1. If God does not exist, objective moral values and duties do not exist
2. Objective moral values and duties do exist
3. Therefore God exists.
I am using Aristotle’s definition of God as ‘a first uncaused cause, intelligent, immaterial, eternal and most good being who is the source of order in the universe.’ ‘Objective’ means what is independent of people’s opinion, like the laws of physics. It must transcend all races, cultures, religions and situations. Moral value refers to the worth of a person or action, while moral duty refers to our obligation to act in a certain way.
Is moral behaviour restricted to believers in God? No. Moral behaviour does not require belief in God but it does require the existence of God to make any kind of sense.
I am no theologian but it is my understanding that in the case of Christianity at least Jesus specifically says that the Holy Spirit 'will convict/convince the (unbelieving) world concerning sin and righteousness' See John 16:7-11. And Romans 2:14-16 speaks of God's law being written on the hearts of gentiles (non-Jews). The non-believer who wilfully and knowingly rejects God, however good to his fellow man or woman, does, unfortunately, commit a cosmically important sin, setting himself or herself up as judge and too proud to submit to his or her Creator, and therfeore at risk of submitting to a self-appointed god on Earth (e.g. the current leader of North Korea is one of thousands of examples that rear their heads through human history).
For an academic exposition on the existence of God by W.L.Craig (a brillliant Christian philosopher who is also an expert on cosmology, quantum mechanics and elementary particle physics) listen to the podcasts on this website. The students ask searching questions, ones which you may ask yourself, and he attempts to deal with them.
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