An easier explanation is this:
Rather than having a soul, you
are a living soul. When you, a
living soul, die, you are no longer a living soul, but in a sleep state. A living soul, again, consists of a physical body + breath of life. Upon death, the body decays and returns to dust eventually. The breath of life goes back to God. What's left behind is in a sleep state.
This then begs the question. Is the part of you that remains and is in a sleep state what you would call a soul? Not in the traditional sense, I don't think. Even the SDA Fundamental Beliefs make it a point to use the word soul in quotes in the statement, "Our “souls” go to sleep when we die."
I don't think scripture defines exactly what that part of us (in the sleep state) is called, since whenever it refers to a soul it's the living soul before death, still having the breath of life.
I refrain from calling it a soul, mainly because it can be misconstrued as the traditional Christian definition of what a soul is (which confuses the matter further), then leading to the term "soul sleep", which many Christians instantly have negative reactions to.
A living soul has consciousness and awareness, according to scripture. When you die, what scripture calls sleep is more like a pause button. That part of you has no awareness, no part in any thing. I think that's why our website calls it a soul in quotes, for lack of a better term.
Here's some interesting reading. It's an Andrews University article called
Death as Sleep: The (Mis)use of a Biblical Metaphor. It appears I'm not the only one.
Excerpts:
The point is rather simple: the persistent appeal of our pioneers to the sleep metaphor to describe death, giving this metaphor a doctrinal status, and even referring to it in terms of the sleep of the soul, made us vulnerable, leading many people to think that we believe in an intermediate state in which the souls of the dead remain dormant and unconscious in the grave waiting for the resurrection. It is important to highlight that for non-SDAs the soul-sleep concept could still be understood dualistically in connection with the immortality of the soul, and throughout Christian history there have been several immortalists who believed just that.
This was, for example, the case of some early Syrian writers such as Ephrem (or Ephraem), John Wyclif, William Tyndale, and Martin Luther. Many Anabaptists and Socinians apparently also subscribed to this view, which was also fairly widespread in England in the sixteenth and seventeenth centuries. Even to this day “soul sleep” is generally defined as “a kind of temporary suspended animation of the soul between the moment of personal death and the time when our bodies will be resurrected.” In view of such a scenario, our emphasis on the sleep of the soul was in fact doomed to misapprehension (emphasis added).
...
The problem, however, is that the improper use of the sleep metaphor was not confined to our pioneers. In several more recent publications by Adventist authors concerning the state of the dead we find statements—even official ones—that could inadvertently be taken as an endorsement of soul-sleep concept as it has traditionally been understood in some dualistic circles. One thinks, for example, of the following paragraph of Carlyle B. Haynes: “Death is really and truly a sleep, a sleep that is deep, that is unconscious, that is unbroken until the awakening at the resurrection. In death,” Haynes continues, “man enters a state of sleep. The language of the Bible makes clear that it is the whole man which sleeps, not merely a part. No intimation is given that man sleeps only as to his body, and that he is wakeful and conscious as to his soul. All that comprises the man sleeps in death.”
...
Death is not sleep. One may resemble the other, but they are in fact two different things. Andrew T. Lincoln concurs: the NT use of sleep for death “was not meant to indicate the actual state of those who had died as some sort of unconscious existence but was a metaphor that stressed the temporary and reversible effect of death.” Similarly, Bruce Reichenbach insists that “the metaphor ‘sleep’. . . does not describe the ontological state of the dead, but rather refers to the possibility of the deceased: that though they now no longer exist, by the power of God they can be recreated to live again.”
I think one of the biggest problems SDA like myself have is properly explaining exactly what death as sleep actually means. I came to these same conclusions before I found the article. My "on pause" analogy pretty much goes along the same lines of the last paragraph above, in what I was trying to convey; that I don't think you can necessarily call it "soul sleep", because it's a metaphor.