In many evangelical circles I hear people use the terms liberal, progressive and emergent quite a bit, especially recently.
Do these all refer to the same categorization of groups or of thought? Or do these have a more precise meaning?
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"Liberal" Christians generally follow some of the traditional practices and morals of Christianity, but generally reject the notion of the inerrancy of Scripture. As a result, their beliefs can be very diverse. Their emphasis on doctrine is sometimes light, but sometimes very academic.
"Emergent" Christians have adopted a "postmodern" mindset, seeking to synthesize what others regard as contradictory views. Generally, they believe that God is bringing all people and indeed the universe to himself, and that people need to join that effort however they can. The emergent/emerging church is in some ways a reaction against the lack of the supernatural in Liberal theology, and in some ways a reaction against the alleged reliance on human logic in conservative evangelicalism.
"Progressive" Christians may also be "liberal" or "emergent", and they seek to overcome what they think are bad or sub-optimal traditions or practices.
The liberal Christian's main concern is to live for God by participating in a moral church community, without an urgent need to understand the somewhat flawed but good Bible. The emergent Christian's main goal is to live for God by joining God's kingdom wherever they can find it, aside from the supposed futile task of exegeting the Bible.
There are many variations and differences within these, and many of those who would identify as part of one of these might reject my broad strokes above. Nevertheless, they are useful distinctions, and I think the most important difference is in how they regard the Bible.
If you have further questions about a specific area, practice, or belief, I could give my thoughts.
Emerging Christianity is the postmodern influenced evolution of evangelical Christianity. It invites people to deconstruct and re-examine and reinterpret various doctrines and teachings of the church.
It is similar to progressive Christianity which is the postmodern influenced evolution of historic mainline liberal Protestant Christianity. However it is also different in that it has a different starting point and lineage. See https://m.huffpost.com/us/entry/892727
Emergent Christianity is a subset of Emerging Christianity that was an intentional movement involving certain people within that movement. Persons such as Brian McLaren, Doug Pagitt and the late Dr. Phyllis Tickle... http://westsidechristianfellowship.org/articles/1232012-emerging-vs-emergent-churches-clearing-up-the-confusion/
It's really just a way of separating their church and their beliefs from pre-existing churchs and beliefs. The same way political parties do it. In Australia, we have a political party called "The Liberal Party", but when you compare their policies to their name, they're not really Liberal at all, they're quite conservative.
To get an exact understanding of what people mean when they use these names, you really need to ask them what they believe it means. Just because they call themselves "progressive", that's their interpretation of progressive, and you might actually believe they're being conservative, or gone all the way to the other side where their beliefs might not satisfy your definition of a Christian.