existence of God somehow forms an explanation for e, or as Swinburne further explains:

The vital point here is that Swinburne frames epistemic probability in terms of a causal relationship. To state that there is some likelihood of a hypothesis h being true in light of evidence e is another way of saying that we can draw a causal inference – either a partial or a full one – from h to e. Since likelihood, as a probability, is stated on a continuous scale from 0 to 1, this gives us in turn a continuous transition from inductive (P(e|h&k)<1) to deductive (P(e|h&k)=1) inference, and correspondingly, from partial to full causal explanation. Thus, stating a significant likelihood of h conditionally on e is justified if and only if h provides some causal explanation for the existence of e. In our case, the causal relationship is secured by an action of God.

Note that Swinburne briefly suggests an alternative formulation of the hypothesis h and, consequently, an alternative version of the whole argument. We could add the single intentions of God to the hypothesis of his existence, either one by one, which gives us a slightly different hypothesis in every single argument, or as a whole. In the following discussion, I will call the compound hypothesis of God's existence, together with the set of his various intentions, the ‘extended’ hypothesis.

If the hypothesis h, together with some intention of God, provides a ‘full explanation’ of e, this means that we may draw a deductive inference from God's existence together with this intention to e. Further, it follows that the corresponding likelihood-factor, that is, the conditional probability of the joint hypothesis of God's existence, plus his intention, based on a certain piece of evidence, takes the value of 1. Formally stated,

$$[(int_e \& h) \to e] \to [{\rm P}(e|int_e \& h) = 1]$$

where ‘int _e’ stands for ‘God has the intention to bring about e’.

Yet, God is certainly omnipotent – ‘God's intention to bring about δ is [deductively] followed by δ’ (Swinburne (Reference Swinburne2004), 108) – and that implies that we omit h from the formula above, while the likelihood of int _e on e still equals 1:

$$[(int_e ) \to e] \to [{\rm P(}e|int_e ) = 1]$$

One straightforward consequence is that since e is replaceable for int _e, the value P(e|h&k) not merely depends on, but equals the probability that God will have the particular intention:

$${\rm P(}e|h\& k) = {\rm P(}int_e |h\& k)$$

But this (valid!) substitution poses some problems for Swinburne's argumentation. First, it seems to diminish the empirical content of his argument. Whereas the term P(e|h&k) directly refers to some empirical evidence e, there is just an indirect reference to the empirical sphere in P(int _e|h&k). Very obviously, assessing the value of P(int _e|h&k) falls in the realm of theology, while reflecting on P(e|h&k) at least appears to be in part an empirical issue. The empirical part of the argument is thus reduced to weighing the likelihood of e on the sum of the alternative hypotheses. If we substitute P(e|h&k) for P(int _e|h&k), the Bayes's theorem reads as follows:

$${\rm P(}h|e\& k) = \displaystyle{{P(int_e |h\& k)P(h|k)} \over {P(int_e |h\& k)P(h|k) + P(e| \sim h\& k)P( \sim h\& k)}}$$

Second, a principled objection arises which goes back to Mellor (Reference Mellor1969) in his discussion of the design argument of R. F. Tennant:Footnote ⁸ without having experience of an agent's action, how are we to know about his intentions a priori? Mellor's point is that to assess the probability that particular (real-world) evidence has been brought about by a designer, we must have an idea of what kind of action we would expect the designer to perform before we come to inspect the evidence. In other words, knowledge about his preference order must be given independently of and prior to the actual state of affairs.

More recently, Dawes (Reference Dawes2009) has framed the objection in terms of theological scepticism. In accordance with Swinburne (Reference Swinburne2004, 72), Dawes (Reference Dawes2009, 45) emphasizes that a full theistic explanation has to comprise particular divine goals or intentions, yet a theological sceptic denies that it is possible to have access to God's intentions. Sober (Reference Sober2009, 141–147) endorses this view in his criticism of design arguments, which essentially aims to prove that the design hypothesis is unscientific because it is not independently testable. I do not want to comment on Sober's account of scientificity in general, but let us cast a glance at what it means for a hypothesis to be independently testable. Sober stresses that the design hypothesis has empirically observable consequences only if it is conjoined with auxiliary assumptions about the goals and abilities of a putative designer. Now Sober's (ibid., 145) crucial point is, like Mellor's, to propose a constraint on introducing auxiliary assumptions. ‘Proper’ auxiliary assumptions should be justifiable without assuming that the hypothesis in question is true and without reference to the empirical evidence we actually observe. That means, to avoid begging the question, we must not base assumptions about specific goals and abilities of a designer on the theological hypothesis that there is a God, or on our actual experience of the facts we want to explain. Further, Sober (ibid., 143) presumes that the justification of auxiliary assumptions should be based on empirical evidence. It is for this reason that Sober is pessimistic about the scientific character of the design hypothesis: as opposed to human designers, we apparently lack empirical information about the goals, abilities, beliefs, and desires of a divine creator.

Nonetheless, Mellor's and Sober's objections are not necessarily fatal to the argument from design. One could refute their strong theological scepticism by loosening the underlying empiricist stance. This is in fact Dawes's and also Swinburne's position. According to Dawes (Reference Dawes2009, 82–84), we are warranted to believe that any theistic explanation is constrained by a so-called ‘rationality principle’. ‘It assumes that God will act in a way that is consistent with his beliefs and desires and that he will act in a way that would bring about his goal’ (ibid., 83). Notice that in order to make use of the principle, one must further specify God's beliefs and desires and define a particular goal. But for now the main point is that Dawes shows how it may be possible to add content to the theistic hypothesis by introducing non-empirical constraints on God's actions.Footnote ⁹

Swinburne shares Dawes's basic assumption that God is acting rationally. However, there are some slight differences between Dawes's position and Swinburne's, as Swinburne (Reference Swinburne2004, 100–101) strongly emphasizes that God's intentions may be deduced from his moral properties. An important premise underlying Swinburne's ‘moral access claim’ (Philipse (Reference Philipse2012), 150) is that moral values are given objectively and independently of God's free will. In some respect, God is thus restrained by his perfect goodness. As opposed to humans, he is not free to choose between good and evil (Swinburne (Reference Swinburne2004), 338). One may doubt Swinburne's (ibid., 113) assumption that there are in fact moral truths (although our knowledge about this matter may be limited), but this is not my point here. Rather, I would like to stress that by sticking to God's moral properties, Swinburne will consequently have to address the problem of theodicy.

Exploring God's intentions

In the following discussion, my main concern will be with how we get from God's alleged moral properties to assigning likelihoods to the various empirical pieces of evidence. First of all, let us consider the formal framework and the vocabulary provided by Swinburne to reflect on God's proposed intentions:

Basically, we are concerned with ‘actions’ that fall into ‘kinds’ and ‘sets of alternative actions’; and these actions, or kinds of actions, may be ordered by their level of moral goodness. By claiming that God would act in the most moral way possible, Swinburne invokes an ‘optimality condition’ (Dawes (Reference Dawes2009), 84) which places a constraint on what we would expect God to do.

But since God is determined to act as well as possible, it is not sure at first sight why there should be probabilities at all. It is only in the case of several equally best yet incompatible possibilities that uncertainty, and thus probability, arises. Swinburne claims that with n equally best and incompatible alternatives, there is a probability of 1/n for each that God could choose. This is perfectly in line with the standards of probability calculus, according to which the probabilities of a set of mutually exclusive and exhaustive events or hypotheses may be added up to 1 (remember that probability is a normalized and additive measure). To me, it seems a strange notion that God would ‘use some process of random selection or some arbitrary feature of a world to determine which to create’, as suggested by Swinburne in the first edition of The Existence of God (1979, 115); but even if he makes a deliberate choice between equally best actions, the probability 1/n may express our lack of knowledge of his preferences. Technically speaking, two tasks are needed to estimate likelihoods: one must define a set of ‘equally best’ actions or kinds of actions, and one must determine the number of its elements.

In what follows, Swinburne (Reference Swinburne2004, 112–123) identifies God's actions with the creation of particular possible-world states. As suggested by Swinburne in the quotation cited above, a morally perfect being, such as God, acts upon a rule of maximizing possible moral goodness, since he will do the ‘best action’ (if there is one).Footnote ¹⁰ Accordingly, we would expect God to bring about the best possible world, or one out of a set of equally best worlds.

Now, to validate the hypothesis of God's existence, one obviously must assess the relative degree of goodness of our real world as compared to a range of possible worlds. But how do we measure ‘the relative goodness of possible worlds’, as Swinburne (Reference Swinburne and Manson2003, 108) puts it? And, to complicate matters further, since likelihoods have the form of conditional probabilities and hence should lie between 0 and 1, how should one normalize this measure? It is at this point in Swinburne's argumentation that things begin to get muddled.

Accounting for the moral goodness of possible worlds

One of Swinburne's basic assumptions is that ‘[i]f there could be a unique best of all the possible worlds that (it is logically possible) God could create, it would be a unique best act to create it’ (Swinburne (Reference Swinburne2004), 114) – and, we may add, the likelihood that God would do so equals 1. But, contra Leibniz, Swinburne claims that there cannot be a single best world; and even worse, there cannot be a set of n equal best possible worlds, in which case there would be a probability of 1/n that God would have created one of them. Ironically, this is partly due to the need for quantification implied by the probabilistic framework. The relative goodness of possible worlds is presumably a matter of it containing certain numbers of discrete elements. While Swinburne (ibid., 115) thinks of ‘valuable beings’, we might as well plug in valuable events or states of affairs. In any case, this quantitative account leads straight into the fallacy of infinity: Sure enough, no matter how many worthwhile yet discrete elements a world contains, it would always be possible to add more, and hence, to raise its relative value.Footnote ¹¹ There is obviously no way to define a limit or maximum to the possible amount of moral value, and this poses serious problems for normalization.Footnote ¹²

Actually, the fallacy surfaces in a twofold sense: Swinburne recommends dividing the whole set of valuable beings or ‘substances’ into various types that may be ranked in terms of their relative value. The series starts with divine substance, which is thought to be the best imaginable, proceeds to humanly free and animal substance, and ends with the inanimate (ibid., 117–119). But certainly, one could imagine other or intermediate forms, such as angels (ibid., 119), so what should prevent one from conceiving of more and more different types? It follows that one could enhance the relative goodness of possible worlds by raising both the number of types and the number of beings contained within the types.

Provided thus with Swinburne's concept of additive value, the idea of a best possible world seems logically incoherent. It is noteworthy that this conclusion may be turned into an argument both in defence of and against theism. Assuming that there cannot be a best possible world, and assuming further that the idea of a divine creator necessarily implies that there should be such a thing as a best possible world, one could conclude that the very idea of a benevolent and omnipotent divine creator is logically incoherent (Dawes (Reference Dawes2009), 92). The theist might response by rejecting the second premise (that there is a necessary link between the idea of God and the concept of a best possible world). As Dawes (ibid., 93) puts it, ‘you cannot fault God for failing to do what is logically impossible. So God would not be a morally imperfect being by creating a world which is less than the best possible.’

Here arises another dilemma. By adopting the latter approach, Swinburne is able to avoid the invidious task of arguing that our world is in fact the best possible (or one out of several equally best). However, because of his probabilistic explanatory framework, Swinburne is required to specify some level of goodness we would expect God to bring about, and to determine in how many ways this particular level of goodness may be instantiated. If, for conceptual reasons, there cannot be an upper limit, how is this particular level to be defined? Swinburne further explains that

However, to state that God would do ‘some good action’ implies a substantial weakening of the optimality condition initially endorsed by Swinburne. As a consequence, Swinburne is no longer able to provide a formally precise definition of what it means to ‘act in a morally optimal way’. Obviously, this shift makes it hard to sustain the seeming rigour of Swinburne's argument.

Appreciating these conceptual limitations, Swinburne (ibid., 122) concedes that ‘God's perfect goodness does not require him to create any particular amount of such [valuable] beings’. This view is supported by the claim of R. M. Adams,Footnote ¹³ which holds that God has no moral obligations against non-existing creatures (ibid., 115). Applying this argument apparently leads to a dilemma: if God does not wrong potential beings by not creating them, one might conclude that he would not have to create anything at all, but this is surely not the bottom line that Swinburne wants to argue. The solution out of the dilemma comes as a postulate that Swinburne attributes to Aquinas and N. Kretzmann,Footnote ¹⁴ according to which ‘a good being will inevitably try to make other good things’ (ibid., 117). But this evidently brings us back to the start, since we still do not know how to define the level of goodness we should expect God to bring about. However, let us assume for a moment that we have fixed a sufficient level of goodness. The second step in building likelihoods involves determining in how many (mutually incompatible) ways this particular level of goodness may be instantiated. Now that seems to be just another wicked issue. How should we proceed? Dawes (Reference Dawes2009, 95–96) frames the general problem in terms of modal scepticism. The basic question is: (how) are we able to know about the options available to a divine creator? The idea of God's omnipotence endorsed by Swinburne suggests that God can do whatever is logically possible. However, I have argued that the concept of logical possibility resists quantification and normalization.

The manifold meanings of ½

The remaining argumentation of Swinburne (Reference Swinburne2004) rests primarily on one particular value, namely the likelihood that God would create humanly free agents, which he fixes at ½.Footnote ¹⁵ There are several conceptual problems with this value and discussing them may illustrate the difficulties with quantification and normalization in further detail.

First, note that the existence of humans is, as such, not contained within the previous list of eleven fields of evidence. The list includes the existence of a physical universe and subsequent laws of nature, as well as more special issues such as the existence of moral awareness in humans and the occurrence of miracles. Swinburne seemingly conceives of humans (and possibly other kinds of beings) as a purpose, whereas a physical universe guided by laws counts as a means to this purpose.Footnote ¹⁶ In contrast, the more specific kinds of evidence explain why human (and, to some extent, animal) life is purposeful. In its basic character, the argument is thus assuredly an anthropocentric one.

But what does it actually mean to state that the particular likelihood that God would create humanly free agents amounts to ½? Despite my vigorous attempt to envision a sound interpretation, conceptual confusion cannot be avoided.

As mentioned above, it is not at all clear how the notion of ‘elementary events’, or alternatively ‘elementary actions by God’, is defined within Swinburne's possible-world semantics. Is it to create a particular being or kind of being, or is it to bring about a particular, complex arrangement of beings, called a ‘possible-world state’? This information would be crucial for the interpretation of the likelihood-factor value of ½, which might either stand for a single likelihood of one of God's basic actions, or might indicate the overall likelihood of a set of basic actions.

Second, how are ‘kinds’ of actions to be defined? The purpose of introducing ‘kinds’ of actions is obviously to avoid an inflationary growth of members within the set of ‘equally best’ actions, since this is what would happen if every slightly different possible state of affairs that God could bring about were to be taken as a single member of this set. Further, great numbers of equally best possible actions would significantly lower the likelihood of each. Defining the ‘basic elements’ of a set of equally best (or equally good) actions is crucial before we move on to build likelihoods, since their values would alter severely as a function of the number of elements.

According to the rule cited above, we add the likelihoods of a set of equally good but incompatible actions up to 1. It follows then that there are actually two equally likely alternatives, one of which is creating humans. But what about the other? Apparently, Swinburne thinks that there is no other single kind of being as equally good as humans, the existence of which would preclude the other. Rather,

But why should creating other divine or semi-divine beings, or (more) animals or inanimate things be incompatible with creating humans, as suggested by the premise of incompatible actions previously acknowledged by Swinburne? Why should God not bring about all of them?Footnote ¹⁷ Second, why is the alternative to creating humans actually stated as a disjunction? As mentioned before, if we ascribe probabilities to different possible events or actions, it is crucial how these events are defined. Whatever possible action a is, we can of course formulate its negation ∼a and call this disjunction mutually exclusive and exhaustive. But if there are more than just two basic actions, ∼a will consist of a set of other basic actions, which means that it is a disjunction of possibilities not further specified. Now, if creating ‘any number of other divine or semi-divine beings, animals, or inanimate things’ is really a disjunction (a set of four incompatible alternatives), then God can choose just one out of them. Thus, there are on the whole more than two possibilities of possible actions. If the whole disjunction is defined as the alternative set to creating humans and if it bears a probability of ½, then the probability for each of the proposals will be of course a fraction of ½, namely, ⅛. But then it follows that we do not have an overall set of equally good actions, since creating humans was claimed to have a likelihood of ½ and is hence better than each of the alternatives. Now, if there is one single best alternative, Swinburne held that God is obliged to choose it, but then in turn its likelihood would be 1. So this interpretation apparently leads into contradiction.

Maybe Swinburne had in mind that each of the four alternatives to creating humans – other divine or semi-divine beings, (more) animals or inanimate things – has a likelihood of ½, maintaining that humans could be replaced by any of them with equal likelihood. Then we would really have a set of equally good incompatible actions, but the sum of their likelihoods is 2.5, which is greater than 1, and cannot serve as a probability. Normalization would thus require us to lower the single likelihoods to ⅕ if there are five alternatives. Obviously, the more alternatives we take into account, the lower the single values become. And why not imagine a vast amount of different kinds of beings that should all be given equal likelihoods as humans?

So we should possibly stop considering all alternatives as incompatible, and thus as disjoint. It seems more plausible to assume that creating different kinds of beings is, under normal circumstances, not at all incompatible. But then there is no way to calculate with probabilities, since adding up their values only applies to disjoint events or hypotheses.

Maybe there is a range of likelihoods for each of the different kinds of beings that God would create, with 0 indicating that God would never create such a being, and 1 indicating that God would always create such a being. Swinburne (ibid.) claims that it would be equally good for God to create humans as well as not to create them, and thus, both possibilities obtain a likelihood of ½. Here, we really operate with logically disjoint alternatives that add up to 1. Yet on that account, it makes no sense to think of values other than 0, ½, and 1; because according to Swinburne, choice is only given to God if there are in fact equally good alternatives. For a particular kind of being, this is the case if and only if its existence is exactly as good as its non-existence, whereas in any other case, God is determined to choose the better alternative, namely, either to create it, which gives us a likelihood of 1, or not to create it, which gives us a likelihood of 0. Thus, the likelihood-scale is not continuous (the values cannot take up all the real numbers between 0 and 1), but discrete with just three values: 0, ½, and 1. However, with this sort of limitation, why should one actually adopt the concept of probability at all?Footnote ¹⁸

The interpretations discussed so far regard ‘creating a particular kind of being’ as a basic event to which likelihoods can be ascribed. But we could postulate another definition of basic events. Maybe it was Swinburne's intuition that the likelihood that God would create a particular kind of being equals the frequency of its occurrence in different and incompatible, but equally good, world states. Hence, various possible-world states would form the basis of our underlying statistics, and they could be defined as the exact arrangement of beings contained in that world. If there were, say, 100 equally good but incompatible world states, 50 of them would contain humans, and the other 50 would contain whatever else, but not humans. But how do we know about the ‘whatever else’? How do we know about all the unrealized possibilities and their moral values? Why are there exactly 50 of them that are equally as good as all the world states containing humans? Since humans would have to be replaced by some other kind(s) of beings in total instantiating the same amount of value, how is the set of replacements to be normalized and thus restricted? Since logical possibility allows for so much, any restriction is necessarily arbitrary, and the assumption that there are exactly as many world states containing humans of equal value as any replacements seems biased by experience. Hence, this approach is a dead end, as well.

I conclude that neither of the interpretations is compelling, and since Swinburne does not provide a better one, it is not at all clear what is actually meant by saying that the likelihood for God to create human beings amounts to a value of ½. In the following section, I will consider an alternative way of framing the argument that avoids the normalizability problems discussed so far.

Deductive inference and irrelevant confirmation

Recall the alternative version of the hypothesis h that was briefly suggested by Swinburne in the first edition of The Existence of God:

‘We could of course have taken h as “there is a God who had the intention of bringing about a universe”, and then h would provide a full explanation of the existence of the universe and P(e|h&k[&int _e]) would equal 1’ (Swinburne (Reference Swinburne1979), 69). Since God's intention to bring about e entails the existence of e, the likelihood of h&int _e on e is perfectly defined. It amounts to 1 because e follows with certainty from h&int _e. Thus to avoid poorly defined likelihoods, we could postulate the extended hypothesis of God's existence together with his various intentions. This enables us to provide values of the likelihoods by deductive inference.Footnote ¹⁹

Now this strategy seems precarious for another reason. To put it in Sober's (Reference Sober2009, 131; original emphasis) words, the fact that the likelihood of our extended hypothesis amounts to 1 ‘does not embody a victory for the design hypothesis; rather, the result is a stalemate, since it is easy to construct numerous other hypotheses that also entail the observation’. We may explicate Sober's argument in a more technical way. Consider the following reflections on the concept of confirmation:

The tacking problem, as it is termed by Glymour (Reference Glymour1980b), illustrates an unpleasant implication of the concept of confirmation endorsed by Swinburne. Apparently, the sentence A is confirmed by e although A may be completely irrelevant with respect to e. Note that the problem of irrelevant confirmation exemplifies an argumentational fallacy, not a formally invalid inference. The fallacy is also referred to as content cutting in the literature (Earman (Reference Earman1992), 98). In any case, we start from a hypothesis or proposition that consists of ‘parts’, which means that it has the logical form of a disjunction or conjunction.Footnote ²⁰ The crucial point is that one of the parts entails the evidence, whereas the other is entirely irrelevant with respect to evidence. By adopting this double strategy, so the charge runs, any desired hypothesis may be ‘cut into shape’ in order to fit as an explanation of evidence.

Yet Swinburne's argument is vulnerable to exactly this kind of fallacy, if we extend the hypothesis of God's existence with the various intentions:

$$int_e \to e$$

$$(int_e \to e) \to [{\rm P(}e|int_e ) = 1]$$

$$(int_e {\bf \&} {\bi h}) \to e$$

$$[(int_e {\bf \&} {\bi h}) \to e] \to [{\rm P(}e{\rm |}int_e {\bf \&} {\bi h}) = 1]$$

Obviously, the various intentions int _e entail evidence e, whereas the hypothesis h is merely attached to int _e. It follows that if the hypothesis h is irrelevant with respect to the evidence, then it is only trivially confirmed and could be replaced by any sentence whatsoever which would then be trivially confirmed in exactly the same manner. In such a case the argument is obviously of no value. The charge outlined here is common in the literature on design arguments, although usually stated less formally. Sober (Reference Sober2009, 154) indicates that ‘the design hypothesis merely allows our observations … to be folded inside a simple formula’; Dawes (Reference Dawes2009, 141; original emphasis) similarly stresses ‘the danger of accommodation as distinct from prediction’. Schurz (Reference Schurz2011, 124–126) explicitly mentions the pitfall of content cutting with respect to arguments for creationism. Generally spoken, the claim that design arguments lack significant empirical content and the charge of irrelevant confirmation may be seen as two sides of the same coin.

To reject the charge of irrelevant confirmation, Swinburne would have to demonstrate that there really is a strong intrinsic relation between both ‘parts’ of the explanation, between God's existence and his having the particular intentions; he would have to show that intentions arise directly from the definition of God.Footnote ²¹ Although the hypothesis is syntactically stated as a conjunction, the challenge would be to argue that there is a conceptual dependency between the two ‘parts’ such that one part cannot be removed or replaced without affecting the other. Thus, even when using the extended hypothesis exactly the same problem arises that we faced when assessing the likelihood of the previous unextended hypothesis. We need a systematic and normalizable preference order of God before we develop a formally precise idea of his intentions. No matter which version of the argument we choose, the need for a normalized measure of the moral goodness of possible worlds obviously cannot be overcome.

Conclusion: better to adopt a qualitative approach?

I have argued that Swinburne's probabilistic approach fails due to several conceptual pitfalls. First, it is impossible to measure the options available to a divine creator (the space of possible worlds) by means of probability calculus. Second, there is no way to construct a quantitative and normalized preference order of God based on the relative moral goodness of possible worlds. This is because, third, Swinburne's notion of additive value leads straight into the fallacy of infinity. As a consequence, the condition of moral optimality essential to the argument lacks precise formal definition. Further, by modifying the argument in a way that allows the probabilities in question to be defined, we run into the charge of irrelevant confirmation.

In response to many of these difficulties, Dawes (Reference Dawes2009) offers a substantially revised account of Swinburne's argument. In my last section, I shall briefly outline and review his proposals. Note that Dawes's aim is modest as compared to Swinburne's. Dawes seeks to establish a possible way of framing theistic explanations which meets conditions of valid argumentation and does not already fail for conceptual reasons. However, he does not seem to think that theistic explanations are likely to be actually successful as compared to naturalistic alternatives (ibid., 144).

As his main conceptual revision, Dawes supports a qualitative as opposed to a quantitative explanatory framework. More precisely, his point is that Swinburne's overall argument might be reconstructed in terms of inference to the best explanation (IBE) (ibid., 108). Indeed, when Swinburne (Reference Swinburne2004, 339–342) seeks to demonstrate that the entirety of empirical evidence is more likely on the hypothesis of theism than on a disjunction of alternative hypotheses, his general claim might be read as an instance of IBE.

Obviously, one of the most striking shortcomings of Swinburne's argument may be avoided by adopting a qualitative, non-confirmationist approach. We do not have to work with poorly defined probabilities. But there remain further hurdles to take. Dawes (Reference Dawes2009, 116–125) is well aware that the success of proposed theistic explanations crucially depends on meeting the criterion of testability. In other words, theistic explanations should contain significant empirical content (ibid., 117). Thus, the second horn of Swinburne's dilemma, his offering of irrelevant or trivial evidence, does not automatically expire if we substitute confirmation for explanation.

For a hypothesis to be testable, we need to know what to expect if it were true (or, at least, what not to expect). As mentioned before, Dawes (ibid., 82–84) rejects strong theological scepticism – the position that we cannot have access to God's nature or intentions – by claiming that God's actions would be constrained by a ‘rationality principle’. Due to God's properties of omniscience, omnipotence and perfect goodness, this principle implies an ‘optimality condition’ which may be spelled out as follows: ‘What would be the most rational way in which an omnipotent, omniscient, and morally perfect agent could bring about his intended goal?’ (ibid., 85).

I have argued that Swinburne invokes an ‘optimality condition’ in his argument, but fails to define it in a formally precise way. What about Dawes's version? For one thing, Dawes eschews Swinburne's possible-worlds account as he suggests restricting the condition in scope. ‘The optimality condition is not something one applies to the world as a whole’ (ibid., 92). Rather, what is to be explained are ‘particular facts about the world, however broad their scope. (The existence of the world is also a particular fact.)’ (ibid., 93; original emphasis). Further, Dawes's optimality condition bears on means towards some end that God would pursue, not on that end itself. Dawes (ibid., 92) is convinced that by adopting these revisions, ‘[t]he optimality in question is more clearly defined’. Dawes wishes to leave it up to the theist to define some particular plan or goal, yet as a minimal condition, he suggests that God would ‘seek to minimise suffering’ (ibid., 85). With respect to the notion of rationality central to the argument, Dawes (ibid., 89) suggests (without claiming that the list would be exhaustive) ‘three criteria of rational action: consistency, efficacy, and efficiency’.

With these revisions in mind, are we able to work out a sound version of Swinburne's argument? I am not convinced. There is still place for a modal sceptic to claim that we do not have access to the means open to God, and even if we do, there is possibly no easy way to judge what counts as a better or worse realization of some particular plan. However, as it seems to me, the pivotal question to ask is: why should the idea of an ‘optimal means to some end’ be more stringent than the concept of a ‘best possible world’? One of Dawes's replies points at the possibility of making comparative judgements.

We may accept Dawes's reply, given the notable limitations of human knowledge and imagination. However, on strictly logical grounds, a problem arises, of which Dawes is well aware. When discussing an argument by William Rowe,Footnote ²² he states that ‘even if there is no best possible world – in fact precisely if there is no best possible world – there is always a world better than any which God might choose to create’ (ibid., 93; original emphasis). Yet if we substitute ‘possible worlds’ for ‘realizations of some goal’ the bottom line is very similar. If there were no upper limit to how well God might have acted, there would always, in principle, be a better way for God to realize some particular intention, even if we cannot imagine it owing to cognitive constraints. But in case there is a better means, Dawes maintains that the proposed theistic explanation fails. It follows that, for logical reasons, any proposed theistic explanation would have to fail. Hence, Dawes's proposal may be turned into an argument against the bare possibility of theistic explanations. I conclude that while Dawes's qualitative revision of Swinburne's argument may evade some of its most striking failures, an overall conceptually sound framework is not yet reached.

Apart from my formal objections, I suppose that theodicy presents a hard challenge to Swinburne's argument, as well, and may substantially undermine it. Additionally, some may disapprove of the hierarchy of moral values suggested by Swinburne, which places humans above all other kinds of living beings. However, my aim in this article was not to invalidate the position that human life may be considered as having value. Rather, I found Swinburne's probabilistic proof apparatus to be logically misguided. His cumulative-case argument either relies on poorly defined probabilities or amounts to a case of irrelevant confirmation.Footnote ²³