Learn Finnish: Päätös ei perustu luuloon, vaan todellisuuteen. - The decision is not based on a belief, but rather on reality.

We use vaan (but rather / instead) when we are directly correcting a negative statement. The first part of the sentence says what the decision is not based on (ei perustu), and vaan introduces what it is actually based on. If you used mutta, it would sound like you are just adding contrasting side-information, rather than replacing the false premise with the true one.

Why do luulo (belief) and todellisuus (reality) take the illative case (luuloon, todellisuuteen)?

This is because of the verb perustua (to be based on). In Finnish, verbs often require their attached nouns to be in a specific case, known as a verb rection. Perustua always takes the illative case (the "into" case). So, instead of saying a decision is based "on" reality like in English, Finnish grammar treats it as being rooted "into" reality.

Why doesn't the verb repeat after vaan?

Just like in English, you don't need to repeat the verb if it hasn't changed. The full thought is "Päätös ei perustu luuloon, vaan päätös perustuu todellisuuteen." Because vaan links these parallel ideas perfectly, we can drop the repeated words. This is also why todellisuuteen must stay in the illative case: it is still being governed by the invisible second perustuu.

How is the word todellisuuteen formed?