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NOTES ON COUNTERFACTUAL CONDITIONAL


1. Counterfactuals, David K. Lewis, Harvard University Press 1973.

2. Lewis's analysis of counterfactuals is built out of the Kripke
possible world semantics for modal logic, but he shows that the
counterfactual "p implies q" cannot alway be analyzed as N(p⊃q).  To
do this he uses an example on p. 10 with "If Otto had come, it would have
been a lively party; but if both Otto and Anna had come it would
have been a dreary party; but if Waldo had come as well it would have
been lively; but ...".

3. Our approach to a number of questions including "can" and counterfactuals
is to regard the semantics of certain sentences as not solely about the
world, but about the world relative to some theory.  The simplest case is to
parametrize the world as a Cartesian product, i.e. to regard the world
state as an n-tuplet of independent components.  With respect to such a
parametrization, certain counterfactuals are easy, namely those that involve
changing one or more components of the state leaving the others as they are.
In the previous case, if our theory has the coming to the party of Otto, Anna,
and Waldo as independent components, and the liveliness of the party as a
function of state, then all the above counterfactuals have quite definite
meanings.

	Moreover, if we try to achieve the results of the parametrization
treatment of counterfactuals by means of possible world semantics, we see
that the case of changing only one component fits nicely, because we can
regard the worlds with single component changes as possible from the
base world, so that we can interpret the first part of (i) as
"In all possible worlds in which Otto comes the party is lively", but as
Lewis explains, this louses up the possibility of interpreting the second
part.

4. In the case in which our understanding of the world can be regarded as
a Cartesian product representation with independently variable components
and in which the counterfactual sentences can be regarded as about the
results of changing a small finite set of components, I regard the problem
of interpreting counterfactuals as completely solved.  However, this cannot
always be done, so we must consider some other cases.

5. In the first place, consider Lewis's first example, "If kangaroos had
no tails they would topple over".  Our idea that this sentence has a
simple interpretation in a certain theory in which we regard each kind
of animal as being described by a mass distribution and a distribution
of support.  Removing the tail from a kangaroo would produce a different
mass distribution, and it is computable whether in this mass distribution,
the center of mass would be outside the convex hull of the support.
Surely, this is the intended way of trying to answer the question, and
our idea is that the question is implicitly within a theory at that level
of detail.  We humans, and presumably also the robots we shall design,
conduct much of our thinking and much of our conversation within the
confines of such micro-theories, because they enable us to think about
how to improve our behavior.

	However, Lewis goes on to a more detailed theory in which the genetics
and evolution and survival of tailless kangaroos can be discussed.  In
such a more comprehensive theory, the question either has a different answer,
e.g. "If a tailless kangaroo evolved, a different posture that would keep
it upright would also have evolved", or doesn't make sense, e.g. if we
regard the tracks of kangaroos in the sand as a state component, then it
is hard to reconcile their lack of tails with their leaving the same tracks.

	It is natural to assume that the TRUE meaning of the question
about kangaroos refers to an ultimate theory that takes everything into
account, but this doesn't really work.  In the first place, we don't have
any such theory and aren't about to get it.  In the second place, there
may not be any such theory; as I shall explain elsewhere, it may be necessary
to consider our potential knowledge of the world as consisting of an
infinite collection of theories partially ordered by a kind of homomorphism
in which a less detailed theory is an approximation to a more detailed
theory of which it is a homomorphic image.  Ultimate reality would then
be a kind of inverse limit of these theories, but it may not be knowable
and it may not be a theory with well formed formulas as strings of symbols.
Thirdly, even if there were such an ultimate theory, it might have no
counterfactuals, i.e. everything might have to be exactly as it is.  In any
case the counterfactuals of one theory are often meaningless in a more
detailed theory.  Fourthly, a theory may be better for a certain human
purpose than a more detailed theory of which it is a homomorph.  Thus, we
heed the ski instructors statement, "Had you bent your knees more, you
wouldn't have fallen on that turn", and the philosophical haggler who
worries about whether it was really possible to have bent the knees on
that precise occasion, will be a slow pupil, and it will cost him more
to learn to ski.

5. The Cartesian product world model is not the only one in which
counterfactual sentences make sense.  Moreover, not all counterfactual
sentences make sense in a particular Cartesian product model, i.e. if
the components are the latitude-longitude pairs of cities, then the
sentence, "If Los Angeles were 50 miles from New York, then one could
bicycle from Los Angeles to Denver in one day" doesn't have a meaning.
This quite ok in our view, we are not trying to make a single model
in which all counterfactual sentences are meaningful.  Indeed, it is
one can surely think up different preceding paragraphs that will make a given
counterfactual sentence, intuitively true, false or meaningless.

6. A major way of making progress in understanding counterfactuals is
to think up more types of theories that give meaning to counterfactuals.
Special attention should be given to useful counterfactuals like the
ski instructors statement about the reason for the fall.
First of all, the interpretation of counterfactuals as sentences of the
form N(p⊃q) is undoubtedly and important case; it may be a limiting case.

7. Consider the counterfactuals that arise from modeling the world
by a system of interacting automata.  The independently variable components
are the subautomata, and we can ask how the system would act if one
or more of the subautomata were replaced by a different automaton.
Some simpler conditions arise if we merely make an arbitrary change
in the state of a subautomaton or change the signal emitted by one
of its outputs.