Crime and Punishment says that an idea, however well calculated it may seem, does not give anyone the right to take another person's life, and it says so without refuting the idea in argument: it shows what happens to whoever takes it seriously. The novel does not discuss Raskolnikov's theory; it makes the crime happen and follows what remains of the man afterward.
The thesis appears in full force at the beginning, in a tavern: "One death and a hundred lives in exchange, why, that's arithmetic!" Raskolnikov hears from a stranger the sum that had just been born in him, and carries it out. What the sum leaves out enters the scene of the crime: Lizaveta, the meek sister of the old woman, arrives at the wrong moment, and the narrator shows her with her left arm, which "lifted a little," stretched out slowly, as if pushing the blow away.
After the act, the theory does not fall through a counterargument; it falls through the body and through solitude. The very next day comes "a dark feeling of tormenting and endless solitude." Confessing to Sonia, he concludes: "Was it the old woman I killed? It was myself I killed, and not the old woman!" And Razumikhin, who knows nothing of the crime, had already said what the book will show: "a living soul will not submit to mechanics."
What takes the place of the theory are people: Sonia, who accompanies him to Siberia, Porfiry, who tells him that "in suffering there is an idea," and the family. The Epilogue sums it up: "In place of dialectics came life." Readers disagree about what this means. For some, it is Dostoevsky's Christian answer, that of a life that refutes a thesis without argument. For others, it is the book showing the cost of the crime and deferring the renewal to "a new story" that it does not tell, without demonstrating that change is possible.