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Consider these sentences:

Die Katze sitzt auf meinem Schreibtisch.
Wir wohnen auf dem Lande.
Sie fahren auf das Land.

For the first case auf is translated as on, in the second as "in", the third as "to". I figured out these translations from google translate.

What is it that controls the meaning of auf in a sentence?

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    You seem to assume that English prepositions have an exact meaning, whereas German prepositions have not. This is not true.
    – RHa
    Nov 7 '21 at 10:14
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    Indeed, I can easily imagine a German asking on an English language site how to decide whether to use "on" or "in" in sentences like "we live in the country" and "swans live on lakes and rivers" (not to mention "cats live on mice"). Nov 8 '21 at 0:50
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    German, English and Norwegian (and probably other related languages as well) all have weird special usages of the prepositions, and annoyingly these quirks are different in each language. In Norwegian, you go for example “on the cinema” (på kino), where in English you go to the cinema and in German ins Kino. Another example where the languages do it completely different: “we've run out of cake” vs. “wir haben nichts mehr vom Kuchen da” (≈we have nothing of the cake anymore) vs “vi er tom for kake” (≈we are empty for cake). Nov 8 '21 at 16:45
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To expand a bit on the answer given by infinitezero, many common words, especially prepositions, are impossible to translate directly without taking a number of factors into account. Instead, it's best to learn the possible meanings of a word and under what circumstances it can be used. These additional factors are usually referred to as "context". It's easy to miss the fact that word usage is so complicated in your own language because you learned this information a very young age and use it without thinking about it. But when you learn a new language you have to learn a new set of meanings and circumstances for each word. This is not easy, and I hope no one gave you the impression that learning German would be as simple as memorizing the word in German for each word in English. You can get a hint of how complex English prepositions are by looking them up in a dictionary, for example Wiktionary gives 37 meanings for "on" as a preposition; this does not include meanings as other parts of speech and its use in prepositional verbs. German is no less complicated.

To add an additional wrinkle in German, prepositions, such as auf, which describe a location have different meanings depending on the case of the noun that follows. I think it's best to think of auf as two related words; an accusative version and a dative version. This is similar to the way you might use "on" or "onto" in English depending on the circumstances.

The German auf is used (with a dative noun) when something is on top of something else, but in a number of other circumstances as well. In this sense the best translation is usually "on", but this is not always the case and "on" has additional meanings that don't translate to German as auf. For example in English when you say "I'm sitting in the chair" you mean you're sitting on top of the chair, and in German you'd say Ich sitze auf dem Stuhl. If you say "The picture is on the wall" you don't mean the picture is on top of the wall, but that it's attached to the wall, and German uses a different preposition, an, for this meaning: Das Bild ist an der Wand.

But auf as other meanings. It's used when something appears as an image in something else: Ich bin auf dem Bild. – "I'm in the picture." As a location it's used with large open areas as in your second example: Wir wohnen auf dem Lande. – "We live in the country." (As infinitezero mentioned, Land(e) is considered a large open area in German. I gather that the dative Lande is uncommon in modern German, so it would be more usual to say Wir wohnen auf dem Land.) Another example: Die Kinder spielen auf der Straße. – "The children are playing in the street." Large institutional buildings fall under this heading as well: Sie arbeitet auf dem Flughafen. – "She's working at the airport." As with English prepositions, the meaning of auf can be figurative: Ich bin auf einer Reise. – "I'm on a trip."

Your third example uses the accusative version of auf, which is used when the location is used as a destination or direction of movement, hence its use with fahren. In the main meaning of "on top of" this might be translated as "onto" rather than "on": Die Katze springt auf das Bett. – "The cat is jumping onto the bed." But in your example the meaning is as a large open area and "onto" is not appropriate in this case. English tends to use "to" in a variety of circumstances to indicate a destination, which is how you get "to the country" from auf das Land. In this case the meaning of auf is more specific than "to" since it tells you what sort of destination it is, not just that you're going somewhere.

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    Actually, “auf dem Land(e)” means something like “in the countryside” (or is it actually “on”?), as opposed to, say, “in der Stadt”. On the other hand, “in the country”, meaning a specific country that has been mentioned, would be “in dem Land”. Another one of those things that one just has to know.
    – Carsten S
    Nov 7 '21 at 16:46
  • @Carsten S - To me, "in the country" can mean either "in the countryside" or "in the (previously mentioned) country". English is rather ambiguous here while German distinguishes the meanings with different prepositions. The "countryside" meaning was what I had in mind, but I should have been more clear.
    – RDBury
    Nov 7 '21 at 18:29
  • I didn't know that, thanks.
    – Carsten S
    Nov 7 '21 at 20:14
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    "Ich sitze auf dem Stuhl", aber "Ich sitze in dem Sessel" - I guess there's a similar difference in english between sitting in/on an armchair, chair or stool?
    – Lykanion
    Nov 8 '21 at 14:49
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    @Lykanion: I've found a good way to research this kind of thing is by searching images on Google. For example the results of searching on "sitzt Sessel" mostly use (if they use one of the two) in rather than auf, but auf isn't that rare. The results for "sitting armchair" seem to be about even between "in" and "on". It looks like English and German aren't as different as I thought on this point. I wonder if there's a better example of auf translating to "in".
    – RDBury
    Nov 9 '21 at 9:13
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Nothing. Prepositions differ from language to language.

English counter example:

Wir gehen über die Straße.

Wir fahren über Berlin.

Wir fliegen über Berlin.

We walk across the street.

We drive via Berlin.

We fly over Berlin.

In German, Land is regarded as a large open place, that's why auf is needed in both latter cases.

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    This isn't really an English counter-example, but another example of a similar phenomenon (a single German preposition having multiple meanings in English). Nov 8 '21 at 10:14
  • The basic answer is entirely correct. Nothing controls the language, there is no central institution, and that's why arbitrary changes happen. However, "control" has specific meanings in some prominent flavours of syntax theory, and a definite answer may be possible in those schools of thought, as a matter of taste. The fact that they agree on hardly anything and that the common speaker is anyway unaware of them just proves the point.
    – vectory
    Nov 9 '21 at 18:51

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