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pp 1–13 | Cite as

On Paradoxes in Normal Form

  • Mattia Petrolo
  • Paolo Pistone
Article
  • 62 Downloads

Abstract

A proof-theoretic test for paradoxicality was famously proposed by Tennant: a paradox must yield a closed derivation of absurdity with no normal form. Drawing on the remark that all derivations of a given proposition can be transformed into derivations in normal form of a logically equivalent proposition, we investigate the possibility of paradoxes in normal form. We compare paradoxes à la Tennant and paradoxes in normal form from the viewpoint of the computational interpretation of proofs and from the viewpoint of proof-theoretic semantics.

Notes

Acknowledgements

The funding was provided by Agence Nationale de la Recherche (Grant No. ANR-14-FRAL-0002).

References

  1. Bainbridge E, Freyd PJ, Scedrov A, Scott PJ (1990) Functorial polymorphism. Theor Comput Sci 70:35–64CrossRefGoogle Scholar
  2. Blute R (1993) Linear logic, coherence and dinaturality. Theor Comput Sci 115(1):3–41CrossRefGoogle Scholar
  3. Coquand T, Herbelin H (1994) A-translation and looping combinators in pure type systems. J Funct Porgr 4(1):77–88CrossRefGoogle Scholar
  4. Dosen K (2003) Identity of proofs based on normalization and generality. J Symb Log 9(4):477–503CrossRefGoogle Scholar
  5. Gentzen G (1964) Investigations into logical deduction (1934). Am Philos Q 1(4):288–306Google Scholar
  6. Geuvers H, Verkoelen J (2009) On fixed point and looping combinators in type theory. http://www.cs.ru.nl/~herman/PUBS/TLCApaper.pdf
  7. Girard J-Y (1976) Three-valued logic and cut-elimination: the actual meaning of Takeuti’s conjecture. Dissertationes mathematicaeGoogle Scholar
  8. Girard J-Y (1987) Linear logic. Theor Comput Sci 50(1):1–102CrossRefGoogle Scholar
  9. Girard J-Y (1989) Geometry of interaction I: interpretation of system F. In: Ferro R, Bonotto C, Valentini S, Zanardo A (eds) Logic colloquium, AmsterdamGoogle Scholar
  10. Girard J-Y (1990) Geometry of interaction II: deadlock-free algorithms. In: International conference on computational logic, TallinnGoogle Scholar
  11. Girard J-Y, Scedrov A, Scott PJ (1992) Normal forms and cut-free proofs as natural transformations. In: Moschovakis Y (ed) Logic from computer science, vol 21. Springer, Berlin, pp 217–241CrossRefGoogle Scholar
  12. Gödel K (1958) Über eine bisher noch nicht benütze Erweiterung des finiten Standpunktes. Dialectica 12:280–287CrossRefGoogle Scholar
  13. Hurkens AJC (1995) A simplification of Girard’s paradox. In: TLCA 95, second international conference on typed lambda calculi and applications, pp 66–278Google Scholar
  14. Kelly GM, MacLane S (1971) Coherence in closed categories. J Pure Appl Algebr 1(1):97–140CrossRefGoogle Scholar
  15. Kleene SC (1945) On the interpretation of intuitionistic number theory. J Symb Log 10(4):109–124CrossRefGoogle Scholar
  16. Kreisel G (1960) Foundations of intuitionistic logic. In: Proceedings of the 1960 international congress on logic, methodology and philosophy of science. Stanford University Press, pp 98–210Google Scholar
  17. Kreisel G, Takeuti G (1974) Formally self-referential propositions for cut free analysis and related systems. Dissertationes mathematicae 118:1–55Google Scholar
  18. Lamarche F (2008) Proof nets for intuitionistic linear logic: essential nets. Technical report,<inria-00347336>, INRIAGoogle Scholar
  19. Melliès P-A (2006) Functorial boxes in string diagrams. In: Computer science logic. CSL 2006. Lecture Notes in Computer Science, vol 4027. Springer, Berlin, pp 1–30Google Scholar
  20. Mendler PF (1987) Inductive definitions in type theory. PhD thesis, Cornell UniversityGoogle Scholar
  21. Negri S, von Plato J (2001) Structural proof theory. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  22. Petrolo M, Pistone P (2017) A normal paradox. In: Arazim P, Lavicka T (eds) The logical yearbook 2016. College Publications, LondonGoogle Scholar
  23. Prawitz D (1965) Natural deduction, a proof-theoretical study. Almqvist & Wiskell, StockholmGoogle Scholar
  24. Prawitz D (1971) Ideas and results in proof theory. In: Fenstad J (ed) Proceedings of the 2nd Scandinavian logic symposium (Oslo), Studies in logic and foundations of mathematics, vol 63. North-HollandGoogle Scholar
  25. Schroeder-Heister P (2006) Validity concepts in proof-theoretic semantics. Synthese 148:525–571CrossRefGoogle Scholar
  26. Schroeder-Heister P, Tranchini L (2017) Ekman’s paradox. Notre Dame J Form Log 58(4):567–581CrossRefGoogle Scholar
  27. Scott D (1970) Outline of a mathematical theory of computation. Technical report, Oxford University Computing LaboratoryGoogle Scholar
  28. Scott D (1976) Data Types as Lattices. SIAM J Comput 5(3):522–587CrossRefGoogle Scholar
  29. Scott D, Strachey C (1971) Toward a mathematical semantics for computer languages. Technical report, Oxford Programming Research Group Technical Monograph. PRG-6Google Scholar
  30. Sorensen MH, Urzyczyn P (2006) Lectures on the Curry-Howard isomorphism, Studies in logic and the foundations of mathematics, vol 149. Elsevier, AmsterdamGoogle Scholar
  31. Tennant N (1982) Proof and paradox. Dialectica 36:265–296CrossRefGoogle Scholar
  32. Tennant N (1995) On paradox without self-reference. Analysis 55:199–207CrossRefGoogle Scholar
  33. Tennant N (2016) Normalizability, cut eliminability and paradox. Synthese.  https://doi.org/10.1007/s11229-016-1119-8.
  34. Yablo S (1993) Paradox without self-reference. Analysis 4:251–252CrossRefGoogle Scholar

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© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.CNRSUniversité Paris 1 Panthéon-SorbonneParisFrance
  2. 2.Dipartimento di Matematica e FisicaUniversità Roma TreRomeItaly

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