Socrates did it before Gödel
DOI:
https://doi.org/10.12775/LLP.2011.011Keywords
the “paradoxon” of Socrates, Gödel’s Second Incompleteness Theorem, a ramified epistemic logic with propositional quantifiersAbstract
We translate Socrates’ famous saying I know that I know nothing into the arithmetical sentence I prove that I prove nothing. Then it is easy to show that this translated saying is formally undecidable in formal arithmetic, using Gödel’s Second Incompleteness Theorem. We investigate some variations of this Socrates-Gödel sentence. In an appendix we sketch a ramified epistemic logic with propositional quantifiers in order to analyze the Socrates-Gödel sentence in a more logical way, separated from the arithmetical context.References
Kurt Gödel, “Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I”, Monatshefte für Mathematik und Physik 38 (1931): 173–198.
Craig Smorynski, “The incompleteness theorems”, in: Jon Barwise (editor), Handbook of Mathematical Logic, North-Holland, 1977.
Craig Smorynski, Self-Reference and Modal Logic, Springer, 1985.
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2011-11-30
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DEGEN, Josef Wolfgang. Socrates did it before Gödel. Logic and Logical Philosophy. Online. 30 November 2011. Vol. 20, no. 3, pp. 205-214. [Accessed 18 April 2024]. DOI 10.12775/LLP.2011.011.
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