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The Knight's Party website appears to be the official voice for the KKK today. Their FAQ contains the following question and answer:
Q. Why do you burn the cross?
A. Many have been led to believe the Klan burns the cross as an act of desecrating a Christian symbol. Nothing could be further from the truth. The lighted cross is an old symbol used on the hillsides of Scotland. Many churches today use the lighted cross as a symbol. The Methodist church uses the cross with flames at its base sweeping up and around the cross. The Lutheran church uses the symbolism. When the Klan was founded there was not electricity. They could not light up a cross the way we light up a Christmas tree. Fire was used. The fire was used to symbolize the fiery light of Christ – a metaphor often used in the Bible. The Cross obviously is a symbol of Christianity. The Cross lighting ceremony gives honor to Jesus Christ.
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It is no more a desecration of the cross than it is for the American Legion to respectfully lay to peace an American flag by setting it on fire – which is the correct procedure.
Other references echo the sentiment that, instead of destroying, they feel that they are honoring the cross. The reasoning appears to be primarily based on tradition rather than a particular scripture passage.