The Incoherence of the Trinity:A Fitch-style Natural Deduction

So I have been debating a Christian philosopher-friend on the metaphysical possibility of the Trinity recently, and it has helped me clarify my thoughts here. I’m thinking that the only way to remotely parry the claim that it is metaphysically impossible is by appealing to something like trope theory. I tend to think such a solution doesn’t work, but I have to think about it more. In any case, it seems like the Trinitarian is forced to appeal to trope theory, or at least forced to reify the the deity of the persons into a particular. I don’t see how a Platonist, for example, can get out of the conundrum here. I’ll explain and then welcome your thoughts on this.

Here’s a basic Fitch-style deduction I made in LaTeX. It represents the translation of the standard Trinitarian septad we are all familiar with, along with a proof that it is contradictory.

Now, my contention is that at least the translations of (2) - (7) are unassailable. Let’s start with (5) - (7). They can be proved by Leibniz’s law and other Trinitarian propositions like “the Son is begotten” but “the Father is not begotten,” etc. Unless one is going to absurdly say that absolute identity is unintelligible a la Geach, then one has to take (5) - (7) to be true expressions of the Trinitarian dogma.

So how about (2) - (4)? Well, that’s just the standard way to translate predicative statements like “Joe is human.” Creedal statements seem to make it clear to me that the “is” here is an “is of predication.” The “is” here can’t be the “is” of numerical identity because then we can easily derive “the Father is begotten of the Father” and get a contradiction. Besides, however you want to cash out properties, of course each person has the property of <being God>. Even the relative identity theorist – I believe both strong and weak theorists – have to agree that (2) - (4) are faithful expressions of the Trinity. The relative identity theorist affirms propositions like “f is the same G as s,” and this implies Gf and Gs. After all, “x is the same F as y” is true only if Fx and Fy. So I honestly don’t see how a Trinitarian can reasonably object to the translations of (2) - (4).

So that brings us to (1). In my humble opinion, the Trinitarian has to challenge this translation. But this is a completely standard way of translating “there is only one F.” If I wanted to translate “there is only one human” into first-order predicate logic, I would just replace the ‘G’ with a ‘H’, for example. If I wanted to translate “there is only one turtle,” I would just replace the ‘G’ with a ‘T.’ It seems like the most natural translation. Be that as it may, I’m thinking that perhaps the Trinitarian can say that “there is only one God” can be translated as “there is only one divine nature” here. One could say that instead of my translation, of “There is only one thing that is G,” (1) should be translated as “there is only one divine nature.” On Platonism, this would just mean that there is only one property of <being God>. But this wouldn’t be a correct translation. There’s also one human nature on Platonism. There is only one property <being human>. But saying that “there is only one human” means that “there is only one property <being human>” is clearly a mistranslation. When we ask “how many Fs are there?” we are asking how many things instantiate F-ness, not how many properties of <being F> there are. There’s obviously only one such property. So this translation, while avoiding a contradiction, is clearly a mistranslation on Platonism. But what about on nominalism? Well, nominalism will run across the same mistranslation problem as Platonism here unless deity or Godness is reified as a particular. And if it is reified as a particular, then we are talking about something like trope theory. On such a Trinitarian trope theory, the Father’s Godness is a trope which is numerically identical with the Son’s Godness, which is numerically identical with the Holy Spirit’s Godness. (I believe Leftow holds to something like this.) And since a trope is just a proper part of a primary substance, the Godness trope is a proper part of each of the Father, Son, and Holy Spirit. (For those who believe in Divine Simplicity, this will not be an acceptable solution.) I’m not sure this solution is metaphysically possible, but it seems like the best and, quite frankly, the only remotely plausible solution to me – but of course this may just be because the contradiction has been obfuscated by an appeal to strange metaphysics.