# RE: FO semantics for condition language (Action 84)

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>
Message-ID: <001401c6ccf1\$2b765dc0\$a2ca2b8d@informatik.tucottbus.de>
```
> >    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

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[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