Why do some nouns take the genitive case instead of the accusative?

Question

I recently came upon the sentence:

Мы хотим мира.
We want peace.

The first thing that went through my head was, This must be one of those animate masculine nouns that declines for the accusative case. Wanting to be sure, I decided to look it up on a declension table and discovered that my assumption was wrong. The word "мир" does not decline for accusative case and the role "мира" plays in this sentence is that of a noun in the genitive case. So, I started to plug and play various nouns after the phrase "I want" (not just the masculine inanimate), and have come to the conclusion that "мир" declines into the genitive because it is an uncountable noun and thereby conveys the notion of "I want (some) peace." This then also explains these constructs:

Мы хотим воздуха.
We want (some) air.

Мы хотим воды.
We want (some) water.

Мы хотим мяса.
We want (some) meat.

All of the nouns in the sentences above are in the genitive case.

After experimenting with various words, I've come to other observations about use of the genitive in lieu of the accusative:

1. This appears to happen most often, if not exclusively, with uncountable nouns.

2. Just because a genitive declension was chosen doesn't necessarily mean that an accusative declension couldn't have been chosen instead.

3. If you see the use of a genitive declension instead of the accusative, you can assume that some sort of notion of "some" or "some of" is being conveyed.

4. Favoring the genitive over the accusative is not a given for all uncountable nouns. (The word for "mustard" -- горчица -- is a good example.)

5. Using the genitive declension over the accusative may be more common with some words. In addition to my observation that this happens a lot with uncountable nouns, within that subset there may be tendencies to do this with some uncountable nouns but not others which may verge on the point of being something akin to a "fixed phrase." (Can any native or near fluent Russian speakers provide any examples of this?)

6. The gender of an uncountable noun is irrelevant. Uses of the genitive over the accusative can be seen across the genders. In this post, I inadvertently managed to provide examples of all three genders -- мир (masculine), вода (feminine), and мясо (neuter).

I didn't mean to do this when I started drafting up this question, but between what I've already read/learned and intertwining that with some personal observations, I may have answered some of my own questions. Even so, it's always best to get feedback from others when you're trying to make sense of something, so I've decided to leave this as is in the hopes that those more knowledgeable about the Russian language can support/refute/add to what I've observed and/or recommend some helpful resources on the topic.

Answer 1

1. This appears to happen most often, if not exclusively, with uncountable nouns.

Yes, bearing in mind that some uncountable nouns can be used as countable to denote fixed quantities of the uncountable version:

• Два пи́ва, пожа́луйста! - Two beers please!

(In Russian you can ask for two, three or four beers, but not five or six, sadly.)

One notable exception to the 'uncountable exclusivity' is the verb ждать 'to wait' where the genitive case expresses uncertainty about whether you are going to get what you are waiting for and the accusative case expresses expectation:

• Ждать авто́буса. - Wait for a bus (any bus).

• Ждать авто́бус. - Wait for the bus (that you're expecting).

• Жду отве́та! - Write back soon! (set phrase, literally: I'm waiting for a response).

• Жду отве́т че́рез час! - I'm expecting an answer within an hour!

2. Just because the genitive case was chosen, doesn't necessarily mean that an accusative case couldn't have been chosen instead.

Yes, it could but that that would change the meaning slightly:

• Нале́й воды́. - Pour some water (uncountable).
• Нале́й во́ду. - Pour the water (e.g. all water from a bottle, i.e. countable: 1 water = 1 bottle).

3. If you see the genitive used instead of the accusative, you can assume that some sort of notion of "some" or "some of" is being conveyed.

That's right.

4. Favoring the genitive over the accusative is not a given for all uncountable nouns. (The word for "mustard" -- горчи́ца -- is a good example.)

The choice is quite regular for all uncountable nouns and depends on the sense that you want to convey. True, you'll often hear "Дай горчи́цу" (Acc.) because the mustard-pot (горчи́чница) is implied. You could say "Дай горчи́цы", but that would mean you're asking the person to give you some mustard. And how do you expect them to do that? It's probably not what you want. You just want the pot so you can have some of the mustard: попро́бовать горчи́цы.

5. Using the genitive over the accusative may be more common with some words. In addition to my observation that this happens a lot with uncountable nouns, within that subset there may be tendencies to do this with some uncountable nouns, but not others, which may verge on the point of being something akin to a "fixed phrase." (Can any native or near fluent Russian speakers provide any examples of this?)

Not sure. I'd like to see some examples myself!

Ждать у мо́ря пого́ды? It's a set phrase, but grammatically it's in line with what was said about ждать above.

6. The gender of an uncountable noun is irrelevant. Uses of the genitive over the accusative can be seen across the genders. In this post, I inadvertently managed to provide examples of all three genders -- мир (masculine), вода́ (feminine), and мя́со (neuter).

Yes, even in the plural:

• Пое́сть грибо́в. - Eat (some) mushrooms.
• Съесть все грибы́. - Eat (all of) the mushrooms.

Answer 2

The examples in the question are neither in the genitive nor in the accusative case. They are in the lesser know partitive case. Please see this excellent answer: What are the lesser known Russian cases?

Most nouns in the partitive case have the same forms as in the genitive, but sometimes there's a different form for the partitive.

Here are some examples:

Чай / tea — Genitive: Нет чая; Accusative: Я люблю чай; Partitive: Налейте стакан чаю.

Сахар / sugar — Genitive: Нет сахара; Accusative: Я люблю сахар; Partitive: Насыпьте ложку сахару.

Воздух / air — Genitive: Без воздуха; Accusative: Мне нужен воздух; Partitive: Хочу хоть глоток воздуха.

Вода/ water — Genitive: Из воды; Accusative: Включите воду; Partitive: Плесните немного воды.

Горчица / mustard — Genitive: Нет горчицы; Accusative: Купите горчицу; Partitive: Возьмите ложку горчицы.

In examples such as these, where in English one would say: "I want some water/some tea/some sugar/some peace/etc.", in Russian one would use the partitive.

UPDATE

Here are the words in which the partitive is different from the genitive: Русские существительные с разделительным падежом