COMMENT ⊗   VALID 00006 PAGES
C REC  PAGE   DESCRIPTION
C00001 00001
C00002 00002		Here is a mathematical model illustrating the phenomenon.
C00006 00003		Here is a mathematical model illustrating the phenomenon.
C00007 00004		Here is a mathematical model illustrating the phenomenon.
C00009 00005	.cb CARTESIAN COUNTERFACTUALS
C00011 00006	The following example came up in a conversation with Richmond Thomason:
C00015 ENDMK
C⊗;
	Here is a mathematical model illustrating the phenomenon.

	Let ⊗W be a space which we may call the space of situations.
We suppose that ⊗W is very large.
Let ⊗X be a space which we will call the space of states, and we shall
suppose that ⊗X is much smaller than ⊗W.  Let ⊗corr be a relation
of correspondence between elements of ⊗X and elements of ⊗W.
The intended meaning of ⊗corr(x,w) is that ⊗x is a representation
of ⊗w in the domain of some structure ⊗S of a language ⊗L.  We will
not assume that every state ⊗x corresponds to some situation ⊗w nor
will we assume that every situation ⊗w has a corresponding state ⊗x so
we define

	%2∀x.(realized(x) ≡ ∃w.corr(x,w))%1

and

	%2∀w.(represented(w) ≡ ∃x.corr(x,w))%1.

However, we do assume

	%2∀x y w.(corr(x,w) ∧ corr(y,w) ⊃ x = y)%1.

Now let %2qf:W_→_Reals%1 and %2f:X_→_Reals%1 map these domains into
the real numbers.  (Choosing the real numbers is not important; any
other set would do just as well).  Suppose further that

	%2∀x w.(corr(x,w) ⊃ f(x) = qf(w))%1.

Thus the value of %2qf(w)%1 may be computed by computing ⊗f(x) provided
the situation ⊗w has a corresponding state ⊗x.

	We now introduce Cartesian counterfactual statements by
supposing that ⊗X is represented as a cartesian product

	%2t:X → YqxZ%1,

where ⊗t (for theory) is assumed 1-1 onto.

	⊗X may be representable as a cartesian product in many ways,
but we shall define counterfactual statements with respect to the
particular representation ⊗t. 

	Now we fix our attention on a particular ⊗w0 and a corresponding
⊗x0 such that %2t(x0)_=_<y0,z0>%1.  We say "the situation is %2w0%1".

	Let ⊗y1 be an element of ⊗Y different from ⊗y0.  Now consider
the statement "If ⊗y were ⊗y1, then ⊗f(x) would be 7".  We interpret
this as just the sentence

	%2f(<y1,z0>) = 7%1.

If we know the function ⊗f, then its truth value is determined, but notice
that its truth value depends on the particular correspondence ⊗t. 


	Here is a mathematical model illustrating the phenomenon.

	Let ⊗X be a space, let %2f:X_→_Reals%1 and let
%2t:X_→_YqxZ%1 be 1-1 onto, let ⊗x0 be a point of ⊗X,
let %2t(x0)_=_<y0,z0>%1, and let %2f(x0)_=_72.  If ⊗x0 in
some way typifies the present situation, and the function
⊗f gives the value of some physical quantity, say the temperature,
then we may say "The temperature is 72".

	Let ⊗y1 be an element of ⊗Y different from ⊗y0, and
suppose that ⊗f(qt(<y1,z0>))_=_65.   Associated with this we may say
"If ⊗y were ⊗y1 instead of ⊗y0, then the temperature would be
65".
	Here is a mathematical model illustrating the phenomenon.

We begin with the simplest cases and move toward more realistic ones.

	1. Let ⊗m be a quantity defined by

	%2m = f(x,y,z) = x↑2 + y↑2 + z↑2%1.

Suppose further that in some situation ⊗x_=_0, ⊗y_=_3, and ⊗z_=_4.  Then
%2m_=_25%1.  The sentence "If ⊗x were 5, then ⊗m would be 169" would be
accepted as true in an informal mathematical discussion, because
%2f(5,3,4)_=_169%1.  The informal interpretation of the counterfactual is
that "if ⊗x were 5" means that ⊗y and ⊗z retain their "present" values.
and this gives the desired result.

	If the only use for counterfactual conditionals were in
such simple cases, we would regard them as an unnecessary circumlocution.
However, the simple cases often provide a basis for the more
complicated examples that are of real interest.
Notice that the ⊗m is defined on the space of tuples ⊗<x,y,z>, and
that this space is a Cartesian product.

	2. We now change our focus slightly and suppose that ⊗m is
defined by %2m_=_F(qx)%1, where qx takes values in a space ⊗X.
Now suppose that ⊗X is represented as a Cartesian  product
%2X_=_AqxBqxC%1.
.cb CARTESIAN COUNTERFACTUALS

	This contribution to the theory of counterfactual
conditional sentences is based on the following two ideas:

	1. If a quantity is defined on a space provided with
a Cartesian product structure, it is easy to define counterfactuals
based on changing one component and leaving others unchanged.

	2. An approximate theory may have a Cartesian product
structure so that certain counterfactuals may be defined in the theory.
However, the Cartesian product structure may not "lift" to the
world or even to a less approximate theory.  Therfore, entirely
useful counterfactuals may be defined only relative to the
theory.

From our point of view it will turn out that only certain
counterfactuals are meaningful - perhaps fewer than
Lewis or Stalnaker or Turner would allow.
The following example came up in a conversation with Richmond Thomason:

Ed Adams contrasted the following statements:

1. If Oswald didn't shoot Kennedy, then someone else did.

2. If Oswald hadn't shot Kennedy, then someone else would have.

Presumably, the latter is false in our usual interpretation of
history, and it will be false in our Cartesian models.
However, suppose that a year prior to Oswald's birth, his mother
suffered a miscarriage early in a pregnancy.
If she hadn't suffered the miscarriage, the previous pregnancy
would have produced an even nastier fellow who would also have shot
Kennedy - definitely not Oswald, however, since the father would
have been a different person.  Suppose further that the track
of the cosmic ray had to be extremely precise to produce the
miscarriage and any change would have avoided it.  Then according
to David Lewis's notion of counterfactuals, the closest possible
world to the actual one is one in which the cosmic ray missed, and
the counterfactual (2) is true.  If you like, you can change (2) to

2'. If Oswald had never been born, then someone else would have shot Kennedy.

	Thus we see that Cartesian counterfactuals give a different
result than Lewis counterfactuals in this case.  It would seem that
the truth of Cartesian counterfactuals is more determinable, and
they give a result more in accordance with our intuitions.  In fact
given the ability to imagine far-fetched circumstances, it may be
that the truth of a Lewis counterfactual is never determinable.
Cartesian counterfactuals also seem to serve some of the theoretical
purposes of counterfactuals better than Lewis counterfactuals.  For
example, we may want to define an deliberate action as one that
would not have occurred had the actor not intended it.  With
Cartesian counterfactuals the ordinary criteria for the deliberateness
of an action apply, whereas they don't seem to apply with Lewis
counterfactuals.

	Perhaps Lewis counterfactuals can be repaired by taking
notions of closest world corresponding to the Cartesian product
structures, but this won't work in general, because there
is no guarantee that there is a possible world corresponding
to the counterfactual circumstance of the Cartesian product.