Re: Godel's Incompleteness and Nonmonotonic Logic

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

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.

Relevant Pages

  • Re: Godels Incompleteness and Nonmonotonic Logic
    ... > unrelated concept of completeness meaning "whatever follows is also ... The two incompleteness theorems of Godel refer to what ... The "Answer Set" style logics force that everything ...
    (comp.lang.prolog)
  • Re: Godels Incompleteness and Nonmonotonic Logic
    ... >> The incompleteness theorems of Goedel, however, refer to a completely ... The two incompleteness theorems of Godel refer to what ... classical completeness follows from Hilbert ...
    (sci.logic)
  • Re: Godels Incompleteness and Nonmonotonic Logic
    ... >> The incompleteness theorems of Goedel, however, refer to a completely ... The two incompleteness theorems of Godel refer to what ... classical completeness follows from Hilbert ...
    (comp.lang.prolog)
Loading