Do we say it this way because of some connection with French and the "ne . . . pas," "ne . . . ni" constructions? I'm thinking that it might be a direct importation from Old French by the Normans, or perhaps this duality is a general feature of PIE languages? If anyone can clue me in I'd be grateful.
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I'm not quite sure what you're asking. Is it why we have negative-polarity terms for both arms of the conjunction? If so, that is something that many languages have, and certainly strikes me as natural, since both are negated.
Further, many (most?) languages allow you to insert as many negative words into a sentence as seems desirable: it's just English that has in the last few centuries acquired a bizarre rule that limits them. (Though in fact "I haven't got any" still has the negative-polarity "any", it's just not overtly negative).
from old English of "not whether"
Some hold that "neither this nor that" is in error and that the locution should be "neither this or that" because "neither" refers to "this" and "that" both; and that "neither this nor that" is a double negative making "that" be true: "not this but that." Just sayin'.