I am stuck with this problem. I know I have to use propositional logic and truth tables, but I believe that in order to be sure about the right way to get to the Cold Stone Creamery I need to get a tautology, I believe that the table being satisfiable is not enough. My second doubt is if I should do a separate table for have one of the three vendors. I did it like that but none of the three tables gives me a tautology. And last doubt I have is in the last part of the problem. It says that any vendor says the truth, so should my equation should be negated? Please any help will be appreciated... Here is the actual problem:
You find yourself in a maze walking and you reach a 3-way intersection. The roads have signs as follows: -on the left: “bad idea street” -in front of you: “don’t even try it road” -on the right: “you’re lost avenue” On each one of the roads there is a street vendor (hey, it’s Miami, they are everywhere!). You talk to the street vendors and this is what they tell you (after you buy some of their goods of course): -Street vendor on “bad idea street”: “This road will take you straight to Cold Stone Creamery. Also, if “you’re lost avenue” leads you to Cold Stone Creamery, then “don’t even try it road” will also take you to Cold Stone Creamery.” -Street vendor on “don’t even try it road”: “Neither “bad idea street” nor “you’re lost avenue” takes you to Cold Stone Creamery.” -Street vendor on “you’re lost avenue”: “Follow “bad idea street” and you’ll reach Cold Stone Creamery. Furthermore, follow “don’t even try it road” and you will never leave this maze (Muahaha).” It is common knowledge that street vendors never tell the truth. Furthermore, you are dying for some ice-cream, after a whole day of being lost in the maze. Can a road be chosen (with 100% certainty) such that it will take you to your neighborhood Cold Stone Creamery? If it can be done, which road would you choose? Provide formal language and propositional logic facts that formalize the problem and prove(show steps) whether a road can be chose so that you can be sure it will lead you to Cold Stone Creamery(for some yummy ice-cream).
Vendor 1 makes two statements which seem to give the answer:
For the second statement to be strictly false, though, the conditional "if" part has to be true (and the dependent "then" part false). So the way to go is Y.L.Avenue, and this is consistent with the other vendors' remarks being false also.
Written as formulas, define the following claims:
We have the following assertions from vendors U, V and W:
And we are told that these are all false. Therefore: $$ \begin{align} & \sim p \tag{U1 false} \\ & \sim (q \Rightarrow r) \tag{U2 false}\\ \rightarrow \ & \sim (\sim q \lor r) \\ \rightarrow \ & ( q\, \land \sim r) \\ \therefore \ & ( \sim p \land q \,\land \sim r) \\ \end{align} $$