In the Inferno, sinners in Hell are preoccupied with achieving fame and commemoration among the living. The question of how a man is remembered after his death is a topic of serious discussion. The logic goes that if one’s memory fades or is forgotten amongst the living, one truly dies (perhaps expires even from Hell) because not even one’s name whispers on the wind anymore. Despite their crimes, or perhaps because of them, the sinners are willing to exchange virtually anything for the protagonist’s agreement to carry news of their good names back to the living. Like every creature, man fears death, even after his body has expired.
The sinners’ obsession in speaking to Dante can be explained by their desire to inflate or rescue their reputations in the world above; for them, memory is their last hope of living freely.
By falsely promising to honor the sinners’ names, Dante does indeed sin, but such behavior is condoned because God sanctions it.