"Logos" is defined as:
...persuading by the use of reasoning
Logos (Greek for "word") refers to the internal consistency of the message--the clarity of the claim, the logic of its reasons, and the effectiveness of its supporting evidence. The impact of logos on an audience is sometimes called the argument's logical appeal.
The logos, or persuasive reasoning, comes from studying the situations and characters that Huck observes. The unusual thing about Mark Twain's Huck Finn in The Adventures of Huckleberry Finn is that Huck does not come out and tell the reader what to think. The story...
...traces the moral education of a young boy whose better impulses overcome both self-interest and the negative forces of his culture.
Huck gives a multitude of reasons as to why he is low-down and going to hell. The reader can draw inferences that contradict what Huck says or thinks, providing us with what the author thinks (the story's tone)—how the author feels about his subject. Huck is Twain's mouth piece. While he shows little self-confidence, Huck's narrative (and self-criticism) allows us to understand the author's messages regarding morality and religion...about society in general, and slavery in particular.
Twain asks the reader to see beyond Huck's seeming lack of sophistication and social savvy. Twain gives Huck the voice of a kid from the poor side of town who wants to do the right thing, but Huck's actions direct the reader beyond Huck's self-recriminations to the truth we find under the surface of the story—beneath the thin and faulty veneer of civilized society of the post-Civil War South.
Specifically, Huck believes he is wrong for not turning Jim in as a runaway slave. He struggles a great deal over the moral issue of doing what society expects and doing what he believes is right in his heart. (For a good portion of the story, he tells himself that his heart wants what society wants.) Additionally, out of a sense of self-preservation, Huck is certain that by letting Jim go free, Huck will go to hell.
The reader comes to question what Huck says, and see the truths regarding the society in which he lives. We begin to reason not based on what Huck says, but in the difference between what he says and society believes, and our own (and Twain's) perceptions of civilized attitudes and behaviors.
For example, ironically, Huck thinks he cannot measure up to Tom Sawyer. However, Tom is really self-serving and thoughtless. By not telling Huck that Jim's owner (Miss Watson) has died and freed Jim in her will (simply so he can go on an adventure), things get very complicated, especially for Jim. Huck is practical, but Tom is a fanciful and foolish. The reader can see that Tom is not at all representative of the decent and civilized person Huck thinks him to be. By studying Tom, we can infer that Huck is more thoughtful and civilized, and he is a better friend to Tom than Tom is to him. This provides logical reasoning (in an indirect way—because of the way Twain presents Huck) that should convince the reader to draw his or her own conclusions rather than listening to Huck.
All the while Huck is feeling guilty about Jim, the runaway slave treats the boy not only as a friend but also like a son. Jim makes sure that Huck never sees Pap who is dead in the floating house. In Chapter 15, Jim makes Huck see his friend not as a slave but as a person—someone who loves him, mourned for him when he thought Huck was dead, and sad when he realizes that Huck thought little enough of him to play a trick on him.
En when I wake up en fine you back agin, all safe en soun', de tears come, en I could a got down on my knees en kiss yo' foot, I's so thankful. En all you wuz thinkin' 'bout wuz how you could make a fool uv ole Jim wid a lie. Dat truck dah is trash; en trash is what people is dat puts dirt on de head er dey fren's en makes 'em ashamed.
Toward the end of the book, Jim risks his dreams of freedom and reuniting with his family to save Tom. The reader observes that while Huck has seen Jim as simply a slave, Jim's devotion allows the reader to reason that he is a decent and loving man. (Eventually, Huck will come around to this way of thinking, but the reader cannot afford to wait for Huck to see Jim for the man he truly is.) When push comes to shove, Huck thinks upon what Jim has done for him and what society expects—and he goes as far as to write a letter to Miss Watson so she can find Jim:
It was a close place. I took it up, and held it in my hand. I was a trembling, because I'd got to decide, forever, betwixt two things, and I knowed it. I studied a minute, sort of holding my breath, and then says to myself: "All right, then, I'll go to hell" - and tore it up.
Ironically, Huck thinks he is blackguard and a sinner by protecting Jim. When he remembers all the things Jim has done for him, he cannot find a way to harden his heart against the man so he can feel good about the letter to Miss Watson. While Huck struggles, the reader finds an multitude of reasons to understand Twain's inference that Jim is a man deserving of respect and high regard.
In terms of examples of logos, the reader is persuaded by what Huck cannot see. The "internal consistency of the message" does not come from Huck. The "supporting evidence" of this reasoning comes from studying the actions of the other characters in the story—not in what Huck says. We find the message, what Twain is trying to tell us, by looking beyond what Huck says. Miss Watson does not act like a Christian—she does not care for her fellowman in her treatment of Jim. Pap, a racist, is hateful and no true father to Huck, but Jim is. Tom is foolish and self-centered—not a good friend at all. However, Jim is loving and caring: he is the father, he is the friend and he is certainly more an example of a Christian than Miss Watson. Huck, more than anything, carries on a debate with himself as he journeys on the Mississippi—toward enlightenment. Rather than listening to him, we are persuaded by the reasoning presented in the contradiction between what Huck sees as the right thing (from society's standpoint during that era) and the goodness we see in those who act out of true decency. Jim is this kind of person; and Huck eventually comes around.