What is the meaning of 'is' in the context of the Trinity?

When we say "Son is God; Father is God; Spirit is God", the "is" is of predicate for property or essence; the option (2). The three persons in Trinity shares divine property.

On the other hand it is (1) when we say Son is not Father, we use "is" for identity. The three are not a single person sharing different roles or names. They are three persons sharing same nature; they are one being.

To quote a top Christian philosopher and theologian Dr. Craig in his debate with Shabir Ally - The concept of God in Islam and Christianity.

Now, that puts the Muslim in a difficult situation, because the Qur’an says Jesus was a prophet. And therefore, you must believe what he said. But Jesus claimed to be God’s Son, equal to God to sit at his right hand. So if you believe that, you cannot be a Muslim. So, you see, Islam sort of pulls the rug out from under itself. If you believe in Islam, you have to believe what Jesus said, but if you believe what Jesus said, you can’t be a Muslim. So, it’s caught in a vicious circle.

But again, I don’t want to distract the debate to be a debate tonight about the deity of Christ. I want to simply say, let’s assume the Christian doctrine of the Trinity is what it is, and the question is, Is that rational to hold to? And all Shabir actually could say here was that in a triangle, each angle is not a triangle; but according to the doctrine of the Trinity, each person is God.

This is simply based on a misunderstanding, Shabir. The “is” in the statement “Jesus is God” is not an “is” of identity. It’s not like saying “Cicero is Tully,” where those are simply two different names of the same person and the “is” of identity. Rather, this is an “is” of predication. It’s like saying, “The couch is red.” You don’t mean that the couch is a color, you mean that the couch has the property of being red.

Similarly, when it is said, Jesus is God, the Father is God, the Son is God, that is to say that they are all divine; they all share the attributes of deity. This is not an “is” of identity. And unless you understand that, you’re bound to be confused. So, it is simply not the case that according to the classic doctrine of the Trinity that the Godhead as a whole is identical to any one of the persons. It is very much like a triangle, where you have one entity comprised of three angles or one entity comprised of three persons. And if that is the doctrine, then I ask you, “What is rationally objectionable about that?” That is the doctrine I believe, and I see nothing irrational about it.

For further study, read the transcript of his podcast on The Doctrine of Trinity.

I'd like to give you an answer just on one aspect of your question:

If the answer is 1), identity ...
If A = B & B = C then A = C?

What we have here is a problem with language:

First of all Frege identified four different meanings of the word "to be". This does not mean that there cannot be more meanings Frege did not know.

And second you should have a look on the word "identity": If the word "to be" has at least four different meanings - why should the word "identity" have only one meaning?

Obviously this must be the case here:

The word "is" cannot be meant in the way it is used by a mathematician.

As one not raised or well versed in the Christian faith ... the different branches of Christianity.

In a panel discussion at the Kirchentag some speaker said that what I described above is a huge problem:

Catholic and Lutheran theologians (!) use the same words but they have a different understanding what these words exactly mean. Therefore they often are not able to understand each other.

(I myself made the experience that when discussing with atheists this is even worse!)

So it's likely that some of the answers given will be understood incorrectly be readers belonging to a different branch of Christianity.

Short answer is, “identity.” Each Divine Person is perfectly identical to the Divine Essence.

Perhaps the best exposition of this idea is given by St. Thomas Aquinas in the Summa theologiae [S.Th.].

Keep in mind that the Persons, fundamentally, are relations of origin: the Father is Fatherhood; the Son is Sonship or Generation, and the Spirit is Procession. (There is a fourth relation, active spiration, that is not a Person, because it is proper to two of the Persons: Father and Son. Regarding the Divine Relations, see S.Th., Ia q. 28, a. 4, corpus.)

However, in God there are no “accidents;” rather, whatever perfections creatures derive from their accidents, God possesses in their fullness in His very Essence (or Substance).

The same holds true for the Divine Relations. As St. Thomas puts it,

And thus it is evident [after a long demonstration, of course!] that a relation really existing in God is the very same Essence in reality; it does not differ [from the Essence] except according to the notion by which it is known. (S.Th. Ia q. 28, a. 1, corpus; my translation. The original is “Et sic manifestum est quod relatio realiter existens in Deo, est idem essentiae secundum rem; et non differt nisi secundum intelligentiae rationem.”)

In other words, although we creatures need to use two different notions to signify the Divine Essence and each one of its intrinsic relations (including the Persons), in reality, the notions refer to the very same reality.

To use Frege’s terminology, we could say that Divine Essence and the Person have the same Bedeutung (reference) but differ in Sinn (sense).

When speaking of theology, there is the usual caveat that the concept of being or existence (“is”) changes according to the subject, in a way that perhaps Frege did not consider in his logic.

For example, the following statements, although they formally appear very similar, actually have very different meanings:

• The greenness of the tree is (exists).
• The tree is (exists).
• The cow is (exists).
• John (a man) is (exists).
• God is (exist).

(A similar observation could be made for all four of Frege’s meanings of to be. Probably the best exposition of this idea is found in Aristotle’s Metaphysics, especially book 4, in which he talks about the manifold meaning of being, all in reference to one; and in book 5, 1017a10-1017b10, in which he outlines his fourfold division of “being.”)

These statements are arranged in a hierarchy, from the least noble subject to the most noble. This is simply to observe that God can be (exist) in a way that creatures cannot. Strictly speaking, in God, all four of Frege’s senses of being would coincide. For example,

• “God is good”; this is not merely a predicate of God, but is identity. God is His goodness. It is also His very being—He is Goodness Itself.
• “The Son is God”; this is not a “subset,” but rather identity. The Son is God, and He is (so to speak) “all” of God, the “entire” Divine Essence. (And so is the Father and the Spirit.)

Answer to addendum: why this does not violate transitivity.

Remember, “is” changes in meaning depending on the subject to which it is predicated.

The being of God is special: God does not simply “exist,” the way His creatures exist; He is best characterized as Being Itself (Ipsum Esse, in St. Thomas Aquinas’ lingo). What that implies is that the being (existence)* of every creature is a participation in God’s Being.

The Persons, however, don’t merely participate in God’s Being, they possess it completely. And they possess the very same Being. And when we predicate the verb “to be” to the Divine Essence, we mean it in this very special sense: that He is Being Itself.

In this very precise sense of the verb “to be”—which means “to possess the Divine Being in all its fullness” (and note that possessing Being in its fullness means perfect identity with that Being)—there is no problem in saying:

• The Father is God
• The Son is God
• The Holy Spirit is God

However, when we use the verb “to be” to distinguish Persons, the meaning changes, because the only way to compare two relations is by analyzing he terms of those relations.

• The Father is not the Son, because the Father is Unbegotten (a term of the relation of Fatherhood) and the Son is Begotten (the other term). And vice versa, regarding the Sonship of the Son.
• The Father and the Son are not the Holy Spirit, because the Father and the Son are one term of the relation of spiration, and the Holy Spirit is the other (and vice versa, regarding procession).

Hence, when we say

Father = God

Son = God

Holy Spirit = God

we mean “the Father possesses all the Divine Being, which is His Essence” (and so forth with all the Persons).

However, such an affirmation would be false if we tried to apply it between two Persons, because the very meaning of “to be” actually changes:

Father ≠ Son

Father ≠ Holy Spirit

Son ≠ Holy Spirit

If you will, the use of the same symbol (=, or the verb “to be”) fools us into thinking the meaning is the same.

In reality, in proper theological terminology, we would say, “The Father is really distinct from the Son,” etc. The Persons are really distinct, because we are comparing the relations to one another, and, in fact, the terms of the relations are different (unbegottenness vs. begottenness, spiration vs. procession).

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* It should be kept in mind that Aquinas distinguished very carefully between exsistentia (existence) and esse (being), a distinction that was largely lost in St. Thomas’ commentators (Cajetan, Francesco Silvestri, etc.), and certainly by Duns Scotus and Suarez. For Aquinas, Esse means the perfection or—more precisely, the act—of being, which is more than the mere “fact” of being. Exsistentia, on the other hand, means the presence or emergence of being to some kind of observer. A thing “is” whether or not we perceive it. It “exists,” strictly speaking, when we are aware that it is. In English, too, the distinction is largely lost.

_{Divine Relations are depicted in green:} ^{(original image source)}

The Father is God. The Son is God. The Holy Spirit is God.

Here, "is" means "has the same essence as." (cf. St. Thomas Aquinas's Summa Theologica I q. 39 a. 1 question "Whether in God the essence is the same as the person?")

the Father is not the Son, the Son is not the Holy Spirit, the Holy spirit is not the Father.

Understanding this relies on what is called "relative opposition." For example, there is a real opposition between the Father's relation to the Son (paternity) and the Son's relation to the Father (filiation). However, as Parente, Piolanti, & Garafolo say (Dictionary of Dogmatic Theology p. 240):

Only active spiration is not in opposition to paternity and to filiation, and so it has as subject both Father and Son; but it is in opposition to passive spiration, which therefore demands a distinct terminus (the Holy Spirit).

Of the four real relations in God, three are numerically distinct and, thus, constitute the three divine Persons: the Father, who is subsisting Paternity, the Son, who is subsisting Filiation, the Holy Spirit, who is subsisting Spiration of Love.

Think of a human example: a mother and daughter. No matter how united a mother and daughter may be, their relations are still opposed to one another; the mother→daughter and daughter→mother relationships can never meld into such a relationship where the daughter relates to her mother in the same way the mother does to her daughter (e.g., becomes her own mother, vice versa, or both!).

In God, each of the four Divine Relations is God, i.e., has the same Divine Essence (which is His existence). Three of these relations are opposed to each other: paternity, filiation, and passive spiration (cf. this).

In creatures, relations are extrinsic to the essence of the creature. For example, my essence* is humanness, but my relation to my father, mother, wife, friends, or even to God is not humanness; it's not my essence.

^{*Essence or quiddity is the answer to the question: "What is it?" ("Quid est?")}

But in the Holy Trinity, the Divine Relations are the essence of God Himself. This is a consequence of the Trinity being supremely simple.

St. Thomas Aquinas writes, answering the question "Whether relation in God is the same as His essence?,"

…in God relation and essence do not differ from each other, but are one and the same.

^{cf. Fr. Gilles Emery, O.P.'s The Trinitarian Theology of St. Thomas Aquinas ch. 5 "Relations" (p. 78 f.)}

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Also, see St. Thomas's answer to the question "Whether essential names should be predicated in the singular of the three persons?" will shed some light on this. He writes (Summa Theologica I q. 39 a. 3 c.):

Some essential names signify the essence after the manner of substantives; while others signify it after the manner of adjectives. Those which signify it as substantives are predicated of the three Persons in the singular only, and not in the plural. Those which signify the essence as adjectives are predicated of the three persons in the plural. The reason of this is that substantives signify something by way of substance, while adjectives signify something by way of accident, which adheres to a subject. Now just as substance has existence of itself, so also it has of itself unity or multitude; wherefore the singularity or plurality of a substantive name depends upon the form signified by the name. But as accidents have their existence in a subject, so they have unity or plurality from their subject; and therefore the singularity and plurality of adjectives depends upon their "supposita." In creatures, one form does not exist in several "supposita" except by unity of order, as the form of an ordered multitude. So if the names signifying such a form are substantives, they are predicated of many in the singular, but otherwise if they adjectives. For we say that many men are a college, or an army, or a people; but we say that many men are collegians. Now in God the divine essence is signified by way of a form, as above explained (a. 2), which, indeed, is simple and supremely one, as shown above (q. 3 a. 7; q. 11 a. 4). So, names which signify the divine essence in a substantive manner are predicated of the three persons in the singular, and not in the plural. This, then, is the reason why we say that Socrates, Plato, and Cicero are "three men"; whereas we do not say the Father, Son, and Holy Ghost are "three Gods," but "one God"; forasmuch as in the three "supposita" of human nature there are three humanities, whereas in the three divine Persons there is but one divine essence. On the other hand, the names which signify essence in an adjectival manner are predicated of the three persons plurally, by reason of the plurality of "supposita." For we say there are three "existent" or three "wise" beings, or three "eternal," "uncreated," and "immense" beings, if these terms are understood in an adjectival sense. But if taken in a substantive sense, we say "one uncreated, immense, eternal being," as Athanasius declares.

^{A suppositum is an individual substance.}

And that the multiplicity of names applied to God does not contradict His supreme simplicity and oneness (Compendium Theologiæ cap. 24):

This enables us to perceive the reason for the many names that are applied to God, even though in Himself He is absolutely simple. Since our intellect is unable to grasp His essence as it is in itself, we rise to a knowledge of that essence from the things that surround us. Various perfections are discerned in these things, the root and origin of them all being one in God, as has been shown. Since we cannot name an object except as we understand it (for names are signs of things understood), we cannot give names to God except in terms of perfections perceived in other things that have their origin in Him. And since these perfections are multiple in such things, we must assign many names to God. If we saw His essence as it is in itself, a multiplicity of names would not be required; our idea of it would be simple, just as His essence is simple. This vision we hope for in the day of our glory; for, according to Zacharias 14:9, “In that day there shall be one Lord, and His name shall be one.”