According to the question, if the servant is telling the truth one of the signs is true, the other is false.
Let's see which one is true.
Assumption:The sign on the right is true.
Let's see some consequences of this assumption.
Claim
Justification
The sign on the right is true.
Assumption
The dragon is behind the door on the left.
This is part of what the true sign on the right says
The sign on the left is true.
Because the door on the left leads to the dragon
This contradicts the claim of the servant that one of the signs is false.
This means that our assumption is false.
Consequence: The sign on the right is not true.
This also means that the sign on the left is true, so there is a dragon somewhere.
Unless the knight likes the company of ferocious dragons, he wants to avoid it.
Let's see where the dragon is.
It can either be behind the left or the right door.
Assumption:The dragon is behind the left door.
Let's see some consequences of this assumption.
Claim
Justification
The dragon is behind the door on the left.
Assumption
The treasure is behind the right door.
The treasure must be somewhere, and let's hope it is not guarded by the dragon
The sign on the right is true.
What we concluded in the previous two rows is exactly what this sign says
This contradicts our previous conclusion that the sign on the right is not true.
This means that our second assumption cannot be true either.
Conclusion: The dragon is not behind the left door.
This means that it is safe for the knight to choose the left door.
