Recall that the truth value of a sentence is defined relative to an interpretation and a variable assignment . To define the nonmonotonic value of a premise in a rule, we need to select, instead of a single interpretation , a set of interpretations - the interpretations that are considered ``possible''. In the following definition, is a set of interpretations which all have the same universe of discourse , and is a variable assignment with this universe. We consider prerequisites and justifications separately.
The nonmonotonic value of a prerequisite is if and only if it is true at every ``possible'' intepretation.
The nonmonotonic value of a justification is true if and only if its argument is true for at least one ``possible'' intepretation.
be a knowledge base with rules. We define when a set
of interpretations is ``a set of possible worlds'' for
, by means of the following fixpoint construction. Consider a universe of discourse
; by a world we understand an interpretation with the universe
be the set of all worlds that satisfy the sentences in
. Consider a maximal set
of worlds such that, for each rule
and each variable assignment
with the universe
, the following condition holds. If the nonmonotonic value of every prerequisite of
, and the nonmonotonic value of every justification of
is true, then the nonmonotonic value of the consequent of
always exists.) If
is maximal, then we say that
is a set of possible worlds for
. Typically, a knowledge base with rules has many sets of possible worlds; it is clear, for instance, that any two interpretations with different universes cannot belong to the same set of possible worlds.
is a nonmonotonic model of
if it belongs to some set of possible worlds for
. We say that a nonmonotonic knowledge base
nonmonotonically entails a sentence
if and only every nonmonotonic model of
is also a model of
Note that the definition of a model for nonmonotonic knowledge bases is ``nonlocal'' - we cannot check whether an interpretation
is a model by looking at each rule in isolation. This feature of the definition is responsible for the nonmonotonic character in this notion of entailment.