This is what was not explained in Luke's OP: Calvinists pray for the salvation of others.
So what is the explanation when the person continues to reject Christ? God hasn't chosen that person for salvation.
So what is the explanation when the person is regenerated, and accepts Christ? God chose that person for salvation.
The thing not explained by the Calvinist is when God made the decision. If, as some contend, Calvinists believe that God chose the person for salvation at the beginning of creation, what was the point of the prayer?
EW&F answered this on page 6 by saying that Calvinists pray for one's salvation because it gives God the ability to change His mind. If this is correct, then there is a conflict with the statement that God chose the person for salvation at the beginning of creation, and it opens up a whole new line of questioning. If God didn't choose them at the beginning of creation, then when did He choose them?
The other possible answer is: we only pray for someone because of our personal obedience to God--which falls exactly into what Amy said about it affecting the individual.
Somewhere about halfway down page 4, Luke asked about prayers increasing one's chances for salvation, and thus decreasing the chances for one who isn't being prayed for. This is a fallacy. It presumes that God has a finite amount of grace. If God's grace is finite, I'd like to see scripture to support that. Increasing one person's "chance" for salvation doesn't take away any of God's grace offered to any other individual, and therefore doesn't "reduce" anyone else's opportunity for salvation.
Finally, take note of what Amy has been saying about defining "choice." It has been presented that the carnal mind cannot do anything but choose to reject God. If the carnal man understood the things of God, then he could choose to reject them; but as Luke himself pointed out in other threads, 1 Cor 2:14 tells us that the man without the spirit does not accept the things of God because he cannot understand them.
If I understand the position correctly, and I'm not saying I do: Borrowing from Luke and referencing Romans 8, the carnal man cannot understand the things of God until he is regenerated--which is not something he chose, but was done to him. It is now at this point that he chooses to fully accept the things of God, or to reject them. If I've got that correct, then salvation doesn't occur at regeneration, but only after the final choice.
Feel free to tear that apart.
Luke - please stop saying that no non-reformed are answering your OP. Several of us have. You don't agree with our answers, and you think we are wrong. That's not the same as not answering.
Why does the Arminian or non-reformed pray for someone's salvation?
Has it already been pre-ordained that our prayer will be answered? If it has, well, we didn't know it, so the prayer wasn't wasted; we were being obedient and saw the power of obedience. If it hasn't been pre-ordained, then the prayer wasn't wasted, because we were obedient, and saw the power of obedience.
Seems to me that no matter how we try to whittle away at this question, the answer comes out the same....