I'm not going to slam you, I just mean to explain your question, that's it, and you don't have to agree with the conclusions.
The Bible talks of "life" as more than just existing, but as a fullness of experience. Jesus says "I have come that they may have life, and that they may have it more abundantly." (John 10:10). Everyone he was talking to was obviously already alive, so when he says "life", he means something more than that. So in the Bible's language, eternal life doesn't mean eternal existence, but eternal fullness of experience, fellowship with God. Therefore, eternal life begins when you become a Christian, not just when you die.
Christians would agree that eternity in hell is eternal existence, but would call it eternal death - like the word life in the Bible, death means something slightly different than just not existing - it means separation, namely from God.
In short, you would be right in saying it's illogical if eternal life only meant eternal existence, but that's not what it means. You don't have to agree with it, and that's fine, but that's what the Bible means when it talks about "life" and "death"; it's not defined solely in terms of existence. And that's why it's logically consistent. Heaven is eternal life. Hell is eternal death. Both are eternal existence.