What's the difference between "I see nothing" and "I don't see anything"? In what condition each of them is more appropriate?
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There’s no difference in meaning.
Not using an auxiliary verb is
At the Battle of Copenhagen, Nelson may have said [of the signal to retreat, which he could not discern by raising a telescope to his blind eye] “I really do not see the signal.” This has come down to us in popular folklore as “I see no signal”: more poetic, certainly, and possibly more emphatic than even using really.