Because of Alladin's forbidden love for the king's daughter, he has been sentenced to the usual punishment: death by tiger. As the king is a fair man, he has a fighting chance to escape his doom. He is kept in a room with four doors. Behind two doors are tigers. Behind the third a dragon. Behind the fourth is the princess; if he manages to open her door, the king will forgive him and marry his daughter to Alladin. On each door is a number and a plaque. Alladin has been assured that the door leading to the princess has a plaque that tells the truth, and that the doors leading to a tiger have plaques that lie. Alladin does not know whether the plaque on the door leading to the dragon is a lie or the truth.
Door 1: If a tiger waits behind Door 3, then the princess waits behind Door 4.
Door 2: If the princess waits behind Door 3, then the dragon waits behind Door 1.
Door 3: If a dragon waits behind Door 4, then a tiger waits behind Door 2.
Door 4: If a tiger waits behind Door 2, then the princess waits behind Door 3.
I have attached an image of my attempt. I am not sure how to go further or how to approach in a different manner.My approach
Remember that a statement of the form "if $p$, then $q$" is true if $p$ is false. So your first case is checking whether the princess is in door 1. If so, then the door 1 statement is true, which means that a tiger cannot be behind the third door. Therefore we must have the tigers in doors 2 and 4. But this is impossible: since the princess is not behind door 3, the door 2 statement is automatically true, and there cannot be a tiger there. You should be able to evaluate your other cases similarly.