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
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):
- Kurt Gödel.
Eine Interpretation des intuitionistischen Aussagenkalküls.
In Ergebnisse eines mathematischen Kolloquiums 4, pp. 39-40.
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.
- 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