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.CB EXPLAINING COUNTERFACTUALS WITH CARTESIAN PRODUCTS


	Some %2counterfactual conditional sentences%1
(we will call them %2counterfactual%1 for short) can be explained as follows.
Consider a function %2f(x,y,z)%1 and suppose that %2x_=_1, y_=_2%1,
and %2z_=_3%1.  Now consider the assertion %2"If y were 5, then
f would be 7"%1.  The assertion is considered true if and only
if %2f(1,5,3)_=_7%1.

	We first remark that the truth of the counterfactual
is dependent on the particular co-ordinate system.  If we change
to a co-ordinate system %2(x,y,z')%1 where ⊗x and ⊗y are as
before, but %2z'_=_z_+_y%1, then the truth of the above sentence
depends on whether %2f(1,5,0)_=_7%1.  Therefore, we will consider
that this kind of explanation of a counterfactual depends on
the existence of a preferred co-ordinate system.

	Such purely mathematical examples of counterfactuals offer
no problem and therefore little interest.  Let us now consider
a counterfactual conditional sentence about the real world -
an example discussed by David Lewis.

!!a2:	%2"If Otto had come to the party, it would have been a
good party - unless Anna had come too, in which case it would
have been a bad party.  However, if Max had come in addition to
Otto and Anna, it would have been a good party"%1.

	If we denote the presence of Otto, Anna and Max at the
party by Boolean variables ⊗o, ⊗a, and ⊗m respectively, we can
make an abstract model characterized by
a function %2goodness(o,a,m)%1 in which ({eq a2})
becomes the formal assertion

!!a1:	%2goodness(1,0,0) = good ∧ goodness(1,1,0) = bad ∧
goodness(1,1,1) = good%1.

	However, the English language assertions about Otto, Anna and Max
are offered as assertions about the real world, and immediately jumping to
the above abstract model is begging the question.  However, if both the
speaker and the hearer were adherents of a mathematical theory of parties,
then evaluating the truth of ({eq a1}) might be exactly the way they would
settle the truth of ({eq a2})

	Our contention is that this is a common situation
when counterfactuals are used.  The speaker and hearer share
an approximate theory of the phenomenon being discussed,
there is a function dependent on a state, and this state has
a preferred Cartesian product structure.  It is then meaningful
to ask the value of the function in a state in which specified
co-ordinates are given specified new values,
and the rest retain the values of the "base state".
The truth of the counterfactual then poses two requirements.  First,
the statement must be true in the theory, and second, the theory,
including the Cartesian product structure it gives to the situation,
must be acceptable.

	A converse is also possible; namely the counterfactual
may be a way of expressing part of the theory.
Returning to the example, the speaker and the hearer evaluate the truth
of ({eq a2}) within a theory of the goodness of parties and its
dependence on the personalities of the people present.  Within
that theory, the people are structures of objects with properties
and relations.  The elementary terms of that theory are components
of personality and are not atoms, photons and molecules.  In fact,
an attempt to evaluate the truth of ({eq a2}) in terms of atoms,
photons and molecules fails and generates only confusion.

	What are these approximate theories?  How do they approximate
reality?  What are the criteria for their acceptance.
.skip 1
.begin verbatim
John McCarthy
Artificial Intelligence Laboratory
Computer Science Department
Stanford University
Stanford, California 94305

ARPANET: MCCARTHY@SU-AI
.end

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