- From: Gerd Wagner <wagnerg@tu-cottbus.de>
- Date: Thu, 31 Aug 2006 13:32:41 +0200
- To: <axel@polleres.net>, "'Peter F. Patel-Schneider'" <pfps@inf.unibz.it>
- Cc: <public-rif-wg@w3.org>
> > 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 Tel: 0355-69-2397 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