The most simple and straightforward of kaneru's meanings is "to double up," "combine," "be concurrent," "serve two different purposes," or "do two things, simultaneously."
The kanji itself for kaneru suggests this doubleness, twinness, in its shape, being very nearly vertically symmetrical. Its main radical is hachi, 八, the kanji for 8, which itself is symmetrical.
This meaning is usually expressed using the onyomi, which is ken. For example, a study (shosai) that also serves as a bedroom (shinshitsu) is a shosai-ken-shinshitsu - the ken sounding similar to the cum we would use in English: a study-cum-bedroom.
As a verb, kaneru, expresses this multiple-use meaning in such phrases as Risoteki na shigoto wa shumi to jitsueki o kaneru 理想的な仕事は趣味と実益を兼ねる "The ideal job combines pastime and profit."
However, the same kaneru is used to express a completely different meaning - that of reluctance, hesitation, refusal, inability. It is often tacked onto the end of another verb to express this meaning. For example, Sore wa iikanemasu それは言いかねます。"I'm reluctant to say/I can't say/I'd rather not say." Or, Miru ni mikanete, tetsudatte shimatta. 見るに見かねて手伝ってしまった。"I couldn't just stand by and watch, so went and helped out."
Note that in the above examples, kaneru is tacked onto the base form of the preceding, main, verb. The "ii" in "iikanemasu" is from "iu" (say), and the "mi" in "mikaneru" is from "miru" (see, look, watch).
The opposite of kaneru is the negative form, kanenai - and this is what I was referring to at the beginning - the point where it can start feeling non-intuitive. If kaneru expresses reluctance or inability, then kanenai expresses possibility or likelihood.
It parallels, say, the word "unrestrained" in English, where the addition of "un" actually signals letting it all hang out, throwing caution to the wind, and opening up all sorts of possibilities.
Ano onna wa gekido mo shikanenai. あの女は激怒もしかねない。 "That woman is likely to explode (prone to exploding) in anger." The "shi" tacked on before "kanenai" is the root of suru, "to do." Or, Ano kaisha wa jiko ni narikanenai erebeta o shiyo shiteiru. あの会社は事故になりかねないエレベーターを使用している。"That company is using an elevator that could well cause an accident."
So, remember that although kanenai, with its negative "nai" ending sounds like something not happening, it means that something is liable or likely to happen.
Mou kanenai no bunpo o wakatte ite mo, kaiwa suru toki ni machigai shikanenai. もう「兼ねない」の文法をわかっていても、会話するときに間違いしかねないだろう。 "Even though I now understand the grammar of kanenai, I'm still likely to get it wrong in conversation."
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