## 1. Introduction




The most general conception of change is simply difference or
nonidentity in the features of things. Thus we speak of the change of
temperature from place-to-place along a body, or the change in
atmospheric pressures from place-to-place as recorded by isobars, or
the change of height of the surface of the earth as recorded by a
contour map. Contour lines record sameness in quantities (such as 100
metres) from the same quantity-kind (such as height), and the
differences recorded by different contour lines are
quantity-differences (100 metres as opposed to 200 metres). The
philosophical question here is how to construe such statements of
identity and nonidentity, and it seems that the problem of universals
is the main issue.




A narrower usage of "change" is exemplified by change in
the properties of a body over time, that is temporal change. This
essay will focus on temporal change. We begin by separating the
concept of change from several cognate concepts, specifically cause,
time and motion. Then we briefly survey attempts by such thinkers as
Parmenides and McTaggart to deny change. There follows an account of
the problem of the instant of change, where it is concluded that the
problem is too general to admit a single solution, but requires
specification of further metaphysical principles envisaged as
constraints on a type of solution. The final three sections, the bulk
of the essay, consider the question of the consistency or
inconsistency of change, which in one way or another looms over all
our discussions. It emerges that the case for change as an
inconsistent process is stronger than might be expected.


## 2. Change, Cause, Time, Motion




Our interest in this essay will be on the special case of temporal
change. So construed, the notion of change is obviously bound up with
notions of cause, time and motion. Now a distinction between change
and cause can certainly be drawn. It is clear that uncaused change is
conceptually possible, and arguably actual in such things as
radioactive decay. Conversely, the operation of a sustaining cause
results in no change in a thing, if the thing would otherwise be
undergoing a change which the sustaining cause prevents. Hence, the
operation of a cause on a thing is neither necessary nor sufficient
for change in that thing. Accordingly, we put the topic of cause in
the background when discussing change.




Time cannot be so backgrounded. The thesis that time could pass
without change in anything at all has proved controversial, and we
have adopted the usage that change in a thing implies the passage of
time. Aristotle nonetheless argued that change is distinct from time
because change occurs at different rates, whereas time does not
(Physics IV,10). This essay focusses on the topic of change,
while not denying that the topic of time is inseparable from
it. Motion, as change in place, will figure prominently in our
discussion.




One well-known idea is that of Cambridge change. This can be arrived
at by following the well-tried analytical technique of re-casting
philosophically important discussions and concepts in the
meta-language. Thus a Cambridge change in a thing is a change in the
descriptions (truly) borne by the thing. The phrase "Cambridge
change" seems to be due to Geach (1969, 71-2), who so named it
to mark its employment by great Cambridge philosophers such as
Russell and McTaggart. It is apparent that Cambridge change includes
all cases ordinarily thought of as change, such as change of colour,
from "red" to "non-red." But it also includes
changes in the relational predicates of a thing, such as when I
change from having "non-brother" true of me to having
"brother" true of me, just when my mother gives birth to a
second son. It might seem faintly paradoxical that there need be no
(other) changes in me (height, weight, colouring, memories,
character, thoughts) in this circumstance, but it is simply a
consequence of the above piece of metalinguistic ascent. It does
point up, though, that in attempting to capture the object-language
concept, one should take note of the distinction between the monadic
or internal or intrinsic properties of a thing, and its relations or
external or extrinsic features. Thus the natural view of change is
that real, metaphysical change in a thing would be change in the
monadic or internal or intrinsic properties of the thing. We will
return to this point in Section 5.


## 3. Denying Change




It is on the face of it extremely implausible to deny change, but
extreme implausibility has not always deterred philosophers. The
Eleatics (C5th BCE), particularly Parmenides, appear to have been the
first to do so. Parmenides maintained that whatever one speaks about
or thinks about must in some sense exist; if it did not exist then it
is nothing, so one would be speaking or thinking about nothing, which
would be empty. From this thesis, it is deduced that the existing
thing cannot have come into existence, because to say that it could
would be to speak of a time when it did not exist. By similar
reasoning, existing things are eternal because they cannot go out of
existence. It is now a small step to conclude that change is an
illusion, on the grounds that a change in a thing implies that there
was a time when the thing-as-changed did not exist. However, this
argument is not persuasive: the premiss that the non-existent cannot
be thought or spoken about is dubious.




Parmenides' disciples Melissus and Zeno developed this
theme. Melissus argued that motion implies empty space to move into,
but empty space is a nothing and so cannot exist, so that motion is
impossible since it implies a contradiction. This argument requires
the dubious premisses (1) that empty space is a nothing (which is
denied by realists from Newton to Nerlich), and (2) that motion would
have to be change relative to space. Even those who have held that
empty space is a nothing (relationists from Leibniz to Mach and
onward) have not generally denied motion, proposing instead that
motion of a thing is change in the spatial relations between that
thing and other things.




Zeno's brilliant paradoxes are generally accounted as attempts to
defend Parmenides. We will not look at these in detail, but his
paradox of the arrow is relevant to what follows. This is the argument
that an arrow in flight could not really be moving because at any
given instant it would be at a place identical with itself (and not
another place); something at just one (self-identical) place could not
be described as moving, and an arrow which is motionless at every
instant in a temporal interval must be motionless throughout the
interval. Discussion of this subtle argument is deferred until
discussion in a later section of Graham Priest's position, which turns
on similar premisses.




McTaggart's well-known argument (1908) that time is unreal applies
equally to the unreality of (temporal) change it seems. McTaggart
distinguished between two ways of attributing temporal
characteristics to events. The \(A\)-series of events is given
by the descriptions "past," "present" and
"future," while the \(B\)-series is strictly in terms
of the relational concepts "earlier,"
"simultaneous" and "later." Now the
\(B\)-series is insufficient to define change, because
\(B\)-series relations apply unchangingly if they apply at all;
whatever is earlier than something is always earlier than
it. Moreover, the \(B\)-series presupposes the \(A\)-series
since if \(X\) precedes \(Y\) then there must be a time
when \(X\) is past and \(Y\) present. This step in the
argument is not at all absurd: the discovery of spacetime, the
relativistic realisation of the \(B\)-series, has impelled many
from Minkowski on to describe it as a "static" conception
of time. A genuinely dynamic conception of change would thus need to
have things coming into and going out of existence with the passage
of time, whereas spacetime invites quantification over it all
"at once" as it were.




Thus according to McTaggart the source of time and change must be
found in the \(A\)-series. But the \(A\)-series implies a
vicious regress. Any event must have all three properties, pastness,
presentness and futurity, but this is a contradiction. The only way
out of the contradiction is to say that the event is past, present
and future at different times; but the same question arises about the
temporal instants themselves, which would force us to appeal to a
further time series to avoid the contradiction.




Whatever we make of this argument, and much has been written about it,
it highlights the baffling nature of the apparent passage of time. In 
particular, if temporal flux is denied then at the least an explanation 
of its intuitive naturalness is mandatory. For a close analysis, see the 
entry Savitt (2006) in this Encyclopedia. 


 

However, one thing can be said about all the above denials of change:
they all argue against change on the ground that it implies a
contradiction. But the assumption of the consistency of change has
been denied by a number of influential figures, as we will see.


## 4. The Instant of Change




Consider a car moving off from rest at exactly noon. What is its
state of motion at the instant of change? If it is in motion, when
did it start? And if it is motionless, when could it ever begin? This
problem was explored by Medlin (1963), Hamblin (1969), and
others. Put this way, a solution for at least some special cases is
readily available. Locate the time origin \(t = 0\) at noon. If
the car's position function \(f\) is given by, say,
\(f(t) = t^{2}\), then its speed is
\(2t\). If motion is defined as having non-zero speed, then the
car is motionless at \(t=0\). On the other hand, at all
\(t \gt 0\) it is in motion, so there is surely no puzzle about
when it could ever begin: there is no first instant of motion.




However, there are more troublesome special cases. Suppose that the
car's position function is given by: \(f(t)=0\) for all \(t \lt
0\), else \(f(t)=t\). Then speed is zero for all \(t \lt 0\), and
speed is 1 for all \(t \gt 0\). But what of \(t = 0\)? One should
avoid "arbitrary" solutions, which attribute one speed
rather than another, but do not give a reason for so
privileging. There is of course at least one simple solution that is
non-arbitrary, namely that it is neither in motion nor motionless,
since its speed is indeterminate at \(t = 0\). This solution derives
from the fact that according to classical calculus there is no
derivative of such a function at \(t = 0\).




But can we do no better? The present author (1985) proposed to set
aside the problem until more is said about various possible
constraints on the solution. Unless we had some reason to think that
such functions really did describe the world, we might well feel that
a solution was less than imperative and less than unique. For
example, the world might be described wholly with \(C^{\infty}\)
functions \((n\)-th derivatives exist for all \(n\), e.g.,
cos, sin, log, exponential functions). The above function is not
among these, since its derivative is discontinuous. But then it isn't
clear what we might say of it if the example is counterfactual. There
might be different things to say depending on what further principles
describe the possible world. Hence we would need to supplement the
original statement of the problem with an argument to the effect that
we might expect such functions to describe the real world, or
alternatively supply additional metaphysical principles to be
regarded as constraints on the solution.




A related problem is the fracture problem, described by
Medlin. Imagine fracturing a material body such as a piece of wood,
regarded as a plenum (full of matter). What is the state of the two
new surfaces after the fracture? Unless matter is to be created or
destroyed, we seem to have to say that the break is half-open, with
one new matter-surface being topologically closed and the other being
topologically open. But which surface is which? There seems to be no
principle to determine which. In response, it can be asked how
seriously we have to take the postulation of a plenum. If for example
matter is as Boscovich suggested, punctate and surrounded by fields,
then there are no plena, and the problem is no more than
hypothetical. Or again, there might be plena but other principles
might apply. For example, mass-density functions might drop smoothly
to zero at the boundaries between matter and empty space, which would
mean that all surfaces were open. On the other hand, it might be
instead that as a matter of fact all surfaces are topologically
closed. This would need an inconsistent solution (see below, sections
5-7).


## 5. Consistent and Inconsistent Change




If a changing thing has different and incompatible properties then a
contradiction is threatened. The obvious move to make when confronted
with the fact that things change, is to say with Kant (1781) that
they change in relation to time, which avoids the inconsistency. But
then another problem emerges. In what sense can one thing persist
through change? Identity across time and space is the mark of
universals, but we also account particulars such as billiard balls
and persons as having self-identity across time.



 Aristotle's views on the persistence of things are worth noting here.
At the risk of gross oversimplification of what is treated thoroughly
elsewhere (see the entry on 
 Aristotle's metaphysics, it
can be said that early on he took the view that what persists over
time and through change, the *substrate*, can be identified
with *matter*, and that it is the *form* of matter which
is acquired or lost (*Physics* I, 5-7). When he wrote the
*Categories*, it is *substance* which is said to be
susceptible of contrary attributions; and as such substance itself has
no opposites. (Categories 4a10). In the Metaphysics \(Z\), a more
complex doctrine of substance, that which \(is\), is worked
out. Substance is not the substrate, matter, since that lacks
particularity. *Its* substance, what it is to be *that
thing*, that without which it does not exist, is its
*essence*. Aristotle then links essence with his theory of
causes, being identified variously with its final cause and with its
formal cause. 



 Although Aristotle's views about change -- in particular, his
distinction between essence and accident -- have sometimes been
thought to contain a solution to the problem of persistent identity
through change, it seems to this author that they do not really get a
grip on the problem in its most fundamental form. This is perhaps
clearest in the Categories, where the ability of substance to admit
incompatible accidental features is more-or-less definitional.



 The problem can be made sharper by reflection upon the law of the
indiscernibility of identicals. If a thing-at-\(t\_{1}\)
were identical with a thing-at-\(t\_{2}\), then they
should share all their properties. What sort of identity is it, if
not that? But if the properties at different times are incompatible,
then a contradiction follows. Because they emphatically took the view
that contradictions are never true, the great Buddhist logicians
Dharmakirti (C7th CE) and his commentator Dharmottara (C8-9th CE),
who had certainly read their Aristotle, deduced that identity over
time does not exist (see Scherbatsky (1930) vol 2). This is the
Buddhist doctrine of moments, essentially an ontology of
instantaneous temporal slices. The doctrine of the momentariness of
existence is felicitously in accord with the core Buddhist doctrine
of the impermanence of all things. The doctrine of moments might seem
to be an unnecessarily strong application of impermanence, certainly
unnecessary for soteriological purposes, were it not for the evident
strength of the argument in its favour, not to mention its accord
with the spacetime ontology of modern physics. On the other hand, it
is of course psychologically very difficult to believe that one's own
self, as something genuinely self-identical, has not endured from
moment-to-moment in the past. Even so, the thesis of the
momentariness of human existence has had a recent defender in Derek
Parfit (1984), who asks what sort of principle could unify the
temporal stages sufficiently closely to be worth calling identity. He
argues that none could, and proposes that internalising the
momentariness of our lives has a beneficial effect on how we should
face our deaths.



 This theme is echoed in a recent debate on the topic of
'temporary intrinsics', which also connects with the
earlier-mentioned concept of Cambridge change. Cambridge change in a
thing is still change in *something or other*, but it is not
always change in *the thing itself*. Thus we might seek to
isolate change in the thing itself by change in its
*intrinsic* properties. But then we have the problem of in
what sense it continues to be just one thing through a change in its
intrinsic properties. Now obviously this raises the question of how
to define the concept of intrinsicality. We do not address that here,
since it is discussed elsewhere in this *Encyclopedia*, see
Weatherson (2002). So assuming a *prima facie* distinction
between the intrinsic and extrinsic properties of a thing, how does a
thing persist through changes in its intrinsic properties? David
Lewis and others debated this question, e.g., Lewis
(1986), (1988). Several options for a solution were canvassed, three
of which were as follows.



 (1) The basic existents are things indexed by times, that is
time-slices. What primarily exist are things-at-a-time:
"\(a\) is red at \(t\)" is rendered
"\(a\)-at-\(t\) is red". Things that persist over
time are then wholes made up of such parts, and one says that
persisting things *perdure* rather than *endure*. This
is the solution favoured by Lewis, by the present author, and by
space-time theory.



 (2) A second option is to say that, instead of indexing things, one
indexes properties: "\(a\) is red at \(t\)" is
rendered as "\(a\) is red-at-\(t\)". This option
does not seem to have had any defenders, perhaps because those
properties which are universals are supposed to be wholly in each of
their instances, which the indexing apparently denies.



 (3) A third option takes as its basic minimal idea that the index
modifies the whole event: \((a\)'s being red) holds at \(t\). A
variant is to take the index as modifying the exemplification
'relation': \(a\) exemplifies-at-\(t\) redness. Versions
of this position were urged by several contributors: Johnston (1987),
Lowe (1987), (1988), Haslanger (1989). However, the problem for
adverbial-style analyses anywhere is to provide enough semantics,
enough logical structure for the event, to account for the logical
implications of the sentences under analysis, as Davidson (1967)
pointed out. So for example one has things like: \((((Fa)\) at \(t)
\amp a=b)\) implies \(((Fb)\) at \(t)\); or \((((Fa)\) at \(t\_{1})
\amp((Ga)\) at \(t\_{2}) \amp(F\) is incompatible with \(G))\) implies
not \(t\_{1}=t\_{2}\); or \((((Fa)\) at \(t) \amp((Gb)\) at \(t)
\amp(F\) is incompatible with \(G))\) implies not \(a=b\). One thus
cannot rest with a minimalist position. At least Lewis' has the
merit of providing a viable semantics, a direct parallel with
counterpart theory in modal semantics. Of course, the basic ontology
of Lewis' favoured position was Dharmakirti's though Lewis
did not note that fact. More to the point, Dharmakirti's
strategy did not depend on the intrinsic/extrinsic distinction. The
problem of contradictory attributions occurs even if the attributions
are extrinsic, and Dharmakirti's argument is a straightforward
application of Leibniz' law to things-at-a-time. If time-slices
are admitted at all, and it is hard not to do so if they are
sanctioned by relativity theory, then Dharmakirti's argument
goes through. 




Others have taken a different course on the issue of the consistency
of change. Herakleitos (C6th BCE) wrote in a suggestive fashion, with
his doctrine of the unity of opposites. However, his few surviving
sentences are too obscure and fragmentary to give much confidence in
interpretation. He spoke of the same river having different waters at
different times, but there is no development of the
observation. Similarly he spoke of the sea as being at one time both
life-preserving (to fish) and death-dealing (to humans), and
"the path up and the path down are one and the same." These
examples hardly force one to believe in true contradictions,
however.




There is also in Herakleitos the idea that everything is in a state
of flux, always changing, and that it is the struggle between
opposites (opposed tendencies) which drives change. This can be seen
as an early version of the Marxist dynamic of dialectical
materialism. But without a separate argument for the inconsistency of
change, there is no reason to think that it remains anything but a
formally consistent theory.




Hegel was more explicit. In *The Science of Logic* he said
that only insofar as something has contradiction in itself does it
move, have impulse or activity. Indeed, movement is existing
contradiction itself. "Something moves not because at one moment
of time it is here and at another there, but because at one and the
same moment it is here and not here." (Hegel (1812) p. 440).




There is something appealing in this argument. As Priest and Routley
put it, "in change... there is at each stage a moment when
the changing item is both in a given state, because it has just
reached that state, but also not in that state, because it is not
stationary but moving through and beyond that state" (Priest,
Routley and Norman, 1989, p. 7). Think of a body coming to rest at a
given time, and compare it with the same body proceeding on to
further motion. There must be something about the body at that
instant which distinguishes the two scenarios, or there could be
nothing at the time to count as continuing change. Cause cannot do
it, for a body can continue in its state of motion without being
impressed by an external force, as Newton taught us. Nor can mere
velocity do it, since velocity is a relation to surrounding
points. Indeed, there is no difference in velocity between a body
momentarily at rest, and a body at rest for a period around the
instant; yet one is changing and the other not.




We will look more closely at this argument in the next
section. However, here we can remind ourselves of Hegel's
idealism. Just about everyone agrees that contradictions within ideas
are easier to swallow than contradictions in the external world. In
the special case of the phenomenology of motion, it is not such an
absurd speculation that what distinguishes the direct perception of
motion from the mere static memory of difference in position, is that
nearby small variations in the stimulus are read into a kind of
buffer where they are not compared as static memory does so much as
overlapped or superimposed in the way that contradictions are. After
all, we are not at all good at discriminating small intervals of
time, as the success of 25 frames per second makes apparent. Thus,
the mind constructs a kind of contradictory theory which undergoes
constant update. Indeed, this may well be the source of the
troublesome intuition we noted earlier, that it is one and the same
thing which endures through change, even though it is acknowledged
that it has different properties at different (nearby) times. If this
is right, then if one thinks with Hegel that the world is a kind of
idea, then the contradictoriness of ideas such as motion is apt to
spill over to the contradictoriness of their realisations in the
world. Indeed, even without the assumption of full-blown idealism,
there is always the caution that if a theory (consistent or not) can
be made out which describes an epistemic state, i.e., a cognitive
state, then how can we be entirely confident that the world simply
*could not* be that way? 




Taking a far less ambitious view than Hegel, Von Wright (1968)
nonetheless proposed an interesting account of conditions in which
change would have to be regarded as inconsistent. The account
requires two conditions. The first condition is that time is regarded
as structured as nested intervals rather than an assemblage of atomic
point-instants. This is an attractive proposal, if only because
no-one has ever seen a temporal or spatial point. Of course, standard
relativity theory proposes that spacetime is punctate, as does the
usual mathematics of the continuum. But a successful non-punctate
mathematics using intervals instead can be worked out, albeit with
considerably extra complexity. (see e.g., Weyl 1960). Now in the
ontology of intervals, since there are no atomic points to attach a
unique proposition to, the most one can say is that a proposition
holds somewhere in the interval, with the limiting case that it holds
throughout the interval.




Von Wright's second condition was then to suppose that an interval
might be so structured that a given proposition \(p\) and its
negation \(\neg p\) are dense in each other throughout the
interval. This means that no subinterval, no matter how small, can be
found in which just \(p\) holds throughout that subinterval, and
no subinterval can be found in which just \(\neg p\) holds
throughout the subinterval: every subinterval in which one holds, the
other holds as well. From an external point of view admitting
instants, we can see that this is a genuine consistent possibility, if
for example we think of
\(p\) as the proposition that a rational number of seconds has
passed, and \(\neg p\) as the proposition that an irrational
number of seconds has passed. These are dense in each other on the
classical real line regarded as time. Thus, there is no subinterval
which is purely \(p\) throughout and no subinterval which is
purely \(\neg p\) throughout.




This was von Wright's proposed account of a continuous change in
an ontology of intervals. The state \(\neg p\) changes continuously to
\(p\) if there is a preceding interval which is \(\neg p\) throughout,
then an interval with \(\neg p\) and \(p\) dense in each other, then a
succeeding interval with p holding throughout. Von Wright described
this as a kind of inconsistency. Unfortunately it is not clear from
his written words whether he had in mind that the situation was
inconsistent or only possibly inconsistent. His argument seems to be
this. In an ontology of intervals we begin with descriptions like
"It rained here yesterday" which means that it rained
sometime here yesterday. The basic description is thus "\(p\)
holds (somewhere) in the interval \(I\)." The special case where
\(p\) holds throughout \(I\) is noted, where to hold throughout is for
there to be no subinterval in which \(\neg p\) holds. Now
\(p\)'s holding in \(I\) is of course compatible with \(\neg
p\)'s holding in \(I\). But there is no contradiction here, as
long as there is a partition of \(I\) into subintervals such that
\(p\) holds throughout the subinterval or \(\neg p\) holds throughout
the subinterval. Thus if we take in that a disjunction holds in an
interval just in case there is a partition in which each of the
disjuncts holds throughout its subintervals, we can say that if there
is such a partition for \(p\), then the law of excluded middle \(p
\vee \neg p\) holds throughout the interval. Von Wright introduced the
modal operator \(Np\) for "Necessarily \(p\)." If we
define "\(Np\) holds in \(I\)" to mean that \(p\) holds
throughout \(I\), we can say that if there is no continuous change in
the above sense, then Excluded Middle LEM holds necessarily, \(N(p
\vee \neg p)\). However, defining the modal "Possibly" in
the usual way as \(M =df \neg N\neg\) and assuming de Morgan's
Laws, Double Negation and Commutativity, we get the result that in an
interval in which there is continuous change, \(M(p \amp \neg p)\)
holds, i.e.. a contradiction is possible. Presumably it further
follows that in a subinterval which has continuous change throughout,
\(N(p \amp \neg p)\) holds. Needless to say this implies that a
contradiction is true in that subinterval. We might note that the
result that continuous change is a true contradiction follows without
the detour through modal logic, since if LEM is false then \(\neg(p
\vee \neg p)\) holds for some \(p\), and so by de Morgan and Double
Negation, \(p \amp \neg p\) holds (throughout).




This ingenious construction has its problems. It is certainly
dangerous to assume De Morgan's Laws and Double Negation when the
logic of intervals is the case in point. They both fail for open set
logic, which is to say intuitionism, just as they both fail for its
topological dual, closed set logic. On the other hand, what is one to
say if the world is structured as intervals, non-punctate, and if
there are subintervals in which propositions and their negations are
dense in each other, interspersed with intervals where one of the
propositions holds throughout? The latter are clearly periods of
non-change, and the former are reasonably described as intervals of
change. And yet it would seem that the best one can do is to say that
\(p \amp \neg p\) holds in the transition periods: there
appears to be no consistent way of describing what is happening in the
situation which adheres to intervals and eschews points.


## 6. Inconsistent Motion




Many of the above themes come together in Graham Priest's
inconsistent account of motion in In Contradiction (1987). Priest
sets up the opposing consistent account of change as what he calls
the cinematic view of change. This is the view that an object in
motion does no more than simply occupy different points of space at
different times, like a succession of stills in a film only
continuously connected. He attributes the view to Russell and
Hume. It is an extrinsic view of change, in the sense that change is
seen as a matter of a relation to states at nearby instants of time.
The best-worked-out version of this view is the usual mathematical
description of change of position by a suitable function of time; and
then motion as velocity, that is rate of change of position, is given
by the first derivative, which is a relation to nearby intervals.




Priest wishes instead to have an intrinsic account of change, in which
it is a matter of the features of the object solely at the instant
whether it is changing at the instant. He offers three arguments
against the extrinsic account. First there is the
"abutment" argument (p. 203). Taking the usual view of
time as a continuously distributed collection of point-instants, in
any change there must be an interval throughout which \(p\) holds
abutting an interval throughout which \(\neg p\) holds. It makes
no difference whether there is a last instant for \(p\) and no
first instant for \(\neg p\), or no last instant for \(p\)
and a first instant for \(\neg p\); either way there is no room
for a time at which the system is changing. For example, if we said
that the change was at the boundary point, then there would be nothing
about that point to distinguish it from the situation where there was
no change at all because the abutting intervals had the same
proposition holding throughout each. Hence says Priest there is no
change at all in the cinematic view: for change there would have to be
a time when change was occurring, and that is absent in this case.




Priest's second argument (p. 217) appeals to causation. It is at least
imaginable that the universe is "Laplacean," by which he
means that the state at any time is determined by the states at prior
times. But if change is cinematic, then there is no sense to saying
that the instantaneous state of the world at the prior time determines
its state at subsequent times: for example, not even velocity is
determined by the intrinsic instantaneous state of a body. Now a
Laplacean universe is possible, but the cinematic view makes Laplacean
change *a priori* false. Hence Priest concludes by rejecting
the Laplacean view. 




Priest's third argument (p. 218) is his version of Zeno's arrow
argument mentioned earlier. In the cinematic view of change, there is
nothing about the arrow at any instant to contribute to its motion:
it is indistinguishable from an arrow at rest. But then there is
nothing to constitute its motion: an infinite number of zero motions
does not add up to anything but zero motion. In response to the reply
that according to measure theory a (nondenumerably) infinite number
of points of measure zero can have a non-zero measure, Priest argues
that this is just mathematics: "...it does not ease the
discomfort ... when one tries to understand how the arrow actually
achieves its motion. At any point in its motion it advances not at
all. Yet in some apparently magical way, in a collection of these it
advances. Now a sum of nothings, even infinitely many nothings, is
nothing. So how does it do it?" (pp. 218-9)




Setting aside questions about the strength of these arguments for the
present, how then are we to give an acceptable intrinsic account of
motion? According to Priest, the only acceptable answer is Hegel's:
that motion is inconsistent. Support comes from Leibniz' Continuity
Condition (LCC). This is essentially the thesis, suitably qualified,
that whatever holds up to a limit, holds at the limit. Priest's
argument for the LCC appeals to causality. He describes change
violating the LCC as "capricious" (p. 210). Humeans might
be able to accept it, but for them there are no connections, nothing
to constitute past states' determination of future states. He also
argues if the LCC fails, change would occur, but "at no
time" (p. 210): for a proposition switching values
discontinuously at a boundary there would be no instant identifiable
by its intrinsic properties alone as the one at which the change
occurred.




Priest's qualification to the LCC is that it applies only to atomic
sentences and their negations: otherwise we would have to admit the
case where a disjunction \(p \vee q\) held right up to a limit in virtue
of \(p\) holding at the rational points and \(q\) holding at
the irrational points: this would be capricious behaviour in which we
can make no sense of the past determining the future. We would also
admit problems if we allowed the LCC to apply to tense operators:
Future-\(p\) can obviously hold up to a limit without holding at
the limit.




But now we observe that the LCC so qualified implies that continuous
change is contradictory. For consider any particle with equation of
motion \(x = f(t)\). Then at \(t = a\) its position \(x =
f(a)\). However if it is in motion then in the neighbourhood we have
\(\neg(x = f(a))\), so by the LCC at the limit also \(\neg(x =
f(a))\), along with of course \(x = f(a)\) as well. Priest amplifies
this account by proposing that no moving body can be consistently
localised. Rather, in moving at time \(t\) it inconsistently occupies
a small finite (Planck length) lozenge of space, which is made up of
the positions it takes in the corresponding lozenge of time
surrounding \(t\). This gives a natural intrinsic account of
motionlessness at \(t\), namely that there is no contradiction in its
position at \(t\). One can propose an account of velocity, as varying
with the length of the lozenge or spread of position in the direction
of motion. There are applications in Quantum Theory, too. The
Heisenberg uncertainty of position may simply be the size of the
spread or smeared position. Moreover, there is a possibility for
backward causation implicit in the advanced wave front of
inconsistency affecting earlier states in the inconsistently
identified smear of spatial positions; and backward causation may be
the way to go with quantum nonlocality, as Huw Price (1996) has
argued.




One quick objection does not succeed. One might argue that since
motion and rest are not relativistically invariant, neither could the
contradictoriness in motion be part of the absolute character of
reality. This may be so, but it does not prevent the concept being of
use in the analysis of phenomena by means of frames: frame-relative
inconsistencies would still be a (relational) part of the world. More
importantly, the concept may find its natural home in QM rather than
GR. It is well-known that there are deep incompatibilities between
them as they now stand, but the jury is still out on how to resolve
them, and it may well be that absolute motion is a part of the
solution.




In asking how strong are the arguments in favour of this well-crafted
position, we return to Priest's three arguments against the rival,
consistent, extrinsic, cinematic view. We recall the first argument
was the "abutment" argument: consistent change cannot allow
that there is a (single) time at which the change takes place. This
will not sway the opposition, who will reply that it is the nature of
change, even change at a point, that it is relational in that it
requires comparison with nearby points; hence the demand for an
intrinsic conception of change is a mistake.




The second argument was that the cinematic view is incompatible with
the Laplacean view that the past determines the present. The way
Priest puts it is not so plausible: he says that Laplaceanism is
possible, whereas the cinematic view rules it out "*a
priori*" (p. 217). But this is a modal fallacy: the
cinematic view is only ruled out when one adopts the Laplacean view,
and so that is only relatively apriori.




The third argument, Zeno's arrow, has greater force though. How can
any number, even an infinite number, of zeros add up to a nonzero?
The mathematics of measure theory may say that intervals have a
non-zero measure whereas individual points are zero, but so what?
What is needed is a story which makes its application intelligible
and non-arbitrary. If this is not forthcoming, there is the strong
counter-intuition that zero marks the absence of existence; and no
number of absent or non-existent things or quantities makes a
present, existent thing or quantity.




So Zeno's argument after all seems to be the most resilient. But the
Laplacean universe also has appeal. Many philosophers have felt
uneasy about Hume's views on causation: if the past does not
determine the future then the universe is indeed capricious. 




Now one might endeavour to support Russell's contrary view by arguing
that non-zero speed is both necessary and sufficient for motion. But
both sides of this equivalence might be disputed. On the necessity of
non-zero speed for motion, a challenge might be mounted that zero
speed but non-zero acceleration is motion. On the question of the
sufficiency of non-zero speed for motion, Priest in the second edition
of In Contradiction (2006) says that he does not deny this. But this
opens him up a possible objection, namely that if non-zero (speed or
acceleration) is necessary and sufficient for motion, then the extra
element of inconsistency would seem to be explanatorily otiose, since
there is no need to add the extra element of inconsistency in order to
constitute motion. Such an objection does not refute his view, but it
would seem to make it unsimple. Moreover, one might still adopt an
inconsistent view coupled with the denial of sufficiency, which avoids
this objection.




In (2006), Priest extends his account to time itself. Hitherto,
quantities other than times were regarded as changing to the extent
that they were inconsistently smeared out in a small lozenge or spread
of time. In 2006, even the identity conditions for times are smeared
out: if \(t\_1\) and \(t\_2\) are in the same spread then both \(t\_1 =
t\_2\) and not-\((t\_1 = t\_2)\) hold, and in particular not-\((t=t)\)
holds for each \(t\). Priest proposes that this gives explanations of
several perennially puzzling features of time, specifically its flow,
how it differs from space, and its direction. Focussing just on flow,
it is the fact that not-\((t=t)\) is constant for all t which supplies
the intrinsic feature of time necessary in Hegelian terms for its
changingness or flow. The view faces some interesting objections, one
of which is a sorites-like problem that if times in the same spread
are (inconsistently) identical with each other, then since any time
will be identical with others in the same spread, and those others
identical with further times in other spreads, identity will be spread
everywhere. Of course, many replies have been made to the sorites,
but one might also observe that none are particularly appealing. At
least, the arguments need to be worked through for the particular
case. 


## 7. Discontinuous Change and the Leibniz Continuity Condition




If the LCC is to have a chance of being applicable, then it needs
further restriction, beyond atomic sentences and their negations. This
is because it has implausible consequences when applied to certain
atomic sentences. Consider any increasing function \(f(t)\). Then
sentences of the form \(f(t) \lt f(a)\) will hold for \(t \lt a\). By
the LCC then, \(f(a) \lt f(a)\). This is surely a gratuitous
conclusion even before the contradicting sentence \(-f(a) \lt f(a)\)
is taken into account. The present author (1997) therefore proposed to
restrict the application to the atomic sentences of equational
theories, that is to sentences of the form \(f(t) = 0\). This is not
so unreasonable on independent grounds, since the basic laws of nature
are expressed in equational form.




So restricted, we can note that far from being unreasonable, it turns
out that the LCC is satisfied in a large class of reasonable models,
specifically the C-infinity worlds mentioned earlier, in which every
function is continuous. These include all those of GR. Now a
C-infinity world gives us a kind of half-way house for cause. It
might be that all correlations are coincidences, but at least if
functions are continuous then causation is a distinctive correlation
in that it is transmitted locally. This can be applied beneficially
to produce not a general account of inconsistent change, but a
particular account of certain inconsistent changes, as follows.




Quantum measurement has long been problematic, for more than one
reason. One reason has been that it represents an irreducibly
different kind of process from Schrodinger evolution. Another is that
it is change which is discontinuous and yet causal: one can make
things to happen with measurement, even though one cannot determine
the exact outcome. A third reason is nonlocality itself: the nonlocal
is *ipso facto* the discontinuous, and yet the nonlocal is
governed by a kind of statistical causality. But now, to settle at
least some of these issues, it has been proposed to utilise the
theory of inconsistent continuous functions. These arise when a
function is classically discontinuous, but we inconsistently identify
the limit of the function (assuming it has a limit) with its value at
the limit. Such functions, by virtue of being continuous, can be
shown to satisfy the LCC. But granted that the formal details exist,
what reason is there to apply them? It is precisely that we want to
preserve a degree of causality, that is LCC-causality, while yet
retaining the essential discontinuity and unpredictability of the
process. Thus the slogan "nonlocality is inconsistent
locality," which is intended not to apply to change in general
but to discontinuous change which we nonetheless have reason to think
of as causal.


## 8. Cognition of Motion



As we saw earlier, the basic case for the inconsistency of change
turns on the following premisses: (i) \(Fa\), (ii) \(Gb\),
(iii) \(Fx \rightarrow {\sim}Gx\), (iv)
\(a=b\). Up to the last premiss all is consistent, but the
identification of the entities a and b from different times gives one
and the same thing incompatible properties. It was suggested in
Section 5, taking a lead from Hegel, that even if we say that change
is a consistent process, the *perception* of change might well
involve inconsistent representations. It is time to take a closer look
at what cognitive science can tell us about this. It turns out that
the argument is inconclusive, but suggestive. It is useful to
concentrate on the example of the perception of motion. A thorough
survey can be found in Palmer (1999), Chapter 10.



A basic mechanism for perceived motion is the Reichardt detector. It
seems not to be entirely settled whether there are such mechanisms in
the CNS. On the other hand, as we see, it is so simple that it is hard
to imagine that we don't have Reichardt detectors in our brains. 



The original Reichardt detector involved two spatially-separated
retinal transducers responding to intensities \(I\_{1}\), \(I\_{2}\) of
incident light respectively at times \(t\_{1}\),
\(t\_{2}\). Essentially, I measures the number of incident photons on a
single retinal cell at a time. (For convenience we can regard the
input into a given retinal cell as the same as its output, though the
output is electrochemical, as befits a transducer.) One of the two
inputs, say \(I\_{1}\), undergoes a delay, and so is compared at a
later time \(t\_{2}\) with the other input \(I\_{2}\) undelayed. There
are different ways to make the comparison: Reichardt suggested both
addition and multiplication, but it is more reasonable to consider
subtraction of one from the other (a fixed operation for a given
Reichardt detector). The difference between \(I\_{1}\) and \(I\_{2}\) is
then propagated further into the CNS. Thus we have a *change
detector*, there is no change in intensity from \(I\_{1}\) at
\(t\_{1}\) to \(I\_{2}\) at \(t\_{2}\) iff \(I\_{2} - I\_{1}=0\).


 To see better what this means, we notice that the above detector
can be pulled apart into two even simpler mechanisms.



The first of these is a *spatial change* or *spatial
difference* detector. This consists of two transducers whose
outputs *without a delay* are compared by subtraction. If the two
input cells have positions \(x\_{1}\), \(x\_{2}\), then
the *quotient* \((I\_{2} - I\_{1}) / (x\_{2} - x\_{1})\) measures the
difference in *simultaneous* values of I over the spatial
distance \(x\_{2}-x\_{1}\). This can be written as the derivative \(dI /
dx\). It, and its numerator, are zero iff there is no difference in
\(I\) measured by the two input cells at the same time. The temporal
variable \(t\) is fixed here. In passing, we can also see a mechanism
here for the perception of spatial relations as emphasised
by *Gestalt* psychologists.



The second of these simpler mechanisms is a  *temporal change*
detector. It consists of a single input cell, whose output \(I\_{1}\)
at \(t\_{1}\) is split into two, one of which passes through a delay,
then compared with the undelayed signal \(I\_{2}\) from the next moment
\(t\_{2}\). The quotient \((I\_{2}-I\_{1}) / (t\_{2}-t\_{1})\) measures
change in I at a single input over the time period. This can be
written \(dI / dt\). This, and its numerator, are zero iff there is no
change in the intensities presented to the same cell at different
times. The spatial variable \(x\) is fixed here.



These two mechanisms give ways to register and measure spatial and
temporal change. Now it is clear that these two mechanisms, when
coupled together, are exactly the original Reichardt detector. But now
we can see exactly what that measures. Calculus allows us to write
\(dx/dt =(dI/dt)/(dI/dx)\). But \(dx/dt\) is the rate change of
position over time, which is *speed*. Importantly, it is the
intensity variable \(I\) which is fixed here. Thus what this measures
is how a *single intensity*, such as that of a moving dot, an
edge, or a shadow, is varying its spatial position over time. This
is *motion*. For a given pair of input cells, the direction of
the motion is fixed (different detectors for different directions), so
we have a register for *velocity*.



In passing, acceleration detection is also straightforward to explain:
split the output from a motion-detector, delay one signal, and
compare.



We can see that the sense in which information about different times
is compared, is by delaying information from one time. A delay is a
kind of buffer. Then the information is "superimposed" on the
information from the next time, by presenting the two informations
together at the one comparator. A "composite picture" is obtained by
subtracting one from the other. This is conjunction of sorts, since it
puts two pieces of information together. But it does not obviously
make for a state with contradictory content. Indeed, it does not make
for a picture of a thing enduring over time, since the same intensity
\(I\) at different place-times might be due to quite unrelated events. But
there is more to be said about perceived motion



We can learn more by considering a family of phenomena known
collectively as *phi*, which is in turn more generally classified
under the heading of *apparent motion*. The simplest example is
of two intermittently illuminated lights, flashing back-and-forth. We
naturally represent it as a *single, moving light*. This is not
mere sameness of intensity, but representation as an *enduring*
thing. Moreover, it is not surprising that we should evolve to have
this response, if we are to successfully track game, or danger,
through intermittent presentations. Or think of a row of lights, as in
a neon sign, with either a single bulb illuminated but its position
varying, or a single bulb unilluminated, again with systematically
varying position. Or think of a clock dial, with one bulb out, moving
around the circle. The content of our experience is as an enduring
single thing, having different positions at different times. This is
the phi phenomenon. Speed things up (sometimes
called *magni-phi*), and it ceases to look discrete, instead
presenting as a continuously moving thing, which enhances the
effect.



Think of the example of a black dot moving across our visual
field. Recalling our general rubric for change at the beginning of
this section, we can affirm the first three premisses as statements
about the content of our experience: (i) \(a\) is at \(x\_{1}\) at
\(t\_{1}\), (ii) \(b\) is at \(x\_{2}\) at \(t\_{2}\), (iii) if anything
is at place \(x\_{1}\) at a time \(t\_{1}\), then it is not at place
\(x\_{2}\) at \(t\_{2}\). Now add in the content of the phi-experience:
(iv) \(a=b\), and we deduce that \(a\) is also not at \(x\_{1}\) at
\(t\_{1}\).



A consistentist about the phenomenology of motion might support (iii)
and challenge (iv), borrowing from Dharmakirti, or from Lewis'
perdurance/counterpart account. An alternative consistentist position
would be to challenge (iii) and support (iv). In favour of (iii), it
might be said that if the moving black dot is slowed down
sufficiently, and particularly if its positions are discrete, or it
jumps about, then (iii) finds intuitive assent: \(a\) and \(b\) just
are the sorts of things (events) that occupy distinct places at
distinct times. This may be what happens. Again, imagine perception
cut off after \(t\_{1}\) but before \(t\_{2}\) (close the eyes, or
otherwise vanish the dot). Then (iii) holds, the phenomenal dot at
\(t\_{1}\) is a distinct existence from a dot at \(t\_{2}\). Our
cognitive system responds to stimuli both as events, where (iii)
holds; and as enduring objects, where (iv) holds. If nothing moves,
then there is no contradiction, but when we see motion, the
object-recognition mechanism identifies distinct things at different
times and so a contradictory content is born. That, I suspect, is the
phenomenological content of motion.




This model has a number of features, which are only briefly summarised
here. A now box is postulated, distinguished by its distinctive
multi-modal inputs and outputs. Information enters the now box and is
retained for a brief time, the "specious
present". Dismissal from the now box cannot be instantaneous,
but takes some time to "fade" or reduce in intensity. Thus
the now contains new contents and fading, older contents; and a
natural direction of time. The delay and comparisons of Reichardt
detectors are enlisted in this process. The endurance or persistence
of contents suggests an account of apparent motion, phi or beta, in
which identical states close in time and space but not identical, are
counted as identical. The important thing for our purposes is that the
identity of different and changing temporal stages of a perceptual
process (the perceived moving dot), gives the necessary premiss for
inconsistency, as discussed earlier.



In passing, this model gives natural accounts of akinetopsia, and of
the "moving now". Akinetopsia, that is motion blindness,
has a "stop-start" phenomenology where a thing is
perceived as distant, then suddenly up close, without any awareness of
the in-between. This suggests that something has gone wrong with the
fade mechanism, and the contents of the now do not fade, but persist
full strength for a time, crowding out new updates. Eventually it
collapses, and then no in-between has been registered. As for the
moving now, it is noted that the contents of the now change over time,
but it seems to the experiencer as if the now is a single thing
persisting or enduring over time, and moving from event to
event. These matters are expanded on in Mortensen (2013). 



An independent account of the flow of time, with striking similarities
is to be found in Gruber, Bach and Block (2015). The flow of time has
two levels of explanation, and is described as an
"illusion" insofar as experience is in conflict with
measurement (compare Muller-Lyer). It would appear that the levels,
upper and lower, correspond to \(A\) and \(B\) concepts respectively,
so that the upper level, the passage of time, is an illusion. The
authors postulate two mechanisms, perceptual completion and object
persistence, to account for the flow of time. Perceptual completion
involves adding to an experience to complete it some way, for example
apparent motion, that is phi or beta. Object persistence requires that
an object be perceived as the same (thing) now as it was then. It is
essentially the same mechanism that identifies an object at a time,
with itself at a later time. This is conceptually necessary for an
order of past/present/future, the \(A\) series. This account also has
a specious present or now, which is similar to our now box.



We cannot hope to do justice to this complex paper here. However,
focussing on the theme of inconsistency, one question that could be
asked is how much these mechanisms, perceptual completion and object
persistence, are really different, as against being species of the
same genus. It would seem that the illusion is pervasive because of
the mechanism of identification of earlier/later stages. But this is
what we saw was sufficient for contradiction. If these models of
motion perception are right, it follows that contradiction is
pervasive in our experience as the phenomenology of change, which is
to say, everywhere.


## 9. Conclusion


 

There remain many loose ends from our discussion. Still, it
emerges that the connection between change and inconsistency is deep,
and that the case for inconsistencies in motion and other change is
surprisingly robust.