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When you are asking someone whether they are cold, you ask "ist dir kalt" instead of "bist du kalt", as you might expect. I understand this is because the question is dative.
However, when you ask whether someone is sick, you would say "bist du krank". In this case, it is not dative.
Why is this the case? Why is it that some questions of this nature of dative and some are not, and how do you know when to use which case?
"Bist Du kalt?" and "Ist Dir kalt?" are both valid questions, but ask for different things. For simplicity's sake, I'll go back to the declarative sentences.
"Du bist kalt" refers to the physical quality of the person's body. I can grab somebody's hands and say, "Oh, Du bist aber kalt". It can also be used figuratively, to convey that somebody is emotionally cold.
"Dir ist kalt" refers to the person feeling cold, that the person's freezing. That'll often coincide with the person's body being physically cold, but it doesn't necessarily have to.
In English, both variants are expressed as "You are cold".
So basically, the analogous question to "Bist Du krank?" wouldn't be "Ist Dir kalt?", but "Bist Du kalt?", with the difference in meaning as I explained.
The difference between Ich bin krank and Mir ist kalt ist the subject of the sentence.
First, let's have a look at the following sentences:
- Ich bin krank.
(I am sick.)- Es ist kalt.
(It is cold.)
Here, the subjects are ich and it, respectively. If you want to say that it is me who feels the coldness of the abstract subject it, you can add the dative object mir:
- Es ist mir kalt.
or- Mir ist es kalt.
Both sentences might sound a bit strange to a native speaker, but they are correct. In fact, when the verb is a form of sein, then the abstract subject es is often omitted, i.e., it is present only implicitly:
- Mir ist kalt.
The sentence
Ich bin kalt.
is also possible but means that I am the subject being cold (my body). Often the expression jemand ist kalt is used as a metaphor to say that someone is coldhearted (see meaning 3. of kalt).
Finally, have a look at
Er scheint mir krank.
(He appears sick to me.)
Here, er is the subject being sick, and mir is the dative object to whom his being sickness appears. The sentence
Mir ist krank.
does not make sense, for it means that there is an abstract it being sick, and I only feel its sickness.
