Responding to the terrifying events that take place at Castle Dracula, Jonathan Harker begins to use logical arguments—or logos—in his diary entries in Chapter 3:
Let me begin with facts - bare, meagre facts, verified by books and figures, and of which there can be no doubt. I must not confuse them with experiences which will have to rest on my own observation or my memory of them.
This use of logic is Harker's argumentative attempt to persuade readers that he is not insane; given the otherworldly nature of his encounters, it's natural to question his sanity. Harker wishes to persuade both himself and (future) others that Dracula is a very real threat. As an educated man living in England in the late 19th century, Jonathan Harker very likely subscribes to at least some aspects of Enlightenment philosophy—especially the use of logic and scientific reasoning to interpret the world. Though Jonathan is religious, he is not superstitious; and though he has faith in a higher power, his belief does not extend to otherworldly monsters. This appeal to logic is a natural progression in Jonathan's narrative, given his character's cultural context. When he encounters something unknown or inexplicable, he defaults to logic and scientific inquiry instead of immediately jumping to a superstitious conclusion.
At the beginning of Chapter 15, as an attempt to reason through Van Helsing's assertion that Lucy is the one biting the children, Dr. Seward uses logos:
The logic is simple, no madman's logic this time, jumping from tussock to tussock in a misty bog. If it be not true, then proof will be relief; at worst it will not harm.
This logic, or need for physical "proof," parallels Jonathan Harker's response upon first encountering Dracula. Both men require some scientific, non-superstitious methodology to counter the world of myth and legend that seems to have overtaken them. Both men are the product of their time, entrenched more in Freud and Darwin than they are in folk tales.
Though Harker and Seward both exhibit unwavering faith in the power of their God, both are reluctant to accept the presence of evil, supernatural adversaries. The Devil, or Satan, is frequently invoked as a merely abstract antagonist. When Dr. Seward encounters the unknown in the form of Count Dracula, he defaults to logic and, in the process, eventually accepts the supernatural as reality—yet, even then, he must make Dracula into a concrete Satan, fitting him comfortably within his pre-existing understanding of the supernatural world.
In an intense passage towards the end of Chapter 18, Dr. Seward's inmate, Renfield, uses ethos, logos, and pathos as he argues for his freedom. At first, Renfield appeals to Seward's logical side as a man of science:
'You, gentlemen, who by nationality, by heredity, or by the possession of natural gifts, are fitted to hold your respective places in the moving world, I take to witness that I am as sane as at least the majority of men who are in full possession of their liberties. And I am sure that you, Dr. Seward, humanitarian and medico-journalist as well as scientist, will deem it a moral duty to deal with me as one to be considered as under exceptional circumstances.
In this passage, Renfield appeals to both Seward's logic and ethics, asking him to perceive Renfield's sanity with his own two eyes, and furthermore make the ethical choice to comply with Renfield's wishes. After he is made desperate, Renfield resorts to an emotional argument (pathos) to make his case:
'By all you hold sacred - by all you hold dear - by your love that is lost - by your hope that lives - for the sake of the Almighty, take me out of this and save my soul from guilt!
This emotional appeal is the most extreme of Renfield's argumentative techniques, bringing the tension in this chapter to its climax. It is by all means quite extraordinary, given the assumptions Seward and the other men have made about Renfield, that he would express himself so clearly with both logic and emotion. As Renfield becomes more articulate, the sense of foreboding increases, cluing the reader in to the fact that something sinister will soon occur.
Mina Harker's use of logical reasoning/argumentation (logos) to deduce the route Dracula will take to his castle is quite impressive: not on her part, for being "smart for a woman," but on Stoker's part, for writing a woman who holds her own and even exceeds the intelligence of the male protagonists. In Chapter 26, she utilizes a series of well-reasoned exclusions to deduce that Dracula will likely travel by water:
My surmise is, this: that in London the Count decided to get back to his Castle by water, as the most safe and secret way. He was brought from the Castle by Szgany, and probably they delivered their cargo to Slovaks who took their boxes to Varna, for there they were shipped for London. Thus the Count had knowledge of persons who could arrange this service. When the box was on land, before sunrise or after sunset, he came out from his box, met Skinsky and instructed him what to do as to arranging the carriage of the box up some river. When this was done, and he knew that all was in train, he blotted out his traces, as he thought, by murdering his agent.
The male protagonists heed her advice (which turns out to be correct) and pursue Dracula, eventually catching him and destroying him. Crucially, Mina's fellow protagonists are not emasculated by her skilled display of logical reasoning; on the contrary, they celebrate her intelligence and treat her almost as an intellectual equal. Though Mina is the object of sexual purity politics and infantilization throughout the novel, she is given more independence and leeway by its end. Indeed, it could be argued that she is the team's greatest asset in the hunt for Dracula. Such an attitude towards women is quite progressive for a novel written by a man in 1897.