I recently discussed the notion of the "visible Church" with Jeff Cagle, a ruling elder in the Presbyterian Church of America, on his blog Butterfly House. (See the discussion here.) His position is that the Church is the set of all elect persons. As I argue in that discussion, his position entails that the Church is an abstract object, and his position faces a deep hermeneutical tension with respect to Matthew 16 and Matthew 18.
Perhaps it would be helpful to consider the various ecclesiological possibilities, and their implications, in a stepwise fashion.
Either Christ founded an institution, or He did not.
If Christ did not found an institution, then there is not and never was such a thing as a "visible Church". There are only visible Christians. All existing Christian institutions were founded and formed by mere men. Each Christian may simply pick the institution that he thinks best reflects what Christ taught, according to the selection of writings that he thinks best reflects what Christ taught. Or, he may start his own institution. Or he may dispense with institutions altogether. There is no such thing as schism; all Christians are already one in Christ, invisibly. There is and never was any authoritative interpreter of Scripture or authoritative determination of heresy. The determination of what is heresy is relative to the individual interpreter, as is the determination of what is and is not part of the canon of Scripture. The goal of ecumenicism is agreement on some bare essentials (where what counts as bare essentials is determined by each individual), and social collaboration. There is no justification for seeking institutional unity (see here).
If Christ did found an institution, then either that institution has continued to the present day, or it has not.
If the institution that Christ founded has ceased to exist, then there is now no such thing as a "visible Church"; there are only visible Christians. All existing Christian institutions were founded and formed by mere men. Each Christian may simply pick the institution that he thinks best reflects what Christ taught, or start his own institution, or disregard Christian institutions altogether. Whatever canon of Scripture the institution Christ founded was using before it ceased to exist may or may not be reliable, since if the Holy Spirit didn't preserve the institution Christ founded, then there is no reason to assume that the Holy Spirit guided and preserved the formation of the canon. Again, there is no such thing as schism, only branches. Anyone can start his own branch. There is no minimum number of persons necessary to count as a branch; there can even be a "branch of one person". All Christians are already one in Christ, invisibly, regardless of their visible unity or degree of visible disunity. There is now no authoritative interpreter of Scripture or authoritative determination of heresy. Since the cessation of the Church that Christ founded, the determination of what is heresy is relative to the individual interpreter according to his personalized canon of Scripture. The goal of ecumenicism is agreement on some bare essentials (as determined by each individual), and social collaboration. Again, there is no justification for seeking to restore institutional unity (see here).
If Christ did found an institution, and that institution remains to this day, then there is such a thing as a "visible Church", and all Christians should be in full communion with this institution. There is such a thing as schism; to be in schism is not to be in full communion with this institution. There is an authoritative interpreter of Scripture; it is the Magisterium of this institution. There are authoritative determinations of what is heresy; they are the rulings of this Magisterium. The goal of ecumenicism is to bring all Christians into full communion with this institution.
Objection 1: Why isn't there a fourth possibility: Christ founded an institution and it still exists but is apostate? First, because that would entail that Christ put us in a position wherein we are forced to choose between the sin of heresy and the sin of schism. But Christ would never do that. Second, because the gates of hell will not prevail over what Christ Himself founded. And if the institution Christ founded fell into apostasy, the gates of hell would have prevailed over what Christ Himself founded. Third, because the cornerstone of the Church is Christ Himself, and so the Church cannot totter unless Christ totters. But Christ cannot totter. Therefore, this is not a fourth possibility.
Objection 2: Maybe those are the only three possibilities, but why couldn't it be that all the various Christian denominations are parts of the one institution Christ founded? First, because that would require either that there be no difference between many institutions existing within a single institution, and many institutions not existing within a single institution, or it would require that if all the various Christian denominations were not part of the one institution Christ founded, there would be some institutional difference from what we see at present. But the former evacuates the meaning of institutional unity, and there is no conceivable institutional difference in the case of the latter. Second, I have already pointed out the problem with the notion of "mere Christianity" here. But without something like "mere Christianity", there is no principled difference between the 'branches' of the second half of the second millennium, and the schisms of the first millennium, as I have shown here. Third, because in many respects these various Christian denominations contradict each other. But a house divided against itself cannot stand. Therefore these various Christian denominations cannot be parts of the institution Christ founded.
For more on this subject, see the following three posts I wrote earlier this year:
Church and Jesus are Inseparable (January 8, 2008)
The Church: Catholic or Invisible? (March 28, 2008)
Christ Founded a Visible Church (May 10, 2008)
"Let unity, the greatest good of all goods, be your preoccupation." - St. Ignatius of Antioch (Letter to St. Polycarp)