Classifications are Groups fallacy
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Original Question
I see a lot of a particular intellectual error going around in politics, history and media. That is the identification of a classification group based upon a given attribute, which in itself is valid endeavor and then the ascription of a collective organization, relations or attributes to said classifications as though they were all part of an organized group simply based upon the fact they have an attribute in common.
So people talked back in the Cold War about the 'Free World' or the 'Democratic World' fighting against X even though that does not make any logical sense. A set of regimes may all be classified as free or as democratic, but nothing about that inherently implies said regimes are allied at all, they could potentially all be democratic and actually at war with each-other.
Often I find this used to to hide the hierarchical relationships in the actual groups that exist. So the American President is the 'Leader of the Free World' to cover up the fact that what is meant by the Free World is really the American world. The use of the term is therefore an empty phrase, of course the American President is the leader of the American world but by using the term Free World we invoke the above fallacy to nicely hide the hierarchy we are describing while supporting it.
It is also used to mystify conflicts that are essentially mundane into something essentially religious or epic. So the battle of two groups over in effect territory or power or irrational things becomes an epic struggle between contrasting attributes even though such attributes are potentially entirely incidental to the conflict in question. Even if they were relevant, simply being in conflict with something for opposing your attribute does not apply you are in alliance with others of the same attribute.
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Answers
2This seems more like a set of propaganda techniques rather than any specific fallacy. For instance - "simply being in conflict with something for opposing your attribute does not apply you are in alliance with others of the same attribute" is a correct statement, but to the propagandist, the 'enemy of my enemy is my friend', allowing them to oversimplify the conflict by glossing over major differences between the supposed allies in order to unite the people against a common supposed enemy.
"So the American President is the 'Leader of the Free World' to cover up the fact that what is meant by the Free World is really the American world." What you say here is also correct, but the phrase 'Free World' is loaded with positive connotations, so the speaker is priming their audience to believe that America = Free World - and 'freedom' is good, so 'free' is good. Therefore, America represents the 'good guys' and anyone they're against are the 'bad guys'. It's rhetoric used, as aforementioned, to oversimplify the conflict by use of black-and-white morality and garner popular support for political causes.
If I were to pinpoint specific fallacies, there's an implied false dilemma, in addition to prejudicial language. You can also argue that the speaker is card stacking to emphasise their side's goodness while highlighting the other side's badness, while ignoring contrary evidence.
I find it common that discourse is impossible because of the lack of precise definitions. At a cocktail party, that is fine. In a serious discussion, it is not. The situation set forth here is a good example, and the first place to start is to learn exactly what the people in the discussion mean when they use the operative terms in the discussion.
In addition to those fallacies already named in another comment, the following come to mind.
Hasty generalization (this also goes by other names such as the fallacy of insufficient statistics, fallacy of insufficient sample, fallacy of the lonely fact, leaping to a conclusion, hasty induction, secundum quid, converse accident) – it is the basing a broad conclusion on a small sample.
Fallacy of composition – assuming that something true of part of a whole must also be true of the whole.
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The gadarene swine fallacy
appears to be related to what I am talking about.