What is the concept of predestination? Are there different denominations that believe in predestination but have differences in what they believe about it? If so, what are the differences?
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The concept of Predestination in so much as it speaks to the concept of Election is rooted in a couple of different passages in scripture. Romans 8:29-30 being the foremost. It basically says that he knew from the beginning who he was going to choose and he "Predestined" or choose them. There are many other passages that support this concept, outlined in this answer.
The question of whether he has predestined every event that will happen in the world is a slightly more contentious one. However, I believe quite strongly that he has indeed predestined every event that will happen. This does not mean that our actions are meaningless or that we should simply drift through life. Romans 8:28 (NIV) gets to the heart of this
As for denominations that believe in predestination to one degree or another. I know that Reformed traditions (Presbyterian, URC, CRC, Reformed Baptists and countless other "reformed churches") ascribe to some tenet of this belief.
I cannot speak to other traditions right off hand, but those who know of them are welcome to add them.
Single Predestination or Double Predestination?
Now before you start thinking "Aw, man! There are two of them?!?" Let me explain the difference:
God chooses us, solely by his grace, to go to Heaven. God does not choose people for Hell. If we go to Hell it is because of our own sinfullness.
God has chosen some people to go to Heaven, and some people to go to Hell.
Now that we know what they are, what does the Bible teach?
Ok, that seems to indicate that we are Predestined to Heaven, now what about Hell?
Based off these verses, it seems clear to me that the Bible teaches Single Predestination and not Double Predestination.
God has predestined everything. Here's a simple explanation:
A man was put before two doors, and given the free will to choose a door. He chose one and opened it. Behind the door, it said, "Chosen from the beginning of time".
See, God knows what we will do and yet we are given the free choice. It's hard for our finite minds to understand this concept, as we are limited to what we perceive here and now.
Predestination means different things to different people. I saw that one answer on here accurately described the difference between single predestination and double predestination. An alternative would come from what you can call the Arminian perspective (rooted from the ideas of Jacobus Arminius who had a different perspective on how salvation works than John Calvin did). If you believe that God is all-knowing, then you accept the fact that God knows our decisions before we do. If because God is all-knowing, does that mean He put a series of events into motion that eventually caused us to make one decision over another? That is a difficult question to answer because Calvinism and Arminianism both have their problems. It is not necessarily true that because God knows something will happen, He must have caused it. Adam and Eve chose to sin in the garden, but God knew that they would and He created them anyway. The problem that Calvinists face is whether God created sin. How can a perfect and holy God also be the author of sin? On the other hand, those of Arminius' persuasion can lessen God's sovereignty, His all-powerfulness by overemphasizing the human's role in the plan of salvation. God predestined the way in which humans could come back to Him. Therefore, those that accept Him are "predestined" to be saved. In other words, God chose which people would be saved, those that responded by confessing their sins and believing that Jesus died on the cross for their sins. A basic page to highlight a few of the differences between the two views can be found here: http://www.gotquestions.org/Calvinism-vs-Arminianism.html