欠く vs. 欠ける vs. 欠かせる vs. 欠かせない

Question

I can't clearly see the differences between the meanings of 欠く, 欠ける, 欠かせる, and 欠かせない that I get from resources like Google Translate, Deepl, or wwwjdic. For example, if I enter the following in Deepl

彼は時間を欠く。
彼は時間が欠ける。
彼は時間を欠かせる。
彼は時間を欠かせない。

...the translation it produces is

He lacks time.
He lacks time.
He lacks time.
He lacks time.

This can't be right (for one thing, the fourth sentence is the direct negation of the third one, so their translations can't reasonably be identical).

If instead I give the same input to Google Translate, I get (sic)

he lacks time
he lacks time
he misses time
He needs time.

This is only marginally better: the translations for the third and fourth sentences are at least a little different, though much less so than I expected.

(In fairness to these tools, I must admit that some of the original sentences may be malformed, or at least weird. Please let me know if this is the case. In particular, I am insecure about the particles between 時間 and the verbs.)

---

Q: Can someone give me four sentences that clearly illustrate the differences among these four verbs?

Q: Regarding the last Japanese sentence in particular (彼は時間を欠かせない), how does it compare to the following variant¹?

彼は時間を欠かせなければいけない。

---

FWIW, my best guess (and it is just a guess!) is that the literal translations four the original four sentences above are something like

He lacks time.
Time is lacked by him.
He makes/lets time lack.
He does not make/let time lack.

Granted, the first two translations have essentially the same general meaning, though differently worded. The last two literal translations sound particularly stilted. Slightly more idiomatic ones would be:

It is okay/possible for him to lack time. (= He does not require time.)
It is not okay/possible for him to lack time. (= He requires time.)

---

^{1The original version of my post included a second variant that I now think is downright ungrammatical, so I deleted it.}

Answer 1

You should keep in mind that machine translators will try to make a guess even if you enter completely invalid sentences. Unfortunately, among the first four sentences in your question, only the first one is barely acceptable, and the remaining three are nonsensical.

Next, 欠かせる is actually the potential form of another verb, 欠かす. 欠かせない is its negative version. (You seem to be thinking 欠かせる is the causative form of 欠く, but it's almost never used this way, just as the causative version of 動く is always 動かす and never 動かせる. Read this if you're confused.)

• (Xを)欠く (godan): A transitive verb meaning "to lack X". This is a highly stiff and bookish verb, and you should avoid this in ordinary conversations. Besides, this is normally used with something that someone/something should have as an integral part; simply not having X is not enough. Time is not part of 彼, so that's why 彼は時間を欠く is unnatural, if not 100% incomprehensive. You should almost always say 彼には時間がない or 彼には時間が足りない instead.

  • 彼は誠実さを欠く。(bookish)
  • 右の角を欠いた猛獣 (bookish)
  • 厳密さを欠いた理論 (bookish)

• (Xが)欠ける (ichidan): An intransitive verb meaning "(for X) to lack", "(for X) to miss". Usually used in the form 欠けている ("X is lacking/missing"), but 欠けた is okay in a relative clause. Prefer this over 欠く unless you're intentionally writing something stilted.

  • 彼(に)は誠実さが欠けている。
  • 右の角が欠けた猛獣 / 右の角が欠けている猛獣
  • 厳密さの欠けた理論 / 厳密さの欠けている理論

• (Xを)欠かす (godan): This is originally the causative form of 欠く, so it's more like "to allow (someone) to miss X", "to do/go without X", "to (intentionally) skip X", etc. 欠かせない is the negative-potential version of this, so it's like "cannot miss X" or "cannot do without X", "X is indispensable", etc.

  • 1日運動を欠かすだけで筋肉の量は落ちる。
  • 車を運転するときは絶対にガソリンを欠かすなよ！
  • そのドラマは、毎週欠かさずに見ています。
  • 日本語の読み書きができるようになるには、漢字の勉強が欠かせない。

---

Q: Regarding the last Japanese sentence in particular (彼は時間を欠かせない), how does it compare to the following variant1?

彼は時間を欠かせなければいけない。

Both seem almost nonsensical to me. If you insist, the former may mean "He is unable to lack time", and the latter may mean "He has to be able to make time lacking".