A long-standing problem

According to creedally orthodox accounts of the Trinity, the divine persons are genuinely distinct from one another. Indeed, for much of the tradition, it is entirely safe to say that the distinctions are nothing short of real distinctions (in the technical sense). This yields what we can refer to as:

At the same time, however, the traditional formulations of the doctrine of divine simplicity also exert pressure on Christian theology, and one prominent version of that doctrine (ratified at the Fourth Lateran Council) delivers the verdict that the divine persons just are the one ultimate divine reality that is the single divine essence. This yields what can be referred to as:

Some contemporary theologians as well as many theologians in the tradition do not hesitate to affirm both TRIN and SIMP. In our day, Adonis Vidu says that ‘the triune persons are not a different reality from the divine essence’, and he insists that ‘one must speak of numerical identity between person and essence’ because ‘the distinction between the persons is not real but only conceptual’ even though the ‘distinction between the persons is, against Sabellianism, a real distinction’ (Vidu Reference Vidu2021, 98). Vidu's commitments are nothing new. As Thomas Marschler observes, after Lateran IV ‘every subsequent theory of the distinction between the divine essence and the persons of the Trinity had to avoid a real distinction’ (Marschler Reference Marschler, Schartl, Tapp and Wegner2016, 92). But every subsequent theory also needed some kind of distinction.

Here is why. According to classical logic, identity is reflexive, symmetrical, and transitive, and the identity relation yields the indiscernibility of identicals. So claims that the divine persons are really distinct from one another but really identical with the divine essence run the risk of incoherence. Obviously, Christians committed to creedal orthodoxy affirm TRIN; they reject the notion that the Father and Son are distinct only in our way of thinking about things and instead affirm that the Father and Son (and Holy Spirit, of course) are really distinct from one another. But if they also affirm SIMP then they are faced with a serious problem. For if the Father is identical to the divine essence, and the Son (and Spirit) is identical to the divine essence, then, given the transitivity of identity, it follows logically that the Father is identical to the Son (and also to the Spirit). But such a conclusion directly contradicts TRIN – according to TRIN, the Father is not identical to the Son, but, on the entailment that comes with SIMP, the Father is identical to the Son. And since, according to (classical) logic, contradictions are necessarily false, any (non-glutty) theology that includes both TRIN and SIMP (either by affirmation or by entailment) is not only false but even necessarily false.

The conclusions follow logically. According to classical logic (or, indeed, any system of logic hospitable to the transitivity of identity), the conclusions – problematic as they are for orthodox Christian theology – are inevitable.

The problematic nature of these conclusions has not exactly been a secret, and all manner of manoeuvres were employed in late medieval and early modern scholasticism in efforts to avoid them. Some theologians, even after Lateran IV, are daring enough to insist upon nothing less than a ‘real distinction’ (distinctio realis) between person and essence. Real distinctions are independent of mind and marked by logical and ontological separability. More precisely, real distinctions (commonly and standardly – the God–world relation is a notable exception) yield two-way or mutual separability; as Richard Cross puts it, ‘it was very commonly supposed that the mark of such a real distinction is mutual separability’ (Cross Reference Cross2022, 64). While unusual after the condemnations associated with it by Lateran IV, appeal to the real distinction between person and essence is one way forward (particularly for some Protestant theologians; the Reformed divine Francis Turretin (Reference Turretin, Giger and Dennison1992, 278) says that ‘some maintain that it is real’).

Other theologians, especially those in the Thomist tradition but also including some Jesuits, Scotists, and Protestants, work to address the challenge by developing rational or conceptual distinctions (distinctio rationis). Rational distinctions are mind-dependent and, strictly speaking, are inseparable. As Suarez puts it, such distinctions conceive of ‘things which are not distinct as though they were distinct’ (Suarez Reference Suarez and Vollert1947, 19). Many early modern scholastics, tacitly admitting that the treatment offered by Aquinas is insufficient, further distinguish between the ‘distinction by reason reasoning’ (distinctio rationis ratiocinantis) and the ‘distinction of reason belonging to the object of reasoning with a foundation in reality’ (distinctio rationis ratiocinatae quae habit fundamentum in re). Recognizing that the former will do nothing to turn back the obvious threat, these theologians adopt the latter and press it into service to distinguish between the persons and the essence.

Many theologians recognize the need for some further distinction ‘between’ those that are real and those that are merely rational. Some theologians, such as Turretin, opt for the modal distinction (distinctio modalis). As Jorge J. E. Gracia explains, ‘modes are not realities independent of the things they modify, but nevertheless are intensionally distinct from them’ (Gracia Reference Gracia and Gracia1994, 13). Things that are modally distinct have a kind of objective reality that goes beyond the rational but not as far as real distinction, and one-way separability is the central mark or identifying characteristic of a distinction that is modal: as Suarez says, ‘if two things are actually separated in such a way that one continues in existence but the other does not, they must be at least modally distinct’ (Suarez Reference Suarez and Vollert1947, 42).

But other theologians, especially those of the house and lineage of John Duns Scotus (but also including some Dominicans, Jesuits, and Protestants), defend the use of the ‘formal’ distinction (distinctio formalis). Such a distinction obtains when there is a genuine distinction that is inherent in the entity in question rather than in the intellect of the cognizer (and thus is not a rational distinction) and inseparability (and thus neither a real nor a modal distinction). As David Bradshaw puts it, the formal distinction applies to entities that are genuinely distinct but ‘cannot exist separately (and thus are not “really distinct”)’ – and thus, in this case, yielding an account of real identity which ‘clearly is not transitive, and so does not threaten the distinction of the persons’ (Bradshaw Reference Bradshaw2019, 31).

But none of these options can claim anything like unanimous affirmation or even common consent (much less creedal or conciliar affirmation). All extant options continue to raise concerns and attract critics. To posit a real distinction between person and essence is to invite charges that the doctrine of simplicity is compromised and that Trinitarian monotheism has been traded in for polytheism. And certainly it is to run afoul of the dictums of Lateran IV (and, while Orthodox and Protestant theologians are not beholden to that standard in the same way that Roman Catholics may be, nonetheless the more traditionally minded among them generally seem reluctant to trespass the boundaries it stakes). At the other end of the spectrum, critics aver that merely rational distinctions are insufficient to block transitivity and thus offer no way out. In between, the modal distinction (as applied to the Trinity) has long attracted fierce criticism for compromising both divine simplicity and Trinitarian orthodoxy (e.g. Montoya Reference Montoya1625, 108–112), and even as redoubtable a defender of the distinction (in general) as Suarez denies its applicability here on the grounds that it is ‘greatly opposed to the divine simplicity and perfection’ (Suarez Reference Suarez and Vollert1947, 43). Meanwhile, despite its obvious promise, the formal distinction largely languishes in obscurity in contemporary theology.

I have rehearsed this story to provide a bit more context for Beall's proposal. The upshot of this discussion should be clear: there is an obvious problem, it is a serious problem, and there is nothing like a clear consensus on the right solution to that problem. Enter Beall's contradictory Trinitarianism.

The promise of Beall's contradictory Trinitarianism

The problem is obvious, and it is severe. It is a problem produced by the conjunction of orthodox Trinitarian theology and the doctrine of divine simplicity together with classical logic. Given the problem, as we have seen, many theologians develop various strategies in search of solutions. But other theologians and philosophers, concluding that they have looked in vain for adequate solutions, reject either Trinitarian orthodoxy or the doctrine of simplicity. Beall takes a different way out; he argues that we should reject classical logic in favour of his preferred non-classical logic. Beall affirms Trinitarian orthodoxy and does not deny the doctrine of simplicity (e.g. Beall Reference Beall2023, 84–86). To be clear, he is not giving up on Trinitarian orthodoxy; on the contrary, he says that ‘Trinitarian identity is an intricate relation at the core of divine reality – at the very core of God’, and it is a relation that ‘reflects the oneness of God given that each of Christ, Father, and Spirit is truly identical to the one and triune only being (viz, God)’ (Beall Reference Beall2023, 39–40). The ‘oneness of God’, he says, is ‘only part of the full truth’, for the full truth includes ‘the threeness of God given that each of Christ, Father, and Spirit are pairwise non-identical to each other’ (Beall Reference Beall2023, 40). Thus ‘Trinitarian identity exemplifies unity in trinity and trinity in unity’ – and it ‘does so via contradiction’ (Beall Reference Beall2023, 40). Indeed, he says that ‘the triune god (viz., God) to whom Son is identical, to whom Father is identical, and to whom Spirit is identical, is truly described only via contradiction’ (Beall Reference Beall2023, 37, emphasis mine).

So Beall affirms that the doctrine is contradictory, and he also affirms that it is a true contradiction. To do so, he relies upon the system of logical consequence known as First Degree Entailment (FDE). Notably, this system ‘contains no predicates; it contains only sentential connectives and two first-order quantifiers’; accordingly, it ‘contains no necessity predicate, no possibility predicate, no truth predicate, no falsity predicate’, and ‘not even a validity predicate’ (Beall Reference Beall2023, 40). And, importantly for our purposes, it contains ‘no identity predicate’ (Beall Reference Beall2023, 40). No, logic cannot do all that, and it is mistaken to expect so much of it. Instead, we should understand that logical consequence or validity amounts to ‘absence of counterexample’ (Beall Reference Beall2023, 43). Beall's FDE thus rejects most of what are commonly taken to be valid patterns of inference – notably modus ponens (‘detachment’), modus tollens, and disjunctive syllogism – as logically invalid. Instead, the proper notion of entailment is centred on DeMorgan behaviour – and, crucially, it allows for the presence of both ‘gaps’ (where a predicate is neither true nor false) and ‘gluts’ (where a predicate is both true and false). And, finally, the payoff: FDE includes neither reflexivity nor transitivity.

To recap, the problem is generated by the conjunction of TRIN and SIMP along with classical logic; given the transitivity of identity, we are left with the unfortunate conclusion that the Father is identical with the Son with whom he is not (supposed to be) identical. Theologians in the tradition have worked to locate a solution, but theologians and philosophers who do not find those solutions acceptable often reject either TRIN or SIMP or even both. Beall urges us instead to reject classical logic and to adopt FDE as the preferred account of logical consequence. By doing so we solve – or, rather, simply avoid – the problem.

Further reflections, with some questions

Beall relies on FDE to allow for the possibility of ‘gluts’ – and thus to account for the doctrine of the Trinity as glutty theology. He seems non-committal about the possibility of ‘gaps’; as he puts it, ‘the extent to which the true theology is gappy is largely an open question’ (and thus would seem not to rule out LP, which rejects gaps but allows for gluts, and thus retains reflexivity and symmetry while rejecting transitivity) (Beall Reference Beall2023, 30; cf. Beall and Logan Reference Beall and Logan2017). It is important to be clear, however, that Beall's account does not completely rule out the validity of modus ponens, modus tollens, disjunctive syllogism, and other familiar patterns of reasoning (including exclusion and exhaustion). He draws a ‘critical distinction’ between ‘logical invalidity and extra-logical or theory-specific invalidity’ (Beall Reference Beall2023, 31). And he says that while (on FDE and K3) there are portions of reality where there may be gaps and while (on FDE and LP) there are similar portions of reality where there may be gluts, so also there are portions of reality – ‘indeed, by far the most familiar’ chunks of reality – where ‘true theories have neither gaps nor gluts’ (Beall Reference Beall2023, 51, emphasis original). Such true theories with neither gluts nor gaps are ‘shrieked’, and ‘to shriek a theory is to shriek all predicates in the language of that theory’ (Beall Reference Beall2023, 33). Note that if the theory is shrieked, then ‘all predicates’ in the theory are shrieked; the shrieked predicates in a given theory all come shrink-wrapped together as a package deal.

So how are we to think about the orthodox Christian doctrine that God is both Triune and simple? Even granting Beall's overall case against classical logic and for FDE, we still ask: is the doctrine of the Triune God to be shrieked or is it not? Beall's answer is clear: no, it is not to be shrieked.

But if it is not to be shrieked, several consequences seem to follow. First, it becomes much harder to argue either for or against the traditional doctrine. Consider an old argument against the creedal doctrine. In the late fourth century, the Eunomians famously argued against pro-Nicene theology from the doctrine of simplicity. As Lewis Ayres observes, for the Eunomians ‘the Son has a clearly subordinate status; Eunomius assumes that ingenerate defines God in a unique way: God's unity and simplicity imply that ingenerate is the only characteristic of God’ (Ayres Reference Ayres2004, 147). Eunomius's argument, (over-)simplified, amounts to

1. (1) If the doctrine of simplicity is true, then there can be no distinction – and thus no distinction between generate and ingenerate – in the divine nature.

2. (2) The doctrine of simplicity is true.

3. (3) Therefore, there can be no distinction between ingenerate and generate in the divine nature.

Whatever one does with such an argument (Gregory of Nyssa drew a distinction between the divine nature which is simple and the persons who subsist in that nature, thus defining simplicity so that it does not include or entail SIMP), it is obvious that it is a modus ponens argument. Such arguments, which have abounded among critics of orthodoxy both ancient and modern, would seem to be complete non-starters if we accept FDE and deny shriekage. And, because such arguments rely upon familiar notions of detachment, identity, exhaustion, exclusion, etc., so would all (or at least most) arguments against the doctrine of the Trinity, the doctrine of simplicity, or their conjunction (at least in my judgement).

Defenders of the traditional doctrine might welcome this consequence, but loss of access to such familiar patterns will also impact arguments for the orthodox account. Consider Gregory of Nyssa's counterattack against the Eunomians. Again (over-)simplifying, his argument can be summarized as:

1. (4) If the doctrine of simplicity is true, then there can be no gradations of divinity or divisions within the divine nature.

2. (5) The doctrine of simplicity is true.

3. (6) Therefore, there can be no gradations of divinity or divisions within the divine nature (for further discussion and references, see McCall Reference McCall, Crisp and Sanders2014, 44–51).

Again, it should be clear that this argument proceeds by modus ponens (however one judges its soundness). And so (by my lights at least, but I do not think that this is controversial) do many arguments in favour of classical orthodoxy and against its rivals in the Christian tradition; such arguments routinely assume identity, detachment, exclusion, and exhaustion and regularly employ modus ponens, modus tollens, disjunctive syllogism (and their classical-logical and shrieky cousins).

Taking the route of FDE allows the Trinitarian to avoid transitivity and the problems that it brings – and thus to defend orthodoxy. But if taking the route of FDE offers protection to the Trinitarian, doing so also cuts off the Trinitarian from important traditional lines of argument for the doctrine – and thus undercuts the case for orthodoxy. Without the classical-logical (or theory-specific shrieked) arguments, it is much harder to know how we are to evaluate the case for and against various theological proposals. It may be too much to say that we are left in the dark without a light, but it is not clear just how we are to proceed in assessment of competing doctrinal proposals. How is the case for the orthodox account – or, for that matter, the Eunomian alternative – to be evaluated? Sans such arguments, why should we conclude that the orthodox doctrine is glutty rather than gappy? Are we left on a one-way route to theological ‘agnostaletheism’ (cf. Parsons Reference Parsons1990; Restall Reference Restall, Caert and Hjortland2015)?

Those who accept the orthodox doctrine but who also either deny classical logic or accept FDE but then deny that the orthodox doctrine should be shrieked find themselves in the very odd and awkward position of holding that the orthodox conclusions of the faith are correct while the very arguments that brought the church to those same conclusions were faulty and flawed. What Philip-Neri Reese says about Christology seems entirely appropriate here too: ‘For all the arguments that laid the ground for, directly contributed to, and subsequently defended conciliar teaching presupposed that the correct Christology requires consistency’ (Reese Reference Reese2023, 13). As Reese argues,

Indeed, Reese says,

Mutatis mutandis, we can say the same for the doctrine of the Trinity. Moreover, it is not merely the case that the major patristic, medieval, and early modern theologians used such modes of argumentation; as I have argued elsewhere, such patterns are already evident in the teachings of Paul and Jesus (see McCall Reference McCall2021, 198–205).

Of course to point out that something is odd and awkward is not to raise a fatal objection. But it does seem weird to say that some theory X is true but deny that the extant arguments for X hold any water. This raises some questions: without the search for consistency and the traditional arguments that motivated the development of the orthodox doctrine, would we even have that orthodox doctrine? Why should we hold to that doctrine as true – why not, instead, leave open the possibility that it is gappy (rather than glutty) and default to theological agnostaletheism?

Second, it seems that Beall's proposal would leave us unable to say all that we should say about God and will have us struggling to move forward in the further development of doctrine. Meghan Page has argued that Beall's Christological proposal, as it stands, is faced with a dilemma ‘between heresy and inadequacy’ (Page Reference Page2021). It may be that the non-classical proposal is simply underdeveloped, but it seems to me that Page's point about Christology applies to the doctrines of Trinity and simplicity too. Christians have not been content merely to affirm the creedal and confessional doctrine of God, for part of the task of Christian theology is to move forward. Thus Christian theologians have also made affirmations (and denials) based upon that venerable doctrine of the simple and Triune God, divine simplicity provides resources for thinking about the relation between God and morality (e.g. Stump Reference Stump2003, 127–128), the doctrine of the Trinity rules out various purported explanations of the atonement (e.g. McCall Reference McCall2012), and so on. But many of these developments have proceeded by way of classical logic (or, perhaps, appropriately shrieked non-classical logics) – and, indeed, in at least some cases such theological development would seem to require such logic. This leaves us with another question: should theological developments that have occurred ‘downstream’ of orthodox Trinitarianism be vacated if those developments are based upon arguments made via classical logic (or, alternatively, via theory-specific shrieked accounts)?