Here is Charles Hodge on the nature of the unity of the church in the presbyterian view, from What Is Presbyterianism? (taken from the online version found at the blog Semper Reformanda). Hodge points out clearly and admirably how the church is one body, in contrast to the schismatic errors of congregationalism/independency (and semi-congregationalism).
As then presbyters are all of the same rank, and as they exercise their power in the government of the Church, in connection with the people, or their representatives, this of necessity gives rise to Sessions in our individual congregations, and to Presbyteries, Synods, and Assemblies, for the exercise of more extended jurisdiction. This brings into view the third great principle of Presbyterianism, the government of the Church by judicatories composed of presbyters and elders, &c. This takes for granted the unity of the Church in opposition to the theory of the Independents.
The Presbyterian doctrine on this subject is, that the Church is one in such a sense that a smaller part is subject to a larger, and the larger to the whole. It has one Lord, one faith, one baptism. The principles of government laid down in the Scriptures bind the whole Church. The terms of admission, and the legitimate grounds of exclusion, are everywhere the same. The same qualifications are everywhere to be demanded for admission to the sacred office, and the same grounds for deposition. Every man who is properly received as a member of a particular church, becomes a member of the Church universal; every one rightfully excluded from a particular church, is excluded from the whole Church; every one rightfully ordained to the ministry in one church, is a minister of the universal Church, and when rightfully deposed in one, he ceases to be a minister in any. Hence, while every particular church has a right to manage its own affairs and administer its own discipline, it cannot be independent and irresponsible in the exercise of that right. As its members are members of the Church universal, and those whom it excommunicates are, according to the Scriptural theory, delivered unto Satan, and cut off from the communion of the saints, the acts of a particular church become the acts of the whole Church, and therefore the whole has the right to see that they are performed according to the law of Christ. Hence, on the one hand, the right of appeal; and, on the other, the right of review and control.
This is the Presbyterian theory on this subject; that it is the scriptural doctrine appears, 1. From the nature of the Church. The Church is everywhere represented as one. It is one body, one family, one fold, one king dom. It is one because pervaded by one Spirit. We are all baptized into one Spirit so as to become, says the apostle, one body. This indwelling of the Spirit which thus unites all the members of Christ’s body, produces not only that subjective or inward union which manifests itself in sympathy and affection, in unity of faith and love, but also outward union and communion. It leads Christians to unite for the purposes of worship, and of mutual watch and care. It requires them to be subject one to another in the fear of the Lord. It brings them all into subjection to the word of God as the standard of faith and practice. It gives them not only an interest in each other’s welfare, purity, and edification, but it imposes the obligation to promote these objects. If one member suffers, all suffer with it; and if one member is honoured, all rejoice with it. All this is true, not merely of those frequenting the same place of worship, but of the universal body of believers. So that an independent church is as much a solecism as an independent Christian, or as an independent finger of the human body, or an independent branch of a tree. If the Church is a living body united to the same head, governed by the same laws, and pervaded by the same Spirit, it is impossible that one part should be independent of all the rest.
2. All the reasons which require the subjection of a believer to the brethren of a particular church, require his subjection to all his brethren in the Lord. The ground of this obligation is not the church covenant. It is not the compact into which a number of believers enter, and which binds only those who are parties to it. Church power has a much higher source than the consent of the governed. The Church is a divinely constituted society, deriving its power from its charter. Those who join it, join it as an existing society, and a society existing with certain prerogatives and privileges, which they come to share, and not to bestow. This divinely constituted society, which every believer is bound to join, is not the local and limited association of his own neighbourhood, but the universal brotherhood of believers; and therefore all his obligations of communion and obedience terminate on the whole Church. He is bound to obey his brethren, not because he has agreed to do so, but because they are his brethren—because they are temples of the Holy Ghost, enlightened, sanctified, and guided by Him. It is impossible, therefore, to limit the obedience of a Christian to the particular congregation of which he is a member, or to make one such congregation independent of all others, without utterly destroying the very nature of the Church, and tearing asunder the living members of Christ’s body. If this attempt should be fully accomplished, these separate churches would as certainly bleed to death, as a limb when severed from the body.
3. The Church, during the apostolic age, did not consist of isolated, independent congregations, but was one body, of which the separate churches were constituent members, each subject to all the rest, or to an authority which extended over all. This appears, in the first place, from the history of the origin of those churches. The apostles were commanded to remain in Jerusalem until they received power from on high. On the day of Pentecost the promised Spirit was poured out, and they began to speak as the Spirit gave them utterance. Many thousands in that city were added to the Lord, and they continued in the apostles’ doctrine and fellowship, and in breaking of bread and prayer. They constituted the Church in Jerusalem. It was one not only spiritually, but externally, united in the same worship, and subject to the same rulers. When scattered abroad, they preached the word everywhere, and great multitudes were added to the Church. The believers in every place were associated in separate, but not independent churches, for they all remained subject to a common tribunal.
For, secondly, the apostles constituted a bond of union to the whole body of believers. There is not the slightest evidence that the apostles had different dioceses. Paul wrote with full authority to the Church in Rome before he had ever visited the imperial city. Peter addressed his epistles to the churches of Pontus, Cappadocia, Asia, and Bithynia, the very centre of Paul’s field of labour. That the apostles exercised this general jurisdiction, and were thus the bond of external union to the Church, arose, as we have seen, from the very nature of their office. Having been commissioned to found and organize the Church, and being so filled with the Spirit as to render them infallible, their word was law. Their inspiration necessarily secured this universal authority. We accordingly find that they everywhere exercised the powers not only of teachers, but also of rulers. Paul speaks of the power given to him for edification; of the things which he ordained in all the churches. His epistles are filled with such orders, which were of binding authority then as now. He threatens the Corinthians to come to them with a rod; he cut off a member of their church, whom they had neglected to discipline; and he delivered. “Hymeneus and Alexander unto Satan, that they might learn not to blaspheme. As a historical fact, therefore, the apostolic churches were not independent congregations, but were all subject to one common authority.
In the third place, this is further evident from the Council at Jerusalem. No thing need be assumed that is not expressly mentioned in the record. The simple facts of the case are, that a controversy having arisen in the church at Antioch, concerning the Mosaic law, instead of settling it among themselves as an independent body, they referred the case to the apostles and elders at. Jerusalem, and there it was authoritatively decided, not for that church only, but for all others. Paul, therefore, in his next missionary journey, as he “passed through the cities, delivered to them,” it is said, “the decrees for to keep, which were ordained of the apostles and elders which were at Jerusalem.” Acts xvi. 4. It matters not whether the authority of that Council was due to the inspiration of its chief members or not. It is enough that it had authority over the whole Church. The several congregations were not independent, but were united under one common tribunal.
4th. In the fourth place, we may appeal to the common consciousness of Christians, as manifested in the whole history of the Church. Everything organic has what may be called nisus formatavus; an inward force, by which it is impelled to assume the form suited to its nature. This inward impulse may, by circumstances, be impeded or misdirected, so that the normal state of a plant or animal may never be attained. Still, this force never fails to manifest its existence, nor the state to which it tends. What is thus true in nature, is no less true in the Church. There is nothing more conspicuous in her history than the law by which believers are impelled to express their inward unity by outward union. It has been manifested in all ages, and under all circumstances. It gave rise to all the early councils. It determined the idea of heresy and schism. It led to the exclusion from all churches of those who, for the denial of the common faith, were excluded from any one, and who refused to acknowledge their subjection to the Church as a whole. This feeling was clearly exhibited at the time of the Reformation. The churches then formed, ran together as naturally as drops of quicksilver; and when this union was prevented by internal or external circumstances, it was deplored as a great evil. It may do for men of the world to attribute this remarkable characteristic in the history of the Church, to the love of power, or to some other unworthy source. But it is not thus to be accounted for. It is a law of the Spirit. If what all men do, is to be referred to some abiding principle of human nature; what all Christians do, must be referred to something which belongs to them as Christians.
So deeply seated is this conviction that outward union and mutual subjection is the normal state of the Church, that it manifests itself in those whose theory leads them to deny and resist it. Their Consociations, Associations, and Advisory Councils, are so many devices to satisfy an inward craving, and to prevent the dissolution to which it is felt that absolute Independency must inevitably lead.
That then, the Church is one, in the sense that a smaller part should be subject to a larger, and a larger to the whole, is evident. 1. From its nature as being one kingdom, one family, one body, having one head, one faith, one written constitution, and actuated by one Spirit; 2d. From the command of Christ that we should obey our brethren, not because they live near to us; not because we have covenanted to obey them; but because they are our brethren, the temples and organs of the Holy Ghost; 3. From the fact that during the apostolic age the churches were not independent bodies, but subject in all matters of doctrine, order, and discipline, to a common tribunal; and 4. Because the whole history of the Church proposes that this union and the subjection is the normal state of the Church towards which it strives by an inward law of its being. If it is necessary that one Christian should be subject to other Christians; it is no less necessary that one church should be subject in the same spirit, to the same extent, and on the same grounds, to other churches.
For more, see here and here.