B10. CHANGE OF STATE
1. The "change" Predicate
It is hard to imagine a universe without change. It is even harder to
imagine cognition in such a universe. Change of state is very nearly
the most basic concept human beings are equipped with. We could not
survive very long without recognizing the changes in our environment.
Our very sense of time arises from our awareness of change.
We will represent change of state with the predicate "change" relating
two eventualities. The expression "(change e1 e2)" says that
eventuality e1 changes into eventuality e2. Moving is a change from
being at one place to being at another. Growing up is a change from
being small to being larger. Learning is a change from not knowing
something to knowing it. And so on, for all the processes we are
familiar with.
We cannot define "change". It is too basic a concept in our minds.
But there are a number of constraints we can express concerning the
eventualities that are the arguments to "change".
The arguments of "change" are eventualities.
(forall (e1 e2) (1)
(if (change e1 e2)(and (eventuality e1)(eventuality e2))))
They can be eventuality types or eventuality tokens. For example, the
change may be from the door being open 3 inches to the door being open
2 inches, where both are eventuality tokens of the door being open.
There is no assumption in the "change" predicate that the
eventualities e1 and e2 precisely define the conditions before and
after the change; we will see below that the two derivative predicates
"changeFrom" and "changeTo" will incorporate this assumption, and
hence will apply primarily to eventuality types.
One might think there could be a magical change of a person into a
cat, and this would be a change relation between entities that are not
eventualities. But we view this as a change in properties of a single
individual. There is an x such that first x is a person and then x is
a cat.
A change of state is a change of state of something. It would be
strange to say that there was a change from Bill Clinton's being
president to the Red Sox winning the World Series. They have nothing
to do with each other. So a further constraint on "change" is this.
(forall (e1 e2) (2)
(if (change e1 e2)
(exists (x)(and (arg* x e1)(arg* x e2)))))
The eventualities e1 and e2 have to involve some common entity x.
Change is defeasibly transitive. If e1 changes into e2 and e2 changes
into e3, then generally we can say there is a change from e1 to e3.
(forall (e1 e2 e3) (3)
(if (and (change e1 e2)(change e2 e3)(etc))
(change e1 e3)))
One reason "change" is only defeasibly transitive is that the first
change may involve one entity and the second another. For example,
suppose e1 is John's being married to Mary, e2 is John's being married
to Susan, and e3 is Susan's being married to Bill. It seems strange
to say that there is a change from John's being married to Mary to
Susan's being married to Bill. Part of the "etc" condition for this
axiom is that the two changes involve the same entity.
We would like to say that when e1 changes into e2, they are somehow
contradictory or inconsistent with each other. Something has actually
changed. We should not get direct changes from an eventuality e into
itself. However, this cannot be expressed as a constraint on the
arguments of change, because cyclic change is possible. We can have a
change from state e1 to state e2 and then a change back to state e1
again. By transitivity, there defeasibly has been a change from e1 to
itself. But the only reason this has been possible is because there
was an intermediate state that was different. We can state this by
saying that if the states are not inconsistent, then the change must
be a composite of two changes, one to and the other from an
inconsistent state.
(forall (e e1 e2) (4)
(if (change' e e1 e2)
(or (inconsistent e1 e2)
(exists (e3 e4 e5)
(and (change' e4 e1 e3)(change' e5 e3 e2)
(and' e e4 e5))))))
Since change is not normally cyclic, we can make it a defeasible
inference that the start and end states are inconsistent, by thinking
of the negation of the second disjunct of axiom (4) as an "et cetera"
condition.
(forall (e e1 e2) (5)
(if (and (change e1 e2) (etc))
(inconsistent e1 e2)))
2. Predicates Derived from "change"
It will be useful in avoiding verbosity later to introduce here several
predicates defined in terms of "change".
We will say that there is a "changeIn" something when there is a
change in its properties.
(forall (e x) (6)
(iff (changeIn' e x)
(exists (e1 e2)
(and (arg* x e1)(arg* x e2)(change' e e1 e2)))))
We know from axiom (2) that there is at least one such x in every
change.
Very often we are interested not in both the start and end states of a
change, but only one. For this, we will define "changeFrom" and
"changeTo" predicates. We would like these to be more powerful than
the "change" predicate, however. We would like it to be the case that
when there has been a "changeFrom" some eventuality e1, where p'(e1,x)
holds, then after the change, p(x) is not true. That is, no
eventuality corresponding to p(x) really exists. The following axiom
accomplishes this.
(forall (e e1 e3) (7)
(if (subst e3 e3 e1 e1)
(iff (changeFrom' e e1)
(exists (e2)
(and (change' e e1 e2)(inconsistent e3 e2))))))
Recall from Chapter B3 that the expression "(subst e3 e3 e1 e1)" means
that e1 and e3 have the same predicate and same arguments, other than
the self argument. So "(open' e1 d)" and "(open' e3 d)" may be two
distinct instances of the door d being open, but if there is a
"changeFrom" e1, it can't be into e3 or any other situation in which
the door is still open.
The "changeTo" predicate is defined similarly.
(forall (e e2 e4) (8)
(if (subst e4 e4 e2 e2)
(iff (changeTo' e e2)
(exists (e1)
(and (change' e e1 e2)(inconsistent e4 e1))))))
Here the start state of the change must exclude any other instance of
the type which e2 instantiates. We can't "changeTo" a state of the
door being open from a state in which the door is already open.
If we want to make the arguments of the change more specific, we can
do so by having them be explicitly conjunctive. Thus, we can have a
"changeFrom" a state of the door being open two inches by saying
something like
(and (changeFrom' e e1)(and' e1 e2 e3)
(open' e2 d)(measure' e3 e2 2in))
In chapter B6 we introduced the notion of an external figure being
"at" a location in a structured ground. We can combine this with the
notion of "change" to get a very abstract sense of "move". An entity
x moves from y to z exactly when there is a change from x's being at y
to its being at z.
(forall (e x y z) (9)
(iff (move' e x y z)
(exists (e1 e2 s)
(and (at' e1 x y s)(at' e2 x z s)(change' e e1 e2)))))
As we introduce various specializations of the "at" relation, we will
inherit corresponding specializations of the "move" event. For
example, a change in the measure of something is a "move".
Among the many possible scales, some are conceived of as being really
or metaphorically vertical. Altitude is obviously vertical in
reality. The scale of numbers is viewed metaphorically as vertical,
and resident on this, the scale of probabilities is viewed as
vertical. Scales that provide values for human concerns tend to be
viewd as vertical. Two scales that are not normally viewed as
vertical in our culture are horizontal distance and time.
We will not attempt to analyze why some scales are vertical and others
are not. We will simply stipulate of the appropriate scales that they
are vertical. For example, a non-negative numeric scale is vertical.
(forall (s) (if (nonNegNumericScale s)(vertical s))) (10)
The argument of "vertical" has to be a scale.
(forall (s) (if (vertical s) (scale s))) (11)
If an entity moves from a point on vertical scale to a higher point,
we say that there as been an increase.
(forall (e x y z s) (12)
(iff (increase x s)
(exists (y z)
(and (at' e1 x y s)(at' e2 x z s)(lts y z s)
(vertical s)(change e1 e2)))))
Similarly, if an entity moves from a point on a vertical scale to a
lower point, there has been a decrease.
(forall (e x y z s) (13)
(iff (decrease x s)
(exists (y z)
(and (at' e1 x y s)(at' e2 x z s)(lts x y s)
(vertical s)(change e1 e2)))))
Predicates Introduced in this Chapter
(change e1 e2): There is a change from eventuality e1 to
eventuality e2.
(changeIn x): There is a change in some property of x.
(changeFrom e1): There is a change out of eventuality e1.
(changeTo e2): There is a change into eventuality e2.
(move x y z): x moves from y to z.
(vertical s): Scale s is a real or metaphorical vertical scale.
(increase x s): x increases on scale s.
(decrease x s): x decreases on scale s.