RE: FO semantics for condition language (Action 84)

> >    I satisfies Neg a        iff    I does not satisfy a
> >    I satisfies Naf a        iff    I does not satisfy a
> 
> Hmmm, this looks strange to me... It makes no difference 
> between naf and neg, hmmmm...

It's just Peter's funny insistence (or should I say
stubbornness?) that classical FOL is all you need.
 
> Actually, as far as I know, the difference depends on what 
> interpretations 'I' looks like, if you allow only classical 
> models, then there is not really a difference...

Indeed, classical logic has the most idealized concept 
of an interpretation. It supports only one kind of 
negation (Boolean complement). But there are many other 
(non-classical) notions of an interpretation supporting 
two (or more) kinds of negation and other distinctions 
among connectives.

> Particularly, LEM law of the excluded middle which is an axiom 
> in standard first-order semantics) does not necessarily hold, 
> i.e. models are not necessarily 'total'.
> 
> The "least commitment" semantics where LEM doesn't hold is AFAIK 
> intuitionistic logics... 

There are many logical systems where LEM does not hold,
including intuitionistic logic and fuzzy logic, but the 
simplest refinement of classical logic that supports two 
kinds of negation is *partial logic* [1], which assigns both 
a truth and a falsity extension to a predicate. This allows 
for "truth value gaps" (partiality) and for "truth value 
clashes" ("paraconsistency"). 

Thus partial logic defines a small family of three- and 
four-valued logics (depending on the requirements that 
are imposed on interpetations and the choice of definition 
clauses for connectives). It also allows to distinguish
between different kinds of predicates, which may be
partial or total. Classical logic can be viewed as the
special (overidealized) case of partial logic where all 
predicates are total (an assumption that seems to be 
justified for mathematics, but not for knowledge 
representation).

In partial logic, NEG corresponds to *strong* negation
(also called "Kleene negation"), and NAF corresponds
to weak negation under the preferential semantics of
minimal/stable models. Both negations can be found
in various computational logic systems [2].

> >    I satisfies Naf Neg a    iff    I satisfies a
> 
> Rather:  I satisfies Naf Neg a  iff  I does not satisfy neg a

Yes, indeed, there is no such double negation law in 
partial logic.

-Gerd

---------------------------------------------
LS Internet-Technologie
http://oxygen.informatik.tu-cottbus.de/IT
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Email: G.Wagner@tu-cottbus.de
 

[1] H. Herre, J. Jaspars and G. Wagner: Partial Logics with 
    Two Kinds of Negation as a Foundation for Knowledge-Based 
    Reasoning. In D.M. Gabbay and H. Wansing (Eds.), What is
    Negation?, Kluwer Academic Publishers, 1999.
http://www.informatik.tu-cottbus.de/~gwagner/papers/PartialLogics.pdf

[2] G. Wagner: Web Rules Need Two Kinds of Negation.
In F. Bry, N. Henze and J. Maluszynski (Eds.), Principles 
and Practice of Semantic Web Reasoning, Proc. of the 1st 
International Workshop, PPSW3 '03. Springer-Verlag LNCS 2901, 2003
http://www.informatik.tu-cottbus.de/~gwagner/myRuleML/WebRules2Neg.pdf

Received on Thursday, 31 August 2006 11:33:00 UTC