The main character Ivan Ilych is obsessed with propriety and formality. As a way of expressing this formality, the author uses the full names of the characters. This style prohibits the reader from getting too close to the characters, and keeps the reader at arms distance. It heightens the experience of being indifferent, cold and formal.
The characters, even the family members have no genuine emotional ties; they are polite strangers, detached and indifferent. This is the way that Ivan Ilych acted before he was sick as well.
"He treated people impersonally and was indifferent to their vital interests. This was most evident in his relationship with his wife, with whom he talked at times only when a third person was present."
All the people around Ivan Ilych were treated the same way, with impersonal decorous behavior, always proper, but with no sincerity.
Single name characters, especially Gerasim, the Russian peasant, who gives great comfort to Ivan Ilych in his final days and his son, Vladimir, who grieves for his illness and expresses real love for his father, are more vivid to the reader.
The reader experiences the emotions that they go through as they care for and watch Ivan Ilych die.
"Gerasim smiled again and turned to leave the room. But Ivan Ilych felt his presence such a comfort that he did not want to let him go." (Tolstoy)