Presbyterian (or presbyteral) polity is a method of church governance typified by the rule of assemblies of presbyters, or elders. Each local church is governed by a body of elected elders usually called the session or consistory, though other terms, such as church board, may apply. Groups of local churches are governed by a higher assembly of elders known as the presbytery or classis; presbyteries can be grouped into a synod, and Presbyteries, along with synods nationwide often join together in a general assembly. Responsibility for conduct of church services is reserved to an ordained minister or pastor known as a teaching elder, or a minister of the word and sacrament.
Presbyterian polity was developed as a rejection of governance by hierarchies of single bishops (episcopal polity), but also differs from the congregationalist polity in which each congregation is independent. In contrast to the other two forms, authority in the presbyterian polity flows both from the top down (as higher assemblies exercise limited but important authority over individual congregations, e.g., only the presbytery can ordain ministers, install pastors, and start up, close, and approve relocating a congregation) and from the bottom up (e.g., the moderator and officers are not appointed from above but are rather elected by and from among the members of the assembly). This theory of governance developed in Geneva under John Calvin and was introduced to Scotland by John Knox after his period of exile in Geneva. It is strongly associated with French, Dutch, Swiss and Scottish Reformation movements, and the Reformed and Presbyterian churches.
Presbyterian polity is constructed on specific assumptions about the form of the government intended by the Bible:
Presbyterianism uses a conciliar method of church government (that is, leadership by the group or council). Thus, the presbyters and "elders" govern together as a group, and at all times the office is for the service of the congregation, to pray for them and to encourage them in the faith. The elders together exercise oversight (episcopacy) over the local congregation, with superior groups of elders gathered on a regional basis exercising wider oversight.
Presbyterians typically have viewed this method of government as approximating that of the New Testament and earliest churches. However, sometimes it is admitted that episcopacy was a form of government that was used very early in the church for practical reasons.
Presbyterianism is also distinct from congregationalism, in that individual congregations are not independent, but are answerable to the wider church, through its governing bodies (presbyteries, synods and assemblies). Moreover, the ordained ministry possesses a distinct responsibility for preaching and sacraments. Congregational churches are sometimes called "Presbyterian" if they are governed by a council of elders; but the difference is that every local congregation is independent, and its elders are accountable to its members, and congregationalism's wider assemblies are not ordinarily empowered to enforce discipline. Thus, these are ruled by elders only at the level of the congregations, which are united with one another by covenants of trust. Reformed Baptist churches are organized to be governed by elders, on the congregationalist model.
Depending upon the specific denomination, teaching elders may also be referred to with terms such as "minister of Word and Sacraments".
The elders are persons chosen from among the congregation and ordained for this service. Beyond that, practices vary: sometimes elders are elected by the congregation,  sometimes appointed by the session, in some denominations elders serve for life, others have fixed terms, and some churches appoint elders on a rotation from among willing members in good standing in the church. However in many churches, ruling elders retain their ordination for life, even though they serve fixed terms. Even after the end of their terms, they may be active in presbyteries or other bodies, and may serve communion.
In addition to sitting on the session and other church courts, ruling elders have duties as individuals. Again, Miller (1831) explains, 
Presbyteries are responsible for the ordination of ministers and for preparing candidates for that office.  In some denominations they are called Ministers of Word and Sacrament, and in others they are called Teaching Elders. Ministers called to a particular congregation are called pastors, and serve a function analogous to clergy in other denominations. (Because ruling elders are often ordained in a fashion nearly identical to teaching elders, the distinction between lay and clergy is not as clear under the Presbyterian system as in others.)
Ministers may be considered equal in status with the other elders, but they have a distinct ordination and distinct function. They are the primary preachers and teachers, celebrants of sacraments. There are sometimes further distinctions between the minister and the other elders. Some Presbyterian denominations enroll ministers as members of their respective congregations, while others enroll the minister as a member of the regional presbytery.
Until the 20th century, only men had been eligible for ordination as elders or ministers of the word and sacrament. This is widely not the case any longer; although it is usually considered a demarcation issue, distinguishing "liberal" from "conservative" presbyterian denominations.
The general assembly of a denomination often decides on what grounds a person may be ordained, but the ordination of ministers is the right of the presbytery, and the right to extend a call to a minister is the privilege of the members of the parish or congregation.   
The office of deacon has different meanings among different presbyterian churches. In some churches, deacons exercise responsibility for practical matters of finance and fabric, either separately or together with the elders. In some cases deacons administer the welfare matters of the congregation, while a separate board of management or trustees administers the other material business of the congregation, such as its endowments, salaries and buildings.
Elders make decisions for the local parish through an elected council called the Session (Latin. sessio from sedere "to sit"), sometimes the Kirk session, church session, or (in Continental Reformed usage) consistory. The members of the session are the pastor(s) of that congregation (sometimes referred to as a teaching elder), and the installed ruling or canon elders (ruling or canon because they are responsible for measuring the spiritual life and work of a congregation). In some Continental Reformed churches, also deacons are members of the consistory.
In most denominations, the pastor serves as Moderator and thus presides over the session (primus inter pares). All elders have an equal vote in the session. In some denominations, the pastor is given no vote, however in a sitting body of an even number or with a quorum of the session counted she or he can break a tie with a casting vote.
In the Polity of the Presbyterian Church (U.S.A.), the pastor and associate pastor(s) have a vote as members of the session on any and all matters; however, oftentimes she or he refrains from voting except in tie situations. The Pastor is not a voting member of the congregation.
In presbyterianism, congregations are united in accountability to a regional body called the presbytery, or, among Continental Reformed bodies, the classis. Presbyteries are made up of the minister and an elder 'commissioned' from each parish, as well as other clergy such as theological college professors, chaplains, and retired ministers. When there is a larger number of ordained ministers than ruling elders, additional ruling elders are appointed to redress the imbalance. The commissioners of the presbytery are expected to exercise their own judgement and are not required to represent the majority view of their congregations.
The officers of a presbytery are a moderator and a clerk. The moderator acts as chair of presbytery meetings and has a casting, but not deliberative, vote. As with the moderators of synods and assemblies, the moderatorship is a primus inter pares position appointed by the presbytery itself. The moderator is addressed as "moderator" during meetings, but his/her position has no bearing outside of the presbytery meeting and affords him/her no special place in other courts, although typically the moderator (especially if a member of the clergy) will conduct worship and oversee ordinations and installations of ministers as a "liturgical" bishop, and other ordinances which are seen as acts of the presbytery.
The clerk takes minutes and deals with the correspondence of the presbytery, and is often appointed for an indefinite term. Presbytery Clerks are generally regarded as substantially influential due to their greater experience of the governance of the church and their ordering of the business of the presbytery. They are thus very much more than secretaries and often in fact are the lynch pin of the organisation.
Presbyteries meet at a regularity between monthly and quarterly, some half-yearly.
In denominations too large for all the work of the denomination to be done by a single presbytery, the parishes may be divided into several presbyteries under synods and general assemblies, the synod being the lower court of the two. In the United Church of Canada, this is referred to as "conferences" and "General Council."
Often all members of the constituent presbyteries are members of the synod. Like the commissioners to presbyteries, the commissioners to synods do not act on instruction from their congregations or presbyteries, but exercise their own judgement. A synod also has a moderator and clerk, and generally meet less often than the presbytery.
Some presbyterian churches have no intermediate court between the presbytery and the general assembly.
The general assembly (or general synod) is the highest court of presbyterian polity. Each presbytery selects a number of its members to be commissioners to the general assembly. The general assembly is chaired by its own moderator, who is usually elected to a single term. He or she is addressed as moderator during meetings, but like the other moderators, his/her position has no bearing outside of the assembly meeting and affords him/her no special place in other courts. He or she presides over meetings of the assembly, and may be called on in a representative function for the remainder of the year.
The stated clerk and deputy clerk of the general assembly administer the minutes, correspondence, and business of the assembly. In some cases a separate business convenor is appointed to deal with the agenda. General assemblies meet less regularly than their subordinate courts, often annually, or in the case of the Presbyterian Church (USA), every other year.
The powers of the general assembly are usually wide-ranging. However, they may be limited by some form of external review. For example, the rules of the Church of Scotland include the Barrier Act, which requires that certain major changes to the polity of the church be referred to the presbyteries, before being enacted by the general assembly.
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