It's a Biblical doctrine. Sinners are dead but still alive.
Genesis 2:17 (NKJV)
17 but of the tree of the knowledge of good and evil you shall not eat, for in the day that you eat of it you shall surely die."
Well they didn't die physically, the died spiritually.
It's not a Biblical doctrine. It's an assumed doctrine. The other passage you posted and this one say nothing about a spiritual death. You're simply imposing that on the text. Like this passage, I suppose your argument is that since Adam didn't die in that 24 hour period it must mean that he died some other way. Thus spiritual death. However, maybe there's another understanding, one that fits with the Scriptures. In the Bible a prophetic day is 1000 years. David said,
For a thousand years in thy sight are but as yesterday when it is past, and as a watch in the night.1 (Ps. 90:4 KJV)
Peter, when responding to the question of Christ's delay alluded to this passage when he said,
But, beloved, be not ignorant of this one thing, that one day is with the Lord as a thousand years, and a thousand years as one day. (2 Pet. 3:8 KJV)
A day with the Lord is as a thousand years, and God told Adam he would die in the day he ate from the tree. Adam lived 970 years, thus he died in the day he ate from the tree.
This is both the ancient Jewish and early Christian understanding of this passage.
Ancient Jewish understanding. Book of Jubilees
29. And at the close of the nineteenth jubilee, in the seventh week in the sixth year thereof, Adam died, and all
his sons buried him in the land of his creation, 8 and he was the first to be buried 9 in the earth. 30. And he lacked seventy years of one thousand years; for one thousand years are as one day in the testimony
p. 56
of the heavens and therefore was it written concerning the tree of knowledge: "On the day that ye eat thereof ye will die." 1 For this reason he did not complete the years of this day; for he died during it.
Early Christian understanding. Irenaeus, Against Hereies, Book 5, Chapter 23.
For by summing up in Himself the whole human race from the beginning to the end, He has also summed up its death. From this it is clear that the Lord suffered death, in obedience to His Father, upon that day on which Adam died while he disobeyed God. Now he died on the same day in which he did eat. For God said, “In that day on which ye shall eat of it, ye shall die by death.” The Lord, therefore, recapitulating in Himself this day, underwent His sufferings upon the day preceding the Sabbath, that is, the sixth day of the creation, on which day man was created; thus granting him a second creation by means of His passion, which is that [creation] out of death. And there are some, again, who relegate the death of Adam to the thousandth year; for since “a day of the Lord is as a thousand years,” he did not overstep the thousand years, but died within them, thus bearing out the sentence of his sin. Whether, therefore, with respect to disobedience, which is death; whether [we consider] that, on account of that, they were delivered over to death, and made debtors to it; whether with respect to [the fact that on] one and the same day on which they ate they also died (for it is one day of the creation); whether [we regard this point], that, with respect to this cycle of days, they died on the day in which they did also eat, that is, the day of the preparation, which is termed “the pure supper,” that is, the sixth day of the feast, which the Lord also exhibited when He suffered on that day; or whether [we reflect] that he (Adam) did not overstep the thousand years, but died within their limit, — it follows that, in regard to all these significations, God is indeed true. For they died who tasted of the tree; and the serpent is proved a liar and a murderer, as the Lord said of him: “For he is a murderer from the beginning, and the truth is not in him.”[
Early Church Fathers - – Ante-Nicene Fathers: The Writings of the Fathers Down To A.D. 325.
As you can see, we have a perfectly logical and Biblical interpretation that doesn't impose on the text