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C00002 00002	anomal[s86,jmc]		A non-monotonic logic using anomalies
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anomal[s86,jmc]		A non-monotonic logic using anomalies

p cancels Nq implies that if  p  is assigned true, then  Nq ∧ ¬q
doesn't count as an anomaly.

	The logic contains a modal operator  N  read ``normally''.
(There are plenty of alternative notations for ``necessarily'').
We shall consider the propositional version first, and so the
wffs are built up from propositional letters using the usual
propositional connectives and the modal operator  N.

	An interpretation is an assignment of truth values to all
wffs satisfying the propositional rules, e.g. if  \pi and \rho
are assigned  T, then  $\pi ∧ \rho$ must also be assigned T.
An anomaly is any assignment in which  $N \pi$ is assigned $T$,
and $\pi$ is assigned $F$.  A model of a formula is an interpretation
in which that formula is assigned  $T$.

	We are interested in models of a formula that minimize the
set of anomalies.  We have the following conjectures.

	1. All anomalous wffs in a minimal model of a formula $\pi$
belong to a finite number of logical equivalence classes.

	2. These formulas are equivalent to subformulas of $\pi$.

	3. Universally quantified theories, i.e. with all quantifiers on
the outside of the wffs, should introduce no complication.

	The above normality logic doesn't seem capable of handling
cancellation of inheritance.  In particular, it can't express the fact
that objects normally don't fly but birds normally do.
In order to fix this we add the notation $\pi cancels \rho$ and allow
this as an additional rule for forming formulas.  Its semantics is that
if $\pi cancels \rho$ and $\pi$ are assigned $T$, then $N\rho ∧ ¬\rho$
doesn't count as an anomaly.

	Flying-birds is then axiomatized as follows:

	$$N ¬flies x$$

	$$bird x cancels ¬flies x$$

	$$bird x ⊃ N flies x$$

	$$ostrich x ⊃ bird x$$

	$$ostrich x cancels N flies x$$

	$$ostrich x ⊃ N ¬flies x$$.

Here we have gone beyond propositional logic, but the intent is merely
that arbitrary individual constants may be substituted for the free
variables.

Query: Do we need $\pi cancels \rho$ or can we get by with just
$cancelled \rho$ and write $\pi ⊃ cancelled \rho$?

	The bird axioms are then

$$N ¬flies x$$

$$bird x ⊃ cancelled ¬flies x$$

$$bird x ⊃ N flies x$$

$$ostrich x ⊃ bird x$$

$$ostrich x ⊃ cancelled flies x$$

$$ostrich x ⊃ N ¬flies x$$.

	Oops, it looks like Vladimir is right, and the lack of $ab$s makes
trouble.  Namely, once $N ¬flies x$ has been cancelled by $bird x$,
$ostrich x$ can't reinstate it.

What about making it

$$N (bird x ⊃ flies x)$$

and adding

$$N(ostrich x ⊃ cancelled (bird x ⊃ cancelled ¬flies x))$$

But then maybe all we need is $¬N \rho$, 

April 28

val
making normality logic do the bird problem
	It looks like we need to tinker with the notion of minimization.
Suppose we have N ¬Np.  We want to prefer having ¬Np true to having p
true.  The reason is that we use the outer normality statements to control
the inner ones, so we might as well explicitly give them higher priority.
Thus inner anomalies are better than outer anomalies.  How can we express
this in a general way?

s86.in[let,jmc]/213p contains a message from Joe Halpern clobbering the
logic in its present form.
msg.msg[1,jmc]/213p