The Ad hoc Nature of Impure Relative Identity Trinity Theorizing.
A couple days ago, I was honored to be given a chance to speak to the Muslims at the Exploring Answers Podcast; and their audience. The topic was on the Logical Problem of the Trinity. It’s a project that I’ve decided to take more time to study. The issue can be roughly stated as a dilemma. Either, when we say each person of the Trinity ‘is’ God, we either mean the ‘is’ of predication (such as ‘Bill Clinton is old’) or the ‘is’ of identity (Bill Clinton is the 42nd President). The former seems to result in the idea that there are three gods (tritheism) the latter that there is no difference between the three persons (that they’re all just synonyms for the same being).
For more of an in-depth explanation, you can watch my video here where I provide my own answer for the dilemma. My answer is to posit that God has relative identity relations, and not the absolute ones we would attribute to Bill Clinton and his various personas. This is the answer Peter Van Inwagen provides in his essay And Yet There are Not Three Gods But One God.
One of the issues is how can one show that such identity relations are metaphysically possible? There are roughly two approaches one can take, the first approach is to deny (or at least give possible doubt) as to whether or not there are absolute identity relations. This is the approach taken taken by pure theorists such as Peter Geach . Pure theorists, according to Michael Rae, deny any and all inferences that follow from absolute identity.
However, impure theorists are okay with classical identity so long as you can make the following sorts of inferences in certain cases,
x is an F, y is an F, x is a G, y is a G, x is the same F as y, but x is not the same G as y 
Impure theorists offer metaphysical scenarios that are suppose to be beyond the ability of absolute identity to decipher. One example of this would be Rae himself, and J.E. Brower. They offers the following example,
In the region occupied by a bronze statue, there is a statue and there is a lump of bronze; the lump is not identical with the statue (the statue but not the lump would be destroyed if the lump were melted down and recast in the shape of a disc); but only one material object fills that region .
The idea of this view is to explain why a statue can be the same lump of bronze as the disk, but not the same shape and provide some basis for thinking these RI moves can be valid. They go on to note that one of the benefits of this view is And they go into why their impure method is superior to pure-RI and Social-Trinitarian strategies,
It should also be clear how our solution meets the other desiderata. Unlike (pure) Relative-Identity solutions, ours is compatible with the claim that classical identity exists and is as fundamental as any other sameness relation... Moreover, it supplies an explanation for why ‘x = y’ does not follow from’ x is the same God as y’. Unlike Social-Trinitarian strategies, on the other hand, ours is clearly compatible with the view that God is an individual rather than a society, and that the Persons are not parts of God (and hence satisfies 03) 
Unlike Rea and Brower who offers a commitment to RI given our theorizing of composites, or Geach and Inwagen who are either agnostic or hostile to absolute identity, I think, at this part of my Trinitarian journey, the best approach — following my discussion on the EA show — is to let mystery have some explanatory role. This is not to appeal to a sort of mysterianism where the propositions are an apparent contradiction despite seeming like one. I think that we want to understand what we are saying when we recite the creeds. We should show that our claims are self-consistent in a system of logic.
Rather, by mystery, I mean the view that the human intellect is not one which could, in principle, discover on its own, The persons of the Trinity are the same infinite being as one another. Christians should claim that the persons of the Trinity have a relative identity relation which is Sui Generis, that is, of its own kind. While this seems to run into issues of ad hoc reasoning — positing a proposition added to a thesis to save it from scrutiny — I think we need to consider a forgotten part of the Trinity doctrine.
The Trinity is a mystery, in some sense, and must be accepted as a mystery as a matter of doctrine for every Catholic. Ludwig Ott says,
Human reason cannot fathom the mystery of the Blessed Trinity even after the dogma has been revealed by God (sent. Fidei Proxima) .
Ott explains that a belief which is Fidei Proxima is,
A Teaching proximate to Faith…[it] is a doctrine, which is regarded by theologians generally as a truth of Revelation. but which has not yet been finally promulgated as such by the Church .
The denial of which ends in one accepting a position which is,
“Proposition Proximate to Heresy (propositio heresi proxima) which signifies that the proposition is opposed to a truth which is proximate to the Faith (Sent. fidei proxima) .
But why? Why is this something to be regarded as a mystery? In principle, there needs to be some reason why certain truths are ones which we could not reason to a priori or from the nature of the world. Human beings reason about God through their senses, reason and God’s creation. But if human beings seem committed to classical identity relations, then it would stand to reason that even positing such an argument could not in principle work without revelation.
The Relative Identity appeal needs to be presented as not just an explanation for the Trinity’s coherency, but its very mystery. The appeal is not ad hoc because it is not just seeking to explain the doctrine’s coherency, but its very allusiveness of the human intellect to deduce. If one is purposing the Logical Problem of the Trinity as an internal critique, then one has to critique the whole of the doctrine, and not merely the identity relations. As it stands, relative identity should be chosen as the best option because it explains both consistency and inaccessibility.
While there are more metaphysical issues to deal with, I think the objection that impure and God-only relative identity theorizing is an ad hoc appeal can be reasonably addressed without subscribing to a pure RI theory. Nor do we need to providing some metaphysical argument — such that Rea and Brower do — that requires others to accept the possibility of objects fulfilling RI relations.
However, I think this approach is limited. The notion that the Trinity is a mystery, one only known through revelation, is true given that it is taught as Catholic doctrine. If a Protestant or an Eastern Orthodox Christian were to take this approach, then they would have to provide a justification for why the doctrine is per se mysterious.
Furthermore, one could never, on this reading, make any Trinitarian arguments for the doctrine, without appeal to special revelation. But it seems to me a fair price worth paying. I’ll be bringing this point up in a future video, but I think it’s a point I’d be interested in hearing feedback about.
Did I avert the charge of ad hoc reasoning? Please, let me know.
 Geach, P. T. (1972). Identity Theory. In Logic Matters (pp. 238–247). Essay, Basil Blackwell.
 Rae, M. C. (2003). Relative Identity and the Doctrine of the Trinity. Philosophia Christi, 5(2) (p. 433) https://andrewmbailey.com/trinity/Relative%20Identity%20and%20the%20Doctrine%20of%20the%20Trinity.pdf.
 Brower, J. E., & Rea, M. C. (2009). Material Constitution and the Trinity. Faith and Philosophy, 22(1), 57–76. (p. 62)
 Ibid, (p. 70).
 Ott, L. (1954). Fundamentals of Catholic Dogma. (J. C. Bastible, Ed., & P. Lynch, Trans.) Fort Collins, CO: Roman Catholic Books. Retrieved from https://cornellcatholic.org/documents/2020/12/fundamentalsofcatholicdogma.pdf. (p. 74).
 Ibid, (p. 10)