The question of the persistence, or identity through time, of material objects may be put thus: under what conditions does a succession of object-stages constitute a single, persisting material object? A natural first attempt at an answer to this question is to say that it involves spatio-temporal and qualitative continuity of the stages. This may be stated more rigorously as what I will call "The Simple Continuity Analysis":
A succession of object-stages S constitutes a single persisting object just in case
(1) S is spatio-temporally continuous, and
(2) S is qualitatively continuous.
(3) S is not a proper part of any spatio-temporally, qualitatively continuous succession of object stages S2.
(Herein I will presume that for some continuous interval I, and for every time t in I, some stage of S exists at t. That is, S does not occupy a "broken" interval of time. I will not herein deal with questions about objects that cease to exist, and then come back into existence at some later time.)
We should distinguish between two sorts of continuity. A succession is weakly (spatio-temporally or qualitatively) continuous just in case the change that S undergoes can be divided into a series of small changes. A succession is strongly continuous just in case the change that S undergoes can be divided into a series of changes as small as you like.
Although it might at first be thought that strong spatio-temporal continuity is necessary for a succession to constitute a single persisting material object, counter-examples involving gain or loss of parts show that this cannot be correct. Suppose that a tree loses a branch at time T, going from being 30 cubic feet in volume prior to T to being 28 cubic feet in volume after T. There is a succession of tree-stages that constitutes a single persisting tree, but the change in volume which this succession undergoes cannot be divided into a series of changes as small as you like. We cannot divide its change in volume into a series of changes each of which is less than one cubic foot of change, for there is no time at which the tree is between 30 and 28 feet in volume.
We may try to make the concept of weak (spatio-temporal) continuity involved in the proposed analysis more precise as follows (this is Hirsch's definition of "moderate spatio-temporal continuity):
A succession of object-stages S is weakly continuous just in case, for any time t (during which some member of the succession exists), there is an interval I about t such that for any t2 in I, the place occupied by the member of S at t2 overlaps the place occupied by the member of S at t by more than half, i.e. the extent of overlap is greater than the extent of non-overlap.
Satisfying the simple continuity analysis is not sufficient for a succession of stages to constitute a single, persisting material object, however. Hirsch first notes that sometimes we judge that a car goes out of existence when crushed, and is replaced by a block of scrap metal. The succession made up of car-stages and block-stages, however, is weakly continuous. I don't think these examples have much force, for I think that we also judge that the block of scrap metal used to be a car. But if the block used to be a car, some car survived the crushing (albeit not as a car!), and there's only one car that could participated in the crushing, so no car ceased to exist in the crushing.
But there are stronger counter-examples to the sufficiency of the simple continuity analysis. Consider a succession made up of all the stages of a particular tree that exist on odd-numbered days of the month, and all the stages of its trunk on even days of the month. Even though this succession satisfies the simple continuity analysis, it clearly does not constitute any single persisting material object. In general, for any single persisting material object, there will be an indefinite number of successions that satisfy the continuity analysis, but which lead from that object at one time to some proper part of it at another, or from a proper part of the object to the object itself.
What seems to have gone wrong with the simple continuity analysis, according to Hirsch, is that successions satisfy it that contain stages of different sorts, e.g. tree-stages and trunk-stages. Thus Hirsch proposes a sortal analysis of persistence:
A succession of object-stages S constitutes a single, persisting material object just in case
(1) S is spatio-temporally continuous,
(2) S is qualitatively continuous,
(3) For some sortal term F, each stage of S is an F-stage, and
(4) S is not a proper part of any succession of F-stages that satisfies (1)-(3)
The third condition rules out such deviant succession as the one that contains tree-stages on odd-numbered days and trunk-stages on the even-numbered days, for although not all the stages in the succession are tree-stages, nor are they all trunk-stages.
A sortal is just a term F such that any succession of F-stages that satisfies (1)-(4) above constitutes a single persisting F-thing. Such terms as `tree', `trunk', and `car' are sortals, according to Hirsch. Terms that typically apply to overlapping things ("dispersive" terms, in Hirsch's terminology), such as `mass of wood', or `brown thing' are not, for if we were to trace objects under these terms, we could trace deviant successions, such as the one composed of tree-stages and trunk stages (since both tree-stages and trunk-stages are mass-of-wood-stages).
Hirsch holds, however, that we do have a pre-sortal concept of continuity, for presumably even a person who had never seen a tree, and did not have the sortal concepts "tree" and "trunk," would nevertheless be able to trace the career of the tree. If we pointed to the tree (using "the sweeping gesture of ostension," as Quine calls it), and asked him to describe what happened to that thing over time, we would not expect him to trace the career of the object consisting of both tree-stages and trunk-stages. Hirsch thinks that underlying our sortal-laden concept of persistence, we have a concept of persistence that follows that following "basic rule":
A sequence of stages S constitutes a single, persisting material object if and only if
(1) S is spatio-temporally continuous,
(2) S is qualitatively continuous,
(3) S minimizes change at every time t occupied by some member of the sequence, and
(4) S is not part of any longer sequence that satisfies (1)-(3).
Thus the succession leading from tree to trunk and back violates the basic rule, and will not be identified as a single, persisting material thing, even by someone unacquainted with the sortal concepts "tree" and "trunk." Hirsch thinks that the basic rule does not suffice, however, for our full-fledged concept of persistence, for the basic rule sometimes requires us to continue to trace the career of an object where the sortal rule requires that we stop tracing the career of that object. Suppose we have a car that is crushed at time t. According to Hirsch, the car does not survive the crushing: the car ceases to exist, and is replaced by a block of metal. According to Hirsch, this is because there is no succession, including elements prior to and after t, to which we can apply the sortal term "car." The basic rule, however, would license us to regard the block of metal as identical with the car (even though the block of metal is not a car, it used to be a car), for we can continue to trace a change-minimizing, continuous succession that includes both car-stages and block-of-metal-stages. (This would probably be the career naturally traced by someone who doesn't have the sortal concept "car.")
It has been pointed out by Armstrong and Swoyer, that even the sortal rule does not manage to state sufficient conditions for a succession constituting a single, persisting material object. Suppose that someone has a machine that can make cars disappear from existence instantaneously, and that someone else has a machine that can make cars appear out of thin air. By an utter coincidence, at the very moment the first person annihilates one car, the second brings a car into existence in the same spot, a car just like the one that the first person caused to disappear. If this were to happen, we would have a succession that satisfies the sortal rule, but this succession would not constitute the career of a single persisting object (although we might well be fooled into thinking that it did!). This example purports to show that some causal relation between successive stages is required for a succession to constitute a single, persisting thing. The problem with the succession in the counter-example is that the car-stages prior to time t have no direct causal connection with the car-stages after t.
Hirsch, Eli. The Concept of Identity. Oxford, 1982.
Swoyer, Chris. "Causation and Identity." Midwest Studies in Philosophy IX (1984), 593-622.
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