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I am trying to understand the difference in usage between 혼자서 and 혼자. In particular, I have come across this sentence "아이가 혼자서 잘 놉니다" which is translated as "The girl plays well by herself".
What would be the difference in meaning in comparison to "아이가 혼자 잘 놉니다"? as I understand it, 혼자 is already an adverb that means alone or by oneself, so wouldn't that sentence carry the same meaning? is there an additional nuance I'm missing? and what's the grammar behind this 서 particle?
In your sentence, 혼자 and 혼자서 have the same meaning. If anything, 혼자서 might make it clearer and more emphatic since the longer version of any two equivalents generally has this effect. For all intents and purposes though, you can treat the two as the same.
The dictionary defines 혼자 as both noun and adverb, but 혼자서 is just an adverb. (A noun example from the dictionary: 나 혼자라는 생각 = the thought that I am alone). It might be that 혼자 started as a noun hundreds of years ago, and then 혼자로서 (-로서 is like "as") was used as an adverb, and it gradually contracted to 혼자서 and then just 혼자 over time.
서 at the end of a verb form or a noun phrase creates an adverb. It is usually 에서 (particle indicating a place, as in 학교에서, 외국에서) or -아/어/여서 (verb ending showing a reason or manner, as in 먹어서, 같아서) in full form, but both are often contracted to just 서. Other -서 forms (로서, -면서, etc) are not shortened in general but certain frequently used phrases can, as we see with 혼자서.
One thing to note is that 서 at the end is not always the less important, expendable part. For example, the particles 에 and 에서 have important distinction (에 indicating destination, like "to/toward" and 에서 an originating place or a venue for an action, like "in"). So 에서 doesn't become 에 but 서 instead when it contracts, to keep the distinction clear (e.g 학교에 가다, 학교(에)서 돌아오다). In certain individual cases, though, it can be dropped because the surrounding context make the meaning clear (e.g. 친구한테 전화를 했다, 친구한테(서) 전화가 왔다).
In normative grammar (school grammar), -서 is a subjective case marker dedicated to cardinal number subjects. To that end, the sentence "아이가 혼자서 잘 놉니다" is a double-subject phrase, and some might find it a bit unnatural, if not ungrammatical.
Note that the above structure is different from the following
, in that this is a subordinate sentence with two subjects and two predicates corresponding respectively.
However, in the real world, one might find lots of usage of this type of "double-subject" sentence. You'll find many, for example, when you try googling "가 둘이서".
Studies proposed different interpretations of the grammatical function of -서/이서, while attempting to explain such adverbial usages of number + -서/이서 phrases. The majority of linguists agree that it is more than a subjective case marker, and some suggest that -서/이서 is not a subjective case marker at all, but an adverbial derivational suffix. For example, here's a recent survey paper: (paywalled) https://www.kci.go.kr/kciportal/ci/sereArticleSearch/ciSereArtiView.kci?sereArticleSearchBean.artiId=ART002419075
Back to original questions,
wouldn't that sentence carry the same meaning? is there an additional nuance I'm missing?
Yes, they carry pretty much the same meaning, and I don't think they make any significant difference in nuance. But a school grammar enthusiast might disagree with me. Compared to 혼자 as an adverb which is more neutrally describing "the state of being alone", though, there is a minor shift in the nuance to "actively performing an action as a single person/entity" when -서 is added. This doesn't make a big difference in the original sentence where a kid is playing, but, for example,
what's the grammar behind this 서 particle?
This is a longstanding research topic in the field, and unfortunately, there's no clear answer to this question.
This is an excellent question. 혼자 is both a countable noun AND an adverb. But when you have a particle 서 suffixed to 혼자 then 혼자서 becomes a noun and noun only. So in this case:
아이가 혼자 잘 놉니다. 혼자 here describes the action of 잘 놉니다. Translates into 'the kid plays alone just fine.'
아이가 혼자서 잘 놉니다. 혼자서 here is a noun and refers to 아이. It's a way of emphasis that the child is alone. Translates into 'the kid by herself plays just fine.'
Expression-wise, the way to say this more naturally than either of your examples would be 아이가 혼자서도 잘 놉니다. The kid even by herself plays just fine.
