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"Logic is the anatomy of thought."
– John Locke

"Logos" is the appeal to logic. Logos isn't logic like the formal logic in math, philosophy, or even computer science; it is the consistency and clarity of an argument as well as the logic of evidence and reasons.

In formal logic, in abstraction, the following is the case: if A is true and B is true and A is an instance of B, then the repercussions of B will always be true. The problem, however, is that this kind of logic doesn't work for real-life situations. This is where argument comes into play. Formal logic would say that speeding, for example, is a violation of traffic laws. A repercussion of violating a traffic law is a ticket; therefore, every person who speeds gets a ticket.

I've always wondered why candidates have to "approve this message"; I mean, if President Obama is on camera talking about himself, then can't I assume he approves the message? Why does he have to state that he approves it at the end?

There's certainly a law that governs what must be said at the end of a political advertisement, or else President Obama wouldn't say exactly the same thing as every other politician at the end of an ad, but there's also an element of persuasion at work here. 

Why use rhetorical appeals in persuasive writing? Using rhetorical appeals in persuasive writing increases a writer’s chances of achieving his or her purpose. Any rhetorical purpose must be connected to an audience, and rhetorical appeals have been proven to successfully reach and persuade audiences.

Successful writers write to win. Whether a writer wants to achieve a particular grade on a paper, persuade a specific audience to adopt an argument, or obtain an interview with a company, a writer writes with a purpose that he or she aims to fulfill. Using rhetorical appeals, particularly in persuasive writing, is a powerful way to persuade an audience.