by Corry Shores
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A Study of the Transconsistent
Part III. Applications
Ch.11. The Metaphysics of Change I:
The Instant of Change
11.2 The Instant of Change
Things in the physical world change states through time. There are instants of such change when we have no more reason to say that it is in the prior or the following state. In such cases, it is true that the thing is both in a state and not in that state. This is a true contradiction, a dialetheia, that is a real and common situation in the physical world.
Priest will illustrate a problem with the instant of change.
As I write, my pen is touching the paper. As I come to the end of a word I lift it off. At one time it is on; at another it is off (that is, not on). Since the motion is continuous, there must be an instant at which the pen leaves the paper. At that instant, is it on the paper or off? 
[We have things and their interactions here operating in a describable way, so we will call it system s. There is a moment in time in question, the instant when the pen leaves the paper, which we will call t0. We also have different states the pen can be in, either on or off the paper, called s0 and s1. We will describe the state of being on the page in the statement called α (which reads “The pen is on the paper”). Otherwise, the pen would be off the paper, and then it will be in the next succeeding state, s1. After the pen leaves the paper, it is not the case that “the pen is on the paper”, which is a proposition we would then write as ¬α. Now we want to know, at the moment it leaves the paper, what state is it in? There are four possibilities. Either 1) it is only on the paper, 2) it is only off the paper, 3) it is neither on the paper nor off, or 4) or it is both on the paper and off the paper in the same instant.]
We may formulate the problem more generally. Before a time t0, a system s is in a state s0, described by α. After t0 it is in a state s1, described by ¬α. What state is it in at t0? A priori, there are four possible answers:
(A) s is in s0 and s0 only.
(B) s is in s1 and s1 only.
(Γ) s is in neither s0 nor s1.
(Δ) s is in both s0 and s1.
Also, there may not be an answer which holds for all cases, because “Different changes may be changes of different kinds.” (160) If we just assume classical logic, then all changes will have to be of types either A or B. But since we are testing the viability of classical logic for explaining change, we cannot assume from the beginning it is the only means to find the answer. Priest will argue that there are changes of type Δ, that the pen is both on the page and off it the instant it makes that transition.
Previously in section 4.7 Priest ruled out the possibility of Γ type changes, so at least one of α or ¬α must hold. [His argument there has to do with liar paradoxes, for example, ‘This sentence is false’. If it is false, then it is true, but if it is true, then it is false. Therefore it is both true and false. That is not a problem for dialetheians. However, those who reject this need to explain what is wrong with the argumentation that leads to the paradox. One solution is to say that there are truth gaps, that is, that there are sentences which are neither true nor false. Priest in that section showed how their arguments failed, and thus in this current section, he does not find the third option, that it is neither in one state nor the other, convincing.] Now we must argue that not all changes are of types A or B (being either on or off the paper exclusively). So recall that at t0 the pen leaves the paper. Is it on or off? It seems we do not have a better reason to say one over the other. This means we might prefer a symmetrical answer like Γ or Δ (the pen is neither on nor off the paper, or the pen is both on and off the paper). We might break this asymmetry by identifying being on with being zero distance from. [It seems Priest might be saying something like the following here. Consider its motion coming off the paper. Because of infinite divisibility, there is no first point when it is off the paper. Thus maybe we might say that it cannot be both, because the final on point is determinable, but the first off point is not. However, this does not apply for objects falling to the ground for example. This is perhaps because there is a final terminating point for the motion without there being a continuation after it. Priest writes: “There is, however, a way of breaking the asymmetry in this case. Since the motion is continuous, there is, presumably, a last instant at which the distance between the point of my pen and the paper is zero, but no first point at which it is non-zero. (Perhaps more precisely, there is a last point at which the electrical repulsion between my pen and the paper is equal to the weight of the pen, but no first point at which this is not the case.) If we identify being on with being zero distance from, this makes the change of type A. But the identification is highly suspect. An arrow is fired into the ground. At the instant of impact, before the point of the arrow penetrates the ground, is the arrow on the ground?” (160)]
Priest says that even if we can preserve asymmetry in the above example, it will not work in all cases. If we discover some solution, then we have a symmetrical relation between the state before of not having it and the state after of having it.
A particularly striking example of this is a phenomenological one. For days I have been puzzling over a problem. Suddenly the solution strikes me. Now, at the instant the solution strikes me, do I or do I not know the answer? The situation is, again, symmetrical. Before, I did not know the answer; after, I did. Moreover, one cannot suppose that in this case there is some tie-breaking ulterior fact. My epistemological state is all there is, and that is symmetrical. It makes little sense to suppose that I either did or did not determinately know the answer at the instant of change, though I am unaware which. (161)
In the next example, Priest has us consider us walking into a room. There will be a point when we have no more reason to say we are in than we are out. Thus there are cases where we have enough reason to give answers of type A or B.
I am in a room. As I walk through the door, am I in the room or out of (not in) it? To emphasize that this is not a problem of vagueness, suppose we identify my position with that of my centre of gravity, and the door with the vertical plane passing through its centre of gravity. As I leave the room there must be an instant at which the point lies on the plane. At that instant am I in or out? Clearly, there is no reason for saying one rather than the other. It might be suggested that in this and similar cases we are free to stipulate that I was, say, in. Unfortunately this is not a solution, but simply underlines the problem. I am free to stipulate in this way only because neither being in nor not being in has a better claim than the other: I am neither determinately out rather than in, nor determinately in rather than out. Thus, intrinsically, the change is symmetrical, and therefore not of type A or type B. (161)
So we are arguing for type Δ changes (both on and off). Someone might argue against them by opening the possibility for Γ type changes (neither on nor off). They might do this by rejecting the exhaustion principle which says that if α is not true then ¬α is true. [Perhaps then in the case of the pen, it can both be that the pen is not on the table but also not-not on the table, or in other words, that the pen is neither on nor off the table.] This is possible if the arguments from section 4.7 can be met. Nonetheless, the above example still gives us ample reason to suppose that there are type Δ changes.
There is another issue to address. Instead of instants of time, there might only be intervals. If so, then there are no instants of change, and thus no contradictions [because it is never at the same time that the pen is on or off the paper]
There are some problems however with arguing that time is not composed of instants. Science operates as though time can be represented by the real line. Calculus’ application in physics presupposes this. Thus to say that they are wrong about instants is to say that much of their work [or at least methodology, maybe conclusions] are wrong. [[However, there is also a way to conceive of the instant as an infinitesimal interval. This would make it both a combination of states without any passage of duration between them.]]
a good part of science is based on the | assumption that physical continua have a structure that can be represented by the real line, and therefore that we can speak of instants of time. In particular, any science that uses the differential and integral calculus presupposes this. Therefore, this proposal, if adopted, would cause the demise of a good part of science. Or, to put it more tellingly, the proposal flies in the face of well corroborated scientific theories. Its correctness is, therefore, highly suspect. (161-162)
Another problem with the theory that time is composed of intervals and not units is that it fails to account for how (or when) change happens. If two successive intervals have different states, where does the change take place? It cannot be between them, because there are no instants between intervals. But it cannot be in either one, because there is only one state and not a change of states. Thus tie would just be a sequence of still moments. This is the cinematic account of change [we discuss it further in sect.12.2].
suppose that during a certain time a system, s, changes discretely from state s0 to states s1. Then there must be two abutting intervals, X and Y, X wholly preceding Y, such that s0 holds throughout X and s1 holds throughtout [sic] Y. Now, given that there is no instant dividing X and Y, we cannot ask what state s is in at it. However, just because there is no such instant, there is no time at which the system is changing. X is before the change. Y is after it. Thus, in a sense, there is no change in the world at all, just a series of states patched together. The universe would appear to be more like a sequence of photographic stills, shown consecutively, than something in a genuine state of flux or change. We might call this the cinematic account of change. As we will see, it has a habit of surfacing in consistent accounts of change. I will discuss it in more detail in section 12.2. For the present, let us just note that the cinematic account is highly counter-intuitive. (162)
Also, intervals would have to be divisible, which means at one line of subdivision there could be a case of α ∧ ¬α.
it is not even clear that dialetheism can be avoided by eschewing instants of time in favour of intervals; for, unless there are atomic intervals, a possibility that raises the shades of Zeno and exacerbates both the previous problems, intervals must be indefinitely subdivisible. Now, note that the fact that a holds at an interval, X, does not necessarily imply that it holds at every subinterval of X (or else the sun’s shining on a certain day would imply that it shone during every part of the day). There is therefore nothing, in principle, to rule out the possibility of an interval such that every subinterval where a holds has a subinterval where :a holds and vice versa. What holds at this interval? What could it be but α ∧ ¬α?
Priest, Graham. In Contradiction: A Study of the Transconsistent. Oxford/New York: Oxford, 2006 [first published 1987]