Folks, (01)
I'd just like to summarize a few points, which reinforce the claim I
made earlier: There is an open-ended number of different variations of
nonmonotonic logic, and it's impossible to adopt a one-size-fits-all
solution for nonmonotonic logic. (02)
To paraphrase Tolstoy, every happy logic (i.e., classical) is happy
in the same way, but every unhappy logic (nonmonotonic) is unhappy
in its own way. (03)
The solution I recommend is to treat all nonmonotonic operators
as metalevel predicates about some proposition or some proof.
In IKL (or any other logic that supports metalevel statements),
predicates such as is-provable(p), is-not-provable(p), is-default(p),
has-fuzzy-value(p,x), or probability-of(p,x), are metalevel
statements about some proposition p. (04)
If you assume a closed world (such as a database of all airline
reservations or all employees), you can write metalevel axioms
saying that anything not provable is false. If you have an open
world with incomplete information, you can write metalevel axioms
that say what to do about such cases. If you have a mixed DB
with complete info about some things and incomplete info about
other things, you can write axioms to say what to do in each case. (05)
Professional database administrators and authors who have studied
the issue for a long time (such as Chris Date), know how to
design and use DB systems in order to achieve predictable
results. Many casual users manage to avoid trouble by using
a database as a convenient way of storing and accessing positive
data, and they assume that the "not" operator is shorthand for
"not found". (06)
In short, a logic with a classical semantics, such as CL or IKL,
is an ideal foundation for defining the semantics of any and every
version of nonmonotonic logic that has ever been invented. (07)
John (08)
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