How can one explain the Trinity simply to a new believer?
It may seem like a daunting question at first, but when we think about it the truth of God's triune nature is meant to be a joy, a comfort, and a wonder to us!
One place to start is to realize that any analogy or comparison we try to use (and I'll suggest one) will always fall short of the truth of the triune nature of God. That is due in part to our sin, of course. We are not able to see or understand clearly because sin affects all of us. But also we are creatures and the triune God is the creator. His thoughts are not our thoughts and his ways are not our ways (Isaiah 55:8-9). It is to be expected that there are things about God our creator that we, as creatures, simply cannot understand. One further thing is that the doctrine of the Trinity is taught in the unfolding of God's revelation about himself rather than in a specific chapter. If we approach the Bible with an eye open to teaching about the Trinity we will find it from Genesis 1:26-28 to Matthew 28:19-20 to 2 Corinthians 13:14.
One thought I've found very helpful is that because God reveals himself as one God in three persons, equal in substance, power, and glory, the Trinity teaches us that relationship, communication, and love are central to what God tells us about himself. You probably know that Islam denies the Trinity and emphasizes the oneness of God. Dr. Bob Letham has pointed out that this means that for Islam you cannot say "Allah is love." Allah is power, Allah is will exercising power, but Allah is not love because love means another or others are central to his being and Allah is alone. But the God of Scripture is love (1 John 4:7-8). The Father, the Son, and the Holy Spirit love one another. The Son delights to please his heavenly Father, Jesus is called the beloved Son, and the Spirit loves to bring honor and praise to the Son. There is mutual service and self-giving by the members of the Godhead to one another and that spills over into creation and redemption.
In explaining the Trinity to your friend, begin, perhaps, with the work of salvation. Each member of the Trinity is involved in the salvation of sinners. God the Father determines to save sinners by sending his beloved Son. God the Son, out of loving obedience to his Father and love to lost sinners, comes to die in the place of sinners. The Holy Spirit, sent from the Father and the Son, comes to reveal the Son, draw sinners to Christ, and bring glory to the Father. God reveals his trinitarian nature in the work of salvation which, again, demonstrates that God is love (John 3:16, etc.).
Of all the attempted illustrations of the Trinity I've seen the one that comes closest to showing biblical truth is that of the apple. The apple is unique, it is not a pear, banana, or orange, but the apple consists of various parts. The seeds are apple seeds and will produce no other kind of fruit. Without the seeds there are no more apples. The peel is there to protect the seeds and the flesh. It is not banana skin and is uniquely suited for the apple. And when you bite into the apple you know that it is not a pear, blueberry, or anything other than an apple. All three parts are uniquely apple, all are separate from other fruits, all are necessary for the apple to be what it is. In the Trinity the three persons are all of the same kind (God/apple), all have unique roles they play in the Godhead, and they would not be the God of the Bible if they were not triune. The example isn't perfect, of course, but it reminds us that we must hold on to both the "one-ness" of the Godhead with the plurality or "three-ness" of the persons.
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