ある is listed in dictionaries as having ラ行五段活用, which would suggest a negative form of あらない. However, that form does not exist. Why not?
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1About `おっしゃる`: once a combination of words has been reanalyzed as a single new word, it can evolve differently than the words it's derived from. – Aug 27 '13 at 10:17
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1It is in Osaka! Or あらへん, anyway. – Kaji Jun 03 '14 at 16:54
2 Answers
In modern Japanese, instead of the conjugation [未然形]{みぜんけい}+[無]{な}い, another word is used to express the plain negative, namely 無い.
This a process called suppletion, supplying a certain conjugational form with a different word. It exists in English as well. You don't say good and gooder, you talk about better, which comes from Proto-Indo-European *bhAd- good. Further examples include bad - worse, and be - is - was.
However, this kind of conjugation exists in Classical Japanese. Here the negative is expressed via 未然形+ず, and ある is no exception: あらず. ず , or perhaps better known in its form ぬ, is the helper verb of negation.
「[舟]{ふね}は[水]{みず}に[非]{あら}ざれば[行]{ゆ}かず。[水]{みず}、[舟]{ふね}に[入]{い}れば[則]{すなわ}ち[没]{ぼっ}す」
Even in modern Japanese, you can still say あらへん (Kansai dialect).
Lastly, although rare, there are attested instances of the form あらない (see 日本国語大辞典, entry あらない)
せく事はあらない
土佐【とさ】とて鬼の国でも蛇【へび】の国でもあらないものを
There is also あらなくに, but the な here is analyzed as the く-nominalization of ず.

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As in 幸あらん, it is ある+む, but I was thinking about 行かぬ・行かん. And while I'm certain you could find some examples of this usage, googling didn't give me much to justify calling it "used in modern Japanese", so I changed it to あらへん. – blutorange Aug 27 '13 at 05:37
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This is a good answer. However, it leaves me wondering why ある is (to the best of my knowledge) the _only_ Japanese verb to experience any kind of suppletion whatsoever. In English, as you point out, suppletion is relatively common, so it doesn't really come as much of a surprise, but the existence of precisely _one_ suppleted Japanese verb seems unusual enough to merit further explanation. – senshin Aug 27 '13 at 13:47
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Are there really that many word that experience suppletion in English? Also, how about polite conjugations: だ becomes です, 行く becomes 参る, 見る->お見えになる but する->なさる etc. – blutorange Aug 27 '13 at 14:43
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@senshin Suppletion is unusual in any language and is usually limited to relatively frequent words. In Japanese, besides ある→ない and the exceptional humble and honorific forms (e.g. 言う→申し上げる), there's also する→できる. (I won't argue whether these are true suppletion or not, but I think it's debatable.) – Aug 27 '13 at 15:07
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In the game [Castlevania: Order of Ecclesia](http://en.wikipedia.org/wiki/Castlevania:_Order_of_Ecclesia), if you change the games's dialogue to Japanese, one of the NPC's says to the protagonist Shanoa, `あなたには光があらん`. – istrasci Aug 27 '13 at 15:13
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1As pointed out above, it is most likely 光があらん(ことを), meaning _May/Let the light be with you_. It comes from another helper verb む and is unrelated to the negative. – blutorange Aug 27 '13 at 16:20
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1Hmm. So when was あらず supplanted by なし/ない then, or were they coexistent? なし for sure seems to exist in Classical Japanese, though my Classical Japanese *really* sucks. What is the difference? – ithisa Aug 27 '13 at 16:39
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@blutorange: Maybe, but if I recall (it's been a while since I played it), the context in which that was said during the game was not a "well wishing" one. – istrasci Aug 28 '13 at 04:36
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@user54609, regarding the shift from ず・ぬ to なし・なき, have a look at [this comment](http://japanese.stackexchange.com/questions/16226/is-verb-ending-%E3%81%AA%E3%81%84-shortened-to-%E3%82%93/16227#comment36605_16227) and the rest of that thread. – Eiríkr Útlendi Jun 03 '14 at 16:53
To my knowledge, this is historically shrouded in mystery, so there is no authoritative answer. (I'd be very interested in hearing one myself.)
This page speculates (in, unfortunately, a very authoritative tone, yet with no citations...) that the negation of ある (あらない) did at one point exist, but was discarded for the antonym of ある (ない). (N.B., we do at least know that another negation of ある (あらず), did exist historically, so the first part of this claim isn't too hard to believe.)
Their reasoning goes something like...
word negation antonym
開く 開かない 閉める
大きい 大きくない 小さい
ある あらない ない
In these cases, none of the negations match the antonym ("to not open" != "to close", "not big" != "small") except for ある, and since ない is shorter, the あらない form was discarded.
I don't know if I entirely buy their argument or not. It seems logical but honestly anything could have happened historically for us to end up in our current position, so I'd personally be reluctant to actually subscribe to this theory without any evidence.

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2There are a couple instances of `あらない` here: http://www.aozora.gr.jp/cards/000120/files/626_22343.html – Aug 27 '13 at 05:16
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1In classical japanese, あらぬ appears at the beginning of the famous [源氏]{げんじ}[物語]{ものがたり}[桐壺]{きりつぼ}, いとやむごとなききはにはあらぬが. – jovanni Aug 27 '13 at 06:42