15-816 Modal Logic
Lecture 14: Reconciliation

In this lecture we will attempt a reconciliation of the classical and intuitionistic points of view on logic. There are two major parts. The first, due to Kolmogorov [Kol25], shows that an intuitionst can understand the classical logician by systematically translating every statement he makes, prefixing every subformula by a double negation. The second, due to Gödel [Göd33], shows that a classical logican can understand an intuitionist by systematically translating every statement she makes, prefixing every subformula by the modal necessity operator and interpreting the result in the classical modal logic S4.

• Reading: 14-reconcil.pdf
• Original sources (accessible from at CMU only):

  [Göd33]

  Kurt Gödel.
  Eine Interpretation des intuitionistischen Aussagenkalküls.
  In Ergebnisse eines mathematischen Kolloquiums 4, pp. 39-40. 1933.
  Reprinted in English translation as An interpretation of the intuitionistic propositional calculus in ``Collected Works, Kurt Gödel'', Vol. I, pp. 296-301, Oxford University Press, 1986.

  [Kol25]

  Andrey N. Kolmogorov.
  On the principle `tertium non datur'.
  Matematicheskii Sbornik (Mat. Sat.), 32(4):646-667, 1925.
  Reprinted in English translation as On the principle of the excluded middle, in ``From Frege to Gödel'', J. van Heijenoort (editor), pp. 465-479, Harvard University Press, 1971.

• Key concepts:
  • Classical (modal) sequent calculus
  • Intuitionistic logic in classical S4
  • Double-negation translation
• Previous lecture: First-Order Logic and Quantified Modal Logic
• Next lecture: Intuitionistic Kripke Semantics

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Frank Pfenning, André Platzer