COMMENT ⊗   VALID 00003 PAGES
C REC  PAGE   DESCRIPTION
C00001 00001
C00002 00002	.<<counte[s83,jmc]		A cartesian product theory of counterfactuals>>
C00016 00003	.skip to column 1
C00019 ENDMK
C⊗;
.<<counte[s83,jmc]		A cartesian product theory of counterfactuals>>
.require "memo.pub[let,jmc]" source;
.cb A CARTESIAN PRODUCT THEORY OF COUNTERFACTUALS

.cb "by John McCarthy, Stanford University"

	This note proposes a theory of certain counterfactual conditional
sentences based on informal approximate cartesian product theories
of aspects of the world.  Don't confuse my theory of counterfactuals
with the theory that I claim a counterfactual refers to.

	Counterfactual conditional sentences are important in making
computer programs that can do common sense reasoning.  For example, an
action is considered intentional only if the action would have been
different had the intentions been suitably different.

	Our theory has several aspects.

	1. Within a theory that refers to a cartesian product structure
for a state space, a counterfactual conditional sentence that involves an
alternate value of one component has a definite meaning, because a
definite state of the system is referred to.  Thus if a state has the
components (3,4,7) called ⊗x, ⊗y, and ⊗z, then "if  ⊗x  were 5" refers
to the state (5,4,7).

	2. We propose to refer other counterfactuals to this easy
case by using such cartesian product theories as "approximations"
to more general situations.

	3. At first sight such approximations may seem arbitrary,
but such approximate theories often enjoy an objective preferred
status.

	We base our considerations on an example.

	Suppose two ski instructors observe their student fall.
The first ski instructor says, "If he had bent his knees he
wouldn't have fallen".  The second instructor replies, "No, he
would still have fallen, but if he had put his weight on his
downhill ski, he wouldn't have fallen".  Then
they look at a videotape of the event, and the second
instructor agrees that the first one was correct.  If he had
bent his knees, he wouldn't have fallen, and putting his weight
on his downhill ski wouldn't have helped.
How can we explain their ability to agree on the truth of these
counterfactual conditional sentences?

	We believe that the instructors share a theory of skiing,
and when this theory is applied to the information obtained from
the videotape, it gives a definite answer.  A suitable computer program
equipped with the theory and the information on the videotape
would come to the same conclusion.

	The theory models the skier as a stick figure with joints
that he operates.  The limbs have masses and moments of inertia.
The hill has a shape.  The outcome is a function of the motions
of the skiers joints and the slope of the hill.
Built on this theory there is a theory of discrete actions: bending
the knees or not, putting the weight on one ski or the other or both,
swinging the hips, and all the other actions described in the books
written by ski instructors.

	Notice that this theory does not say what the student
will do; it merely gives the consequences of the sequences of
actions.  In the language of automata theory, the automaton is
not autonomous but has inputs from the outside.  If whether
the student bends his knees
is one of the inputs to the system, then the theory determines
what will happen if this input is altered and the others are
left unchanged.

	If we take the theory as given, the
truth of the counterfactual conditional is determined.  Notice
the cartesian product structure of the space of the skier's actions.
Since the action
of the student is described by a collection of independent
inputs to the system, %2within the theory%1 the notion of changing
one input and leaving the others fixed is well defined.

	While interpreting counterfactuals relative to a cartesian product
theory is relatively unproblematic, the more usual philosophical goal is
to assign truth values to counterfactuals regarded as statements about the
world.  A statement relative to a particular theory can be regarded as a
statement about the world provided it can be argued that this theory has a
preferred position among theories of the world.  Thus we argue that the
student wouldn't have fallen if he had bent his knees by claiming that the
theory used by the two instructors best fits the whole phenomenon of
skiing.  We can weaken this condition slightly by requiring merely that
the statement be true in all good theories of skiing.

	What makes a good theory of skiing.  Like a formal scientific
theory, a common sense theory becomes preferred by accounting for
a wide range of phenomena - in this case the experience of ski
instructors.  Since the theory of skiing used by ski instructors
depends on the vast experience of generations of ski instructors and
writers about skiing, we should not be surprised that it is difficult
to give a definition of "if the student had bent his knees"
that does not depend on a theory.  Indeed I would argue
that many common sense statements resemble
statements in the natural sciences in being meaningful only
within elaborate theories based on large fragments of experience.

	We proceed formally as follows:

	1. Relative to a space  A  described as a cartesian product, we
can define changing one component of an element of the space and leaving
the others fixed.

	2. When a component of  A  changes, the value of a function  f 
defined on the space changes in a definite way.

	3. Therefore, the truth of a counterfactual conditional of
the like "If the %2i%1th component of  x  were 3, the value of  f(x) 
would be  7" relative to some initial point  x0.

	4. When a description of the state of an aspect of the
world has a cartesian product structure in a theory  T,  then we can interpret
counterfactual conditional sentences involving a changed value
of a function of the on the aspect caused by a changed value of
a component.  This interpretation is relative to the theory  ⊗T. 

	5. If we have good scientific or common sense reasons for
preferring the theory  ⊗T,  then we can regard such counterfactuals
as objectively true or false.

	The motivation for proposing this theory of counterfactuals
comes from research in artificial intelligence.  We would like robots
and computer programs to use counterfactuals and concepts based on
them in the same circumstances as do people.  The cartesian product
theory seems sufficiently computational that we can imagine programming
the robot to use it to generate appropriate counteractuals.

	Contrast this explanation of counterfactuals with
David Lewis's (1973).  He assigns truth to a counterfactual provided
the consequent is true in the closest possible world to the
present world in which the antecedent is true.  Since
the world is substantially deterministic, we have problems with
what is the closest world in which the student bent his knees.
Perhaps his knees are too stiff to bend because of a childhood
accident caused by a mosquito annoying his father while driving.
However, if the accident hadn't occurred he would have worried
less and eaten more and would have fallen anyway because of
being too fat.  There doesn't seem to be any way to program
our robot to generate appropriate counterfactuals using
Lewis's theory.

	The cartesian model of counterfactuals avoids such problems
and sticks to the theory used by the ski instructors, who can
only be persuaded to think about the childhood accident and the
mosquito after they have had several beers.

	Consider Lewis's example of the party.  If Otto had come
it would have been a good party but not if Anna had also come.
This is quite understandable in terms of a suitable theory of what
makes a good party.

	Cartesian counterfactuals also seem to agree with intuition
in cases where the sentences don't seem to have definite truth values.
For example, "If wishes were horses beggars would ride"
is not associated with any approximate theory.  Sentences
beginning "If the South had won the civil war . . ." seem
to be meaningful or not according to how much theory of history
we imagine them imbedded in.

	I don't know whether all counterfactuals fit the
cartesian model.  Proposed counterexamples will be welcomed.

Challenge for the reader:  Construct a theory in which if someone
had said to Fermat,  "If 2%532%*+1 were a prime, twice it would be
a prime", Fermat would have correctly replied, "False".
Note that 2%532%*+1 is divisible by 641, but Fermat didn't know it.

.bb REFERENCES

%3Lewis, David%1 (1973), %2Counterfactuals%1, Harvard University Press.
.skip to column 1
  We here propose a new theory
based on the approximate theories of the some aspect of the world

	Our object is to propose a new theory of counterfactual
conditional sentences.

	Suppose our space  X   of possibilities is a cartesian product.
X = X1 x . . . Xn.  Suppose that  f : X → A  is a function whose
value interests us, and the point

	x↑0 = (x1↑0, . . . ,xn↑0)

is the current value of  x.  Consider the counterfactual sentence:
%2If  xi  were  b,  then  f  would have the value  c%1.  We
consider it synonymous with

	f(x1↑0, . . . x[i-1]↑0, b, x[i+1]↑0, . . . xn↑0) = c.

Notice that the meaning of this kind of counterfactual depends on
the particular representation of  X  as a cartesian product.
Our idea is that in many (perhaps most) cases where counterfactual
conditional sentences are intuitively meaningful, there is a
distinguished representation of a certain relevant space as a cartesian
product, so that the above definition is meaningful.

	The problem is made more complex by the fact that the
space in question is not "the world" but a space that approximates
the relevant aspect of the world.  The world does not have the
distinguished cartesian product structure, but a certain distinguished
approximating space does.  The counterfactual is then only approximately
meaningful.