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C00002 00002	defaul[w82,jmc]		General theories of defaults - non-monotonic
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defaul[w82,jmc]		General theories of defaults - non-monotonic

	Non-monotonic systems have the property that if a collection
A of assertions is given, other assertions beyond what can be deduced
from A are also asserted.  We can consider then that we have a map
from consistent collections  A of assertions to larger consistent
collections  conseq(A).

	The map  conseq  should have the following properties:

	1. If A is consistent, so is conseq(A).

	2. conseq(A) is closed under deduction.

	3. A ⊂ conseq(A).

	4. conseq(conseq(A)) = conseq(A).

A regular  conseq  will have the additional property

	5. A ⊂ B ⊂ conseq(A) ⊃ conseq(B)=conseq(A).

Of course we don't want monotonicity which would be

	A ⊂ B ⊃ conseq(A) ⊂ conseq(B).

	Another interesting property might be called completeness.  NM
is complete if

	6. For all sentences p and sets A, either p ε conseq(A)
or ¬p ε conseq(A).

If there are enough constants, completeness amounts to choosing for
each  A  a preferred model pm(A).  Property 5 tthen translates into

	5'. If A ⊂ B and the sentences of  B  are true in  conseq(A),
then  conseq(B) = conseq(A).