# Algebraic Problems in Propositional Logic

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Problem 1. On the island of Knights and Knaves, knights always tell the truth and knaves always lie. A logician visits the island and meets an inhabitant. The logician wants to know whether there is gold on the island. Is there a statement such that from its truth the logician can infer that gold is on the island and from its negation that there is not? Let mean that the native is a knight, and that there is gold. If the native answers "yes" to the question "Is true?", the logician knows . Is it possible that (i.e. infers )? Simultaneously, this should hold: . So we must find a propositional expression in variables and such that and are both tautologies. This is equivalent to the statement that and are simultaneously inconsistent (unsatisfiable), that is, ) is inconsistent. The DNF (disjunctive normal form) of the last expression is .

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Contributed by: Izidor Hafner (April 2014)

Open content licensed under CC BY-NC-SA

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References

[1] R. H. Cowen, "Solving Algebraic Problems in Propositional Logic by Tableau," *Archive for Mathematical Logic*, 22, 1982 pp. 187–190.

[2] R. Smullyan, *The Lady or the Tiger?*, New York: Alfred A. Knopf, 1983.

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