4 ideas
In "The Invisible Man," the inventor Isidore Smythe is killed in his flat. A commissionaire, a policeman, a chestnut seller and a porter all watch the entrance and swear that no one went in, yet Smythe's body is missing. Father Brown shows that the killer was the postman, who walked past all four and carried the body out in his mail sack, because none of them counted a postman as a person entering.
When a witness says "nobody came," they mean nobody they expected or considered relevant came. People in routine service roles, such as postmen, waiters and porters, are filed as part of the background rather than as agents. An investigation should therefore ask who was present but not counted, not just who was present.
In "The Blue Cross," Father Brown knew his companion was the thief Flambeau in disguise. The man had sneered at reason, and Brown held that attacking reason is "bad theology" no real priest would voice. The point is that a genuine belief system has internal commitments that an impostor, copying only its surface, will get wrong.
Brown solves crimes by imagining his way into the criminal's motives and temptations. His years of hearing confessions give him a working knowledge of real human sin that a scientific detective's external method lacks. The mechanism is moral self-knowledge: recognizing that you could have done the crime lets you see how and why it was done.