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In which contexts the usage vary?

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Though they both imply there is some kind of mistake, they are pretty different kinds of mistakes.

  • a fallacy is an error in inference. You have some facts and do some reasoning but come to a faulty (or unsubstantiated) conclusion. Then you've committed a 'fallacy'. Coming to a wrong conclusion is a fallacy.

  • a misnomer is simply a wrong name for something, in the sense that the given accepted name is at odds with the nature of the things itself. A 6'7" basketball player named 'Shorty' has a misnomer (the name 'Shorty' is a misnomer).

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A fallacy is a common misconception. A logical fallacy is one type of example.

Misnomer refers, more specifically, to a wrong name or inappropriate designation -- for instance, calling Native Americans "Indians."

An easy way to remember the difference is to remember that misnomer is derived from the the Latin word for "name," nominare.

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    ...Or calling them Native Americans. :) Some would point out that thee folks were here before the European explorers got around to renting boats. The term I've heard recently that I liked was "First Nations".
    – Hack Saw
    Commented Apr 7, 2011 at 18:53
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I think Wikipedia has one of the better succinct definitions: "...a fallacy is incorrect reasoning in argumentation resulting in a misconception." It should be noted that there are a number of distinct logical fallacies, and that an entire section of philosophy is devoted to them. The referenced link is a good start for exploring the various flavors of fallacies.

@Mitch's answer to a misnomer is well stated.

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A fallacy has a technical and a common usage, but both could be loosely summarized as "an error in reasoning" or a "logical fault".

Poor reasoning can also be called fallacious reasoning.

"Begging the question" is an example of a fallacy. From nobelief's fallacy list:

begging the question (or assuming the answer): (e.g., We must encourage our youth to worship God to instill moral behavior.) But does religion and worship actually produce moral behavior?

Here's an example usage from a comment on Serendipity: Or, What has Software Engineering got to do with Climate Change?

I only bother to bring it up because, IMHO, an appreciation of this fallacy is necessary to appreciate the real differences between technical software validation and its verification.

Something being a 'fallacy' tends to indicate an error of process.

A misnomer is simply misnaming something, or calling something by the wrong name or a misleading name.

This can be minute and technical or broad and ideological or philosophical:

Calling that little hill Mount Brawn is an egregious misnomer.

Calling that a budget plan is a misnomer -- there's no plan, it's just budget fraud.

It is incorrect to use misnomer to mean an incorrect assumption, poor reasoning, or more general kind of error where another word should be used instead:

Don't use it this way, please:

It's a misnomer that all dogs bark.

Ugh. From the another comment found in the wilds of the Internet:

Again, please, no:

There's still a misnomer that single women pushing thirty are fatally flawed or intimidating[...]

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