Aristotle, in his Rhetoric, argues that there are three types of intrinsic proofs (Greek: singular "pistis"; plural "pisteis" ) or persuasive techniques used in speeches. He defines them as:
Ethos: Persuasion from the character of the speaker
Pathos: Persuasion by manipulating the emotions of the audience
Logos: Reasoning or logic
Note that "logos" here refers to logic, not to facts, as Aristotle considered the facts (Greek: "pragmata") of a case extrinsic to the art of rhetoric.
"A Modest Proposal" begins by using a pathetic appeal, drawing a portrait of poor mothers accompanied by their starving children begging on the streets, to arouse the emotions of the readers to sympathy at their plight. As the issue of Irish famine and poverty was widely discussed in England at this period, Swift did not need to elaborate this pathetic appeal, as the problem he was addressing was widely known.
Swift uses logos, or logical reasoning, in a paragraph in which he does elaborate mathematical calculations concerning the number of poor children in Ireland and the impossibility of supporting them by other means:
[Of] one million and a half [Irish people], ... there may be about two hundred thousand couple whose wives are breeders; from which number I subtract thirty thousand couple, who are able to maintain their own children ...
The most striking argument from ethos occurs at the end of the essay, in which the author makes the point that he is writing out of a sense of pure benevolence and civic duty, and unable to personally benefit from the scheme he is advancing:
I have not the least personal interest in endeavouring to promote this necessary work, having no other motive than the publick good of my country, ... I have no children, by which I can propose to get a single penny...
Ethos appeals to moral philosophy or credibility. It demonstrates or challenges the author or reader's credibility and moral philosophy with the use of fair minded presentations.
Logos is appeal based on logic or reason. It attempts to provide sufficient evidence from sources and to sound reasoning to the readers through the use of factual data and statistics.
Pathos is the appeal to emotions. It is a rhetorical appeal to persuade readers with an emotional charge through vivid descriptions.
An ethos example would be
As to my own part, having turned my thoughts for many years, upon this important subject, and maturely weighed the several schemes of our projectors, I have always found them grossly mistaken in their computation.
for Swift tries to prove his credibility and maturity.
A logos example would be
I do therefore humbly offer it to publick consideration, that of the hundred and twenty thousand children, already computed, twenty thousand may be reserved for breed, whereof only one fourth part to be males; which is more than we allow to sheep, black cattle, or swine, and my reason is, that these children are seldom the fruits of marriage, a circumstance not much regarded by our savages, therefore, one male will be sufficient to serve four females. That the remaining hundred thousand may, at a year old, be offered in sale to the persons of quality and fortune, through the kingdom, always advising the mother to let them suck plentifully in the last month, so as to render them plump, and fat for a good table. A child will make two dishes at an entertainment for friends, and when the family dines alone, the fore or hind quarter will make a reasonable dish, and seasoned with a little pepper or salt, will be very good boiled on the fourth day, especially in winter.
There are many examples of logos around this area, showing statistical data.
A pathos example would be
There is likewise another great advantage in my scheme, that it will prevent those voluntary abortions, and that horrid practice of women murdering their bastard children, alas! too frequent among us, sacrificing the poor innocent babes, I doubt, more to avoid the expense than the shame, which would move tears and pity in the most savage and inhuman breast.
These harsh and emotional words are used to provoke the emotions out of the readers.