The popular answer is, Yes! The Scriptural answer is undoubtedly, No.
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1. Because eternal life is a matter of promise.
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"This is the promise that he hath promised us, even eternal life" (1 John 2:25). "Eternal life, which God, who cannot lie, promised before the world began" (Titus 1:2). "According to the promise of life which is in Christ Jesus" (2 Tim. 1:1).
The fact that eternal life is the subject of a promise is proof that it is not a present possession, for what a man has in his possession is no longer "promised" to him.
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2. Because it is in the world or age to come that eternal life is to be received and enjoyed.
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"He shall receive ... in the world to come, eternal life" (Mark 10:30). "He that hateth his life in this world shall keep it unto life eternal" (John 12:25). "God will render ... to them who ... seek for glory, honour, and immortality, eternal life ... in the day when God shall judge the secrets of men by Jesus Christ" (Rom. 2:7, 16). "The righteous shall go into life eternal" (Matt. 25:46).
If eternal life is not given until the age to come, it is evident that it is not possessed in the present age.
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3. Because "eternal life" means strictly "the life of the Age to come"; the word in the New Testament being derived from the Greek word for "Age". It is truly life which will never end, and therefore is also translated "everlasting life".
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"They who shall be accounted worthy to obtain that world ... neither can they die any more" (Luke 20:36). "I give unto my sheep eternal life, and they shall never perish" (John 10:28). "The wages of sin is death; but the gift of God is eternal life through Jesus Christ our Lord" (Rom. 6:23).
If eternal life is everlasting life, it follows that in the present state we lack it, seeing our life is not everlasting, but comes to an end when we die.
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4. Because eternal or everlasting life results from a change of this corruptible and mortal body into an incorruptible and immortal one.
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"He shall change our vile body, that it may be fashioned like unto his glorious body" (Phil. 3:21). "This corruptible must put on incorruption, and this mortal must put on immortality" (1 Cor. 15:53). "Clothed upon, that mortality may be swallowed up of life" (2 Cor. 5:4).
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Our present mortal and corruptible state is therefore evidence that we do not currently possess eternal life.