I tried to solve the following logic puzzle. It seems that $(A \vee \neg A) \wedge B$ is the answer, but I'm not sure. Can anyone give me an answer?
A certain country is inhabited only by truth-tellers (people who always tell the truth) and liars (people who always lie). Moreover, the inhabitants will respond only to yes or no questions. A tourist comes to a fork in a road where one branch leads to the capital and the other does not. There is no sign indicating which branch to take, but there is a native standing at the fork. What yes or no question should the tourist ask in order to determine which branch to take?
Hint: Let $A$ stand for “You are a truth-teller” and let $B$ stand for “The left-hand branch leads to the capital.” Construct, by means of a suitable truth table, a statement form involving $A$ and $B$ such that the native's answer to the question as to whether this statement form is true will be yes when and only when $B$ is true.