I have three children: Sally, Robert, and Andrew. Sally is older than Robert, and Robert is older than Andrew. You will naturally conclude that Sally is older than Andrew. Similarly, if I name three numbers, a, b, and c, and tell you that a > b, and b > c you will know that a > c.

These are both examples of transitivity – both being-older-than, in the case of people currently alive, and being greater than, with natural numbers, are transitive relations. But consider if I were to say, I am friends with James, and James is friends with Anna. Can you conclude that I am friends with Anna? Not necessarily. ‘Being-friends-with’ does not have transitivity built in. We might like it to – we might wish that our friends could be friends with each other, and often they are. But not always.Footnote ¹

Some relations are transitive and some are not, in other words. One relation that is clearly transitive, in the familiar world of arithmetic, is ‘equals’: if a = b and b = c, then a = c. Students rely on this a lot when doing basic forms of algebra.

At the heart of Jc Beall's Divine Contradiction are cases of non-transitivity where we might feel inclined to expect transitivity, a bit in the way we are strongly inclined to expect it when we see ‘=’. The Father is God. The Son is God. The Father is not the Son. Or to bring the sense that there is something troubling out a little more strongly, the Father is identical with God; the Son is identical with God; the Father is not identical with the Son.Footnote ²

Anyone who has spent some time thinking about the doctrine of the Trinity will probably have felt the pressure of this issue in some way or other. So the novelty of Beall's book is not to have noticed the difficulty. The novelty is in the response that is offered. You can make sense of the non-transitivity, Beall suggests, if you accept that you need to move away from the logic that we are all familiar with – which prohibits contradictions – and instead adopt a logical system in which it is allowable that the same statement can be both true and false.

According to classical logic, from a contradiction you can prove anything: once you admit a contradiction into a system, everything becomes true. But classical logic is not the only option – there are other systems of logic in which contradiction (the same proposition being both true and false) is permitted, and within which such contradictions don't have the ‘explosive’ effect of leading to the conclusion that anything and everything is true. Beall proposes FDE (first degree entailment logic) as that which we ought to opt for in place of classical logic. We can then affirm the doctrine of the Trinity in what Beall takes to be a straightforward way, and acknowledge the contradictions it involves, without abandoning either logic or clarity.

It's rare to come across a new proposal in trinitarian theology, and Beall's book does in fact offer an approach I don't think I have quite seen before. There's something refreshing to it: a clear challenge to think again. To my mind it is a virtue that the book doesn't claim to offer too much – it doesn't claim to school us in how to imagine the Trinity, how we can ‘hold in mind’ what is believed, and there's no drawing of conclusions from this approach for social or political or even homiletic purposes. What's on offer instead is a kind of clarity and simplicity: we should look what is going on in the doctrine straight in the eye, and acknowledge that God involves contradictions,Footnote ³ and then we won't have to engage in evasions, obfuscations, or unnecessary complexity. We can understand it to be part of the otherness of God that patterns of logical entailment which hold in most or all other intellectual domains don't hold in relation to God, just as, in Beall's previous book, it turned out that they don't hold in relation to the Incarnation.

I was asked to review the book because I am a theologian who has written on the doctrine of the Trinity, and in that capacity I need, however, to register certain reservations: two reservations relating to the way the book is written, and one in relation to the central proposal it makes.

Beall has the admirable intention of writing not only for ‘theologically informed philosophers’ (viii) but also for theologians and indeed church leaders and ‘many non-academic readers’ (ix). My first critical point is that I don't think the book will in fact be accessible to these groups. In my case the ability to read it was supported much less by any knowledge I may have of trinitarian theology than by a long-ago immersion in undergraduate and MA level pure mathematics. I can no longer remember the details of, for instance, the course in mathematical logic I once did, but I am at least not panicked or distracted by abstract notation: there is at least a certain amount in the text (symbols for ‘and’, ‘or’, ‘not’, phrases like ‘if and only if’ and so on) that I can read and interpret without stumbling or slowing down.

Beall is meticulous in defining all the technical terms and symbols used, but this is not the same as writing for a broader audience: we do not pick up a new language by seeing each element in it abstractly defined. At the start of this review, I introduced the concept of transitivity with a care that was unnecessary if it is only to be read by those already interested in analytic theology. I did so partly to give an example of the more human and humane way of presenting concepts that I think is necessary if one wishes to reach beyond those already within the analytic fold.Footnote ⁴

In short, I don't think many of my non-analytic-theological colleagues will make it far in reading this book. Any who do, however, will encounter another kind of difficulty in the language, and this is in the way that God is rather casually described as a portion or fragment of reality (cf. e.g. 50, 74, 96). According to the Christian tradition, as I understand it, God is not a portion or fragment of anything else whatsoever: God is certainly not something rare which is located ‘in an otherwise consistently described created world’ (75). For me and those with a similar theological training, these sorts of phrases are the intellectual equivalent of fingernails on chalk board.

But no doubt somewhere in my brief summary of Beall's position I have said something imprecisely or made some other sort of error to make logicians wince. We've each been schooled into immense care in certain spheres and not so much in others. That some of the language of this book seems to treat the Trinity as one item among others, one thing in the world in the midst of others, is I suspect a superficial rather than a deep problem. The deeper design of the book is to treat the Trinity as something so other to everything else we speak of that it requires that we reconsider our understanding of logic itself.

This proposal – to take so seriously what we learn of the Trinity that we rethink logic itself – is something I find initially very appealing. However, when I attend to what it means a little more specifically, I have to say that the appeal disappears. According to Beall, the doctrine of the Trinity means not only that the Father is God, but also that this is not the case: it is false that the Father is God.

Can this really be said by a Christian believer? Can I maintain, as part of Christian belief, that ‘it's false that the Father is God’: can I deny the Father's divinity? I would then also have to believe not only that Son and Spirit are divine, but also that they are not, namely that it is false that the Spirit is divine and it is false that the Son is divine.

Beall's aim is to work within the bounds of the Athanasian creed, taking it as axiomatic. This is where one ends up, Beall is proposing, if you really accept what you find in that creed. This just doesn't seem plausible: if the denial that the Father is God, the denial that the Son of God, and the denial that the Spirit is God, are somehow part of the package that the Athanasian creed bequeaths, they are a puzzlingly hidden, puzzlingly secret part of its message.

Beall considers and rejects the idea that it might be heretical to advance the falsity of the Father's divinity (and the Son's and the Spirit's). He maintains that what is really important is that one can affirm ‘It is true that Father is divine’. If this were missing, there would be a problem. But the ‘presence of negation’ (that one can affirm ‘It's false that Father is divine’) is not an issue. It only seems to indicate a problem to those not used to the territory of contradiction, Beall maintains, because usually – outside this territory – the presence of the negation would mean the absence of the affirmation of the Father's divinity.

This strikes me as arbitrary – no grounds are given that I can see (either here or in the predecessor volume) for the asymmetry, as regards heresy, between negation and ‘nullation’ (i.e. affirmation). And indeed, if anything, the history of doctrine points in precisely the opposite direction: it is almost always easier to identify what has been ruled out as a result of a doctrinal controversy than to understand what has been affirmed. How then can one be persuaded by Beall's announcement that certain forms of statements are not really heresies even though they give that appearance?

It's really hard, I have to say, to imagine a Christian context in which part of the formal teaching is that the Father is not God, the Son is not God, and the Spirit is not God. Perhaps one who follows Beall would have to conclude that these propositions are implicitly woven into Christian belief, but that one is not encouraged to go around saying them in non-specialist company? They're a sort of secret, hidden bit of the package? All this seems so far away from Christian creeds, Christian practice, and ordinary Christian piety as I understand it that there is, it seems to me, a deep implausibility attached to saying that this is in fact a part of what was always the meaning of the doctrine of the Trinity.

Overall, then, a fascinating but ultimately unpersuasive proposal.Footnote ⁵