Smullyan Logic Puzzle: Either Tiger behind Door 1 or Gold behind Door 2

Question

"The prisoner is presented with two doors. In a room behind each door is either gold or a tiger. The sign on the doors are either both true or both are false.

Door 1: Either there is a tiger behind this door or gold behind the second door.

Door 2: There is gold behind this door.

Which door should the prisoner open?"

I have been trying to figure out the answer to this question. If I interpret Sign 1 as an inclusive or, then I end up getting 3 "no contradiction" cases, but it doesn't tell me which Door is the best choice (unless we actually care about probability such as 2/3 chance of getting Gold). To mitigate this, the best I could come up with is to interpret Sign 1 as an exclusive or.

Solution attempt

EDIT: Here are other places on the internet where the same problem appears.

http://pi.math.cornell.edu/~araymer/Puzzle/PuzzleNights.html

(Page 4) http://www2.gcc.edu/dept/math/faculty/BancroftED/teaching/handouts/MATH213_rules_of_inference.pdf

https://www.ibtimes.co.uk/mathematician-puzzle-maker-raymond-smullyan-dead-97-1605912

Answer 1

Smullyan, like all of math, uses the inclusive or as the default.

If door 1 is false, there is gold behind door 1 and a tiger behind 2. Then 2 is also false, and there is a tiger behind 2. This is consistent.

If door 1 is true so is 2 and there is gold behind 2. We do not know what is behind 1 because the gold behind door 2 is enough to make it true.

As stated there is no solution to the problem. I don't have this book (is it The Lady or the Tiger?) but the ones I have of his are very carefully proofread. Do you have the problem right?

Answer 2

Did Smullyen say there was a solution?

If the "or" is inclusive then, if they are both true there is Gold behind door 2 and we don't know about Door 1; it could have either. And if they are both false then there is a tiger behind Door 2, and gold behind door 1.

So are we to assume that the likelihood truth or false is 50-50? What about Gold or tiger given any flexibility?

Id say go with Door 1-- it definitely has gold in one case but doesn't definitely have a tiger in the other... but keep your fingers crossed.

....

If its an exclusive or, which I doubt as Ross Millikan points out why I think that, then

If they are both true then there is gold behind both.

If they both are false then there is gold behind door one and a tiger behind the door two.

......

But just because a problem can be stated, doesn't mean it can be solved. If it's an inclusive "or" it can't be resolved, but who ever said it could?