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the success of an action can then be dealt with along the lines of our action
theory.
Default Logic [85], the formalism in which our Fluent Calculus axiomati-
zation of Chapter 2 has been embedded in view of the Qualication Problem,
was among the earliest nonmonotonic frameworks, the key reference to many
of which is the special volume [11]. Numerous modications and extensions
to classical Default Logic have subsequently been developed in order to cope
with a number of ontological aspects missing or arguably being counter-
intuitively dealt with in the original approach; a variant which uniformly
addresses many of these aspects is presented in [19], just to mention one.
However, the default rules occurring in our axiomatization bear a very spe-
cic structure so that, fortunately, none of the criticisms of classical Default
Logic applies (speaking in technical terms, our defaults are, without excep-
tion, both \prerequisite-free" and \normal", which means they all are of the
form
: !
! ). Prioritized Default Logic has been introduced in [15] to support
the notion of preference among default rules. We have adopted the more ele-
gant reformulation proposed by [87], who also points out some problems with
the original approach for certain classes of default theories|but again our
particular default theories are not subject to this criticism due to their special
structure. The approach [13] provides the rst formal framework for priorities
among unrestricted default rules. A variety of approaches to the automation
of reasoning in default logics exist, e.g. [85, 7, 97, 96], just to mention a few.
Those calculi which perform proof search in a local fashion, that is, which do
not necessarily require the generation of entire extensions of an underlying
default theory, are usually, however, restricted to so-called credulous reason-
ing. In contrast to skeptical reasoning, the latter entails any formula that
belongs to (at least) one extension. A general method for extending credu-
lous proof procedures to skeptical entailment without loosing the property of
being local and goal-oriented has been developed in the second part of [106].
For rounding o this chapter on the Qualication Problem some remarks
seem appropriate on the topical debate about nonmonotonic vs. probabilistic
logics. While the former amounts to qualitative reasoning about small like-
lihoods, the latter is concerned with quantitative reasoning on the basis of
exact knowledge of probability values. The strongest argument undermining
the fundamental of all nonmonotonic frameworks is that sometimes counter-
intuitive, if not contradictory, conclusions are unavoidable. This is best illus-
trated with the well-known Lottery Paradox introduced in [62] (see also [84]):
\In a fair lottery with 100 tickets the chance that any given ticket will
lose is 99 per cent. It therefore seems reasonable, for any given ticket, to
believe, or accept as a basis for action, the statement `This ticket will lose.'
Yet the conjunction of such statements for all tickets must be false, since
some ticket will win, so we can hardly accept the conjunction. We seem
to have to accept each statement separately, but not the conjunction of
them." 13
13
[63], pp. 187{8
 
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