"Subsistit in" in Lumen Gentium
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The Latin phrase subsistit in appears in the eighth paragraph of Lumen Gentium, a landmark document of the Second Vatican Council of the Catholic Church, with important implications for how the Catholic Church views its relations with other Christian Churches and other religions. The words, which in the official English translation appear as subsists in, describe the relationship between the Church of Christ (the one, holy, catholic and apostolic Church, the authentic body of Christians founded and led by Jesus), and the Catholic Church. Lumen Gentium defines that the one subsists in the other, which was a reformulation of the longstanding Church position, which stated that the Church of Christ is (Latin est) the Catholic Church.
This statement has led to a broader interpretation of the doctrine extra Ecclesiam nulla salus (Outside the Church there is no salvation), recognizing that other Christian Churches are not completely excluded from the Church of Christ, and that they too, to some degree, are instruments of salvation.
 Sacramental language
The interpretive key to the document is in the first paragraph:
"Since the Church, in Christ, is in the nature of sacrament--a sign and instrument, that is, of communion with God and of unity among all men--she here purposes, for the benefit of the faithful and of the whole world, to set forth, as clearly as possible, and in the tradition laid down by earlier Councils, her own nature and universal mission." (Lumen Gentium, paragraph 1)
This is the sacramental part. In saying that "the Church is a sign and instrument of communion with God and of unity among all men." the document takes itself out of a merely literal world view into a mystical one. Mystical language can only be analogical because it deals with what in itself cannot be fully grasped. This is a flag to the theologian to look for arguments by analogy, both implicit and explicit. Such analogous language makes up the entire first third of the document. 
 Christological language
The most subtle of these analogies is in paragraph 8b. After describing the Church of Christ as one, holy, catholic and apostolic, the council fathers goes on to say that this one church “subsists in” (subsistit in) the Catholic Church. In all of the Church’s body of teaching this verb is only used in describing the relation of the one person of Christ and the dual nature of the Incarnation: the one person of Christ subsists in two natures.
This is the sole church of Christ which in the Creed we profess to be one, holy, catholic and apostolic, which our Savior, after his resurrection, entrusted to Peter's pastoral care (John. 21:17), commissioning him and the other apostles to extend and rule it (cf. Matthew. 28:18, etc.), and which he raised up for all ages as "the pillar and mainstay of the truth" (1 Tim. 3:15). This Church, constituted and organized as a society in the present world, subsists in the Catholic Church, which is governed by the successor of Peter and by the bishops in communion with him. Nevertheless, many elements of sanctification and of truth are found outside its visible confines. Since these are gifts belonging to the Church of Christ, they are forces impelling towards Catholic unity. (emphasis added)
Subsistit in is a fairly limited term in that it makes no claims about how this is so. None of the Christological councils, Nicea, Chalcedon, etc, would dare tattempt to explain literally how the Incarnation was possible, but philosophically needed to explore the relationship between the second Person of the Trinity and the person of Jesus Christ. "Subsistit in" literally means "stands under," which doesn’t make a lot of sense. We might say "dwells in" or "is made concrete in." This language avoids making any claims of how this is true in any physical, or we might say, scientific way while still defining the relation of one person, to human and divine natures (as opposed to Nestorius’ formula akin to two persons in two natures, etc.). 
 Church of Christ and Catholic Church distinct realities
To say the Church of Christ "subsists in" the Catholic Church, does two things both of which were novel in Catholic theology up to that point. First, it makes a distinction between the Church of Christ and the Catholic Church. Second, the statement relates the Church of Christ to the Catholic Church but without making any specific claims regarding how the two are precisely related.
"Subsists in," then being an analogous term is saying something like: the Church of Christ dwells in the Catholic Church in a way similar to the incarnation of the Second Person of the Trinity dwells in (or is made concrete in) the person of Jesus Christ. That is to say that while God in his fullness was active in all times and places during the period of Jesus' life, this same God dwelled in a mysterious and particular way in the person of Jesus Christ. In making this analogy, Lumen Gentium does not confine Christ's Church to the visible confines of the Catholic Church. 
This analogy having been made now assists the reader in interpreting more clearly the preceding paragraph:
“The one mediator, Christ, established and ever sustains here on earth his holy Church, the community of faith, hope and charity, as a visible organization through which he communicates truth and grace to all men. But, the society structured with hierarchical organs and the mystical body of Christ, the visible society and the spiritual community, the earthly Church and the Church endowed with heavenly riches, are not to be thought of as two realities. On the contrary, they form one complete reality which comes together from a human and a divine element. For this reason the Church is compared, not without significance, to the mystery of the incarnate Word. As the assumed nature, inseparably united to him, serves the divine Word as a living organ of salvation, so, in a somewhat similar way, does the social structure of the Church serve the Spirit of Christ who vivifies it, in the building up of the body." (Lumen Gentium, paragraph 8a)
 Elements of sanctification in other churches
To give a fuller interpretation of this statement, the document states further:
“Nevertheless, many elements of sanctification and of truth are found outside its visible confines. Since these are gifts belonging to the Church of Christ, they are forces impelling towards Catholic unity.” (Lumen Gentium, paragraph 8b)
 Ongoing impact
"It follows that these separated churches and communities, though we believe that they suffer from defects, have by no means been deprived of significance and value in the mystery of salvation. For the Spirit of Christ has not refrained from using them as means of salvation which derive their efficacy from the very fullness of grace and truth entrusted to the Catholic Church." (Unitatis Redintegratio, para. 3, quoted in Ut Unum Sint, para. 10)
"To the extent that these elements are found in other Christian communities, the one church of Christ is effectively present in them. For this reason the Second Vatican Council speaks of a certain, though imperfect communion. The Dogmatic Constitution Lumen Gentium stresses that the Catholic Church "recognizes that in many ways she is linked" with these communities by a true union in the Holy Spirit." (Ut Unum Sint, para. 11).
With the expression subsistit in, the Second Vatican Council sought to harmonize two doctrinal statements: on the one hand, that the Church of Christ, despite the divisions which exist among Christians, continues to exist fully only in the Catholic Church, and on the other hand, that “outside of her structure, many elements can be found of sanctification and truth”, that is, in those Churches and ecclesial communities which are not yet in full communion with the Catholic Church. But with respect to these, it needs to be stated that “they derive their efficacy from the very fullness of grace and truth entrusted to the Catholic Church”. (Dominus Iesus, para. 16)
 Conservative reaction
Certain Traditional Catholic and Sedevacantist groups consider Lumen Gentium to be the demarcation of when the Roman Church fell into heresy, pointing to the use of "subsistit in" rather than "est" as an abdication of the Church's historic (and to them compulsory) identification of itself alone as God's church.
In an interview with Frankfurter Allgemeine Zeitung, then-Cardinal Ratzinger (now Pope Benedict XVI) responded to this criticism as follows:
- "The concept expressed by 'is' (to be) is far broader than that expressed by 'to subsist'. 'To subsist' is a very precise way of being, that is, to be as a subject, which exists in itself. Thus the Council Fathers meant to say that the being of the Church as such is a broader entity than the Roman Catholic Church, but within the latter it acquires, in an incomparable way, the character of a true and proper subject."