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Is there a word for an example that is wrong?

That is, using an example in an argument that is either a misrepresentation or does not illustrate what the arguer suggests it does.

Is there a word for this kind of false example?

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I don't know of a word specific to "bad examples," but "irrelevant" might work in this case. – Kurt E. Clothier Apr 5 '14 at 19:08
Contradiction sounds like the term you are looking for. – R Sahu Oct 7 '14 at 1:46

For the "invalid argument" itself, I'm quite partial to...

specious - superficially plausible, but actually wrong

And for the "invalid example" supporting what may in fact be a valid argument,...

spurious - not being what it purports to be; false or fake

As this NGram for argument, and this one for example show, many others do the same.

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I believe that sophistry, meaning a clever but flawed argument, is applicable here.

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I think that fallacy fits really well here.

a deceptive, misleading, or false notion, belief

You could use illusion too but many people associate that with magic/wizardry.

an erroneous mental representation

The speaker was definitely committing an act of prevarication.

a statement that deviates from or perverts the truth

But most people would just say lie.

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If your example is practicing an "adherence to or persistence in an erroneous use of language, memorization, practice, belief, etc., out of habit or obstinacy", it is a mumpsimus

If your example is of "Incorrect doctrine or opinions", it is a cacodoxy

If they are making a point but failing or ignoring to address the topic, it is referred to as a 'ignoratio elenchi'

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If the example was an honest attempt at refuting my position, I would simply point out how the example was flawed:

Your example is flawed, because it makes a false assumption.

If the example was a deliberate attempt to misrepresent my position, I would say that the example is a fabrication.

fabricate: to create or make up (something, such as a story) in order to trick people
(Source: Merriam-Webster)

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I would say in my field that it is an invalid example. Nothing more fancy than that.

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