The Reformed tradition has never maintained, as far as I know, that only pastors have the power to baptize. The concept of ordination conferring special metaphysical powers is not a Reformed one. However as a matter of church order, baptism is done in a public service, by a pastor or an elder who has been authorized to conduct sacraments for a church. (Elders are sometimes authorized in rural areas with few pastors available.) We think that particular order is consistent with what Scripture has to say with about church organization.
The PCUSA normally accepts baptisms from other churches when done in the way authorized by their church, as long as the baptism is a Christian baptism, in the name of the Father, the Son and the Holy Spirit, done in accordance with that church’s rules. We have typically not accepted Mormon baptisms, for example, because their rejection of the Trinity makes it unclear that they are actually baptizing in the name of the Father, the Son and the Holy Spirit.
For this reason “who can baptize” really has two meanings: In a PCUSA Church, only the pastor, in public worship, authorized by the Session. (However I've known of an unusual situation in which a pastor was prepared to baptize with an elder representing the congregation. This wasn't a typical "emergency baptism," but a more complex situation.) But if you’re asking about baptisms in other churches, things are more flexible, because this is a matter of church order, and other churches have made different decisions on that. In reference to a posting above, I would consider a self-ordained pastor to be authorized to conduct baptisms as long as he is accepted as pastor by a Christian congregation. The PCUSA (though not AMR’s PCA) would accept emergency baptisms done by a lay person in accordance with Catholic rules, though we wouldn’t do that ourselves. We have a formal agreement with the Catholic Church to accept each other’s baptisms.
The PCUSA normally accepts baptisms from other churches when done in the way authorized by their church, as long as the baptism is a Christian baptism, in the name of the Father, the Son and the Holy Spirit, done in accordance with that church’s rules. We have typically not accepted Mormon baptisms, for example, because their rejection of the Trinity makes it unclear that they are actually baptizing in the name of the Father, the Son and the Holy Spirit.
For this reason “who can baptize” really has two meanings: In a PCUSA Church, only the pastor, in public worship, authorized by the Session. (However I've known of an unusual situation in which a pastor was prepared to baptize with an elder representing the congregation. This wasn't a typical "emergency baptism," but a more complex situation.) But if you’re asking about baptisms in other churches, things are more flexible, because this is a matter of church order, and other churches have made different decisions on that. In reference to a posting above, I would consider a self-ordained pastor to be authorized to conduct baptisms as long as he is accepted as pastor by a Christian congregation. The PCUSA (though not AMR’s PCA) would accept emergency baptisms done by a lay person in accordance with Catholic rules, though we wouldn’t do that ourselves. We have a formal agreement with the Catholic Church to accept each other’s baptisms.
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