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The word なし, of course, means ない, and it is defined as such in dictionaries. But why does this word exist? Are there even any situations where you can say なし but you can't say ない? Is it a remnant of something from classical Japanese? It seems similar enough in use and meaning to ない that it doesn't seem that it can be dismissed as its own word entirely and etymologically separate from ない. I checked etymology sites and dictionaries and tried googling, but I can't find any explanation for this word.

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I would say that 無し is the counterpart to 有り and 無い is the counterpart to 有る. 無し/有り is different from 無い/有る, e.g. compare "今日の夕飯は無し" and "今日の夕飯は無い". Of course, etymologically 無い and 無し share the same origin... Grammatically speaking, I think that 無し is the (古文) 終止形 of 無い. – Earthliŋ Jan 20 '13 at 4:04
なし is actually an example on the Wikipedia page for 終止形. – snailboat Jan 20 '13 at 4:29
Indeed, as @user1205935 says, this is the old 終止形{しゅうしけい} (terminal form) of 無い. I think using it in the modern day gives a little more emphasis than ない. Like "not at all" or something to that effect. – Darius Jahandarie Jan 20 '13 at 4:42
Some other 終止形 commonly heard or seen in modern Japanese are あり and よし. And many nouns are really old 終止形s, like 仲良し, 証{あかし}, 寿司{すし} etc. – dainichi Jan 20 '13 at 8:34
@dainichi akasi (cognate with 灯) is the nominalized form (via 連用形) of the verb akas-u (明かす, but also written 証す). It is not the adjective akasi (赤し) -> akaki > akai "red". That said, it is generally now accepted that akas-u, aka-i, akaru-i all share the same etymology. – Dono Jan 20 '13 at 14:22
up vote 7 down vote accepted

nasi and nai are the same word. Like all adjectives, nasi is the conclusive form (終止形), while nai is the attributive form (連体形). More specifically, the attributive ends in naki, but the medial -k- drops out in modern Japanese becoming nai. This is true of all adjectives: atusi -> atuki > atui, takasi -> takaki > takai, muzukasi -> muzukasiki > muzukasii etc. You can still see this medial -k- in the adverbial form (連用形) -ku.

Note that in modern Japanese, the original conclusive was replaced by the attributive, so it may now act as a conclusive. This is true in verbs as well.

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