I need help with this propositional logic problem

Question

I study compound statements, and I encountered this problem in the book: The problem

I tried a solution: Let p be proposition "The first door leads to freedom" and let q be proposition "The warrior tells the truth". The question I am going to ask should help me to understand which door to choose. I can choose the correct door by knowing the value of p. So, I want to ask a question which results in yes (T) when p is true, and in no (F) when p is false regardless of which warrior am I asking. I built this truth table to understand what the question should be: My table

Then I tried to construct a compound statement with p and q according to my truth table. But then I realized that it has the same truth table as p, i.e. equivalent to it. I know that it's wrong because if I ask a such question, it is same as asking the warrior "Does the first door lead to freedom?", and obviously he can lie or tell the truth, so it doesn't help me.

The book provides a solution, but I can not understand it: Book solution

I do not understand what do columns "Desired Answer" and "Truth table of Question" mean and where do they come from. I also do not understand why my solution doesn't work.

Can you please explain the solution to me in easy language?

Answer 1

The "desired answer" column in the book's solution is the same as your right-hand column: it's what we're trying to construct our question so that the warrior will say. Of course it's the same as column p because that's exactly the information you're trying to get.

But of course what the warrior says isn't the same thing as what's actually true -- that's the point, after all. So the "truth table of question" column indicates what needs to be true for the warrior to give the "desired" answers. When the warrior is truthful, they will agree. When the warrior is untruthful, they will disagree.

So we've started by asking "what do we want the warrior to say in each case?", and then moved to "what do we want the true answer to our question to be in each case?".

And then the final step is to find a question whose true answer is what's shown in the "truth table of question" column. The book-answer shows a clumsy way to do that by just translating that column of the truth table into English, and then a less-clumsy equivalent.

Your solution doesn't work because you tried to build a question whose true answer is the thing you care about, but you need a question whose answer as given by the warrior is the thing you care about.