Life & Death
The Bible describes mortal life as a coming together of spirit & body (Genesis 2:7), and it describes mortal death as a separation of spirit & body (Eccl. 12:7).
The Bible describes eternal life in terms of a relationship with God (John chapter 17, see esp. vss. 3, 19-26), and it describes eternal destruction in terms of separation from God (2 Thess. 1:9).
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Consciousness
The Bible does not provide a clinical definition of consciousness; rather, it describes the actions of conscious agents. They think, they feel, they make decisions, etc. If we wanted to be extremely precise, instead of contrasting death & consciousness, we could contrast death & thinking, feeling, deciding, etc. For purposes of simplicity, I will refer to these traits as "conscious".
The Bible repeatedly demonstrates a distinction between death and being unconscious.
A few examples:
- The Prodigal Son is described as dead (Luke 15:24,32). The son never physically dies in the story--but his father describes him as having been dead. His father isn't killing the fatted calf for a dia de los muertos celebration & dressing the boy up for a viewing at the morgue--he knows his son is alive. His son was separated (from his family, from his faith) and he has now rejoined. The son's anguish during the story clearly demonstrates that he's conscious.
- In 2 Cor. 12:1-4 Paul describes one or more visions in which he acknowledges uncertainty about whether the person experiencing the vision (possibly himself) was in or out of the body. Ergo, one can be conscious without a body.
- In Colossians 2:13 Paul describes conscious, active people as having been dead because of sin. They were simultaneously dead & conscious.
- In Luke 16 Jesus offers the parable of the rich man & Lazarus, which describes Hades as a conscious intermediate state for the dead. See my arguments for the realistic setting of the parable in this post.
- In 1 Peter 3:18-20, 4:6 we are told that Jesus (while dead) preached to the dead spirits in prison. That this was the apostolic understanding of the passage is well-supported by the words of their early disciples (see here).
Other examples along similar lines could be cited.
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Conclusion
The Bible describes physical & spiritual death in terms of joining & separating.
The Bible also describes (and/or acknowledges the possibility of) conscious activity by individuals who are separated from their physical body and by individuals who are separated from God.