This question may be related to [What does で分かる
mean?](https://japanese.stackexchange.com/questions/5986/what-
does-%E3%81%A7%E5%88%86%E3%81%8B%E3%82%8B-mean/5987) (Cf. Tsuyoshi Ito and my
comments to my answer).

(Regular) verbs can be turned into the potential form by attaching
`-((ra)r)e-`:

> tabe-ru vs. tabe-(ra)re-ru  
>  kak-u vs. kak-e-ru

However, the verb `分かる` cannot. Why is that?

> wakar-u  
>  * wakar-e-ru

The verb `理解する`, which has a same/similar meaning, can have the potential
form. `Suru` verbs are known to be irregular and require `-deki-`, and `理解する`
does as well:

> suru vs. dekiru  
>  benkyou-suru vs. benkyou-dekiru  
>  rikai-suru vs. rikai-dekiru

`知る` also does not seem to allow the potential form.

> sir-u  
>  * sir-e-ru (Okay under the spontaneous or the passive interpretations of
> `-((ra)r)e-`, but not under the potential interpretation).

But for `知る`, you can have a lengthier expression to turn it into a potential
form. Nevertheless, `分かる` even resist this form:

> 知ることができる  
>  理解することができる  
>  * 分かることができる

This seems to be indicating that the problem is not morphological or
phonological (for example along the lines of crash with homonymous/homophoneme
expressions that Flaw suggests).
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potential-form
Why doesn't 分かる have a potential form?
A devil's-advocate, non-rigorous argument from etymology:

(1) /wakaru/ is morphologically /wak.ar.u/, and so in /ar/ already contains a
spontaneous/passive morpheme that is equivalent (in some ways, different in
others of course) to modern /((ra)r)e/.

(2) Constructions like /nihongo ga wakaru/ are often explained as equivalent
to "[somebody] understands Japanese" not only in meaning but also in
(underlying) structure. That is, the "nihongo ga" is interpreted as a direct
object, marked with "ga" instead of "o" for reasons that aren't relevant here.
But if we take the construction at face value, based on the etymology outlined
in (1) above, it actually means "Japanese is understood".

(3) Therefore it is unnatural to say /*nihongo ga wakareru/ for the same
reasons that it would be to say *"Japanese is understood-ed" in English. There
is no passive version because the original phrase is already passive, or at
the very least contains no direct object. Since the passive and the potential
are closely linked in Japanese, there is no potential version either ( _this
is probably the most hand-wavey part of this entire argument_ ).

Thoughts:

  * We cannot say /*kazoku ga areru/, *"a family is had" either (with the intended meaning of "able to have a family"). But we can say /kazoku ga sonzai dekiru/, "a family can exist". I think the important distinction here is that /aru/ implies an attachment to some sort of /X ni/ while /sonzai suru/ does not.

  * Words like /umareru/ "be born" have similar etymology, but you still see /umarerareru/ sometimes (e.g. "悪人でも極楽浄土に生まれられる" "Even bad people **can be reborn** in the Pure Land"). I have no good counter-argument against this and don't really have good judgment of its naturalness either.

  * We _do_ see similar patterns in words like /sirareru/ "be known"; I do not think that /*sirarerareru/ "be able to be known" is allowed. This is perhaps evidence for the hand-wavey portion noted above.

  * One way to test my proposal would be to investigate Japanese speakers whose idiolects allow/require the construction /o wakaru/, and find out if they _do_ in fact use constructions like /wakareru/, or at least find them less unnatural than speakers who only allow /ga wakaru/.
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Shogakukan's _Kokugo Dai Jiten Dictionary_ (JA-JA) contains an entry for 分かれる,
for which sense 4 is:

> 〔自ラ下一〕  
>  区別がつく。差別ができる。  
>  (Intransitive, ラ, _shimo ichidan_ conjugation)  
>  To have a clear distinction. To be discernable.

As to why this form might not be used that much, a brief survey of Google
Books searching for
["が分かれる"](https://www.google.com/search?tbo=p&tbm=bks&q=%22%E3%81%8C%E5%88%86%E3%81%8B%E3%82%8C%E3%82%8B%22)
shows lots of hits where 分かれる is being used in the intransitive sense of "to
split into different pieces". 分かれる appears to be used more often for this
sense than for the potential of 分かる.

Moreover, both Shogakukan's and Daijirin's definitions for 分かる include
phrasing like 「知れる」 and 「理解できる」, suggesting that a sense of potentiality is
inherent in the verb 分かる.

These two factors combined might lead speakers to be overtly explicit about
potentiality, resulting in constructions like 「分かる **ことができる** 」.
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As pointed out in other answers, わかる is etymologically an
potential/intransitive/passive verb of わける/わく. Then there are two questions:

  * why cannot potential/passive/intransitive verbs have potential form.
  * why the intransitive わかる is always used.
  * how the potential form works

## Why cannot potential verbs have potential forms

I have had this questions for years and my explanation is: making the object
the subject is just the way to express “result” in Japanese. A “result” is
something you can assert it happens/will happen/happened only after if
actually happens. It is not something can be controlled by the agent.

It is common to see people say お茶が入った, 風呂が沸いた, ご飯ができた, etc. Some people think
using intransitive verbs are more polite and being polite is a feature of
Japanese. However, I hardly think so.

It is interesting that in Chinese, we say neither 1~3a nor 1~3b. Instead, we
say something like 1~3c. The point is, the transitive verb and passive verb
are used to report the whole process.

> 1a) お茶が入った  
>  2a) 風呂が沸いた  
>  3a) ご飯ができた  
>  
>  1b) お茶を入れた  
>  2b) 風呂を沸かした  
>  3b) ご飯を作った  
>  
>  1c) 茶沏好了 (お茶が、入れられて、うまくなった)  
>  2c) 水烧开了 (風呂が、沸かされて 、沸いた)  
>  3c) 饭作好了 (ご飯が、作られて、できた)

In a context where some action is supposed to be taken / have been taken, if
you just want to report the result, which has no duration, you must use a
different expression, that is, the intransitive form.

The difference between Chinese and Japanese is that in Japanese you can just
make the original object the subject of verb without explicitly mentioning its
cause, while in Chinese the relation between the consequence and its cause
must be explicitly conveyed. I am not good at English and I do not know how
what expressions will be used in this context in English.

Because the result is not something we can control, we might be interested in
what the result is supposed to be, ie its potentiality. When there is an
unstated agent and his effort in the context and no adverbs, it can be usually
assumed that an adverb うまく (successfully/skillfully) is implied.
(Historically, 能く{よく} (well, skillfully), which was similar to うまく, was
another way to express potentiality)

> 4a) ご飯が作れる  
>  5a) 漢字が読める  
>  6a) 運転ができない

They roughly equal their more verbose conditional versions.

> 4b) ご飯を作りたければ、作れる  
>  5b) 漢字を読めたければ、読める  
>  6b) 運転をしようとしても、できない

**In other words, the difference between “will” and “can” mainly depends on
the context -- whether there is an implied effort or not.**

In the last sentence 6, the distinction between “will” and “can” are
neutralized in negative statements, which is unexceptional because
historically, the “potential” function of られる only appears in negative
sentences.

For example, 得ず{えず} means both “will not get” or “cannot get”, 能わず might be
the intransitive verb of あたえる, which means “will not get” or “cannot get”,
too.

## Why わかる is always intransitive

I do not have a confident answer. But I think, there are several verbs of
perception that are usually intransitive, such as 見える, 聞きこる, わかる.

I think it reflects the process of human mental activity. We are always
looking, listening, observing the circumstance around us, and thinking about
the information we perceived. So what concerns us is not whether we performed
the action or not, but the result we got.

I looked around, but we do I see? The result will be 靴が見えた.

I though about it, but we do I realize? The result will be お客さんが来ていることがわかった.

Maybe because their intransitive versions are frequently used, they are
irregular. The active wakeru or waku might have never been used to mean " to
understand". I think I have read a paper about the etymology of wakaru, but I
cannot find it.

## The weirdness of the potential form

Although I consider the potential form is as a special case and productive
form to express “the result”, it do not mean it always works as you may
expect. Compare:

> 10a) ご飯ができる  
>  10b) ご飯が作れる  
>  10c) ご飯ができない  
>  10d) ご飯が作れない  
>  
>  11a) 富士山が見える  
>  11b) 富士山が見られる  
>  11c) 富士山が見えない  
>  11d) 富士山が見られない

We can see a clear difference in how these action succeed or fail. Some forms,
such as ご飯が作れる and ご飯ができない are not likely to be used.

However, adding some adverbs will make a difference.

> 12a) ご飯がおいしくできる  
>  12b) ご飯がおいしく作れる  
>  12c) ご飯がおいしくできない  
>  12d) ご飯がおいしく作れない

We can even use adjectives.

> 13a) おいしいご飯ができる  
>  13b) おいしいご飯が作れる  
>  13c) おいしいご飯ができない  
>  13d) おいしいご飯が作れない

It is a little hard to determine the naturalness and differences between all
these expressions, but the meaning of potential form is not easy to explain.

With adverbs expressing desirable resultative state, the difference between
the potential form and the active/passive form can sometimes be more or less
neutralized.

> 14a) 絵がうまく書いてある  
>  14b) 絵がうまく書かれている  
>  14c) 絵がうまく書けている
>
> 15a) 英語を話す  
>  15b) 英語を話せる  
>  15c) 英語を流暢に話す  
>  15d) 英語を流暢に話せる

Since うまく and 流暢 are desirable state, it can usually be safely assumed that
they are the results of the agent's effort.

For some verbs, the passive form can be used instead of the (regular)
potential form.

> 17) 残念に思われる  
>  18) 残念に思える

Sometimes, non-volitional verbs can have passive forms

> 30) 得る <> 得られる (得る has become a volitional verb now)  
>  31) 知る <> 知られる

Some potential/passive forms become volitional verbs and can further be put
into the potential form.

> 32) 痩す <> 痩せる  
>  33) 生まれる <> 生まれられる  
>  34) 覚える <> 覚えられる

For some verbs, they are primarily passive/non-volitional, the potential forms
are not often used.

> 35) 教わる <> ?教われる, but 教わることができる  
>  36) 知る <> ?知れる, but 知ることができる
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If you look in a Japanese-Japanese dictionary, the primary definition of 分かる
is 明白になる (Sanseido) "become clear" or something equivalent. It gets translated
in English as "understand" not because that is what it actually means but
because that is what an English speaker usually says in cases where a Japanese
speaker uses 分かる.

So "(watashi wa) hon ga wakaru" gets translated as "I can understand the book"
but what it actually says is "The book is clear (to me)". Note that the ga-
marked shugo (subject) is _not_ me, as in the standard English non-literal
rendering. It is the book.

Since the word really means to be clear, we can see that a thing either is or
isn't clear. It can't be in a state of "possible-to-be-clear".

By contrast, rikai suru _does_ mean to understand, so naturally it can take a
potential form _rikai dekiru_ - to be capable of understanding.
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