COMMENT ⊗   VALID 00004 PAGES
C REC  PAGE   DESCRIPTION
C00001 00001
C00002 00002	%normal[f88,jmc]		Normally with two arguments
C00006 00003	\smallskip\centerline{Copyright \copyright\ \number\year\ by John McCarthy}
C00007 00004	Notes:
C00011 ENDMK
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%normal[f88,jmc]		Normally with two arguments
\input memo.tex[let,jmc]
\title{$normally$ should have two arguments}

\section{Introduction and a Preliminary Example}

	Some time ago I studied a modal operator called $normally$.
The idea was that a theory using $normally$ would expand into
a theory with $ab$ predicates that could then be circumcribed.
I couldn't get it to work, i.e. to give the intuitively right answers
to some test problems.  Using $normally(p,q)$, taking $p$ and $q$
as {\it properties} seems to work better.  We use the usual
flying birds problem as a paradigmatic example.

	We write
%
$$normally(birds,fliers)$$
%
to mean that birds are normally fliers.  Here $birds$ is the property
of being a bird, and $fliers$ is the property of being a
flier.  The actual set of birds is denoted by $set-of(birds)$ and
we similarly use $set-of(fliers)$.  The reason for using properties
as arguments to $normally$ rather than the sets themselves is to
avoid identifying the properties associated with sets that just
happen to have the same members, e.g. humans and featherless
bipeds.  We propose to consider $normally(birds,fliers)$ as
an abbreviation for
%
$$(∀x)(x \in set-of(birds) ∧ ¬ab(Normally(birds,fliers),x)
 ⊃ x \in set-of(fliers)).$$
%
If we allow the operator $\in$ to apply to properties we can simplify
this to
%
$$(∀x)(x \in birds ∧ ¬ab(Normally(birds,fliers),x) ⊃ x \in fliers).$$
%
The cancellation of inheritance for penguins is expressed by
%
$$abnormal(Normally(birds,fliers),penguins),$$
%
and it is considered an abbreviation of
%
$$(∀x)(x \in set-of(penguins) ∧ ¬ab(Abnormal(Normally(birds,fliers),penguins)$$
$$⊃ ab(Normally(birds,fliers),x).$$
%
The fact that penguins normally don't fly is expressed by
%
$$normally(penguins,non(fliers)),$$
%
which abbreviates
%
$$(∀x)(x \in set-of(penguins) ∧ ¬ab(Normally(penguins,non(fliers),x)
	⊃ ¬(x \in set-of(fliers))).$$
%
\section{Preliminary Remarks}

	1. Using $normally$ this way with two arguments
makes it inapplicable to formulas in general, where it doesn't
seem to have a clear meaning.  Instead we are restricting it
to assert that an object with one property normally has
another.

	2. It isn't yet clear whether this formalism improves matters.
I hope it will help with the Yale shooting problem, and I hope it
is closer to a notation suitable for the common sense knowledge
base.  It is also closer to a form that can be the result of an
inference.
\smallskip\centerline{Copyright \copyright\ \number\year\ by John McCarthy}
\smallskip\noindent{This draft of NORMAL[F88,JMC]
TEXed on \jmcdate\ at \theTime}
\vfill\eject\end
Notes:

VAL comments Nov 23

1. normally is not transitive.  Indeed we can't even prove normally(p,p).
I didn't quite understand what he said might go wrong if we took those
as axioms, but also didn't think it was necessarily a good idea.

2. He thought that using  normally(true, p) would have all the
same power, but then we'd be back to using  normally  as a
modal operator.

3. I need to look into where I got stuck before with unary  normally.