In this paper, we present a framework in which we analyze three riddles about truth that are all (originally) due to Smullyan. We start with the riddle of the yes-no brothers and then the somewhat more complicated riddle of the da-ja brothers is studied. Finally, we study the Hardest Logic Puzzle Ever (HLPE). We present the respective riddles as sets of sentences of quotational languages , which are interpreted by sentence-structures. Using a revision-process the consistency of these sets is established. (...) In our formal framework we observe some interesting dissimilarities between HLPE’ s available solutions that were hidden due to their previous formulation in natural language. Finally, we discuss more recent solutions to HLPE which, by means of self-referential questions , reduce the number of questions that have to be asked in order to solve HLPE . Although the essence of the paper is to introduce a framework that allows us to formalize riddles about truth that do not involve self-reference, we will also shed some formal light on the self-referential solutions to HLPE. (shrink)

Rabern and Rabern (Analysis 68:105–112 2 ) and Uzquiano (Analysis 70:39–44 4 ) have each presented increasingly harder versions of ‘the hardest logic puzzle ever’ (Boolos The Harvard Review of Philosophy 6:62–65 1 ), and each has provided a two-question solution to his predecessor’s puzzle. But Uzquiano’s puzzle is different from the original and different from Rabern and Rabern’s in at least one important respect: it cannot be solved in less than three questions. In this paper we solve Uzquiano’s puzzle (...) in three questions and show why there is no solution in two. Finally, to cement a tradition, we introduce a puzzle of our own. (shrink)

We present the simplest solution ever to 'the hardest logic puzzle ever'. We then modify the puzzle to make it even harder and give a simple solution to the modified puzzle. The final sections investigate exploding god-heads and a two-question solution to the original puzzle.

If you think that a proposition can have more than one contradictory, or can have none, then you need read no further. What I will show is that if then It is not obvious that this must be so. If p1 and p2 are distinct but logically equivalent, it might appear that their contradictories, q1 and q2, should also be distinct though logically equivalent. In traditional logic, a pair of propositions which satisfy (3)(i) alone are called contraries, and a pair (...) which satisfy (3)(ii) alone are called sub-contraries.2 The relation of contradictoriness defined by (3) is clearly symmetrical: Let p1 and p2 be two propositions which are logically equivalent, and assume (1). Then p1 has a contradictory, q. We first show that Suppose p2 and q were both true. Then, since p1 and p2 are logically equivalent, p1 and q would both be true. But p1 and q are contradictories, so, by (3)(i) p1 and q cannot both be true. We next show that By (3)(ii) at least one of p1 and q must be true. So, since p1 and p2 are logically equivalent, at least one of p2 and q must be true. From (5) and (6) it follows that p2 and q are contradictories, so by (4) From (7) and (1) we have that p1 and p2 are identical; and since all that was assumed about them was that they are logically equivalent, (2) follows. (shrink)