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For example, one person may define capitalism as in private ownership of production and distribution while another may consider that there is no true capitalism until a market free of government interference is achieved.

This is convenient in debates and is also somewhat absurd as anything that isn't pure market is socialism or whatever.

Along with perhaps a false dichotomy, this could be described as moving the goal posts to make sure that one's position is always true/valid.

Is this a fallacy in itself and does it have a name?

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    when people argue and use ambiguous terminology which is not explicitly defined for the purpose of argument, they are usually just talking past each other. – Mr. Kennedy Oct 31 '16 at 3:11
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This sounds like an example of equivocation, which Wikipedia defines as follows

the misleading use of a term with more than one meaning or sense (by glossing over which meaning is intended at a particular time). It generally occurs with polysemic words (words with multiple meanings).

However, the term equivocation is mainly used for using two senses of the same word within a single argument, rather than a debate. See also equivocation on LogicalFallacies.info.

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You are describing the No True Scotsman fallacy, which you might understand as a special form of equivocation (as in the good suggestion in another answer). NTS involves insulating a claim from counterexamples by appearing to use one of its terms (e.g. Capitalism) in a descriptive way, as describing and being made true by actual cases, but actually using it in a very restrictive prescriptive way. In this prescriptive sense, a writer allows that the term applies only when it makes the claim it is part of true.

For example, for the term Scotsman, the categorical claim might be "No Scotsman eats porridge with sugar." Consider the objection, "but my friend Angus MacDonald from Edinburgh eats porridge with sugar!"

For a reasonable person, that should count as a counterexample to the categorical claim. The writer should modify it or give it up entirely.

The writer commits the New True Scotsman fallacy when instead of giving up or modifying the claim in the face of the counterexample, he or she sticks to it by rejecting the counterexample – here by rejecting that Angus MacDonald from Edinburgh counts as a true Scotsman, as he can't be one (the reasoning goes), precisely because he eats porridge with sugar!

That is, the writer is actually using the term in such a way that it automatically excludes all counterexamples. It is being treated as true by definition. What's fallacious or misleading is that the claim is presented as a descriptive claim, when really it's offering a prescription for how to define its central term.

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You might be thinking of the fallacy of four terms, where a syllogism arrives at a fallacious conclusion by using two different meanings for the same word. Wikipedia also has an entry about this fallacy.

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