Here is something from the web from someone who believe that annihilationism is false:
In one of his discourses Jesus declared: “Do not fear those who kill the body but are unable to kill the soul; but rather fear Him who is able to destroy both soul and body in hell” (Matthew 10:28 NASB).
The Greek word for “destroy” is apolesai, which derives from the verb form apollumi. The verb occurs about ninety times, and a noun form is found some eighteen times. I do not know of a single reputable English translation that renders the term in any instance, “to go out of existence.” It is translated by such common English words as “perish,” “destroy,” “lose,” or “lost.”
The term is employed of physical items that lose their usefulness. A wineskin that cracks open and is no longer usable is said to “perish” (Luke 5:37). A sheep that wanders away from the safety of the fold is described as “lost” (Luke 15:4, 6), i.e., separated from the shepherd. The wayward prodigal son was “lost” to his father (Luke 15:24), though certainly not annihilated. Food that spoils is said to have “perished” (John 6:27).
The basic argument appears to be:
1. We have cases where "apollumi" does not entail annihilation;
2. Therefore, this casts doubt that the term, when applied to the human "soul" implies annihilation.
I think this argument is quite problematic, despite its obvious superficial appeal. Yes, clearly some things can be "apollumi-ed" without being annihilated. But clearly there is something suspicious in this argument. Consider the following argument about the word "perish" as used in 21st century American English-speaking culture:
1. When food perishes, it is certainly not annihilated;
2. Therefore, when people perish we cannot assume they are annihilated either.
But clearly we can, at least in our present culture where we use the word "perish" (when referring to an end of life scenario) to mean "lights out for good". It appears that the person making the argument I have pasted above is, intentionally or otherwise, ignoring the role the referent (the "thing" that is destroyed, or perishes) plays in determining the sense we apply to the concept of "destroy" or "perish".
In short, the fact that perishing food is not annihilated might 'open the door' to the possibility of a "non-annihilation" sense of "perish". But that door is slammed shut by considering the effect of the referent, as illustrated by the fact that, in our culture at least, we would never say a person perishes and expect our audience to imagine their soul lives on.
So we really do need to ask what the words "destroy" and "perish" (and other similar words) meant in the culture in which in which the relevant Bible texts were penned and, specifically, what they meant as applied to human beings in end of life scenarios. I am not qualified to answer this question, but I will bet a flagon of October ale that, just as in our culture, to say a person "perishes" is to say they are annihilated.