## "Unsolvable problem"

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What is the correct construction of "unsolvable problem"? Is it "解{と}けられない問題{もんだい}"？(Question 1)

Plain(intransitive): 解{と}ける "To be solved".

Potential form(of intransitive): 解けられる "can be solved".

Potential + Negated(of intransitive): 解けられない "cannot be solved"

Plain(transitive) 解{と}く "(Some agent) solves".

Potential form(of transitive): 解ける "(Some agent) can solve".

Potential + Negated(of transitive): 解けない "(Some agent) cannot solve"

Does 解けられない問題 mean it intrinsically cannot be solved? (Question 2)

Does the transitive version "解けない" implies that some agent is unable to solve, but does not mean it is absolutely unsolvable? (Question 3)

*(Agent: one who initiates and/or completes an action or event)

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Neither of the current answers sit well with me at the moment, so I'm going to risk adding to the confusion by posting another.

### Question 1 (grammar)

First, let's clarify the two verbs in question:

(These are not the only definitions, but for the sake of brevity and on-topic-ness we'll go with these.)

The difference lies in the subjects. With 解く, the subject is the person (or agent) who actually and intentionally does the solving, but with 解ける, the subject is the problem itself, and no agent is given:

Note that 問題が解けた could also mean "[I] was able to solve the problem", since 解けた is both the past potential form of 解く (解く→解ける→解けた) and the past form of 解ける (解ける→解けた).

This shows why the potential form of 解く (解ける) is valid, but the potential form of 解ける (解けられる) is not: 解けられる personifies an inanimate subject. (sawa says this is ungrammatical; I don't know if that's the right term, since it's a syntactically valid form of 解ける, but it's certainly not used.)

○ （私は）この問題が解けない。 [I] can't solve this problem.

× この問題は解けられない。 This problem can't solve itself.

The second sentence might sound fine at first (from a grammatical standpoint), but when you think about it, assigning an ability to an inanimate subject doesn't work here. (Even in English, the better way to express the idea behind sentence #2 is, "This problem will not solve itself.")

Because of this, only 解けない問題 is correct.

### Questions 2 and 3

This is an issue of context, as sawa and istrasci mentioned. For example:

だれでも解けない問題 a problem no one can solve

istrasci also mentioned the ～にくい suffix. I was about to agree with this, but then I saw sawa's edit which reminded me that 解く can be read as both く and ほどく, and 解ける can be read as both ける and ほどける. (The meanings differ between readings.) ～にくい can be attached to both ほどく and ほどける, such as in this way:

ほどきにくいくつひも a shoelace that is difficult to untie

ほどけにくいくつひも a shoelace that won't come undone easily

…and to とく, but not to とける:

○ ときにくい問題 a problem that's difficult to solve

× とけにくい問題

However, the ～にくい suffix does not convey the same level of "impossibility" as だれでも解けない does.

### Bonus (more "unsolvable")

Generally, once you start throwing out kanji compounds like 解決 and 不可能, you're in the realm of written Japanese, but these might be good to know:

In 問題が解けた, 解けた is the potential form of the verb 解く created by the affix -e-, and still has a subject. Since it is potential, the subject apprears with に, not が: (彼に)問題が解けた. Implicit or not, there is always the agent who solves a question. Animacy is irrelevant. -rare- is another affix meaning potential, and adding it to 解ける is redundant. It is completely ungrammatical. It is ungrammatical in the same way the English expression more better is ungrammatical. The meaning 'undone' belongs to a different verb 解ける (hodokeru), written the same but pronounced differently. – None – 2011-07-29T16:24:32.390

@sawa: 解けた is both the past potential form of 解く as well as the past form of 解ける. 問題が解けた is therefore ambiguous. I will edit to clarify this point. – Derek Schaab – 2011-07-29T16:31:41.173

2Why is ほどけにくい　acceptable while とけにくい unacceptable? Following the logic of "I cannot assign ability to an inanimate subject", it should cause both to be unacceptable since both are intransitive and by extension force the existence of the "inanimate subject" – Flaw – 2011-07-29T16:53:42.230

I do not think that ほどけられない靴ひも (that is, intransitive ほどける + られる) is correct, and I wonder why ほどけにくい靴ひも does not have the same problem. Because of this, I doubt that とけにくい問題 has the same problem as とけられない問題. とけにくい問題 still sounds strange to me, but I think that the reason is something different. – Tsuyoshi Ito – 2011-07-29T17:03:15.257

@Flaw: ～にくい, in addition to the meaning of "difficult to do" (potential), also has the meaning of "won't happen easily" (non-potential). The second is used when attaching to intransitive verbs such as ほどける. (Another example: 燃えにくい材質 "material that won't burn easily".) I don't have a solid explanation on hand for why ほどけにくい works and とけにくい doesn't. – Derek Schaab – 2011-07-29T17:23:26.393

@Tsuyoshi: Could it be related to whether the verb expresses a continuous or discrete change? ～にくい, when it means なかなか～しない, seems to imply that the verb may happen only gradually over a period of time, but never quickly and easily. Because of the gradual transition, a continuous-change verb (such as ほどける or 燃える) is required. But 解(と)ける expresses a discrete change: a problem is either solved or it isn't, and the change from unsolved to solved takes place instantly. Perhaps this is why 解(と)けにくい sounds odd, but other combinations, such as 溶けにくい, work. – Derek Schaab – 2011-07-29T17:32:58.157

Maybe, but it is hard to tell whether that is the reason or not. – Tsuyoshi Ito – 2011-07-29T18:01:32.940

@Tsuyoshi: Sounds like a research paper just begging to be written. :) – Derek Schaab – 2011-07-29T18:13:47.213

2But you can say 割れにくい, and that is instantaneous. Perhaps 〜eにくい has some sort of restriction relating to concrete vs abstract subjects? – Matt – 2011-07-29T22:53:56.000

@Matt: That's a good counterexample; back to the drawing board, I guess. :) This is definitely a puzzler. – Derek Schaab – 2011-07-29T22:58:20.547

Should I start a new question comparing ほどけにくい and とけにくい ? It seems to have exceeded the scope of this question. This question was primarily based on the ~られる construction. Should there be a entirely different question to treat ~にくい ? Or should I edit this question to contain ~にくい as well? – Flaw – 2011-07-29T23:52:09.317

@Flaw: Starting a new question is definitely the way to go. I'm very curious to see if we can come up with a good answer for this ～にくい conundrum. – Derek Schaab – 2011-07-30T12:15:11.480

Will do, i'm drafting the question based on the discussion here. – Flaw – 2011-07-30T14:21:54.987

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There is no such thing as intransitive form of 解ける used in the sense as in 解ける問題. 解けられる and 解けられない are ungrammatical. The three forms that you have under the transitive version are the correct one.

To answer Question 3, They are just a single usage. If the subject is implicit and is to be interpreted as 'anyone', then that will mean that it is absolutely unsolvable.

Edit Maybe, you are confusing 解ける (tokeru) with 解ける (hodokeru), which is an intransitve verb, does not mean 'solve', and is a completely different verb.

"There is no such thing as intransitive 解く" - But there is intransitive 解ける, just like Flaw wrote in the question. And its potential form is 解けられる, is it not? – Lukman – 2011-07-29T13:46:27.317

@Lukman It is still transitive in the sense that it takes an object. The object of 解ける takes が due to the potential morpheme -e-. – None – 2011-07-29T14:14:34.303

the potential form of 解く yields 解ける. Is this "解ける(potential transitive)" equivalent to the "解ける(intransitive)" ? – Flaw – 2011-07-29T14:32:59.543

@Flaw As I wrote, I have no idea what you mean by intransitive version. Do you have an example? – None – 2011-07-29T14:37:09.137

I'm not sure too but searching on dictionaries yield "(v1,vi) to be solved". v1 for 一段, and vi for intransitive. – Flaw – 2011-07-29T14:44:36.273

1@sawa: “There is no such thing as intransitive 解く.” You are the only person who has mentioned “intransitive 解く.” – Tsuyoshi Ito – 2011-07-29T15:44:58.463

@Tsuyoshi_Ito Thanks for pointing out my mistake. As you may have guessed, I should have written 解ける. – None – 2011-07-29T16:00:41.127

(1) I did not have any guess what you meant. (2) I do not agree with your claim “There is no such thing as intransitive form of 解ける read as 'tokeru'.” 誤解が解ける seems an example of intransitive verb 解ける read as とける, and it is hard to claim that it is a different verb from 解ける in 問題が解ける. – Tsuyoshi Ito – 2011-07-29T16:47:43.730

@Tsuyoshi_Ito In 誤解が解ける, the affix -e- is used under a different usage (自発) than the -e- in 問題が解ける (可能). So even though the verb root tok- is the same, the 解ける here is a different one. – None – 2011-07-29T16:57:22.357

(1) Do you at least agree that there is an intransitive verb 解ける which is read as とける? (2) According to Keisuke Onoe, both 可能 and 自発 can be understood as two aspects of the same core meaning 出来 (しゅったい), at least in the case of れる・られる. So I am not sure if it is fair to claim that the two とける are different words. – Tsuyoshi Ito – 2011-07-29T17:06:56.067

@Tsuyoshi_Ito (1) I agree. And I corrected that part right after your first comment to be more accurate. (2) I disagree with that. – None – 2011-07-29T17:10:53.697

Your answer still contains the claim “There is no such thing as intransitive form of 解ける read as 'tokeru'.” (I guess that you know the usual distinction between “which” with a comma and “which” without a comma.) – Tsuyoshi Ito – 2011-07-29T17:13:28.063

@Tsuyoshi_Ito You're right. I meant restrictive relative, but didn't write it. I will correct. – None – 2011-07-29T17:16:33.307

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Question 1: Either 解けない問題 or 解けられない問題 is correct. The subtlety comes in what the exact semantics are. 解けられない問題 means the problem itself it unsolvable. 解けない問題 means an Agent (as you denoted) cannot solve the problem, although the Agent is omitted. Would be clearer to use something like 私が解けない問題 ("A problem I can't solve") or whoever the Agent happens to be.

Thus I would say that 解けられない問題 is stronger because it implies that nobody can solve it.

Question 2: Yes (based on what I just said), but something like 解けにくい or 解けがたい might work as well or better without the possible confusion.

Question 3: Again, it would probably be better to specifically to include the agent, otherwise まぎらわしい. If you wanted to use this form to say "absolutely unsolvable", you'd have to say だれも解けない問題.

3解けられない問題 sounds undoubtedly ungrammatical to my native ear. I do not know if there is any situation where form 解けられる can be used. – Tsuyoshi Ito – 2011-07-29T15:36:34.220

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