IN DEFENSE OF VAGUE EXISTENCE AND IDENTITY

Ted Everett

(Philosophy Colloquium, 9/08)

 

            Vague predicates are those that suffer from sorites.  That is, (1) vague predicates apply to objects to different degrees or extents, (2) at extreme degrees it is determinate whether the predicates apply to their objects, but (3) there is no intermediate degree fixed in the language that determines when the predicates in question start to apply.  The most common examples of vague predicates are adjectives like "tall".  We can imagine a man growing taller continuously from, say, five to seven feet in height.  At the beginning of this process none of us would say that the man is tall, and at the end of the process we would all say that he is tall.  But there is no one place in the middle where we would say that he suddenly "pops" into being tall.  Any attribute of things that we can imagine gradually growing or accumulating seems to yield the same problem.  And we do not need to involve the idea of change in every case – it would be enough, for example, to line up men of slightly different heights in order, with a five-foot man on the left and a seven-footer on the right, and note that there is no adjacent pair of men in the middle, one of whom is definitely tall while the other is not.  Either scenario can produce the famous paradox:  If the five-foot man is not tall, and adding just a hundredth of an inch, say, never turns a not-tall man into a tall one, then the growing (or sequential) man never becomes tall, even at seven feet.  Yet the seven-foot man certainly is tall – so plainly, something has gone wrong.  The basic problem of sorites is the indeterminate extensions of vague predicates; the sorites paradox is just a symptom. 

            Most, if not all, ordinary predicates are vague in this way.  A sortal predicate like "ship" or "restaurant" may seem more crisp than "tall", but imagine an old ship brought onto the beach and gradually, bit by bit, transformed into a restaurant.  At the beginning of the process is a clear case of a ship, at the end is an equally clear non-ship, but in the middle is again no definite point at which the thing stops being a ship or starts being a restaurant (I suppose it could be both, or neither, for a while).  Even natural-kind predicates like "human" have the sorites problem.  You are a clearly human being, but you have distant ancestors who are clearly non-human apes, and there is no specific generation in-between for which the children are human and the parents are not.  We could, of course, in theory, pick out some generation among the early hominids to call by convention the last pre-human generation, but any such decision would be largely arbitrary.  For it is not part of the meaning of the word "human" that it should start applying only to related creatures with this or that precise cluster of attributes, or to evolving creatures at precisely this or that stage. 

            Relational predicates can be just as vague.  Imagine a pair of strangers gradually getting to know one another, and, as sometimes happens, turning insensibly over time into close friends. There need be no one moment at which they stop being non-friends and start being friends.  Rather, the term simply applies better and better to the state of their relationship as time goes by. They may start actually using the term "friend" towards each other at some specific point, but they would probably not deny that they had been friends in fact one minute earlier, and for some indefinite time before that. 

            "Exists" is a vague predicate along these lines.  Set good fire to the restaurant on the beach, and this begins a process at one end of which we'd all say that the restaurant exists, and at the other end of which we'd all say that it does not.  But again, there is no point in the middle, no particular weight or volume or length of remaining timber where the restaurant just pops out of existence. Some objects, like the Western Roman Empire, say, can spend more time passing away then they have spent clearly existing, as historians argue over when and how, between the third and eighth or even ninth centuries A.D., we ought to say the Western Empire really fell.  A thing's existence can have vague spatial boundaries as well as temporal ones.  Does the ordinary, unofficial concept of New York City fully include all five of its boroughs?  Does it fully exclude all of its suburbs?  A man visiting China might reasonably say that he is from New York City when he comes from Yonkers.  A man visiting Times Square might deny that he is from the city when he comes from Queens. 

            "Identical" or "same" is, along similar lines, a vague relation.[1]  Take two samples of medium blue color, one only imperceptibly more green than the other, so that we would all call them the same color on sight.  Now vary them a little more, a little more, a little more, and by the time the second sample is undoubtedly green, we'd say that the two colors are not the same but definitely different.  Yet there is no precise point in the process at which we must say that the previous color pair was still the same, and the following pair was not.  This is as true about identities among physical objects. After the beached ship has been turned into a restaurant, we are liable to say that the restaurant is not the same thing as the ship.  It cannot be the same ship, certainly, since it is not a ship at all by the end of the process.  It is arguably not even the same material object, if we allow that the physical pieces of the onetime ship have all been replaced at one point or another, and all of its inner structures altered.  If a stronger case is needed, let the restaurant then be turned gradually into an abstract metal sculpture, retaining no parts or structure of the original ship, the sculpture then melted and shaped into hundreds of little metal bars, and the bars distributed among the nations of the world.  Surely, by the end of all this there will result a thing or set of things that is not identical to the original ship under any normal description.  But again, there will not be any definite one of the many finely-distinguishable intermediate objects that is the last one to retain the ship's identity, or the first one that does not. 

            It can be argued that existence and identity as not really as vague as all that.  There are contrary intuitions.  In one way of speaking, at least, we often treat both existence and identity as what Peter Unger (1971) once called "absolute" properties.  These are properties that do have "cut-offs" – degrees or extents at which the property begins to apply to an object – that are fixed in the language, but at one extreme or the other, rather than somewhere in the middle.  Unger raised the issue in his famous discussion of skepticism (with an intended application to the predicate "knows"), while basing his argument on more homely absolutes like "empty" and "flat", which he distinguished from relatives like "tall" that have no upper limit.  I would usually say that this table here is flat, but is it really flat?  Looked at sufficiently closely, there will be bumps of various sorts; it will be easy to imagine something flatter.  But if something could be flatter, then this thing cannot really be flat, because "flat" just means "flat all the way".  In the same way, I might call for another glass of beer on the grounds that this one here is empty, but it is not truly empty, Unger would say, because it is not absolutely empty: though I have done my best, one or two drops of beer remain.  Perhaps what I mean in calling the glass empty is that it is almost empty, in which case I have probably gotten my point across well enough to the bartender, even while saying something false. 

            In the same way of speaking, we could say that anything that exists to any positive extent exists entirely, no matter how little there is of it when it is coming to be, or how much of a mess it becomes as it passes away.  Thus, the Byzantine Empire existed simpliciter even on May 29, 1453, as the Turks were pouring through Constantinople's shattered walls, despite its having lost almost all of its territory, power, and political integrity by that point.  Even the Western Roman Empire, despite its more protracted and complex demise, ought in this way to be seen as having popped into non-existence at some point or other, perhaps at the moment in 476 when Romulus Augustulus, the final Western Emperor, formally handed his crown to the barbarian Odoacer (or whatever event of that sort may have actually happened). And you can trace your own existence backwards to the moment of your conception when you popped into existence, and forwards to the final cessation of all your brain activity, at which point you will pop back out, without your ever having existed to a greater or lesser extent in the meantime.[2]  Identity is often held to be an absolute as well, so that the only pairs of things that should be counted as identical at all are those that are entirely identical in every way.  If there is any difference whatsoever between this ship and that restaurant, or this ship and that ship, or this ship and…(pointing in the same direction) this ship, then they are absolutely different things, however similar they might remain.  And though we often call a pair of colors identical or the same that are only slightly or imperceptibly different, we never tell the truth in doing so: if things are different at all, then they are just plain different.  If we want to say that two colors are very similar, or almost the same, then we can say so truly in those words – or say what is strictly false, that they are the same simpliciter, and take our chances.  

            Perhaps it is best to say that all such predicates are ambiguous.  It would at least be awkward, after all, to suppose that we are almost always wrong in ordinary speech, whether we are calling a table flat, a glass empty, or two colors the same.  Whose language is it, anyway?  We certainly count such ordinary statements as often being true, not false – though I suppose one could apply the same sort of semantic argument to "true" as Unger does to "flat" and "empty".  This will only end by pushing the absolutist out of ordinary conversations altogether, as somebody who just doesn't understand what everyday predicates really mean.  It may be better, then, to say that such predicates function in two senses, one ideal and absolute and the other commonsense and relative, so that my ordinary statement that my beer-glass is empty will not be contradicted by a bartender who genuinely understands the language.  I am not speaking falsely, and I am not really saying that my glass is almost empty.  I am saying that it is just plain empty, despite the one or two warm drops that remain, and I am telling the truth.  Similarly, when I say that these two colors are the same, I am not necessarily saying that they are perfectly or absolutely identical – just that they are the same enough to be the same, for ordinary purposes.  If I had wanted to say that they were identical absolutely, then that is what I would have said, just as I could have said that this table is absolutely, perfectly flat, if that were point that I had wished to communicate, though such a statement would have puzzled my listeners, for nothing is perfectly flat.  And if I had wanted to say that the Western Roman Empire absolutely ceased to exist in 476 A.D. I could have said that as well, though historians would surely count this as a foolish statement.

            Having allowed, then, for possible absolute or non-vague uses of many ordinary predicates, including "exists" and "identical", I still want to say that these predicates are sometimes vague in that they apply to things in varying degrees, and that, in at least some uses, they have no fixed cut-off standards in the language.  There is still something subjective or arbitrary, then, in what counts as minimally flat or empty, minimally existing, or minimally the same.         There is nevertheless great resistance to accepting this view, even limited to one sense or use of the predicates involved.  Some people see a core logical problem, usually couched as an objection to vague identity, as rendering illegitimate any vague use of these predicates.  The problem stems from Leibniz's Law, which says that two things are identical only if they have exactly the same properties.  Thus, from Iab and Fa, where I stands for "identical" and F is any predicate or open sentence, we can derive Fb – or, as we sometimes say, identicals are intersubstitutable salva veritate.  Gareth Evans (1978) has a well-known argument from Leibniz's Law to the conclusion that vague identity is incoherent.  Here is the argument.  Assume that a is identical to b, but only vaguely, meaning less than completely or determinately, identical.  We know that a is determinately, i.e. non-vaguely, identical to itself.  Therefore, substituting b for a in the true claim that a is determinately identical to a, we derive the statement that b is determinately identical to a.  But this contradicts the assumption that b is less than determinately identical to a.  It follows that this assumption cannot be true.  The whole idea of indeterminate identity is therefore incoherent. 

            This reductio argument has the general form:

 

                                                Iab

                                                Fb

                                                ~Fa                             

                        Therefore,        ~Fb  

 

where F stands for "is vaguely identical to a".  The same form of argument can be used to show that vague existence is also incoherent.  Consider anything that passes away.  Call it "a" when it still fully exists, and "b" once it has started to pass away, and let F stand for "vaguely exists".  By hypothesis, Iab, since both a and b are the thing that is passing away.  Fb is true because b is in the process of passing away, hence no longer fully existent.  And ~Fa is also true, because by hypothesis a has not yet started to pass away.  But Leibniz's Law permits the substitution of b for a in ~Fa, which then yields a contradiction.  Thus the notion of vague existence is as incoherent as that of vague identity. 

            The problem with this form of argument is that clearly, any predicate can be substituted for F, and the same absurdity derived, as long as the identicals a and b are said to differ in respect of F.  So, for any example of a thing's changing with respect to a property F, we can name the thing "a" at the beginning of the process, while it is F, and "b" once it has changed to being not-F, though Iab must be true because a and b are names for the same thing.  Then we can apply Leibniz's Law, substituting b for a in Fa, to derive the conclusion that b is also F.  Therefore, it looks like not just vague existence and identity, but also change of any sort is rendered incoherent by this form of argument. 

            What is going on most deeply, here, I think, is that Leibniz's Law is simply false when it is applied to the changeable things of the empirical world.  It is correct for pure geometry, arithmetic, and other such fields of Platonic objects; this is perhaps what motivates its application to the world at large.  But the things of the material world are things that survive changes in their properties.  Some people, most famously Plato, have concluded – from essentially the argument above – that, because change is absurd, and because things change in the material world, the material world is therefore not the real world.  It seems to me that dispensing with Leibniz's Law is a less drastic course to take. 

            There are, of course, a number of intermediate positions.  One general way of preserving Leibniz's Law for the material world is to adopt one or another form of momentarism.  One version replaces ordinary objects, like this person or that ship, with a long series of distinct momentary objects, each of which is absolutely self-identical.  The changing, time-extended objects of common sense must then be reconstructed as pseudo-objects, with the momentary objects counted as their parts or elements.  So, instead of saying at t1 that S is not bald, and at t2 that S is bald, we can say that the momentary object S-at-t1is not bald, while the different object S-at-t2 is bald, without producing any contradiction.  Alternatively, we can keep the enduring objects of common sense while momentarizing ordinary properties.  This lets us say that S is not bald-at-t1, and the same S is bald-at-t2, which is a wholly different property – hence, again, no violation of Leibniz's Law.  A third form of momentarism indexes neither objects nor properties, but whole facts, so that the pairs <[S is not bald], t1> and <[S is bald], t2> can both be held as true, while both S and baldness are persistent things.  In all three versions of this approach, then, the simple, ordinary statement "S is bald" is held to be incomplete, and this is thought to be a reasonable price to pay for the preservation of Leibniz's Law.  Analogously, we can restrict the spatial vagueness of ordinary objects by replacing them with particular points or definite regions of space or space-time, or perhaps definite sets of atoms or molecules – again, leaving the problem of how to reconstruct the things of common sense from these more precise objects.  All these approaches – momentarism, pointillism, atomism, four-dimensionalism – have some appeal, under the circumstances.  But philosophers would not be motivated to pursue any of them, I think, if we did not think something of the sort is necessary to maintain coherence under Leibniz's Law.  It would surely be better, more intuitive, if we could keep on speaking as we usually do of ordinary objects, their properties, and the facts that they make up as all extended and persistent things.

            Another basic strategy to handle change while preserving something of Leibniz's Law is essentialism (or, more broadly, the demand for specific criteria of identity, short of absolute identity in all ways).  For as long as an object exists, on this approach, it must retain all its essential properties (or continue to satisfy whatever identity criteria are specified), but its other properties, the accidental ones, can be replaced without destroying the object or turning it into something else.  Thus, Socrates can be white one day and brown the next – Leibniz's Law will be suppressed for inessential changes of this sort.  But Socrates cannot be a man one day and a horse or a table the next, for being a man is part of his essential nature.  This is a plausible sort of compromise between the seeming demands of logic and the vivid fact of change in the material world.  But the distinction between essence and accident, while clear enough in theory, is very hard to draw in practice.  Is it really true that a man could not become a horse while still being the same object – surely not the same man, but perhaps the same thing?  I do not see why such change could not in principle take place.  Certainly, there are many examples of such transformations in fiction – not real life, we know, but an apparently coherent alternative.  For example, people have been turned into a pillar of salt (Lot's wife), a tree (Daphne), and a dung beetle (Gregor Samsa)[3].  Humans have also come to be out of non-human beings, such as a puppet (Pinocchio), a statue (Pygmalion's lover), and even a deity (the Christian God).  In the real world, horses are turned into dog food, thence into dogs, and whole sharks and cows have been cut up by Damien Hirst into sculptural cubes, which I suppose could then be turned into different sculptures, or buildings, or possibly ships.  And there is at least one example of a real man, the computer scientist Ray Kurzweil, who has publicly announced that he intends to turn himself into a machine.  Once the predicted technological "singularity" occurs, he says, he plans to have his memories and personality downloaded into a very advanced computer, which will effectively make him immortal.  He won't be a man anymore, or even an organic being, but he will still be the same individual thing, Ray Kurzweil.[4]  If this sort of transformative change is even logically possible, then the essence of a man (at least, a man like Kurzweil) cannot be to be a man.  Perhaps we have no essence, as the existentialists believe.  In any case, essence as properties a thing has necessarily – on pain of not existing or no longer being the same thing – is a hard concept to apply to ordinary objects.

            A final option is the approach that I am suggesting here.  I say that we should simply renounce Leibniz's Law as fully binding on ordinary objects.  Allow that things will often change in unrestricted ways, but that nevertheless we will say it is them, those very things, that do the changing, up to some indefinite, subjective point where we no longer find it reasonable to speak of the changed things being identical to the original ones.  In other words, we should accept the vagueness of existence and identity regarding ordinary things, just as we accept the vagueness of predicates like tall or flat or large. 

            This does no damage to the world itself, for the categorical application of vague predicates depends not only on objective facts, but on our judgments as well.  This table is exactly as flat as it is, without anyone's having to judge whether it is flat simpliciter.  To say that it is flat simpliciter is just to say that it is flat enough to be called flat – that it is reasonable for us to treat it as though it were completely flat, though we know that nothing is.  And this room is exactly as large as it is, in the ways that it is large, and this is an objective fact, not something up to us.  But it is up to us to judge whether the room is large simpliciter, and this is an optional, pragmatic judgment, one that depends on our purposes in classifying rooms by size.  This solves the general sorites problem, to say the objective world can be described entirely in terms of how much each thing is each vague way, without any statements as to whether each thing is each vague way.  There is no threat of paradox when the subjective element of judgment is taken into account.  The five-foot man is not tall to anybody, the seven-foot man is tall to everybody, and at various points in-between, under various conditions, speakers of English will become disposed to start asserting that men in the middle are tall.  Thus, in a way, there really is a hundredth of an inch that makes the difference – but separately for each of us, on each occasion of judgment.  Which hundredth of an inch it is depends on context and the speaker's momentary disposition, not just on the meaning of the word "tall" and the underlying objective facts.

            The predicates "exists" and "are identical" work very normally in this regard.  Things exist in the ways they exist, to the extents that they exist, and things are the same in the ways they are the same, to the extents they are the same, all independently of human judgment.  But when we say that yes, this falling empire still exists or that this changing ship and that one are the same thing, we are imposing a decision on the facts. Somehow, the empire and the pair of ships meet our criteria and standards for existence and identity simpliciter, within the context of the present, usually informal discussion. 

            The world is in flux, but it is slow flux.  Ordinary objects are vague because they vary and change and meld into other things and come to be and pass away.  But most things that we are interested in tend to be reasonably stable and distinct through moderate extents of time and space.  If we say that something exists here and now, we mean that those of its constituents and properties that we most care about for present purposes are, around here and for the time being, in place.  If we call this and that ordinary thing identical, this is in virtue of their being the same in all or almost all of the properties that we care about for present purposes.  Should we wish or need to be more precise about the edges of things, so as to run into fewer potential contradictions in our conversations – or should we need to work around a contradiction that has already appeared – we can reduce the temporal or geographical scope of our vague objects, and speak about downtown Manhattan rather than New York City, Rome under the Antonines rather than the whole Roman Empire, or the Early Wittgenstein rather than Wittgenstein simpliciter – or further to Greenwich Village this week, Cisalpine Gaul in 150 A.D, or Wittgenstein in the Tractatus.  Coherence is thus maintained, but usually on the fly.  And there is seldom if ever a compelling reason to divide our objects all the way down into moments or points, or to time-index our facts with absolute precision – as if that were even possible.  We may also treat this or that property of an object as essential to it, or apply other identity criteria to objects of this or that sort, insofar as we determine ourselves to say the thing exists or stays the same thing only for as long as it retains that property or continues to satisfy those criteria.  This is a reasonable thing to do, often enough, especially in legal matters.  We need formal criteria to enforce the lawful differences in treatment between a child and an adult, a soldier and a spy, a living person and a corpse.  But these are still decisions that we make, as means of keeping track of things that change and vary by degrees, while maintaining adequately coherent discourse.

            What, then, finally, is left of Leibniz's Law?  I think that Leibniz's Law functions for us as a kind of ideal, a little like speed-limits on the highway.  It is true enough to say that being the same means not being different.  But things do change, at least a little, whenever anything happens, and we need to be able to describe this.  So we use the same names again for changing or extended things, implicitly asserting that they remain the same despite their being different in some way, hence not the same things absolutely.  When too much happens, when an ordinary object changes too many of its properties, it becomes better to say that the object has turned into something else, or simply that it no longer exists.  But for the most part, things stay the same for the most part, so Leibniz's Law remains true for the most part.  Over the short run, in a local situation, it defines things very well.

 

Ted Everett

SUNY-Geneseo

 

 

REFERENCES

Evans, G.: 1978, 'Can There Be Vague Objects?', Analysis 38, 208.

 

Unger, P.: 1971, 'A Defense of Skepticism', The Philosophical Review 80, 198-219.

 


NOTES



[1] I am treating these predicates as equivalent, but I should note that "the same" is sometimes used as a weak or vague alternative to "identical".  Are these two types of car (a Ford and a Mercury, say) the same?  We might say that they are the same but they are not identical, meaning that they differ only in a few details  We would never say that they are identical but not the same.

[2] Though even if we legally define a person's death as the final cessation of all brain activity, this has a vagueness of its own when looked at sufficiently closely, due to our still inexact concepts of the brain, cessation, and activity.

[3] Dung beetle is the conventional translation. I gather it is controversial just what kind of awful "vermin" Kafka intended as the result of Samsa's metamorphosis, supposing he had anything very specific in mind.

[4] In the meantime, he is taking 180 pills a day, in hopes of surviving organically for the necessary few more decades.