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C00002 00002 Certain concepts, e.g. %2X can do Y%1, are meaningful as
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Certain concepts, e.g. %2X can do Y%1, are meaningful as
statements in rather complex theories. For example, suppose we
denote the state of the world by %2s%1, and suppose we have functions
%2f%41%2(s)%1,...,%2f%4n%2(s)%1 that are directly or indirectly
observable. Suppose further that %2F(s)%1 is another function of the
world-state but that we can approximate it by
%2F"(s) = F'(f%41%2(s),...,f%4n%2(s))%1.
Now consider the counterfactual
conditional sentence, "If %2f%42%2(s)%1 were 4, then %2F(s)%1 would be
3 - calling the present state of the world %2s%40%1." By itself, this
sentence has no meaning, because no definite state %2s%1 of the world
is specified by the condition. However, in the framework of the
functions %2f%41%2(s),...,f%4n%2(s)%1 and the given approximation to
%2F(s)%1, the assertion can be verified by computing ⊗F' with all
arguments except the second having the values associated with the
state %2s%40%1 of the world.
This gives rise to some remarks:
&. The most straightforward case of counterfactuals arises when
the state of a phenomenon has a distinguished Cartesian product
structure. Then the meaning of a change of one component without
changing the others is quite clear. Changes of more than one
component also have definite meanings. This is a stronger
structure than the %2possible worlds%1 structure discussed in
(Lewis 1973).
Our treatment of such concepts as %2intention%1, %2ability%1, and
%2freedom of will%1 all depend on this treatment of counterfactuals
in terms of a Cartesian product structure.
&. The usual case is one in which the state %2s%1 is a substantially
unknown entity and the form of the function %2F%1 is also
unknown, but the values of %2f%41%2(s),...,f%4n%2(s)%1 and
the function %2F'%1 are much better known.
Suppose further that %2F"(s)%1 is known to be only a fair approximation
to %2F(s)%1. We now have a situation in which the counterfactual
conditional statement is meaningful as long as it is not examined too
closely, i.e. as long as we are thinking of the world in terms of
the values of %2f%41%2,...,f%4n%1, but when we go beyond the
approximate theory, the whole meaning of the sentence seems to
disintegrate.
Our idea is that this is a very common phenomenon. In
particular it applies to statements of the form %2"X can do Y"%1.
Such statements can be given a precise meaning in terms
of a system of interacting automata as is discussed in detail in
(McCarthy and Hayes 1969). We determine whether Automaton 1 can put Automaton
3 in state 5 at time 10 by answering a question about an automaton system
in which the outputs from Automaton 1 are replaced by inputs from
outside the system. Namely, we ask whether there is a sequence of
inputs to the new system that ⊗would put Automaton 3 in state 5 at time 10;
if yes, we say that Automaton 1 ⊗could do it in the original system
even though we may be able to show that it won't emit the necessary
outputs. In that paper, we argue that this definition corresponds
to the intuitive notion of %2X can do Y.%1.
What was not noted in that paper is that modelling the
situation by the particular system of interacting automata is
an approximation, and the sentences involving
⊗can derived from the approximation
cannot necessarily be translated into single assertions about the
real world.
I contend that the statement, %2"I can go skiing tomorrow,
but I don't intend to, because I want to finish this paper"%1 has the
following properties:
1. It has a precise meaning in a certain approximate theory of
the world
in which I and my environment are considered as collections of interacting
automata.
2. It cannot be directly interpreted as a statement about the world
itself, because it can't be stated in what total configurations of the
world the success of my attempt to go skiing is to be validated.
3. The approximate theory within which the statement is meaningful
may have an objectively preferred status in that it may be the only theory
not enormously more complex that enables my actions and mental states to
be predicted.
4. The statement may convey useful information.
Our conclusion is that the statement is %3true%1, but in a sense that
depends essentially on the approximate theory, and that this intellectual
situation is normal and should be accepted. We further conjecture that
the old-fashioned common-sense analysis of a personality into %2will%1
and %2intellect%1 and other components may be valid and might be put
on a precise scientific footing using %2definitions relative to
approximate theories%1.
Consider the statement %2"If Stanford University were 500
miles north of where it is, it would be in Oregon"%1. The naive
way to test the statement is to obtain a Mercator projection map,
mark off 500 miles along the meridian through Stanford University,
and check whether the resulting point is shown on the map as
being in Oregon. Using the Mercator projection maps of a given
date we can form a theory of moving universities. Within this
theory all similar questions have definite answers.
However, the original naive answer may be challenged by someone
who says, %2"How do you know that Leland Stanford didn't have
enough political power to have changed the borders of California and
Oregon so that his university would have been in California after all.