What is the meaning of 'is' in the context of the Trinity? The Philosopher Gottlob Frege distinguishes between four different meaning behind the word 'is':

1) 'is' as in identity; Saul is Paul.

2) 'is' as in predicate; Paul is Christian.

3) 'is' as in subset; Paul is a Man. (Paul is a member of the class 'Men').

4) 'is' as in existence; God is.

As one not raised or well versed in the Christian faith, I would like to know, when Christians say

"The Father is God. The Son is God. The Holy Spirit is God, and yet the Father is not the Son, the Son is not the Holy Spirit, the Holy spirit is not the Father."

what meaning they ascribe to the word 'is'. Please feel free to give the different answers according to the different branches of Christianity.

Addendum: If the answer is 1), identity, how do Christians get around the problem of the Transitivity of Identity' If A = B &amp; B = C then A = C?
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, &amp; 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.)

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.”