Regretfully unable to: 〜かねる

〜かねる expresses a polite inability: “I'm afraid I'm not in a position to”. The obstacle isn't physical but moral, social or a matter of rules: you can't, because it wouldn't be appropriate or isn't allowed. It's the phrase of customer service and official replies (お答えしかねます). Careful: かねない (“could well, might”) means almost the opposite.

• その件については、私からはお答えしかねます。そのけんについては、わたしからはおこたえしかねます。On that matter, I'm afraid I'm not in a position to answer.
• 申し訳ございませんが、ご要望には応じかねます。もうしわけございませんが、ごようぼうにはおうじかねます。We're very sorry, but we're unable to accommodate your request.
• 部長の意見には、さすがに賛成しかねる。ぶちょうのいけんには、さすがにさんせいしかねる。I really can't bring myself to agree with the department head's view.
• 困っている様子を見るに見かねて、声をかけた。こまっているようすをみるにみかねて、こえをかけた。Unable to stand by any longer watching him struggle, I went over and spoke to him.

かねる originally means “to find it hard to, to be unable to bring oneself to”. Attached to a verb stem, it says you can't do something, not for lack of ability, but because rules, your position or morals prevent it. So it's never “I can't swim”: it's “I'm not in a position to, I can't bring myself to”.

In service settings, it's the tool of polite refusal: instead of a blunt できません, staff say いたしかねます, “I'm afraid we're unable to”. The regret is built into the form.

• こちらの商品は、返品をお受けいたしかねます。こちらのしょうひんは、へんぴんをおうけいたしかねます。I'm afraid we're unable to accept returns on this item.
• お電話では、詳しいご案内はいたしかねます。おでんわでは、くわしいごあんないはいたしかねます。We're not able to give you full details over the phone.

Take the verb stem (the ます-form without ます) and add かねる, which conjugates like a る-verb. In service settings, it nearly always attaches to a humble form: お答えする → お答えしかねます, いたす → いたしかねます.

• その考えには、どうしても同意しかねる。そのかんがえには、どうしてもどういしかねる。I really can't go along with that idea.
• 引っ越すかどうか、まだ決めかねている。ひっこすかどうか、まだきめかねている。I still can't decide whether to move or not.

できません is clear but curt: it states an impossibility. かねます wraps it in apology: we'd like to, but we can't. In a shop, a hotel or a bank, it's the expected form.

On the other hand, かねる isn't used for a physical inability or a skill: “泳ぎかねる” (can't swim) is wrong. You say 泳げない.

• 子どもは三歳なので、まだ泳げない。こどもはさんさいなので、まだおよげない。My child is three, so they can't swim yet. (ability: no かねる)
• 個人情報のため、お教えしかねます。こじんじょうほうのため、おおしえしかねます。As it's personal information, we're unable to share it with you.

かねない (N2, line 5), in the negative form, doesn't mean “can” but “could well, risks”: an unfortunate consequence is possible. So it isn't the logical opposite of かねる; it's a different tool.

A trick to avoid mistakes: かねる (affirmative) = I can't; かねない (negative) = it could well happen.

• スピードを出しすぎると、事故を起こしかねない。すぴーどをだしすぎると、じこをおこしかねない。If you drive too fast, you could cause an accident.
• その日はご予約をお受けしかねます。そのひはごよやくをおうけしかねます。We're unable to take bookings that day.