Reading Henry's question about 了 below, I thought I would look up 了。
You will have to enter 了。
To my mind, a child with no arms, or its arms cut off, is 'unfinished' or 'incomplete' not '完成'.
How did 了 acquire a meaning of 完成?
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了 means 尦 [ ㄌㄧㄠˋ | liào ].
尦 means the shanks cross each other while walking. Sometimes female models walk in this way.
尦 means the rear shanks cross each other when an ox is walking.
了戾 means one or two strands of a matter are twined, bound and not straight.
紾 [ ㄓㄣˇ | zhěn ]：扭轉；拗折。 to twist; to bend. 縳 [ ㄓㄨㄢˋ | zhuàn ]：捲；束。 to wrap; to roll up; to bind; to tie up. 戾 [ ㄌㄧˋ | lì ]：這裡是「彎曲」的意思。crooked; curved.
It depicts the twined and bound shape of two feet.
Everything related to "collect and bind" is called 結. That is why we have the term, 了結. 《廣雅》 interprets 了 as "finish, end, stop, complete, etc.".
From the above information, we can see that the original meaning of 了 is 尦. And, 尦 means that the legs cross each other when walking. The tangled shape of the two feet is similar to the appearance of 收束. It is thus classified as 結, and then extended to the word, 了結. Furthermore, 結 also means 終, 止 or 成, so 了 may mean "ended, stoped or complete".
In short, the original meaning of 了 is "to cross the legs when walking", and "finished or complete" is its extended meaning.
Due to my poor search skills, this is the only classical article I found, which connects 了 to 結束.
The following information is made by the modern people.
In my opinion, the credibility is lower if it is not explained based on the classical text.
*象(symbolize) 其(it's) 足了(enough/satisfied) 戾(return) 之形(form of)
*symbolize 'satisfied and return'
When someone is playing, searching or gathering (whatever kids do), he would stretch his arm out.
After he satisfied and finished, he would return home, no longer stretching his arms. That's where the meaning of "completed /finished" for 了 came from.
First notice that 了 is a sign for perfective aspect(完成时貌).
Modern Mandarin Chinese doesn't have tense(时态), but rather aspect(时貌) instead. For more about this distinction, you can check this, which briefly listed several aspects in Mandarin Chinese including perfective aspect, imperfective aspect and experiential aspect.
For the question how 了 got the meaning "finish", you can check this paper. I will lists the highlights from the paper here:
了 gets the meaning "finish" since the age of Northern and Southern Dynasties (AD 420-589), though this use is rare. And 了 doesn't become the sign of perfective tense (verb+object+了) during this period.
During the time between Northern and Southern Dynasties and Tang Dynasty (AD 618-907), the structure "verb+object+perfective aspect modal verb" persists. But the modal verb used in this structure is evolving over time. In Tang Dynasty, 了 become the most common modal verb for perfective aspect.
(The paper also mentioned that they cannot explain the origin of the structure "verb+object+perfective aspect modal verb" before the Northern and Southern Dynasty. There are two hypotheses trying to explain this. The first is that Chinese languages adopted this structure from Sanskrit, which is proposed by other researchers while this paper says it's not well supported. Another theory is that this structure evolves from more ancient Old Chinese grammar.)
The next is to explain how did the "verb+object+perfective aspect modal verb" evolved to "verb+perfective aspect modal verb+object" structure as in Modern Mandarin Chinese.
Before Tang dynasty, the structure is "verb+object+perfective aspect modal verb". But sometimes there are verbs or adjectives acting as complement to the verb of main clause. These verbs or adjectives have meaning related to perfective aspect.
Between middle Tang Dynasty and Song Dynasty (960-1297), the perfective aspect modal verb 了 moves forward (to form the structure "verb+perfective aspect modal verb+object").