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.

Let

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

. Let

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

for

and

is

, and the nonmonotonic value of every justification of

for

and

is true, then the nonmonotonic value of the consequent of

for

and

is

. (This

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.

An interpretation

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.

Wed Dec 7 13:23:42 PST 1994