In the depth of his heart he knew he was dying, but not only was he not accustomed to the thought, he simply did not and could not grasp it.
The syllogism he had learnt from Kiesewetter's Logic: "Caius is a man, men are mortal, therefore Caius is mortal," had always seemed to him correct as applied to Caius, but certainly not as applied to himself. That Caius – man in the abstract – was mortal, was perfectly correct, but he was not Caius, not an abstract man, but a creature quite, quite separate from all others. He had been little Vanya, with a mamma and a papa, with Mitya and Volodya, with the toys, a coachman and a nurse, afterwards with Katenka and will all the joys, griefs, and delights of childhood, boyhood, and youth. What did Caius know of the smell of that striped leather ball Vanya had been so fond of? Had Caius kissed his mother's hand like that, and did the silk of her dress rustle so for Caius? Had he rioted like that at school when the pastry was bad? Had Caius been in love like that? Could Caius preside at a session as he did? "Caius really was mortal, and it was right for him to die; but for me, little Vanya, Ivan Ilych, with all my thoughts and emotions, it's altogether a different matter. It cannot be that I ought to die. That would be too terrible." (6.2-3)
This brings out the difference between understanding intellectually that people die, and understanding what death means. Ivan has known that everybody dies since he was a kid, but it's never sunk in that he, Ivan Ilych, with all of his memories, joys, and loves will completely cease to exist at some point. It's significant that he thinks back to his "Vanya" days. This is the first time we get a suggestion that Ivan was once happy as a child. Perhaps he was happy because he was loved by his parents. Their care and love gave him a sense of himself as something unique, important, and valuable. But nothing can save him now from dying.