Re: Godel's Incompleteness and Nonmonotonic Logic

From: Student (jagasian_at_mailinator.com)
Date: 08/23/04

  • Next message: Dirk Mittler: "Attempting to tet for the consistency of an unbound variable again." 
    Date: 23 Aug 2004 14:37:23 -0700

    > The incompleteness theorems of Goedel, however, refer to a completely
    > unrelated concept of completeness meaning "whatever follows is also
    > provable".

    You are wrong. The two incompleteness theorems of Godel refer to what
    you called "Hilbert completeness". Godel's theorems imply that you
    cannot have both completeness and consistency. The incompleteness
    Godel refers to is that there exist (closed first-order, not
    necessarily ground or atomic) logical formulas that cannot be proven,
    even though they are
    definitely true.

    The "Answer Set" style logics force that everything
    or its negation is provable. Hence a simple encoding of arithmetic
    would result in an inconsistent system, as for example, the formula
    "F" of Godel's first incompleteness theorem cannot be proven, and so
    by answer set logics, "not F" is provable. Since "not F" is provable,
    the logic is simply inconsistent.

    Digging a little on the net, I have found at least one notable
    detractor to nonmonotonic logics, Jean-Yves Girard. In "Girard, J.-Y.
    : Locus Solum, Mathematical Structures in Computer Science 11, pp.
    301-506, 2001." Girard mentions in brief that Nonmonotonic logics fall
    into the trap of inconsistency/incompleteness.

  • Next message: Dirk Mittler: "Attempting to tet for the consistency of an unbound variable again."

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