In The Coherence of Theism [hereafter CT] Richard Swinburne writes, ‘it is definitely logically impossible that there be an omniscient person who is himself perfectly free’(CT, 177).Footnote ¹ In The Existence of God [hereafter EG] Swinburne writes, ‘I take the proposition “God exists” … to be logically equivalent to “there exists necessarily a person without a body (i.e. a spirit) who necessarily is eternal, perfectly free, omnipotent, omniscient”’[my emphases] (EG, 7). Prima facie, the two passages imply that God cannot exist, whereas Swinburne argues that God most likely exists. The key to reconciling the tension between the two passages is to note that ‘omniscient’ is being used in two different ways.

In order to distinguish these two meanings let me introduce some terminology. Let us understand by ‘maximal’ that the modified word is as strong as possible while remaining coherent. Let us understand by ‘restricted’ that the modified word is used in a weaker sense than its maximal sense. When Swinburne writes that God cannot be omniscient and perfectly free (CT, 177), Swinburne is working with a maximal omniscience. When Swinburne writes that God is omniscient and perfectly free (EG, 7), Swinburne is working with a restricted omniscience. The same holds for ‘omnipotence’. As Swinburne writes: ‘The argument of Part II has been that it is coherent to suppose that there exists eternally an omnipresent spirit, perfectly free, the creator of the universe, omnipotent, omniscient, perfectly good, and a source of moral obligation – so long as “omnipotent” and “omniscient” are understood in somewhat restricted senses’ (CT, 243). The crucial point is that God is not maximally omniscient or omnipotent, but rather is restrictedly omniscient and omnipotent. God is perfectly free, and due to this, cannot be maximally omniscient or omnipotent. These complex relationships between the properties are a main topic of The Coherence of Theism. Note that, for Swinburne, the properties freedom, knowledge, and power are crucial because all other properties of God (Swinburne claims) follow from these three properties (EG, 99). Let us now turn to Swinburne's argument in The Existence of God.

In The Existence of God Swinburne employs Bayes' Theorem to argue that God likely exists. Bayes' Theorem is as follows:

h is the hypothesis ‘God exists’, e is evidence for or against the existence of God, and k is ‘mere tautological evidence’. (k and e change as Swinburne feeds evidence into Bayes' Theorem, but I am following Swinburne in chapter 14 of The Existence of God where the final accounting of probability is done. Here k is mere tautological evidence and e is all of (the conjunction of) the evidence.) Swinburne argues that P(h|e&k), the ‘posterior probability’ or ‘the probability of God's existence on the evidence’, is greater than 50 per cent; that is, Swinburne argues that God likely exists. P(h|e&k) varies directly with P(h|k), which is ‘the intrinsic probability’ of God's existence. This intrinsic probability, P(h|k), is higher the simpler God is. Swinburne argues that P(h|k) is relatively high, thereby raising P(h|e&k), by arguing that God is very simple. Why, according to Swinburne, is God simple? Swinburne claims that the infinite is simple, and that God is infinite on the properties freedom, knowledge, and power. I call this argument ‘Swinburne's argument to the simplicity of God via the infinite’. As Swinburne writes, ‘the supposition that there is a God is an extremely simple supposition; the postulation of a God of infinite freedom, knowledge, and power is the postulation of the simplest kind of person that there could be … . The divine properties fit together, and they are properties of infinite degree’ (EG, 151).

Before going on to argue that Swinburne's argument to the simplicity of God via the infinite is undermined by God's being merely restricted in omnipotence and omniscience, note that I am granting Swinburne his claim that the infinite is simple, though his argument that the infinite is simple seems unsuccessful. In both The Existence of God (EG, 55) and in the article, ‘Prior probabilities in the argument from fine-tuning’, he writes, ‘For we can understand, for example, the notion of an infinite velocity … without needing to know what the googleplex is’.Footnote ² But we can also understand the googleplex without understanding the infinite, thus neither the googleplex nor the infinite would be simpler than the other on Swinburne's own view of simplicity.Footnote ³ It is then not clear how this argument can show that the infinite is simpler than finite values, though, to reiterate, I grant him this claim.

To return to the main theme, Swinburne's argument to the simplicity of God via the infinite requires that God be infinite in freedom, knowledge, and power. To be infinite in these properties is the same as God's being maximal in these properties. As is clear from Swinburne's writing and as Quentin Smith noted, ‘“infinite” [can] mean the maximum degree of a degreed property. This is the sense in which God is infinite’.Footnote ⁴ (I henceforth use ‘infinite’ and ‘maximal’ interchangeably, and also ‘perfect’ with ‘freedom’. That is, ‘infinite’, ‘maximal’, and ‘perfect’ will all mean the maximum degree of a property.) But we have seen that two of the three properties, knowledge and power, are not possessed by God to an infinite degree. God is merely restrictedly omniscient and omnipotent. Thus Swinburne's argument to the simplicity of God via the infinite fails. Then P(h|k) is not relatively high, and so P(h|e&k) is far lower than Swinburne claims. That is, Swinburne's argument in The Existence of God fails to show that God likely exists.

Another way to put (briefly summarize) my argument: Swinburne claims that God is omniscient and omnipotent in The Existence of God. As The Coherence of Theism makes clear, these properties must be restricted omniscience and omnipotence, but for Swinburne's argument to the simplicity of God via the infinite to succeed, the properties must be maximal omniscience and omnipotence. Thus Swinburne's argument to the simplicity of God via the infinite fails, which undermines his overall argument.

Let me now discuss two potential responses that I see available to Swinburne. The first response maintains that God is restrictedly omniscient and omnipotent, and begins from the claim that God is perfectly free, and that God then has as much knowledge and power as is consistent with perfect freedom. But if this is the move Swinburne wishes to make, I do not see how he can argue from the simplicity of the infinite to a relatively high value for P(h|k). For God would not be maximal on knowledge and power. God would be maximal on these two properties relative to perfect freedom, but as The Coherence of Theism made clear, this is not the same as being maximal on these two properties. God's being less than maximally (that is, restrictedly) omniscient and omnipotent would ‘cry out for an explanation’ (EG, 97). There would be an explanation of course, namely an explanation in terms of God's being perfectly free, but, to reiterate, God would not be maximally omniscient or omnipotent, and thus the argument from the simplicity of the infinite would fail. Swinburne would have to claim that God is infinitely powerful (the same would hold for knowledge, though for the following example I only consider power) relative to perfect freedom and that this relative infinite power is, in fact, simple.

Let me indicate why I believe that this argument fails. Consider the following scale of minimum to maximum power: Min has minimal power. Max has maximal power. God has slightly less power than Max. For example, Max can perform morally bad actions whereas God cannot. The move under consideration is the claim that, since God is infinite on power relative to having perfect freedom, then such a relative maximal power is simple. But this move opens the door to claims of simplicity at any level of power. For example, imagine Alexander, a mortal, who devotes his entire life to the acquisition of power. Alexander is then maximally powerful relative to his physical and mental limitations. If God's power is considered maximal and thus simple, so too would Alexander's power be considered maximal and thus simple.Footnote ⁵ The point is that any argument claiming that relatively maximal power is simple makes all levels of power simple (for any level of power is maximal relative to some constraint or other), and thus all levels of power would be equally simple. Such an argument could not show that God is simple compared to, e.g. Alexander. But this is precisely the conclusion that Swinburne needs (that is, Swinburne needs to show that God is simpler than any other being by virtue of God's level of power). And so an argument that relatively maximal power is simple cannot succeed and thus is of no help to Swinburne.

Note also that the question arises: why is priority given to freedom? That is, why is God maximally free (and then given as much knowledge and power as is consistent) as opposed to say, being maximally powerful (and then given as much freedom and knowledge as is consistent)? The concern is that saying that a person is infinite on freedom, knowledge, and power (as Swinburne does repeatedly in The Existence of God) does not specify a unique being; the order of priority of properties must be specified. I return to this issue of the priority of properties below, following the second potential response.

The second potential response is to argue that God is simple in virtue of possessing one simple property. In The Christian God [hereafter CG] Swinburne argues that God possesses ‘pure, limitless intentional power’ and that this property is simple.Footnote ⁶ This response runs into two main problems. First, Swinburne does not argue to the simplicity of God via pure, limitless, intentional power in The Existence of God, rather he argues from freedom, knowledge, and power, which he claims God possesses to an infinite degree. If Swinburne wishes to argue to the simplicity of God from pure, limitless, intentional power, then this should be made explicit in The Existence of God (even if mainly by way of footnote).

Second and more importantly, though Swinburne states that this property of pure, limitless, intentional power is one simple property, it is not clear that anything is gained in simplicity relative to the discussion of God in The Coherence of Theism, where the divine properties are complex and had to be ‘knocked into shape’ (CT, 165). For Swinburne's discussion of pure, limitless, intentional power in The Christian God makes frequent reference to The Coherence of Theism; indeed it seems impossible to fully understand pure, limitless, intentional power without understanding Swinburne's discussion of freedom, knowledge, and power in The Coherence of Theism. Additionally, ‘limitless’ is a misnomer in the following sense. Because God's power is ‘pure’, God is perfectly free (CG, 152); yet God's power is not limitless, but rather is limited by God's perfect freedom. That is, God's perfect freedom still places complex limitations on God's power (and knowledge). When fully described, pure, limitless, intentional power does not appear to be one simple property.

Swinburne has also argued that God is almighty, and that this property implies the others and is simple. In his article ‘Could there be more than one God?’, Swinburne discusses almightiness, which includes the predicates freedom, knowledge, and power. He writes, ‘If I am right in supposing that the predicates do thus fit together, that there is a God becomes a very simple claim and for that reason more likely to be true.’Footnote ⁷ If this is the response Swinburne wishes to give, namely relying on the simplicity of almightiness, more work needs to be done explaining this property of almightiness and its simplicity. As with pure, limitless, intentional power, if this is how Swinburne wishes to argue to the simplicity of God then it should be made explicit in The Existence of God, and it also must be explained why almightiness is a simple property that avoids the complexities of The Coherence of Theism.

Let us now consider the issue of the priority of the properties freedom, knowledge, and power. Swinburne is clear that he gives priority to freedom and briefly addresses this problem of priority by suggesting that giving priority to freedom yields the simplest being (EG, 98);Footnote ⁸ however, this treatment is not complete or persuasive. For a close reading of The Christian God raises the concern that there might be a simpler being than God. Consider an ordered triple representing God's level of freedom, knowledge, and power. Let ‘R’ stand for ‘restricted’ and ‘∞’ stand for ‘infinite’ or ‘maximal’. Then The Coherence of Theism makes clear that God is (∞, R, R); Swinburne's argument to the simplicity of God via the infinite requires that God be (∞, ∞, ∞).Footnote ⁹ In an interesting passage in The Christian God Swinburne considers the issue of the priority of properties and writes:

Recall the ordered triples representing freedom, knowledge, and power. In the quotation above, if God is to have any freedom, then God is (∞, R, R). If the being,Footnote ¹⁰ let us call him ‘g2’, has restricted freedom, then he has no freedom, and so is (0, ∞, ∞). But Swinburne wrote that both the infinite and 0 are simple (EG, 97). Thus g2 of no freedom and infinite knowledge and power, or (0, ∞, ∞), would be simpler than God with perfect freedom, but restricted knowledge and power, or (∞, R, R). It would then seem that on Swinburne's own view of simplicity he should argue that g2 is most likely to exist. Note too that if it is impossible to derive g2's perfect goodness, then this would seem to be a further advantage as the problem of evil would no longer be so acute a problem. In the language of Bayes' Theorem, P(e|h _g2&k) and P(h _g2|k) are higher than P(e|h&k) and P(h|k), respectively, when h _g2 is the hypothesis that g2 exists and h is the hypothesis that God exists.Footnote ¹¹ Then, by Bayes' Theorem, the evidence confirms g2's existence more than God's existence, which implies that God cannot exist with a probability greater than 50 per cent. That is, on Swinburne's own view of simplicity, it seems that priority should be given to knowledge (yielding g2) and not to freedom (yielding God); some argument to the contrary is needed if Swinburne's argument is to successfully show that God most likely exists.

I conclude by reviewing my main argument. God is omniscient and omnipotent in a restricted sense (a less than maximal sense); however, Swinburne's argument in The Existence of God relies on God's being maximally omniscient and omnipotent. Why? Because Swinburne argues that P(h|k) is relatively high by arguing that God is simple in virtue of being infinite on the properties knowledge, power, and freedom, where infinite means maximal. Thus Swinburne's argument to the simplicity of God via the infinite in The Existence of God fails. The failure of this argument greatly lowers the value of P(h|k), which in turn lowers P(h|e&k). That is, I have suggested that Swinburne's argument to the simplicity of God via the infinite does not succeed, which means that Swinburne's argument that God likely exists does not succeed.Footnote ¹²