There are a lot of comments here that say that the jester is unjustified in assuming that there is a correlation between the inscriptions and the contents of the boxes. This is, in my opinion, complete and utter nonsense. Once we assign meanings to the words true and false (in this case, “is an accurate description of reality” and “is not an accurate description of reality”), all other statements are either false, true or meaningless. A statement can be meaningless because it describes something that is not real (for example, “This box contains the key” is meaningless if the world does not contain any boxes) or because it is inconsistent (it has at least one infinite loop, as with “This statement is false”). If a statement is meaningful it affects our observations of reality, and so we can use Bayesian reasoning to assign a probabilty for the statement being true. If the statement is meaningless, we cannot assign a probabilty for it being true without violating our assumption that there is a consistent underlying reality to observe, in which case we cannot trust our observations. Halt, Melt and Catch Fire.
The statement “This box contains the key” is a description of reality, and is either false or true. The statement “Both inscriptions are true” is meaningful if there exists another inscription, true if the second description is true and false if the second description is false or meaningless. The statement “Both inscriptions are false” is meaningless because it is inconsistent—we cannot assign a truth-value to it. The statement “Either both inscriptions are true, or both inscriptions are false” is therefore either true (both inscriptions are true, implying that the key is in box 2) or meaningless. In the latter case, we can gain no information from the statement—the jester might as well have been given only the second box and the second inscription. The jester’s mistake lies in assuming that both inscriptions must be meaningful—“one is meaningless and the other is false” is as valid an answer as “both are true”, in that both of those statements are meaningful—the latter is true if the second box contains the key, and the former is true if the second box does not contain the key. The jester should have evaluated the probabilty that the problem was meant to be solvable and the probability that the problem was not meant to be solvable, given that the problem is not solvable, which is an assessment of the king’s ability at puzzle-devising and the king’s desire to kill the jester.
It is also provable that we cannot assign a probabilty of 1 or 0 to any statement’s truth (including tautologies), since we must have some function from which truth and falsity are defined, and specifying both an input and an output (a statement and its truth value) changes the function we use. If a statement is assigned a truth-value except by the rules of whatever logical system we pick, the logical system fails and we cannot draw any inferences at all. A system with a definition of truth, a set of thruth-preserving operations and at least one axiom must always be meaningless—the assumption of the axiom’s truth is not a truth-preserving operation, and neither is the assumption that our truth-preserving operations are truth-preserving. Axiomatic logic works only if we accept the possibility that the axioms might be false and that our reasoning might be flawed—you can’t argue based on the truth of A without either allowing arguments based on ~A or including “A” in your definition of truth. In other words, axiomatic logic can’t be applied to reality with certainty—we would end up like the jester, asserting that reality must be wrong. As a consequence of the above, defining “true” as “reflecting an observable underlying reality” implies that all meaningful statements must have observable consequences.
The argument above applies to itself. The last sentence applies to itself and the paragraph before that. The last sentence… (If I acquire the karma to post articles, I’ll probably write one explaining this in more detail, assuming anyone’s interested.)
There are a lot of comments here that say that the jester is unjustified in assuming that there is a correlation between the inscriptions and the contents of the boxes. This is, in my opinion, complete and utter nonsense. Once we assign meanings to the words true and false (in this case, “is an accurate description of reality” and “is not an accurate description of reality”), all other statements are either false, true or meaningless. A statement can be meaningless because it describes something that is not real (for example, “This box contains the key” is meaningless if the world does not contain any boxes) or because it is inconsistent (it has at least one infinite loop, as with “This statement is false”). If a statement is meaningful it affects our observations of reality, and so we can use Bayesian reasoning to assign a probabilty for the statement being true. If the statement is meaningless, we cannot assign a probabilty for it being true without violating our assumption that there is a consistent underlying reality to observe, in which case we cannot trust our observations. Halt, Melt and Catch Fire.
The statement “This box contains the key” is a description of reality, and is either false or true. The statement “Both inscriptions are true” is meaningful if there exists another inscription, true if the second description is true and false if the second description is false or meaningless. The statement “Both inscriptions are false” is meaningless because it is inconsistent—we cannot assign a truth-value to it. The statement “Either both inscriptions are true, or both inscriptions are false” is therefore either true (both inscriptions are true, implying that the key is in box 2) or meaningless. In the latter case, we can gain no information from the statement—the jester might as well have been given only the second box and the second inscription. The jester’s mistake lies in assuming that both inscriptions must be meaningful—“one is meaningless and the other is false” is as valid an answer as “both are true”, in that both of those statements are meaningful—the latter is true if the second box contains the key, and the former is true if the second box does not contain the key. The jester should have evaluated the probabilty that the problem was meant to be solvable and the probability that the problem was not meant to be solvable, given that the problem is not solvable, which is an assessment of the king’s ability at puzzle-devising and the king’s desire to kill the jester.
It is also provable that we cannot assign a probabilty of 1 or 0 to any statement’s truth (including tautologies), since we must have some function from which truth and falsity are defined, and specifying both an input and an output (a statement and its truth value) changes the function we use. If a statement is assigned a truth-value except by the rules of whatever logical system we pick, the logical system fails and we cannot draw any inferences at all. A system with a definition of truth, a set of thruth-preserving operations and at least one axiom must always be meaningless—the assumption of the axiom’s truth is not a truth-preserving operation, and neither is the assumption that our truth-preserving operations are truth-preserving. Axiomatic logic works only if we accept the possibility that the axioms might be false and that our reasoning might be flawed—you can’t argue based on the truth of A without either allowing arguments based on ~A or including “A” in your definition of truth. In other words, axiomatic logic can’t be applied to reality with certainty—we would end up like the jester, asserting that reality must be wrong. As a consequence of the above, defining “true” as “reflecting an observable underlying reality” implies that all meaningful statements must have observable consequences.
The argument above applies to itself. The last sentence applies to itself and the paragraph before that. The last sentence… (If I acquire the karma to post articles, I’ll probably write one explaining this in more detail, assuming anyone’s interested.)