Stress-testing the
mind of Christ

Where a recovering ex-atheist rams the Bible into other worldviews to see what breaks (note: Scripture cannot be broken)

Square circles and the Trinity, part 4: Steve’s argument

In this series, I interact with the criticisms of the Trinity forwarded by Steve Zara in our recent debate, using them as a springboard to examine this important doctrine and demonstrate that it is not intrinsically self-contradictory.

This is part 4 of 4. It dissects the argument Steve makes against the Trinity, showing where it fails and why.

« continued from ‘Square circles and the Trinity, part 3: the law of identity’

Having now looked in a little detail at being and identity, and argued that, for a Christian, these terms must be at least potentially non-univocal, I think we’re in a good position to examine Steve’s argument.

IV. The formal argument against the Trinity

The quote below has been slightly corrected for grammar and readability—

[…] The Trinity is a logical contradiction. This is a mereological (part/whole) ontological error. Let’s look at this formally:

  1. D is defined as { A , B , C } (God defined as {Father, Son, Holy Spirit})
  2. ( A = D ) AND ( B = D ) AND ( C = D ) (God wholly present as Father and wholly present as Son and wholly present as Holy Spirit)
  3. A = B = C = D (from (2), because “=” is a transitive relation)
  4. D := { A , A , A } (from (1) and (3))
  5. D := { B , B , B } (from (1) and (3))
  6. D := { C , C , C } (from (1) and (3))
  7. D := [ D , D , D ] (from (1) and (3))
  8. D := { { A , B , C } , { A , B , C } , { A , B , C } } (from (7) and (1))

Conclusion 1: statements (4)–(8) contradict statements (1) and (2).

So, 1 and 2 lead to contradictions—which means that tri-unity is logically impossible. In statement (8), the elements of the sets included in D are defined again as { A , B , C }, in which the “A”s, “B”s, “C”s are again said to be equal to D and thus defined as { A , B , C }—it’s an infinite regress.


Steve prefaces this argument with the explanation that the Trinity is “a mereological (part/whole) ontological error”. I don’t know much at all about mereology, except that it’s a theory in mathematics and ontology which describes parts and their relationships to wholes. It replaces talk of “sets” of objects with talk of “sums” of objects, where the objects themselves are no more than the various things which make up wholes. I also know that mereology is not necessarily accepted by those who study these things; so from the start it seems like a suspect basis for formulating an argument against the Trinity. Even if the argument is valid, this might just prove that mereology does not comport with reality, as opposed to proving that the Trinity does not.

Furthermore, if mereology is about describing parts and wholes, and if it describes wholes as the sum of various parts, then from the outset there is likely to be a problem. God is not comprised of parts. Specifically, as I discussed in the previous article, the persons of God are not parts of God; they are fully God. So there appears to at least be a natural antipathy between mereology and Christian ontology; and potentially a significant disconnect. The way that mereology models reality may well be incapable of modeling trinitarian ontology. Trying to do so will result in contradictions and errors within the mereological framework. But that’s no reason to assume that trinitarianism is faulty; quite the opposite. Given that we’re under no compulsion to accept mereology in the first place, since it is just one man-made theory of ontology; and given that we’re under every compulsion to accept God’s own testimony about his being, any argument against that testimony which relies on the man-made theory is—well, rather weak to say the least.

Thus, given how mereology models reality (as I understand it) I think there is good reason to reject Steve’s argument without even looking at it. Still, I’m going to look at it anyway.

Premise (1)

D is defined as { A , B , C } (God defined as {Father, Son, Holy Spirit})

As I start considering this, I find myself wondering exactly what it means. For all its formality, this premise is very ambiguous. (This is a problem, because the rest of the argument hangs on it.) It’s hard to be sure exactly what is meant here, or how sound it is theologically. In what way is God defined as the Father, Son, and Holy Spirit? Judging from what little I know of mereology, I assume that A, B, and C are meant to be parts of the whole, D. But this immediately runs into problems because the Father, Son, and Spirit are not parts of God. God is not a whole comprised of three person-parts. Rather, the Father is God, the Son is God, and the Spirit is God. As I said just above, mereology appears to be ill-equipped to describe this.

Further, what is the context of this definition? Other things define God aside from the Father, Son, and Spirit. It’s obviously an ontological definition (that is, it is defining God’s being)—but even ontologically speaking there are other things than the persons of the Trinity which can be said to define God. There is knowledge, for example. There is spirit.

That said, I take it that Steve is couching the argument in the context of persons—so, whatever else defines God, the one essence, God (D) is defined as three persons (A, B, C). To be fair to Steve, he probably thought this was obvious—but I found the ambiguity here, and in the rest of the premises, made it very difficult to accurately interpret the argument.

Premise (2)

( A = D ) AND ( B = D ) AND ( C = D ) (God wholly present as Father AND wholly present as Son AND wholly present as Holy Spirit)

Unfortunately, this premise is as ambiguous as the first. What does the equals sign represent here? Generally, “=” indicates actual equality—that is, identity, such as in 1 = 1 or 2 = ( 1 + 1 ). If so, A = D would be describing equality or sameness of being. The English translation of the notation seems to bear this out, although it’s frustratingly unclear as well. The term “wholly present as” is presumably supposed to mean “shares fully in being”; so the Father shares fully in being God, as do the Son and the Spirit. So it seems that what this premise is saying is that the Father is wholly God, the Son is wholly God, and the Spirit is wholly God; meaning that they are all of the same being. This is true under trinitarian theology, so let me continue—

Premise (3)

3. A = B = C = D (from (2), because ‘=’ is a transitive relation)

Here the argument runs aground. It tries to leverage the transitive relationship of identity, which I mentioned in part 3 of this series, to show that, because for instance A = D and C = D, therefore A = C. (In other words, the Father is the same as the Spirit.) If this were the case, it would actually be quite sufficient to disprove trinitarianism, because it would prove modalism instead. The fact that you can then substitute A for C because A = C, and get a statement like D = { A, A, A }, and so invoke an infinite regress, is really just gravy. Premises (4) onward rely on premise (3) to make their case, but all that’s actually needed to disprove the Trinity is premise (3) itself.

There is a major difficulty with this premise, though: it assumes a univocal theory of being. It assumes that the statement “A = D” is completely unambiguous and describes only one possible relationship. We, as Christians, have good reasons for holding to a non-univocal theory of being, and so we have good reasons for denying that the equals sign represents an exclusive relationship. In fact, since this argument is very much similar to the modalist argument I showed previously, it makes no traction against a Christian who holds to the thesis that being—at least as it relates to God—is not univocal in the way we normally assume.

Therefore, in a sense, Steve’s argument is being made against a strawman. I have no reason to accept premise (3) as is because, under my view, it is an incomplete or non-exclusive identity statement. “A = B = C = D” is not a sufficiently complete statement of identity to avoid committing a category error. That is, “A = B”, for example, implies that A actually is B in such a way that it cannot also be not-B. It assumes that there exists only one category of being; whereas I deny this. To properly represent Trinitarianism, premise (3) should be expanded as follows:

  1. A = B = C = D where “=” represents sameness of essence.
  2. A ¬ B ¬ C ¬ D where “¬” represents sameness of persons.

The potential for confusion here is exacerbated by the fact that D actually represents one of the senses of God’s being (essence), while A, B, and C all represent another sense of being (person). So inserting D into premise (b) above is very confusing, because D is not a person of God at all. This is just a byproduct of trying to use univocal language to describe non-univocal being: you confuse yourself.

Again, let me reiterate that a non-Christian has no reason to accept NUB as a thesis. He has no reason to believe that being is or can be non-univocal, and so he will most likely accuse me of arbitrariness in “inventing” an answer. Bear in mind, though, that it only seems arbitrary if one presupposes that God does not exist and has not testified to his nature through Scripture. If he does exist and has so testified, my defense is not arbitrary at all, and constitutes a sufficient defeater to the objection of self-contradiction—at least inasmuch as that objection must be leveled as an internal critique of Christian doctrine.

In other words, if you presuppose that Christianity is false, this defense is worthless. But since you’ve already presupposed that Christianity is false, your objection to the Trinity on grounds of contradiction is worthless as well, since it’s no more than question begging. If, on the other hand, you presuppose for the sake of argument that Christianity is true, so as to show that, even on its own grounds it contradicts itself, then the NUB thesis defeats this claim. If Christianity is true, then its entire epistemic framework provides fully adequate grounds for believing that (i) God is three and one; and (ii) that our epistemic situation is such that (i) is largely incomprehensible to us.

The other premises

Since the conclusion of Steve’s argument rests on its first three premises, and specifically on the transitive relationship of identity which establishes premise (3), there isn’t any need to examine it further. If NUB defuses premise (3) by showing that it’s equivocal and commits an implicit category error, then the argument fails as an internal critique of Christianity. And, as an external critique, it succeeds only by begging the question against the Christian, and is therefore fallacious. In either case, it is defeated.

1 comment

  1. Frank Armstrong


    I’ve really enjoyed reading the debate on the nature of the Trinity you’ve had with Steve Zara, and while I believe in the Trinity I’d like to share with you the difficulty I have in accepting your illustration in the section of part two titled “Reconciling the square and the circle.” You offer the example of the Flatlanders who cannot understand the concept of a cylinder, and suggest that the difficulty of created beings comprehending God is similar to 2-dimensional creatures in a 2-dimensional world comprehending a 3-dimensional object. I think the problem with the illustration of perceiving the cylinder as a square (from the side) or perceiving the cylinder as a circle (from above) is that it really only explains perception, and not reality (regardless of how it is perceived).

    What I mean is that when you offer the formulation of talking about cylinders in those terms (square or circle), the Flatlanders are only talking about their perceptions, not the thing itself. Because a cylinder is not a square nor is it a circle in fact. It is not 2-dimensional, whereas our perception of it may be. Now, the difference is that God is in fact one, and He is in fact three. Neither His one-ness nor His three-ness are perceptions; they are facts regardless of how we perceive Him/Them. And actually, that’s where the difficulty lies. We have zero ability to perceive such a thing that is one and three at the same time. And I wonder if the three laws don’t break down when we try to apply them to God.

    Thanks for your consideration.

    In Christ,

    Frank Armstrong

  I don’t post ill-considered articles and I don’t sponsor ill-considered comments. Take a moment to review what you’ve written…