Proposed definition of "possible world"
Chris,
Instead of arguing about intuitions, I would like to propose
a definition, which we can discuss at a technical level, and accept
or reject, revise or extend for explicit technical reasons:
Definition: A possible world W is a pair (L,D) consisting of a set
of first-order propositions L and a set of ground-level assertions D.
1. Every assertion in D shall be consistent with the propositions in L.
2. Every proposition provable from L is said to be necessary in W.
3. Every proposition consistent with L is said to be possible in W.
4. Every proposition provable from L & D is said to be true in W.
5. Every proposition inconsistent with L & D is said to be false in W.
Informally, the set L is called the laws of W, and the set D is called
the database of W. Other informal terms might be used: L may be called
the set of axioms or database constraints for W.
I make the following claims about this definition:
1. It is consistent with J. Michael Dunn's (1973) definition of
possible world, which he proved to be a consistent extension
of Kripke's semantics for modal logic.
2. It is consistent with the major theories of possible worlds that have
been defined as extensions to Kripke's, including Montague's. I am
not aware of any versions that are inconsistent with it, and I would
be eager to hear about any that might be.
3. It is consistent with Gabbay's (1995) version, in which he claimed
that defining a possible world as a theory is more suitable for
computational purposes than Kripke's original definition.
4. It is consistent with current practice in relational database systems,
in which L corresponds to the set of DB constraints and D corresponds
to the conjunction of all tuples in all the relations of the DB.
5. It is consistent with the major AI knowledge bases in which people
have implemented collections of propositions called worlds, contexts,
alternative universes, etc. I am not aware of any implementations
that are inconsistent with it, but I would be eager to hear about
any that might be.
6. It is consistent with Tarski's (1935) stratified hierarchy of
metalevels, in which the lowest level L0 consists of the deductive
closure (every proposition provable from) L and D. The level L1
is the metalanguage in which assertions can be made about which
propositions are designated laws (or database constraints). The
level L2 is the metametalevel in which assertions, called policies,
can be made about which kinds of laws or constraints are permissible
and how they might vary from one possible world to another.
Following are the benefits of this proposal:
1. For systems in which the model theory is not visible to the knowledge
engineer or the DB administrator, this proposal is indistinguishable
from Kripke's or Montague's theory of possible worlds.
2. But this proposal also permits a knowledge base or database to support
languages and interfaces in which the KE or DBA can distinguish laws,
constraints, or axioms from simple facts. It also permits the
assertions of policies at the metametalevel L2, which constrain
permissible changes or modifications that the KE or DBA is allowed
to make in the laws or constraints at level L1.
3. For the purpose of ontology design and development, it allows the
designers to discuss questions at a technical level that would be
impossible to state in Kripke's original semantics. For example,
instead of saying that an "essential" proposition must be true in
all possible worlds, it is possible to talk about which laws
imply that proposition or even which policies in L2 constrain the
permissible laws in L1 that make that proposition true in every L0.
4. As an example, the policies at level L2 enforce the system of modal
axioms that may be used. The axiom set S5, for example, corresponds
to a policy at level L2 that the DBA is not allowed to change the
laws or DB constraints at level L1. A more lenient axiom set S4
corresponds to the policy that the DBA is allowed to assert a new
constraint at level L1 provided that it does not make the current DB
at level 0 inconsistent with the new constraint.
5. This proposal also allows a DB or KB system to use the laws for
metalevel reasoning in explanation and help facilities. For example,
a user could ask "Can I perform such and such an update?" and the
system would either reply "yes" or say which law or constraint would
be violated by the proposed update. The same facilities could be
used at level L2 to tell a DBA which policy would be violated by
some proposed constraint at level L1.
There is a lot more that can be said (and has been said) about this
proposal, and I would invite comments and suggestions from all concerned.
John Sowa