Caius's syllogism is the logic example in which Caius is a man, men are mortal, therefore Caius is mortal, and in the novella it serves to show why Ivan Ilyich cannot accept his own death.
In chapter VI, after understanding that he is dying, Ivan Ilyich recalls the example he studied in logic and says that it had seemed to him all his life correct only in relation to Caius, and in no way in relation to him. Caius was man in general; he was the boy with a mother, a father, brothers, toys, and a nursemaid, who knew the smell of the striped leather ball, the silk of his mother's dress, and the pastries at School.
The argument is perfect, and Ivan Ilyich accepted it because it spoke of man in general. Against it he does not set another argument, but memories: what makes one's own death impossible is the particularity of one's own life. It is the same I that, at the wake, the colleagues set safe by thinking that this had happened to Ivan Ilyich, and not to them.
The passage can be read as the insufficiency of abstract reason in the face of the experience of dying, or as the defense of a man who used general forms all his life so as not to think of himself as an individual.