Trinity and Set Theory

Let S and T be mathematical sets. Now consider the following two propositions:

1. S is a proper subset of T.

2. S and T have the same number of elements.

Are (1) and (2) consistent? That is, can they both be true? If yes, explain how.

If you think (1) and (2) are consistent, then consider whether there is anything to the following analogy. If there is, explain the analogy. There is a set G. G has three disjoint proper subsets, F, S, H. All four sets agree in cardinality: they have the same number of elements.

From the Mail: Trinity and Incarnation

Dr. Vallicella,

Thank you for some exceptionally helpful posts lately! Regarding your point

(1) “Is there a clear scriptural basis for the doctrine of the Trinity?”

It would seem that a part of that question, or perhaps a prior question to it is:

 (1*) “Is there a clear scriptural basis for the doctrine of the Incarnation?”

It was because early Christians came to believe that Jesus of Nazareth was divine/God  that the question of how a divine Christ/God-Christ related to the ‘Father God’ to whom he prayed and in whom the Christians also first believed.  As a theologically-trained former minister, who still tries to keep up with contemporary work on the historical Jesus, I must confess that I fail to see that (1*) is the case, therefore I fail to see that (1) is the case.

 Mark Weldon Whitten, PhD   

Continue reading “From the Mail: Trinity and Incarnation”

Some Questions About the Trinity Distinguished

It may help to distinguish the following questions.

1. Is there a clear scriptural basis for the doctrine of the Trinity?

2. Is the doctrine, as formulated in the Athanasian creed and related canonical documents, true?

3. Is it possible for human reason, unaided by divine revelation, to know the doctrine to be true?

4. Is the doctrine of the Trinity possibly true?

5. Is the doctrine thinkable (conceivable) without contradiction?

I have little to say about the exegetical (1) since it is beyond my competence as a philosopher. I cannot pronounce upon (2), either for or against, until I have decided (4) and (5). The same goes for the epistemological question, (3). My present interest is in (4) and (5), which are logically prior to the first three, with (5) being logically prior to (4). 

(4) and (5) are distinct questions. An affirmative answer to (5) does not entail an affirmative answer to (4). This is because conceivability is no sure guide to real (extramental) possibility. Of the two questions, (5) comes first in the order of inquiry: if we cannot think the Trinity without contradiction, how could we advance to the further question of whether it is really possible?

(5) is the question at the center of my interest.

It is difficult to get some people to appreciate the force and importance of (5) because they are dogmatists who accept the Trinity doctrine as true simply because they were brought up to believe it, or because it is something their church teaches.  Since they accept it as true, no question of its logical coherence arises for them.  And so they think that anyone who questions the doctrine must not understand it.  To 'set the objector straight' they then repeat the very verbal formulas the logical coherence of which is in question.  "What's the problem? There is one God in three divine Persons!"  They think that if they only repeat the formulas often enough, then the objector will 'get it.'  But it is they who do not get it, since they do not understand the logical problems to which the doctrinal formulations give rise.

Or the adherent may think that the objector is merely 'attacking' or polemicizing against his faith; it doesn't occur to the adherent that there are people whose love of truth is so strong that they will not accept claims without examination.  Now if one examines the creedal formulations, one will see that the gist of the Trinity doctrine is as follows:

1. Monotheism: There is exactly one God.

2. Divinity of Persons:  The Father is God; the Son is God; the Holy Ghost is God.

3. Distinctness of Persons:  The Father is not the Son; and the Holy Ghost is not the Father or the Son.

The problem is to show how these propositions are logically consistent, that is, how they can all be true, but without falling into heresy.  If you cannot see the problem, you are not paying attention, or you lack intelligence, or your thought-processes are being distorted by ideological commitments.  Whatever you think of Brower and Rea's solution to the problem, their exposition of it is very clear and I recommend it to you.  My reason for not accepting their solution is here.

 

On the Trinity: A Medievalist Takes Me to Task

 Long-time reader Michael Sullivan e-mails:

In my experience a lot of the problems in modern philosophy of religion come about from not taking enough care to get right the religious position the philosopher is analyzing. Part of this difficulty stems from the way terminology shifts across the centuries, so that the modern philosopher takes for granted an anachronistic understanding of key terms.

I think something like this is happening in your most recent post "Some Water Analogies for the Trinity". You write: " The sense in which water is a substance is not the sense in which God is a substance. Water is a substance in the sense of a stuff; God is a substance in the sense of a hypostasis (that which stands under) orhypokeimenon (that which is placed under), or as I prefer to say, an individual."

From the standpoint of traditional, classical Trinitarian theology, this is incorrect. God is a substance neither in the sense of stuff (hyle) nor in the sense of individual (hypostasis). Here's a representative explanatory snippet from St John of Damascus, showing the universal traditional use of the terms, from "De Fide Orthodoxa" c.48: "Substantia quidem communem speciem et complectivam speciem homoiodon (id est earum quae unum sunt specie) hypostaseon (id est personarum) significat, utputa Deus, homo; hypostasis autem atomon (id est individuum) demonstrat, scilicet Patrem, Filium, Spiritum Sanctum, Petrum, Paulum."

So "substance" here means something like "essence" or "being" (in the sense of ousia) rather than hypostasis; the whole doctrine of the Trinity depends on this distinction between the one nature, substance, being, essence, etc. on the one hand and the three individual persons or hypostases on the other. In most cases where there is one existing human nature (man), there is one individual hypostasis (Peter or Paul); in the case of the Trinity there is one divine nature (God) instantiated in three hypostases (Father and Son and Holy Spirit); conversely, in the Incarnation there are two existing natures (God and man), but only one hypostasis (Christ the Incarnate Logos).

I hope you don't think this too presumptuous; but Christian doctrine really does turn to unintelligible mush in these crucial distinctions are not carefully preserved.

Continue reading “On the Trinity: A Medievalist Takes Me to Task”

Does Trinity Entail Quaternity?

Christianity, like the other two Abrahamic religions, is monotheistic. But unlike Judaism and Islam, Christianity holds to a trinitarian conception of God. The idea, spelled out in the Athanasian Creed, is that there is one God in three divine Persons, the Father, the Son, and the Holy Ghost. Each person is God, and yet there is exactly one God, despite the fact that the Persons are distinct from one another. How is this possible? How can Christians convince Jews and Muslims that their position is logically tenable and does not collapse into tritheism, and thus into polytheism to the detriment of the divine unity and transcendence?

Here is one problem. God is said to be tripersonal: the one God somehow includes three numerically distinct Persons. But none of these Persons is tripersonal. The Father is not tripersonal. The Son is not tripersonal. The Holy Ghost is not tripersonal. Now if two things differ in a property, then they cannot be identical. (This is the irreproachable principle of the Indiscernibility of Identicals expressed in its contrapositive form.) Now God and each of the Persons differs in point of tripersonality: God is tripersonal while no Person is tripersonal.  It follows that God is not identical to the Father, nor to the Son, nor to the Holy Ghost. Therefore, God is not identical to any of the Persons, whence it follows that God is distinct from each of the three Persons. And if God is distinct from each of the Persons, then he is a Fourth. 

Is God a divine person? If you say yes, then we are on our way to the Quaternity, the doctrine that there is one God in four divine Persons. For if God is not identical to the Father, or to the Son, or or to the Holy Ghost, each of which is a person, and God is a person, then there are four – count ‘em – four Persons.

Some doctrines in philosophy threaten to collapse into others. Thus mind-brain identity theory threatens to collapse into eliminativism about the mind. Other doctrines seem to want to expand. How do we keep the Trinity from expanding into the Quaternity? The attentive reader will have noticed that the argument can be iterated. If the Three-in-One becomes a Four-in-One, how avoid a Five-in-One, ad infinitum?

Some Water Analogies for the Trinity

The following is based partially on H. A. Wolfson, The Philosophy of the Church Fathers, Volume One: Faith, Trinity, Incarnation (Harvard University Press, 1956), pp. 359-363.

Hippolytus: The Logos comes from the Father as water from a fountain.

Tertullian: The Father is to the Logos as fountain is to river. One substance assumes two forms.

Lactantius: The Father is an overflowing fountain, the Son a stream flowing from it.

Zeno of Verona: Father and Son are two seas filled with the same water which, though two, are yet one.

Vallicella of Arizona: Water occurs in three distinct states, the gaseous, the liquid, and the solid. One and and the same quantity of water can assume any of these three states. Distinctness of states is compatible with oneness of substance.

Of the water analogies, I like the last one best (!) despite its being as worthless as the others. All four involve an equivocation on ‘substance.’ The sense in which water is a substance is not the sense in which God is a substance. Water is a substance in the sense of a stuff; God is a substance in the sense of a hypostasis (that which stands under) or hypokeimenon (that which is placed under), or as I prefer to say, an individual. Note also that a quantity of H2O can be in the three states only successively not simultaneously whereas God is 'simultaneously' the three Persons.

Of course, there are better physical analogies, light for example, and also nonphysical analogies such as the soul (Augustine). Something on this later. My only point is that these water analogies do nothing to render the Trinity doctrine intelligible, hence no one should be convinced by them.

Trinitarian Reading Material

Trinity Dale Tuggy's Stanford Encyclopedia of Philosophy entry, Trinity, provides a good overview.  The specialist blog, Trinities, to which he contributes will also be of interest.  Here is Richard Cartwright's excellent "On the Logical Problem of the Trinity."  Caveat lector: it contains a few scanning errors. Essential reading nonetheless.  Cartwright rejects Peter Geach's relative identity solution.  For a defense of the relative identity approach, see A, P. Martinich, Identity and Trinity. (Requires access to JSTOR.  What? You say you don't have access to JSTOR?  How do you live without it?)

Jeffrey Brouwer and Michael Rea, Material Constitution and the Trinity.  For William Lane Craig's critique, see Does the Problem of Material Constitution Illuminate The Doctrine of The Trinity? 

That barely scratches the surface, but should keep you busy for a while.

By the way, don't confuse the diagram with the doctrine.  It is a graphic 'aid' which probably does more harm than good.  Philosophical thinking is not picture-thinking.  As Hegel might have said, Philosophisches Denken ist nicht vorstellendes, sondern begriffliches, Denken.

De Trinitate: The Statue/Lump Analogy and the ‘Is’ of Composition

Thanks to Bill Clinton, it is now widely appreciated that much rides on what the meaning of ‘is’ is. Time was, when only philosophers were aware of this. In our Trinitarian explorations with the help of our Jewish atheist friend Peter we have discussed the ‘is’ of identity and the ‘is’ of predication. We saw that ‘The Father is God’ could be construed as

1. The Father is identical to God

or as

2. The Father is divine.

Both construals left us with logical trouble. If each of the Persons is identical to God, and there is exactly one God, then (given the transitivity and symmetry of identity) there is exactly one Person. On the other hand, if each of the Persons is divine, where ‘is’ functions as copula, then tri-theism is the upshot. Either way, we end up contradicting a central Trinitarian tenet.

But there is also the ‘is’ of composition as when we say, ‘This countertop is marble,’ or in my house, ‘This countertop is faux marble.’ ‘Is’ here is elliptical for ‘is composed of.’ Compare: ‘That jacket is leather,’ and ‘This beverage is whisky.’ To say that a jacket is leather is not to say that it is identical to leather – otherwise it would be an extremely large jacket – or that it has leather as a property: leather is not a property. A jacket is leather by being made out of leather.

Suppose you have a statue S made out for some lump L of material, whether marble, bronze, clay, or whatever. How is S related to L? It seems clear that L can exist without S existing. Thus one could melt the bronze down, or re-shape the clay. In either case, the statue would cease to exist, while the quantity of matter would continue to exist. It follows that S is not identical to L. They are not identical because something is true of L that is not true of S: it is true of L that it can exist without S existing, but it is not true of S that it can exist without S existing.   I am assuming the following principle, one that seems utterly beyond reproach:

(InId)  If x = y, whatever is true of x is true of y, and vice versa.

(This is a rough formulation of the Indiscenibility of Identicals.  A more careful formulation would block  such apparent counterexamples  as:  Maynard G. Krebs believes that the morning star is a planet but does not believe that the evening star is a planet.)

Returning to the statue and the lump, although S is not identical to L, S is not wholly distinct, or wholly diverse, from L either. This is because S cannot exist unless L exists. This suggests the following analogy: The Father is to God as the statue is to the lump of matter out of which it is sculpted. And the same goes for the other Persons. Schematically, P is to G as S to L. The Persons are like hylomorphic compounds where the hyle in question is the divine substance. Thus the Persons are not each identical to God, which would have the consequence that they are identical to one another. Nor are the persons instances of divinity which would entail tri-theism. It is rather than the persons are composed of God as of a common material substance. Thus we avoid a unitarianism in which there is no room for distinctness of Persons, and we avoid tri-theism. So far, so good.

Something like this approach is advocated by Jeffrey Brower and Michael Rea, here.

But does the statue/lump analogy avoid the problems we faced with the water analogy? Aren’t the two analogies so closely analogous that they share the same problems? Liquid, solid, and gaseous are states of water. Similarly, a statue is a state of a lump of matter. Modalism is not avoided. If the Persons are like states, then they are not sufficiently independent. But a statue is even worse off than a state of water. Water can be in one of its states whether or not we exist. But a hunk of matter cannot be a statue unless beings like us are on the scene to interpret it as a statue. Thus my little ceramic bust of Beethoven represents Beethoven only because we take it as representing the great composer. In a world without minds, it would not represent anything. The Persons of the Trinity, however, are in no way dependent on us for their being Persons of the Trinity.

It might be counterargued that water is not to its states as lump to statue. Water must be in one of its three states, but a lump of bronze need not be in any statue-state. That is indeed a point of disanalogy between the two analogies. But notice that God and the Persons are necessarily related: God cannot exist without the Persons. A lump of bronze can exist without being a statue. In this respect, the water analogy is better: water must be in one its three states just as God must be composed of the three Persons.

Besides the threat of modalism, there is also the fact that God is not a substance in the sense in which clay and water are substances. Thus God is not a stuff or hyle, but a substance in the sense of a hypostasis or hypokeimenon. And it does no good to say that God is an immaterial or nonphysical stuff since what must be accommodated is the divine unity. The ground of divine unity cannot be matter whether physical or nonphysical. We saw that one and the same quantity of H20 cannot be simultaneously and throughout liquid, solid, and gaseous. Similarly, one and the same quantity of bronze cannot be simultaneously and throughout three different statues. Connected with this is how God could be a hylomorphic compound, or any sort of compound, given the divine simplicity which rules out all composition in God.

In sum, the statue/lump analogy is not better than the water/state analogy. Neither explains how we can secure both unity of the divine nature and distinctness of Persons.

Is The Doctrine of the Trinity Logically Coherent? (Peter Lupu)

In this installment, Peter Lupu, atheist, defends the logical coherence of the doctrine of the Trinity.  My critical comments follow in blue.

It may be somewhat of an astonishment to those who know me well that I should venture to defend the doctrine of the Trinity. I am not a Christian; I am not religious; I am an atheist; and I have at least on one occasion privately expressed to Bill my reservations about the coherence of the Trinity doctrine. Nevertheless, there is a question here that deserves exploring. What is the question?

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Gratry on Trinity and Mystery

Gratry Alphonse Gratry (1805-1872), Logic, tr. H and M. Singer (Open Court, 1944), p. 336:

What does Catholic theology have to say about unity in the Trinity, and of the Trinity in unity? It teaches that the unity and the Trinity are not expressed in the same respect but in two different respects: absolute unity of nature; absolute trinity of persons. The nature of God, which is one, is not triple; that would be a contradiction in terms . . . ; the nature is purely, simply, and absolutely one. The persons, in their turn, which are three, are not one at all; they are purely, simply, and absolutely three. Doubtless the mystery still remains, but reason . . . is completely maintained here, veiled, it is true, but unimpaired: indeed, instead of unimpaired, I might say that it is divinely sustained.

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Imago Dei in Relation to Aquinas and Christology

This just over the transom from Will Duquette:

 A fool rushes in…

In your comment on Peter Lupu's guest post, you say

> Man was not created in God's material image, since he has none; he 
> was created in God's spiritual image.  But this implies that what is 
> essential to man is not his animal body which presumably can be 
> accounted for in the naturalistic terms of evolutionary biology, but 
> his spirit or consciousness.

However, St. Thomas would say that it is man's nature to be a
rational animal, and hence man's animal body most certainly is
essential.  I appreciate that you might be working in a broader
theistic context rather than an explicitly Christian context; but
given that Christ is God Incarnate, and now dwells in eternity,
it seems to me that man now just is created in God's image, body
and soul both.  From the standpoint of eternity God created the
universe, man in it, and become incarnate as a man as one single
act.

I enjoy your blog; it's part of my continuing education.  Thanks
for providing it.

You're welcome, Mr. Duquette.  Your comment is pertinent and raises a number of difficult and important questions. 

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Augustine on an Analogy for the Incarnation

Augustine2 On this, the Feast of St. Augustine, it is fitting to meditate on an Augustinian passage. There is an interesting passage in On Christian Doctrine that suggests a way to think about the Incarnation. Commenting on the NT text, "the Word was made flesh and dwelt among us," Augustine writes:

In order that what we are thinking may reach the mind of the listener through the fleshly ears, that which we have in mind is expressed in words and is called speech. But our thought is not transformed into sounds; it remains entire in itself and assumes the form of words by means of which it may reach the ears without suffering any deterioration in itself. In the same way the Word of God was made flesh without change that He might dwell among us. (Bk 1, Ch. 13, LLA, 14; tr. D. W. Robertson, Jr.)

What we have here is an analogy. God the Son, the Word, is to the man Jesus of Nazareth as a human thought is to the sounds by means of which the thought is expressed and communicated to a listener. Just as our thoughts, when expressed in speech, do not become sounds but retain their identity as immaterial thoughts, so too the Word (Logos), in becoming man, does not lose its divine identity as the second person of the Trinity. Thought becomes speech without ceasing to be thought; God becomes man without ceasing to be God.

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