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I understand that in these expressions the に isn't a particle (I think it indicates the adverbial form of a na-adjective) but I'm not sure what で is:
必死に走る
run like crazy
必死で走る
run like hell
According to this answer it's ungrammatical to use で to make na-adjective adverbial. Still, the English translation is almost the same but the one with で is stronger.
「必死で走る」「必死に走る」 both sound perfectly fine to me, and I believe these are both grammatically correct.
I understand that in these expressions the に isn't a particle (I think it indicates the adverbial form of a na-adjective)
I think you're right. According to 明鏡国語辞典, 「必死」 is a 形容動詞/na-adjective, so it conjugates to its 連用形/continuative form 「必死に」 to modify the 用言/conjugatable word 「走る」.
but I'm not sure what で is... it's ungrammatical to use で to make na-adjective adverbial.
In 明鏡, 「必死」 is also categorized as a noun. So the で here is a case particle/格助詞, and is used in the sense of:
で 🈩〘格助詞〙
❹ 動作を行うときの様態を表す。「急ぎ足で歩く」...「笑顔で答える」「土足で上がる」
（明鏡国語辞典）
the English translation is almost the same but the one with で is stronger.
「必死に～」「必死で～」 don't really sound different to me... I wouldn't find it wrong if the crazy and hell were swapped :D
動作を行うときの様態を表す。 How does this exactly translate to English? Also, does で in this sense create an adverbial?
May 13 '17 at 8:29
According to this answer it's ungrammatical to use で to make na-adjective adverbial.
If you look at it closely, Earthliŋ said in his answer:
的 makes 世界 into a 形容動詞 ("na-adjective"), which, when functioning as adverb, turns into ～的に.
～的では is simply ungrammatical.
The answer didn't say "～的で is simply ungrammatical", but it said "～的では" is simply ungrammatical.
But this discovery doesn't seem to answer your question.
Similar to the word 必死, I found 元気. You can use 元気に走る、元気で走る、元気な姿 just like 必死に走る、必死で走る、必死な形相｛ぎょうそう｝look/expression.
I think 必死で走る is made by omitting some expressions in the original phrase like by omitting な形相 in 必死な形相で走る.
This could be said 元気で走る is made by omitting な姿 in 元気な姿で走る.
If so, it is natural that you think 必死で走る seems stronger than 必死に走る, because the latter phrase is a phrase as it is while the former one hides such the expression like 形相 in it.
Off the top of my head, it seems that nouns that originally modify a other noun/NP tend to be "na-adjective" and always require に when modifying a verb/verb phrase. 必死 is a noun on its own meaning "certain death" and is not applied that rule.
