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The language that I'm working on makes a variety of distinctions among
negation, separating usages that English (and from your comments, many
other languages) group together.

For example, one form of negation is equivalent to the logical 'not',
another to the logical 'anti'.  In English, if we say we're unhappy or not
happy, that's almost always interpreted as meaning we're unsatisfied or
displeased.  But if we describe a color as "not green", no one assumes that
the color must be red.  In my conlang, 'not happy' and 'anti-happy' are
distinct - all concepts are negated in the same way English negates color.
There are also "not anti-state" and "neither state nor anti-state" prefixes.

There's no inherent reason why we'd need words for both ends of any
polarity.  But our 'subconscious' or 'preconscious' minds don't seem to
process 'not' very well - like telling someone not to think of a purple
elephant.  Athletes trying to perform a difficult task are told to think
about the desired outcome rather than telling themselves to avoid mistakes,
because all the wrong associations are activated otherwise.  If a language
lacks a name for one end of a polarity, it's going to change the way its
speakers think.

That was the intention of Newspeak's design, after all - to direct the
associations people had when they used it.

Matt G.