On formalization of logic puzzles textit{`{a}} ${la}$ Smullyan (Logic, Language, Algebraic system and Related Areas in Computer Science)

Abstract

George Boolos (1996) posed the puzzle "The hardest logic puzzle ever" which had been devised by Raymond Smullyan, and gave a solution in the style of if-and-only-if. Later, Roberts (2001) and Rabern-Rabern (2008) provided another solution in the style of embedded questions as a simpler solution. Here, we introduce a simple formalization of the puzzle consisting of questions, answerers, and answers in terms of propositional logic, and show its adequacy by the truth values (0, 1) semantics. Then it turns out that the two solutions in the different forms are logically equivalent under the semantics. Moreover, the hardest logic puzzle can be considered as a natural extension of the puzzles of knights and knaves, i.e., lying and truth-telling by Smullyan.