There's no Dative here at all, "Twoim zadaniem" is Instrumental. Okay, usually it's the other way round and the thing in Instrumental is on the other side of "jest", but this option here is right and putting it differently would sound really strange.
Alternatively, the part in the infinitive ("znaleźć wyjście") could be turned into a gerund "znalezienie", than it takes Genitive "wyjścia" (the finding of the exit), and we arrive at "Twoim zadaniem jest znalezienie wyjścia". Or possibly "Twoje zadanie to znalezienie wyjścia".
Those two alternatives could also be written the other way round: "Znalezienie wyjścia jest twoim zadaniem", "Znalezienie wyjścia to twoje zadanie", but the focus would strongly move to the "your task" part.
Oh sorry, of course I'm talking about the Instrumental and not the Dative – it was my mistake. I think that I understand everything better now, but still not thoroughly. Let's refer to this sentence for example: "Znalezienie wyjścia jest twoim zadaniem". If I were to change the order of it, you say that the sentence should be written like that: "Twoim zadaniem jest znalezienie wyjścia".
My question is why this example (or something similar to it) creates a bad grammer: "Twoje zadanie jest znalezieniem wyjściam". In other words, I am asking how I can know when to apply the Instrumental case in a sentence. Until today I have always been thinking that the rule is to use it after "jest", but now I see that it was a misconception...
Thank you very much for you help!!!
Well, first of all, although you're already quite an experienced learner here, maybe my guide here may still clarify something for you.
And then, your sentence. Let's show it first on easier examples:
"Kot jest zwierzęciem" (A cat is an animal) is okay, but "Zwierzę jest kotem" (An animal is a cat) is wrong. Not grammatically wrong, but semantically wrong. A specific animal can be a cat, but a totally general 'an animal' cannot, only some animals are cats.
"Paweł jest lekarzem" (Paul is a doctor) is obviously correct, but "Lekarz jest Pawłem" (A doctor is 'a' Paul) is really weird.
And on some mathematical (geometrical) stuff:
Każdy kwadrat jest prostokątem, ale nie każdy prostokąt jest kwadratem. (Every square is a rectangle, but not every rectangle is a square.)
So I think I'd say that it's the matter of what defines what. A definition of 'a cat', although a very simplified one, is that it is in fact 'an animal'. 'An animal' cannot however be defined as 'a cat'. Your task can be defined (precised) by saying that it is "to find the exit". It's hard, at least in Polish, to define "Finding the exit" as being "your task". That's why only the task part can be in Instrumental, even if it's at the beginning of the sentence.
I hope it's understandable now, but don't hesitate to ask if something is still not clear ;)
I've read your whole guide, and I must say again that your willingness to help foreigners learn Polish is impressive. :) Now I understand the topic very well. It's important to understand that the problem was about semantics and not about grammer: Yes, finding the exit defines my task, exactly like a cat defines an animal. However it's incorrect to assert that every task defines "finding an exit" nor that every animal defines a cat. I'm happy that by learning Polish I also happen to develop my logical thinking :)
Marek, but if we want to emphasize that finding the exit is your task, nothing else, like "to be or not to be, that is the question", is "być albo nie być, to jest pytanie" And not "bytem albo nie bytem"? Still confused, because polish has no "the" or "a" its inferred from the context.
Actually the real quote in Polish is "Być albo nie być, oto jest pytanie", but I'm not able to explain what "oto" is... "to" is close enough.
The phrase is "To be or not to be" and it's hard to imagine changing it into "Being or not being", unlike the example here ("to find" vs "finding"). And anyway, the gerund form would be "bycie".
"bytem" is the Instrumental form of "byt". But it means "being", "entity" as an abstract concept, it's rather something from philosophy.
I'm confused about your last sentence, as I don't see how articles are relevant here... ok, and frankly about the first one as well. "Finding the exit is your task" translates to "Znalezienie wyjścia jest twoim zadaniem" (or "Twoim zadaniem jest znalezienie wyjścia", but of course that's "Your task is [finding the exit/to find the exit]". "Your task" is the only part that can take Instrumental. You can say that "Finding the exit is a task" but you can't say that "Your task is a finding the exit". I'd compare it to the example where "Paul is a doctor" but you can't say that "The doctor is a Paul". Is that what you were asking about?
everything I wrote got deleted. Twice. In summary: maybe oto is emphasizing - to be or not to be, that (and no other) is the question. finding the exit - that is your task (nobody else's, and no other tasks). so maybe "znalezienie wyjścia oto twoje zadane" is ok?
To your question: It's because Paul and "finding the exit" are informed as names and genitive.
You can say that "Finding the exit is your task", and also "your task is finding/to find the exit".
I hope you understand what I'm getting at.
Thank you for the response!
In these examples, Paul and "finding the exit" are in Instrumental. The only Genitive here is "wyjścia" itself ('finding of the exit').
"oto" - yeah, I guess you could say that... but it's not that easy to put it in a sentence, it cannot just be used instead of "to". In my comment above I used a comma, but actually this should be "Być albo nie być - oto jest pytanie!". So it also should be "Znalezienie wyjścia - oto (jest) twoje zadanie!". Separate clauses.