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\title{Formalizing Open Reasoning}

	The purpose of this article is to discuss how to formalize
what I shall call {\it open reasoning} using the tools of
mathematical logic.  I introduce the new term in order to
contrast {\it open reasoning} with {\it closed reasoning}.
The idea isn't so new, but maybe the new terminology will
make the contrast clearer.

	The primary case of closed reasoning is reasoning
within a formalized scientific theory.  Here are some
examples.

	1. Predicting the motions of the planets under Newton's
law of gravity.

	2. Determining the
probabilities of events or the expectations of variables in
a probabilistic theory in which a measure has already
been assigned on the sample space.

	3. Determining
the response of a linear circuit to an input signal.

	4. Solving a linear programming problem.

	The primary case of {\it open reasoning} is common
sense reasoning.  Here are some examples.

	1. Planning a party for children.  It is not determined
in advance what activities will be considered.

	2. Dealing with an unexpected event.  During the children's
party there is an earthquake.  Depending on the size of the earthquake,
(a) people may have to be rescued and given first aid, or

(b) people may only have to be reassured, the children taken home,
and the objects that have fallen down replaced, or

(c) the children may have to be told that the disturbance was an earthquake
and the party continued.

	What phenomena have to be taken into account and what
activities have to be considered depends on what has happened.
Making a general plan for dealing with earthquakes cannot be
reduced to an operations research problem in which it is decided
in advance what is to be optimized and how it depends on the
values of a finite set of variables chosen in advance.
For example, a kitten may have to be rescued from the top of
a ladder that it climbed in fright.

	While the prime cases of closed and open reasoning are
taken from science and common sense respectively, this is
not the crux of the difference.  Open reasoning occurs in
science and closed reasoning occurs in common sense.  The
crux is whether the reasoning takes place in a predetermined
deductive framework.

	Thus in developing a theory of evolution, a scientist
may not know in advance what phenomena he is going to consider
relevant.  Similarly, common sense reasoning may involve something
as cut and dried as doing arithmetic.

	Many writers have assumed that scientific reasoning is
characteristically closed, and this has led them to despair
of treating social and common sense phenomena scientifically.
People have also assumed that open reasoning cannot be done
with the tools of mathematical logic.  Both are mistaken.

	Part of the distinction between closed and open
reasoning is that the former is logically monotonic.
Each of the examples given above of reasoning within
a scientific theory can be done by logical deductive reasoning
within an axiomatic theory.  The examples of common sense
reasoning cannot be done this way.  Conclusions must be reached
that are based on the absence of certain information.  The
plan for the children's party will not take into account
all possible special circumstances, e.g. it may not take
earthquakes into account.  A plan for dealing with earthquakes
will not take all possibilities into account.

	If open reasoning is to be done with the tools of
mathematical logic, then deduction must be supplemented
by some form of formalized nonmonotonic reasoning, e.g.
circumscription (McCarthy 1980 and 1986), the logic
of defaults (Reiter 1980) or autoepistemic logic (Moore 1985).
(Ginsberg 19xx) is a book of readings and (Lifschitz 19xx)
compares various approaches.
\smallskip\centerline{Copyright \copyright\ \number\year\ by John McCarthy}
\smallskip\noindent{This draft of \jobname\ TEXed on \jmcdate\ at \theTime}
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