Introduction. In my immediately previous post, the “Thomas Aquinas: Beyond Aristotle’s Aristotelian Conception of the Numerical” of April 1, 2014, the discussion of Thomas Aquinas’s understanding that numerical or arithmetical terms such as “three” have application to non-physical reality quite naturally led me to think once more about orthodox Christianity’s doctrine of the Trinity. As I was writing the post, moreover, I recalled having read, a week or so earlier, an essay in the April, 2014, issue of First Things, the “Uneasy Grace” of Dr. Meghan Sullivan, a philosophy professor at the University of Notre Dame, for one of several things in the essay which in particular caught my attention was her statement that (p. 48):
Our best logic might seem to show that doctrines like the Trinity are contradictory, because nothing can be both one and three simultaneously.
There is then, on the one hand, the logical doctrine that nothing can be both one and three, in the same respect, we have to add, and at the same time, or simultaneously. There is also, on the other hand, the orthodox Christian doctrine that one being, God, is both one god and three persons. To make the issue more explicit: God’s being one god and three persons is also God’s being one god and three gods, because each of the three persons is a god. And God’s being one god and three persons is therefore also being one and three, in the same respect and at the same time.
Now, in the course of her essay, Sullivan sets out quite nicely the options which she sees to be available to someone facing the two doctrines. I will here present those options a bit more starkly: There are but four possibilities:
The logical doctrine is true and the doctrine of orthodox Christianity is true.
The logical doctrine is true and the doctrine of orthodox Christianity is not true.
The logical doctrine is not true and the doctrine of orthodox Christianity is true.
The logical doctrine is not true and the doctrine of orthodox Christianity is not true.
Sullivan adopts the first, albeit somewhat provisionally, holding that (p. 51) “it is rationally acceptable to go on believing both until he [sic] finds a way to break the stalemate.” I, however, hold, and not provisionally, that the second option is the one which it is rationally necessary to adopt. I therefore adapt the statement quoted above by substituting a “proves” for her “might seem to show,” while I at the same time restrict its scope so that it includes but the Trinity. Thus:
Our best logic proves that the doctrine of the Trinity is inconsistent with monotheism, because nothing can be both one god and three persons simultaneously.
More precisely, I will in the present post prove that, given the logic of identity and assuming as an hypothesis the thesis of monotheism, one god cannot be two persons; I take it as evident, that is, that a proof that one god cannot be two persons will be accepted as a substitute for the longer proof that one god cannot be three persons, or indeed four or more.
I offer the proof in two versions. The first version sets it forth in what is almost ordinary English, albeit an English “regimented,” as the logicians say, in such a way as to remove the ambiguity of really ordinary English and to make every inferential step obvious, even painfully obvious. The second version is even more fully formalized.
The Proof in a Regimented English. The proof takes as given, i.e., as evident, two basic principles of the logic of identity. The one is the Principle of the Symmetry of Identity, that:
1. For any existent x and any existent y, x is identical with y if and only if y is identical with x.
The other is the Principle of the Transitivity of Identity, according to which:
2. For any existent x, any existent y, and any existent z, if x is identical with y and y is identical with z, x is identical with z.
Those of a traditional training may recognize the Principle of the Transitivity of Identity as but a different expression of the Principle of Triple Identity.
The proof next assumes, along with orthodox Judaism, orthodox Christianity, and orthodox Islam, the “Thesis of Monotheism,” according to which there is at least one and at most one divine existent, i.e., at least one and at most one god. I think I should point out that I do not personally make that assumption, much less take it as a given, for I am not aware that any purported proof that there is at least one divine existent or god has in fact demonstrated that to be the case. Neither, let me hasten to add, am I aware of any proof that there is not at least one divine existent or god. At this point in my philosophical thinking, I am then neither a theist nor an atheist. Yet I am not a skeptic, for I am also not aware of any proof that that it cannot be known whether or not there is a divine existent or god. Finally, in the interest of fullest of disclosure, I will state that I hope, even fervently, that there is a god and that I and we will be at some point able to know that there is.
All that being said, I’ll spell out the assumed monotheist thesis more precisely, so that we have:
3. There is an existent x such that x is a god and for any existent y, if y is a god, then y is identical with x.
The first conjunct, “There is an existent x such that x is a god,” is one way of affirming that there is at least one god, while the second, “for any existent y, if y is a god, then y is identical with x,” presents in the context of 3 as a whole the affirmation that there is at most one god.
The proof then further assumes the Thesis of the (Two) Distinct Divine Persons, the thesis that:
4. There is an existent x and there is an existent y, such that x is a person and y is a person, that x and y are not identical, and that x is a god and y is a god.
It needs to be understood that the proof at hand is one demonstrating that, given the two principles set forth in 1 and 2 and assuming the monotheist thesis set forth in 3, it is just cannot be the case that there are two divine persons. It will do so as a reduction ad absurdum argument, that is, by showing, step by step, that adopting propositions 1 through 4 leads directly to a contradiction and thus that the four propositions cannot together be true.
The premises set, let us turn now to the proof’s inferential steps. From 3, engaging in the act of inference which logicians identify as existential instantiation, we begin by assigning to the existent affirmed to be a god the name “God” and set down the following statement:
5. God is a god and, for any existent y, if y is a god, then y is identical with God.
Next we engage in the act of inference known as commutation. This is the inference, which I will take to be obviously valid, in accordance with which, given the proposition “It is cold and it is cloudy,” we can conclude, “It is cloudy and it is cold.” Similarly, given 5, we can conclude that:
6. For any existent y, if y is a god, then y is identical with God and God is a divine being.
We now make use of the act of inference known as simplification. This is the inference, which I will take to be equally valid, in accordance with which, given the proposition “It is cold and it is cloudy,” we can conclude, “It is cold.” So, using simplification, we can conclude from 6 that:
7. For any existent y, if y is a god, then y is identical with God.
We have covered about one quarter of the steps needed to complete our proof. But now, from 4, using existential instantiation and applying two new names, “God the Father” and “God the Son,” to, respectively, x and y, we derive:
8. God the Father is a person and God the Son is a person, God the Father is not identical with God the Son, and God the Father is a god and God the Son is a god.
Now, using commutation, we derive, from 8:
9. God the Father is a god and God the Son is a god, God the Father is a person and God the Son is a person, and God the Father and God the Son are not identical.
From 9, using simplification, we derive:
10. God the Father is a god and God the Son is a god.
From 10, once again by simplification, we derive:
11. God the Father is a god.
From, again, 10, we derive, via commutation:
12. God the Son is a god and God the Father is a god.
So, then, from 12, using simplification, we derive:
13. God the Son is a god.
Next we’ll make use of the act of inference known as universal instantiation, by which, holding, as we do in 7, that, for any existent y, if y is a god, then y is identical with God, we infer that:
14. If God the Father is a god, then God the Father is identical with God.
With both 14 and 11 established, we’ll now make use of the act of inference known as Modus Ponens. This is the inference involved when, understanding the proposition, “If Peter is in Massachusetts, then he is in the United States,” and the further proposition “Peter is in Massachusetts,” to be both true, we infer that it is also true that “Peter is in the United States.” So too, from 14 and 11, we conclude that:
15. God the Father is identical with God.
Similarly, with respect to God the Son: from 7, that, for any existent y, if y is a god, then y is identical with God, we can infer, via universal instantiation, that:
16. If God the Son is a god, then God the Son is identical with God.
And, given 16 and 13, we derive, via Modus Ponens:
17. God the Son is identical with God.
We are now at just a bit past the half-way point in the argument. Let’s go back now to the first of the two givens given far above, in 1, the Principle of the Symmetry of Identity, according to which, for any existent x and any existent y, x is identical with y if and only if y is identical with x. Using universal instantiation, we derive:
18. God the Son is identical with God if and only if God is identical with God the Son.
This, 18, is logically equivalent to the conjunction of the two implications:
19. If God the Son is identical with God then God is identical with God the Son and if God is identical with God the Son then God the Son is identical with God.
Next, using simplification, we derive:
20. If God the Son is identical with God then God is identical with God the Son.
From 20 and 17, using Modus Ponens, we arrive at:
21. God is identical with God the Son.
Given 15 and 21, we can, by means of the act of inference known as conjunction, conclude that:
22. God the Father is identical with God and God is identical with God the Son.
Recalling the second of the two givens given far above, in 2, the Principle of the Transitivity of Identity expressed in 2, we can use universal instantiation to yield:
23. If God the Father is identical with God and God is identical with God the Son, then God the Father is identical with God the Son.
23 and 22 together imply, via Modus Ponens:
24. God the Father is identical with God the Son.
You undoubtedly have seen the problem coming, for, from 8 and using simplification, it follows that:
25. God the Father is a person and God the Son is a person and God the Father is not identical with God the Son.
Commutation yields:
26. God the Father is not identical with God the Son and God the Father is a person and God the Son is a person.
And simplification yields:
27. God the Father is not identical with God the Son.
Then, using the act of inference known as conjunction, we conjoin the proposition found in 21 with the one found in 27, to yield:
28. God the Father is identical with God the Son and God the Father is not identical with God the Son.
We have, then, in steps 4-28 from premises 1-3, a reduction ad absurdum of the thesis set forth in 4; if, then, we accept steps 4-28 from premises 1-3, as we must, we must also then conclude that:
29. It is not the case that there is an existent x and there is an existent y, such that x is a person and y is a person, that x and y are not identical, and that x is a god and y is a god.
In other words, starting from the necessary truths of the logic of identity, assuming then the thesis of monotheism, a thesis common to orthodox Judaism, Christianity, and Islam, and then assuming the thesis of the (two) distinct divine persons, we have arrived at a contradiction and so the reductio ad absurdum foreseen above; the belief, in sum, that the thesis of the (two) distinct divine persons can be conjoined with that of monotheism has been shown to be absurd, quod erat demonstrandum.
The Proof More Fully Formalized
I ask the students of “formal” logic among us to forgive my non-standard use of “Ex” and “Ex,” etc., in the following formalization of the argument, instead of the standard means of expressing the “existential quantifier.” Understand that I do so only because both I do not know how to get WordPress to allow me to use the standard means and, in the context of the present argument, there should be no confusion.
That being said, I will also use:
“Ixy” to abbreviate “x is identical with y”;
“Dx” to abbreviate “x is a divine being or god”; and
“Px” to abbreviate “x is a person.”
1. (x)(y)(Ixy <-> Iyx) {Principle of the Symmetry of Identity}
2. (x)(y)(z)((Ixy & Iyz) –> Ixz) {Principle of the Transitivity of Identity}
3. (Ex)(Dx & (y)(Dy –> Iyx)) {Thesis of Monotheism}
4. (Ex)(Ey)(((Px & Py) & ~Ixy) & (Dx & Dy)) {Thesis of Distinct Divine Persons; Assumption towards Reductio ad Absurdum}
5. Da & (y)(Dy –> Iya) {3, Existential Instantiation}
6. (y)(Dy –> Iya) & Da {5, Commutation}
7. (y)(Dy –> Iya) {6, Simplification}
8. ((Pb & Pc) & ~Ibc) & (Db & Dc) {4, Existential Instantiation}
9. (Db & Dc) & ((Pb & Pc) & ~Ibc) {8, Commutation}
10. Db & Dc {9, Simplification}
11. Db {10, Simplification}
12. Dc & Db {10, Commutation}
13. Dc {12, Simplification}
14. Db –> Iba {7, Universal Instantiation}
15. Iba {14, 11, Modus Ponens}
16. Dc –> Ica {7, Universal Instantiation}
17. Ica {16, 13, Modus Ponens}
18. Ica <-> Iac {1, Universal Instantiation}
19. (Ica –> Iac) & (Iac -> Ica) {18, Logical Equivalence}
20. Ica –> Iac {19, Simplification}
21. Iac {20, 17, Modus Ponens}
22. Iba & Iac {15, 21, Conjunction}
23. (Iba & Iac) ® Ibc {2, Universal Instantiation}
24. Ibc {23, 22, Modus Ponens}
25. (Pb & Pc) & ~Ibc {8, Simplification}
26. ~Ibc & (Pb & Pc) {25, Commutation}
27. ~Ibc {22, Simplification}
28. Ibc & ~Ibc (21, 27, Conjunction}
29. ~(Ex)(Ey)(((Px & Py) & ~Ixy) & (Dx & Dy)) {4-28, Reductio ad Absurdum}
I invite, nay, I beg for discussion.
Until next time.
Addendum, April 6, 2014. As I was drafting the paragraph immediately following the statement I quoted of Sullivan, I planned but forgot to integrate the passage in which Aristotle says (Physics, Book I, Ch. 2, 185b25-27, in Jonathan Barnes, editor, The Complete Works of Aristotle (Princeton University Press, 1984), Vol. II):
Even the more recent of the ancient thinkers were in a pother lest the same thing should turn out in their hands both one and many.
It was in part because I had that sentence in mind that I hesitated over Sullivan’s “nothing can be both one and three simultaneously.”
After Aristotle 
I think it follows logically given the assumptions. But my understanding of the Orthodox doctrine of the Trinity is that the “Godhead” is one essence in three persons. It is not that somehow if you were to isolate ‘the Son’ or ‘The Spirit’ or ‘the Father’ that each would be God, but rather they are a tripersonal essence that forms one “God”.
The distinction is difficult because of the greek terms used.
Homoiousias is almost universally rejected, particularly by Athanasias, but Homoousias was thought to be confusing because it could make all the parts of the trinity Identical. This led Athanasius, upon further reflection to say:
“Finally, in a synod gathered in Alexandria in 362 CE, Athanasius and his followers declared that it was acceptable to refer to the Father, Son, and Holy Spirit as “of one substance” as long as this was not understood as obliterating the distinction among the three, and that it was also legitimate to speak of “three substances” as long as this was not understood as if there were three gods.”(The Story of Christianity, Revised Edition, by Justo Gonzalez. Chapter 17, “The Arian Controversy and the Council of Nicea.”)
If we understand the three persons then to be “parts”(if you will) of the Trinity(though as Athanasius makes clear, they are not distinct Gods) it is only the Divine Essence that appears when the three are present together and in communion.
Of course Catholics and Orthodox Christians have their splits. I think in Eastern Orthodoxy this is actually a slightly easier problem. In Orthodoxy the “Divine Nature” is possessed only by God the Father, and is shared with the Son and the Spirit. In this sense they are non-identical but due to the sharing of nature, one in nature. (Sort of how all humans have a human nature, but are not identical to each other. You might say that the three persons of the Trinity are not distinct Gods but together make up a “Godness”.)
So you could challenge the assumptions of the logical derivation here, but I am not practiced enough in my logic to catch any missteps here. (Though I don’t think there are any.)
Thank you for your thoughtful reply. I’ll reply in turn to two things said in your first paragraph and then to that which is said in the quoted paragraph about the views of Athansias and his followers.
First, then, the word “isolate,” as it more than suggests separation, or even spatial separation, should not be used in this discussion. Rather, what is at issue is whether or not there is one and only one god, God, who is also two or three distinct persons, “distinct” here being understood as equivalent to “non-identical.”
Second, my argument does not apply to any doctrine which does not affirm both that both or all of the distinct persons are God and that there is but one god, God. But it seems to me, that one holding that the divine persons “are tripersonal essence that forms one ‘God’” does, at least so far as I can understand it, affirm both. And it does not seem to me that the view that “the ‘Godhead’ is one essence in three persons” is any better off: “one essence in three persons” would seem to involve the same logical difficulties as “one god in three persons.”
Then, it appears to me that the view of Athanasius, at least as thus presented by Gonzalez, does little more than tell us that, on the one hand, it is “acceptable to refer to the Father, Son, and Holy Spirit as ‘of one substance’” as long as it is understood that that is not to speak of the “one substance” as really one substance and that, on the other hand, it is “legitimate to speak of ‘three substances’” as long as it is understood that that is not to speak of the “three substances” as really three substances.
I think it is fair to say that they are different and the same, but I’m no scholar on the matter.
It seems to me that if you have a triangle, with nonidentical sides XYZ, and non identical angles XY, ZY, and ZX, that all compose one triangle, you don’t have to show that they’re non identical. They simply aren’t a triangle.
The Trinity as far as I understand it, has never been that there is somehow one God that is three Gods, or Three Persons that are a single person, but that there are three persons that are one God. Just as it is not that three triangles make up a triangle, but rather THREE-ANGLES by definition.
This is not to say that each of the angles, in virtue of their connection to the triangle, don’t have triangularity, they do. (in virtue of their connection to the triangle.) This is closer I think to the Eastern view that God shares the divine nature with the Son and Spirit, which is contrasted against the Catholo-protestant version.
Again, I’m no scholar, but that’s what it seems like in my limited studies on the matter.
(If I am allowed to make a second reply to this, I thought of another example as well. To bolster the orthodox position by making those sorts of distinctions)
Augustine in “De Trinitate” gives an analysis of the Triune nature of God where he compares his own actions of the mind’s understanding itself, willing itself, and remembering itself. The actions cannot rightly be said to be those of “Mind” proper, on their own, but together they are fully “mind”.
In addition (oddly enough) Ibn-Tufayl uses a similar argument to secure Mutiplicity and Unity in his “Hayy Ibn Yaqzan” (in relation to the Necessary Existent)
“”[it] was something sovereign, divine, unchanging and untouched by decay, indescribable in physical terms, invisible to both sense and imagination, unknowable through any instrument but itself, yet self-discovered…at once the knower, the known, and the knowing, the subject and object of consciousness, and consciousness itself. There is no distinction between the three, for distinction and disjunction apply to bodies. But here there is no body and physical predicates and relations do not apply.”” (G.E. Goodman translation)
I’m did not include these in my original reply, because I feel they support the idea that something can be multiple and one at the same time, but are not essential passages to the point either. I might be misinterpreting them.
I also thought the latter might be interesting to you. Ibn-Tufayl was Averroes’ immediate predecessor in the same court, and they were both Aristotelian.
Thank you once (twice) again.
In response, first to the second of your three comments, let me begin by saying that I agree, though I hesitate over the word “never,” that the orthodox doctrine of the Trinity has not been one holding that “there is somehow one God that is three Gods, or Three Persons that are a single person, but [rather one holding] that there are three persons that are one God.”
But orthodox Christianity holds does that God the Father is God, the one god, and that God the Son is God, the one god. Given, however, that God is the one and only god, it follows that God the Father is God the Son. If Clark Kent is the man changing his clothes in a telephone booth and Superman is the man changing his clothes in the same telephone booth, at the same time, Clark Kent is Superman.
The analogy you offer between the three angles of a triangle and the three persons of the Trinity fails. Geometry does not affirm that each one of the three angles is the triangle. Orthodox Christianity does, however, affirm that each one of the three persons is God.
Next, turning to the third of your replies: I cannot say much about Augustine’s doctrine of the Trinity. But I can say that if he holds, not only that the divine mind is identical with the one divine being and that the divine will is identical with the one divine being, but also that the divine mind is not identical with the divine will, he too falls into contradiction.
If, however, Augustine holds, again that the divine mind is identical with the one divine being and that the divine will is identical with the one divine being, but now that the divine mind is further identical with the divine will, he avoids the difficulty I have been raising completely. But this, if he means to further identify the divine mind with God the Father and the divine will with God the Son, would of course be to identify God the Father with God the Son.
Finally, I would not want to assent to Ibn Tufayl’s premise that “distinction and disjunction apply [only] to bodies,” for the primary point of my April 1, 2014, post, “Thomas Aquinas: Beyond Aristotle’s Aristotelian Conception of the Numerical,” was that that is not the case. Given, however, that he holds that “there is no distinction between the three,” i.e., that there are no real, let me add, distinctions among the divine knower, the divine known, and the divine knowing, he too avoids the difficulty I have been raising completely.
Were, then, anyone to uphold a doctrine according to which there are no real, I add again, distinctions among God the Father, God the Son, God the Holy Spirit, he or she too would avoid the difficulty I have been raising completely.
You will have noticed, I hope, that I have just added “real” twice. Thomas Aquinas adopted a distinction between real distinctions and but conceptual distinctions. He held accordingly that there are no real distinctions among the divine knower, the divine known, and the divine knowing, but only conceptual distinctions, on that point avoiding the difficulty I have been raising completely. On the other hand, he held that there are real distinctions among the divine persons, so, on this point, not at all avoiding the difficulty I have been raising.
Very interesting. And definitely food for thought.
I always have trouble with the Trinity(even as one who believes it) and so always love having some deep mystery to chew on.
Thank you for your thoughtful responses. I’m working through the history of the evolution of the idea of the Trinity myself so these were only minor objections I could bring forth, but I do thank you for your time in answering them.
Please keep me posted.