A Christian should vote how they think Jesus would vote, to put it simply. The question really is, "How would Jesus vote?" I don't think Jesus would vote for anyone who would strengthen abortion rights. In America, we should be free to do ANYTHING we want to do, so long as it doesn't infringe on someone else's freedoms. I believe strongly that this is how God wants it. I Peter says that we should not use our liberty as a cloak for vice. God's way is for people to be free to make bad choices, but His hope is for us to choose His ways. It is the Body of Christ's duty to convert people, changing their hearts so they want to please God, not change the law so they must act morally. Abortion clearly violates the unborn's right to life, liberty, and the pursuit of happiness, so it should be illegal. The gay issue, however, is different. Are their relationships really infringing on the rights of others? I don't believe so. By all means it is a sinful lifestyle, but what is the real solution? No law can change their hearts, only the love of Christ and the Holy Spirit convicting them. We as Christians have done much damage to our testimony by creating emnity between ourselves and the world. They should hate us because of our good deeds, not because of our purposefully making enemies of them. Love the sinner hate the sin.
I personally think everyone should vote their Christian beliefs before anything else. I know a lot of "Christians" who have told me that they vote for the person who can do the most for their area whether that be their town, county, state, etc. That to me is nothing short of selfism. We may have to sacrifice some government benefits if we have a person who is against big government but is also against gay rights, abortions, etc. I would rather have less government agencies and handouts if the county was more moral in its activities and leadership.
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A preacher in those days, when he felt God called him to preach, didn't hunt up a college or seminary, he hunted up a good horse, took off across the country and began crying "Behold the Lamb of God, that taketh away the sins of the world"!