The Hardest Logic Puzzle Ever

The Hardest Logic Puzzle Ever 

                                                                                is a title coined by American philosopher and logician George Boolos in an article published in The Harvard Review of Philosophy (an Italian translation was published earlier in the newspaper La Repubblica, under the title L'indovinello più difficile del mondo) for the following Raymond Smullyan inspired logic puzzle:

Three gods A, B, and C are called, in no particular order, True, False, and Random. True always speaks truly, False always speaks falsely, but whether Random speaks truly or falsely is a completely random matter. Your task is to determine the identities of A, B, and C by asking three yes-no questions; each question must be put to exactly one god. The gods understand English, but will answer all questions in their own language, in which the words for yes and no are 'da' and 'ja', in some order. You do not know which word means which.

             Boolos provides the following clarifications:

It could be that some god gets asked more than one question (and hence that some god is not asked any question at all). What the second question is, and to which god it is put, may depend on the answer to the first question. (And of course similarly for the third question.) Whether Random speaks truly or not should be thought of as depending on the flip of a coin hidden in his brain: if the coin comes down heads, he speaks truly; if tails, falsely. Random will answer 'da' or 'ja' when asked any yes-no question.   The solution Boolos provided his solution in the same article in which he introduced the puzzle. Boolos states that the "first move is to find a god that you can be certain is not Random, and hence is either True or False". There are many different questions that will achieve this result. One strategy is to use complicated logical connectives in your questions (either biconditionals or some equivalent construction). Boolos' question was: Does 'da' mean yes if and only if you are True if and only if B is Random? Equivalently: Are an odd number of the following statements true: you are False, 'da' means yes, B is Random? It was observed by Roberts (2001) -- and independently by Rabern and Rabern (2008) -- that the puzzle's solution can be simplified by using certain counterfactuals. The key to this solution is that, for any yes/no question Q, asking either True or False the question If I asked you Q, would you say 'ja'? results in the answer 'ja' if the truthful answer to Q is yes, and the answer 'da' if the truthful answer to Q is no (Rabern and Rabern (2008) call this result the embedded question lemma). The reason it works can be seen by looking at the eight possible cases. Assume that 'ja' means yes and 'da' means no. (i) True is asked and responds with 'ja'. Since he is telling the truth the truthful answer to Q is 'ja', which means yes. (ii) True is asked and responds with 'da'. Since he is telling the truth the truthful answer to Q is 'da', which means no. (iii) False is asked and responds with 'ja'. Since he is lying it follows that if you asked him Q he would instead answer 'da'. He would be lying, so the truthful answer to Q is 'ja', which means yes. (iv) False is asked and responds with 'da'. Since he is lying it follows that if you asked him Q he would in fact answer 'ja'. He would be lying, so the truthful answer to Q is 'da', which means no. Assume 'ja' means no and 'da' means yes. (v) True is asked and responds with 'ja'. Since he is telling the truth the truthful answer to Q is 'da', which means yes. (vi) True is asked and responds with 'da'. Since he is telling the truth the truthful answer to Q is 'ja', which means no. (vii) False is asked and responds with 'ja'. Since he is lying it follows that if you asked him Q he would in fact answer 'ja'. He would be lying, so the truthful answer to Q is 'da', which means yes. (viii) False is asked and responds with 'da'. Since he is lying it follows that if you asked him Q he would instead answer 'da'. He would be lying, so the truthful answer to Q is 'ja', which means no. Using this fact, one may proceed as follows. Ask god B, "If I asked you 'Is A Random?', would you say 'ja'?". If B answers 'ja', then either B is Random (and is answering randomly), or B is not Random and the answer indicates that A is indeed Random. Either way, C is not Random. If B answers 'da', then either B is Random (and is answering randomly), or B is not Random and the answer indicates that A is not Random. Either way, A is not Random. Go to the god who was identified as not being Random by the previous question (either A or C), and ask him: "If I asked you 'Are you True?', would you say 'ja'?". Since he is not Random, an answer of 'ja' indicates that he is True and an answer of 'da' indicates that he is False. Ask the same god the question: "If I asked you 'Is B Random?', would you say 'ja'?". If the answer is 'ja' then B is Random; if the answer is 'da' then the god you have not yet spoken to is Random. The remaining god can be identified by elimination.  Random's behaviour Most readers of the puzzle assume that Random will provide completely random answers to any question asked of him; however, Rabern and Rabern (2008) have pointed out that the puzzle does not actually state this.And in fact, Boolos' third clarifying remark explicitly refutes this assumption. Whether Random speaks truly or not should be thought of as depending on the flip of a coin hidden in his brain: if the coin comes down heads, he speaks truly; if tails, falsely. This says that Random randomly acts as a false-teller or a truth-teller, not that Random answers randomly. A small change to the question above yields a question which will always elicit a meaningful answer from Random. The change is as follows: If I asked you Q in your current mental state, would you say 'ja'? This effectively extracts the truth-teller and liar personalities from Random and forces him to be only one of them. By doing so the puzzle becomes completely trivial, that is, truthful answers can be easily obtained. 1. Ask god A, "If I asked you 'Are you Random?' in your current mental state, would you say 'ja'?" 2a. If A answers 'ja', then A is Random: Ask god B, "If I asked you 'Are you True?', would you say 'ja'?" If B answers 'ja', then B is True and C is False. If B answers 'da', then B is False and C is True. In both cases, the puzzle is solved. 2b. If A answers 'da', then A is not Random: Ask god A, "If I asked you 'Are you True?', would you say 'ja'?" If A answers 'ja', then A is True. If A answers 'da', then A is False. 3. Ask god A, "If I asked you 'Is B Random?', would you say 'ja'?" If A answers 'ja', then B is Random, and C is the opposite of A. If A answers 'da', then C is Random, and B is the opposite of A. Rabern and Rabern (2008) suggest making an amendment to Boolos' original puzzle so that Random is actually random. The modification is to replace Boolos' third clarifying remark with the following: Whether Random says 'ja' or 'da' should be thought of as depending on the flip of a coin hidden in his brain: if the coin comes down heads, he says 'ja'; if tails, he says 'da'.