There is a mixture of ethos, pathos and logos in "The Cask of Amontillado." To appreciate how they work, one must regard the story as a piece of rhetoric, designed to persuade the reader of something.
At the end of the story, Montresor tells us that all this happened fifty years ago. Because of this detail, I have always thought of it as either a deathbed confession or as an account written on the eve of Montresor's death and left to be read after his demise. In either case, his motive is clear: once he is beyond the reach of justice, Montresor wants to tell posterity how clever and ruthless he has been. The message is this: "I, Montresor, am a brilliant man and an implacable enemy. I am the kind of man you do not want to anger. Look at what happened to this one man...
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who dared to insult me."
Given this message, the pathos lies in the depiction of Monresor's own smoldering resentment and Fortunato's complacency turning to terror, culminating in his desperate attempt to regard the whole affair as a joke. By this time, he is a broken man, pathetic in every sense of the word. The logos of the story is in Montresor's immaculately laid plans, his careful execution and his clever use of Fortunato's own vanity and stubbornness to lure him to his doom. The ethos is built up through the story and is the main point of telling it. By the end, we are convinced that this is a man to be respected and feared (as he said of Fortunato at the outset) and that his family motto, "Nemo me impune lacessit," is deadly accurate.
Pathos refers to emotions and logos refers to logic, but ethos cannot be neatly described by one word. It refers to a kind of ethical appeal made by a person in an attempt to convince an audience of their credibility and character. Montresor does make such appeals. In the first paragraph, Montresor explains that he had "borne as [he] best could" the "thousand injuries of Fortunato"; in other words, he wants his audience to understand that he didn't rashly jump to the actions he now takes against Fortunato, that they were a long time coming and that Fortunato had, essentially, been deserving of punishment for some time. He is, here, attempting to establish his character and demean Fortunato's: Montresor is not rash and violent; rather, Fortunato just really had it coming. If Fortunato had not "ventured upon insult," then Montresor would not have been compelled to exact his revenge.
Next, Montresor uses logos in the service of ethos to show that he was careful and exact in his actions, not impulsive and hotly passionate. He explains,
I must not only punish, but punish with impunity. A wrong is unredressed when retribution overtakes its redresser. It is equally unredressed when the avenger fails to make himself felt as such to him who has done the wrong.
Montresor has presented his desire for revenge as deserved, reasonable even, and now he explains how that revenge must work. If the avenger is punished for his actions, then his revenge is incomplete. Further, if the avenger does not make his identity known to the one who has wronged him, the revenge is likewise incomplete. In other words, Montresor must not be caught and punished for his actions and Fortunato must understand that it is Montresor who now makes him suffer in order for the revenge to be a success. To Montresor, these stipulations are logical and just, and they further establish his credibility as someone who has thought this out and not simply taken rash or ill-advised actions. He stresses that he would be avenged "At length"; because of the nature of his reasonable requirements for revenge (logos), it took a long time for him to work out the details of that revenge and so we know that it was truly warranted and not simply decided upon in the heat of some proud moment (ethos).
Normally ethos, logos, and pathos are synonymous with rhetoric and writing or giving speeches. The ancient Greek philosopher Aristotle was first to categorize these three techniques in the art of giving speeches. Ethos refers to credibility; logos is logic; and, pathos provides an emotional or passionate response to the speech. In order to analyze if Poe used any of these rhetorical devices in his short story, "The Cask of Amontillado," one would have to view Montressor's point of view and determine how often, and how effectively, he uses all three. I would say that the short story does not apply logos because the reader never finds out what transgressions Fortunado allegedly passed on Montressor. Ethos certainly isn't found because the credibility of the narrator's voice is devalued by his irrational actions towards Fortunado, the victim of his wrath. The only thing that seems to motivate Montressor, then, is his passion and hurt pride. Revenge is a topic saturated with out-of-control emotions and is more logically paired with pathos than it would be to either Ethos or Logos.