Axiomathes

pp 1–19

Solving Smullyan Puzzles with Formal Systems

Original Paper

Abstract

Solving numeric, logic and language puzzles and paradoxes is common within a wide community of high school and university students, fact witnessed by the increasing number of books published by mathematicians such as Martin Gardner (popular books as old as Gardner in Aha! insight. W. H. Freeman & Co., London, 1978, Wheels, life and other mathematical amusements. W H Freeman & Co., London, 1985), Douglas Hofstadter [in one of the best popular science books on paradoxes (Hofstadter in Godel, escher, bach: an eternal golden braid, Penguin, London, 2000)], inspired by Gödel’s incompleteness theorems), Patrick Hughes and George Brecht (see Hughes and Brecht in Vicious circles and infinity, an anthology of paradoxes. Penguin Books, London, 1993) and Raymond M. Smullyan (the most well known being Smullyan in Forever undecided, puzzle guide to godel. Oxford Paperbacks, Oxford 1988, To Mock a Mockingbird and other logic puzzles. Oxford Paperbacks, Oxford 2000, The lady or the tiger? And other logic puzzles. Dover Publications Inc., Mineola 2009), inter alia. Books by Smullyan (such as Smullyan 1988, 2000) are, however, much more involved, since they introduce learning trajectories and strategies across several subjects of mathematical logic, as difficult as combinatorial logic (see, e.g., Smullyan 2000), computability theory (see Smullyan 1988), and proof theory (see Smullyan 1988, 2009). These books provide solutions to their suggested exercises. Both statements and their solutions are written in the natural language, introducing some informal algorithms. As an exercise in Mathematics we wonder if an easy proof system could be devised to solve the amusing equations proposed by Smullyan in his books. Moreover, university students of logic could well train themselves in constructing deductive systems to solve puzzles instead of a non-uniform treatment one by one. In this paper, addressing students, we introduce one such formal systems, a tableaux approach able to provide the solutions to the puzzles involving either propositional logic, first order logic, or aspect logic. Let the reader amuse herself or himself!

Keywords

Aspect logic Automatic reasoning Paradoxes Proof systems Puzzles Tableaux 

References

  1. Bell JL, Machover M (1977) A course in mathematical logic. North-Holland Publishing Co., AmsterdamGoogle Scholar
  2. Gardner M (1978) Aha! insight. W. H. Freeman & Co., LondonGoogle Scholar
  3. Gardner M (1985) Wheels, life and other mathematical amusements. W H Freeman & Co., LondonGoogle Scholar
  4. Hofstadter DR (2000) Godel, escher, bach: an eternal golden braid, 20 Anniversary edn. Penguin, LondonGoogle Scholar
  5. Hughes P, Brecht G (1993) Vicious circles and infinity, an anthology of paradoxes. Penguin Books, LondonGoogle Scholar
  6. Leblanc H, Wisdom WA (1972) Deductive logic. Prentice Hall, Inc., Upper Saddle RiverGoogle Scholar
  7. Sernadas A, Sernadas C (2008) Foundations of logic and theory of computation. College Publications, LondonGoogle Scholar
  8. Sloane P, MacHale D (1992) Challenging lateral thinking puzzles. Sterling Publishing Company Inc., New YorkGoogle Scholar
  9. Smullyan RM (1988) Forever undecided, puzzle guide to godel. Oxford Paperbacks, OxfordGoogle Scholar
  10. Smullyan RM (2000) To Mock a Mockingbird and other logic puzzles. Oxford Paperbacks, OxfordGoogle Scholar
  11. Smullyan RM (2008) Logical labyrinths. A. K. Peters/CRC Press, NatickCrossRefGoogle Scholar
  12. Smullyan RM (2009) The lady or the tiger? And other logic puzzles. Dover Publications Inc., MineolaGoogle Scholar
  13. Smullyan RM (2010) King Arthur in search of his dog and other curious puzzles. Dover Publications Inc., MineolaGoogle Scholar
  14. Smullyan RM (2011) What is the name of this book? The riddle of dracula and other logical puzzles. Dover Publications Inc., MineolaGoogle Scholar
  15. Smullyan RM (2012) Alice in puzzle-land: a Carrollian tale for children under eighty. Dover Publications Inc., MineolaGoogle Scholar
  16. Smullyan RM (2012) Satan, cantor & infinity: mind-boggling puzzles, New edition edn. Dover Publications Inc., MineolaGoogle Scholar
  17. Smullyan RM (2013) The Gödelian puzzle book: puzzles, paradoxes and proofs. Dover Publications Inc., MineolaGoogle Scholar
  18. Sukach J (1995) Quicksolve whodunit puzzles. Sterling Publishing Company Inc., New YorkGoogle Scholar
  19. Sukach J (1997) Quicksolve whodunit puzzles. Sterling Publishing Company Inc., New YorkGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  1. 1.Department of Mathematics, Instituto Superior TécnicoUniversity of LisbonLisbonPortugal
  2. 2.Centro de Filosofia das Ciências da Universidade de LisboaLisbonPortugal
  3. 3.Department of Mathematics and StatisticsMcMaster UniversityHamiltonCanada

Personalised recommendations