What are the different ways of expressing future tense in Korean? When do I use which way?
For example, I'm not aware of the difference between 먹을게 and 먹겠다.
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There are quite a few future tense constructions in Korean, each with a slightly different nuance.
I like to call this the "assertion future". This is the most general in terms of morphology. It can apply to first, second or third person, and can be used in any speech style. Like many other tenses, it isn't strictly "just future tense use". The breakdown below follows the use in Comprehensive Korean Grammar:
잠시 후 기차가 출발하겠습니다.
The train will be departing shortly.
In English this would be a will-future. Very common in announcements, especially for the immediate future. Naturally this is paired with -습니다 in the formal deferential speech level or the -다 ending in the plain non-polite level.
I will do it by tomorrow / I promise it will be done by tomorrow
The promise is made in the instant of speaking, although the content of the promise may be an action that lasts further into the future. As with all these intention-type constructions in Korean, it refers to the first person. It is a "definitive, solemn and binding" in its tone. Again, it matches the definite quality of the English will-future.
However, when paired with the negative 못 it translates to not being able to do something, and this refers usually to the near future.
Wow that looks delicious!
So this really doesn't have much to do with the future tense of English or other Indo-European languages (although there are parallel uses of the future tense constructions). It just points out what something "is like" based on the speaker's inference, and often corresponds to English "must" or "I'll bet that...", with the speaker being fairly sure of the statement's truth (compared with other "inference" constructions in Korean like -는 것 같다 or -나 보다).
Would you like to sit here?
Again with either formal deferential or non-formal polite endings, these requests are softened by the use of -겠= and often translate to English "would [you] like to... ?".
The morphology of this is restricted to action verbs (not descriptive verbs) and to the speech levels of 해체 "intimate" or "non-polite non-formal" and 해요체 "non-formal polite". It also takes as subject almost exclusively the 1st person.
It is used primarily for:
...just like -겠- above. But its nuance is much lighter: not only is the speech level less formal and less "solemn and bound", the actions to be taken in the promise are usually more directly controlled by the speaker, and the hearer is usually directly affected. The intention is usually accomplished quickly, often immediately...
A: 안 돼! 난 돈 없네!
B: 괜찮아. 형이 살게!
A: Oh no! I haven't got any money!
B: It's OK. Your bro'll buy it (= I'll buy it)
The Grammar refers to these as "spontaneously formed intentions" which is a particularly nice way to describe it. I feel it can be more contingent on others' reactions and replies.
It cannot take -다 (non-polite plain speech level) or -ㅂ니다 (formal polite deferential speech level), hence it is replaced with its more appropriately "solemn" relative above. So for example 먼저 갈게(요) to colleagues but 먼저 가겠습니다 to senior managers (should they let you leave first!!!).
Hence -ㄹ게- would be quite common in everyday conversation because of its usage, but -겠- has its place too and has a greater range of meanings anyway. To this, there's also -ㄹ 것- and -려고 하다 as well, for other types of futurity and intention.
The most common future tense expression is -(으)ㄹ 것이다 (often shortened to -(으)ㄹ 거다). This construct can be used for both predicted/objective future (is going to) and the speaker's intention (will).
-겠- is very similar to English modal verbs. It can add the sense of will (or shall), would, could, should, or might, depending on the subject's person and other factors. As such, it can sometimes suggest a future, insofar as an intention or expectation is about (near) future, but it rarely expresses predicted future like -(으)ㄹ 것이다.
-(으)ㄹ게 is not really about the future. It tells the other person in a friendly way what one is going to do (intention) because they should know about it. So it is like "(Just so you know), I will / am going to [do something]". This constructed is only possible for the first person because it's strictly about intention.
I will list typical uses of the three.