Re: Godel's Theorem under Fuzzy Logic?

Jorma Mattila (Jorma.Mattila@lut.fi)
Mon, 20 Apr 1998 19:38:09 +0200 (MET DST)

Torkel Franzen <torkel@sm.luth.se> writes:

>Christian Borgelt <borgelt@iws.cs.uni-magdeburg.de> writes:
>
> >Completeness means that starting from the axioms and applying only
> >the allowed inference rules you can, in principle, prove any true
> >formula of the formal system.
>
> No it doesn't. Completeness is a syntactic concept: for every A,
>A or the negation of A is provable in T.

This concept is nowadays better known as that of maximal consistency, but
the term "completeness" has earlier also been used in this sense. All
syntactical systems are not maximal consistent (e.g. classical
propositional calculus), if we allow all the correctly formed formulas of
the logic in question to belong to the system. Nowadays the concept
"completeness" is associated with formal semantics and syntax (with
proof-theory). The completeness result of a logic L means that every
theorem of the system of L is valid expression of the language of L
(soundness), and every valid expression of the language of L is a theorem
of the system of L. This concept of completeness unites the two different
approach of logic, namely semantics and syntax, and it is quite different
from that of maximal consistency, which really is a syntactic comcept.
However, the concept of maximal consistency is used sometimes in semantics.
In possible worlds semantics possible worlds are sometimes maximal
consistent sets of atomic formulas.

Cordially, Jorma K. Mattila

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* Laboratory of Applied Mathematics *
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