This essay tackles the paradox of the heap and the problem of vagueness. It's honestly a surprisingly intractable problem for me and one that, if you are interested in it regarding more than just language use, has to be dealt with. Its one of the main reasons I have become interested in process metaphysics as it is the only satisfactory solution I've found to this problem. For those uninitiated to this problem and in need of something tantalising to keep you on the page, I end up denying objects exist - including atoms and subatomic particles - in order to solve it. Thinking deeply about the intrinsic continuity of the world has only further convinced me of its truth.
The only other approach that is a live option (for me) in terms of saying something substantial about what makes the problem interesting is Tim Williamson's epistemic approach which I really like. It's just a very strange bullet to bite. I think my solution leaves an incredibly tidy state of affairs. This is adapted from an essay written for class so it might assume some prior knowledge about some analytic philosophy stuff but I'm quite proud of this one and want it documented. If nothing else, maybe read the first and last section for the problem and my solution and the source of my inflammatory title and heretical views.
Heaps of Paradoxes
The so-called paradox of the heap has and can take, many different forms. It is a puzzle often supposed to expose a flaw in the way we use language. The traditional version of it is said to be given by Eubulides of Miletus in referring to a heap of sand. What this paradox aims to point out is the vagueness present in some of our concepts. This problem seems to be present when a concept, such as ‘heap’, is insufficiently defined or measured. Usually, and why we are able to use them at all, these vague concepts are applied to clear cases in which everyone agrees, the heap and not-heap for example. But at the edges of the definition, a contentious plurality of answers arises. This contentious plurality is what we call the borderline cases. Sainsbury puts forward a definition of what it means to be a vague concept, he writes that: “a vague word admits of borderline cases – cases in which we do not know whether to apply the word or not… [that is,] vagueness gives rise to borderline cases.”[1]
The typical vagueness paradox follows the general form of first pointing to a clear instantiation of the vague concept, followed by a seemingly innocuous principle of tolerance that supposes small changes to the state of affairs are negligible to the right application of the vague concept. Applying this reasoning repeatedly, leads to the absurd conclusion that the given concept applies to something clearly antithetical to our use of the concept itself. Here is an example:
(1) Drinking a sip of beer will not make Rowan drunk.
(2) Drinking one more sip will not make Rowan drunk.
(C) Therefore, drinking 25 beers will not make Rowan drunk.
While it seems a suspicious jump to move from (2) to (C) it does logically follow if you repeat the reasoning of the principle of tolerance in (2). Even changing the quantity to make (2) more plausible we cannot avoid this. For example, reducing a (already small) sip to a drop of beer would make no difference, given we accept the seemingly true premise that a change that small cannot make any substantial difference. Interestingly this same argument can be run in the opposite direction to produce a clear contradiction:
(1) Drinking 25 beers will make Rowan drunk.
(2) Drinking 1 sip less than 25 beers will still make Rowan drunk.
(C) Therefore, drinking a sip of beer will make Rowan drunk.
What these arguments have in common is that they begin from seemingly acceptable premises and reasoning about what we mean when we talk about someone being drunk, a principle of tolerance that seems plausible, and conclusions that even by themselves seem unacceptable but also when put in opposition, directly contradict each other.[2] Thus, a paradox arises.
There are, broadly speaking, three big-picture approaches to solving this paradox. They are linguistic, epistemic, and metaphysical (with obviously much variance in individual theories). While both epistemic and metaphysical are combined with linguistic solutions (language being vague is basically a given) a popular solution posits it as an epistemic problem. The next sections present and offer some problems to this account.
Knowledge in a (Suspiciously Precise) Pinch
The epistemic approach to sorites paradox claims that vague concepts do draw sharp boundaries, there is a fact of the matter when a ‘heap’ becomes a ‘not-heap’. In claiming this, the second premise of the paradox (2), the tolerance principle, is just false. It is false because at one discrete instance of the small change in state of affairs where a vague predicate is said to be applied, definite assertion of whether it applies or not is warranted. In other words, it is always either true or false, under the epistemicist scheme, whether or not some sand is a heap.[3] To return to the drinking example, there would be a precise sip in which I would go from sober to drunk. Another example is if, in measuring a ‘pinch’ of salt, I grabbed one nanogram over the sharp boundary, it would cease to be accurately described as a pinch of salt. The question then arises, how can we come to know these boundaries?
The second part of the epistemicist account is that we are ignorant of where these sharp boundaries lie, and this ignorance is due to inexact knowledge. This does at least partly explain why we find borderline cases compelling in the first place. By this view, we are unable to decide one way or the other because it is not actually possible for us to find boundaries at all, even in principle. This is because of our limited cognitive, perceptual, and psychological capacities as humans. Were we ever to pick out the ‘true’ smallest instance of a heap of sand made up of n grains, it is easy to imagine a counterfactual in which we would pick out a heap of sand made up of n-1 grains with the same confidence as before. Therefore, we could never pick out sharp boundaries and know that we know them (viz. not by accident). Note that this is not a deficiency in our language use but a lack of knowledge. This argument for why it may be that we cannot find these sharp boundaries may be the same reason the paradox is confounding in the first place: even our senses and experiences are organised such that the principle of tolerance seems plausible.[4]
Thirdly, the epistemicist, to give a plausible account, must show what determines sharp boundaries of vague concepts. The vagueness of concepts, and therefore the knowledge of their right applicability arises from language use. [5] As a language develops and is used among a group of competent language users, vague concepts are used in order to express common thoughts, and an absolutely precise boundary emerges from that use. These sharp boundaries are not just the survey of all past uses of said vague concept. They are the product of all dispositions of those aware, or potentially aware, to assent or dissent to their use in a particular state of affairs thus producing a familiar public sense available to each. This sense is volatile in that it can change reference depending on each use, but each use does fix itself precisely upon a certain referent.
Rejecting the Epistemic Account
There are two objections I give here of the epistemic theory that are variations on the same theme. That is, the epistemicist account simply does not give a satisfying account about how the sharp boundaries emerge given the way we seem to use language.
The epistemicist claims that use of a vague concept, while volatile, fixes precise bounds. The proposition ‘Rowan is drunk,’ for example, might refer to Rowan’s blood-alcohol level being n. This just seems implausible. Williamson explains our ignorance of these bounds through appeal to margin of error principles but does not give a satisfying account of how use of language fixes such perfectly calibrated boundaries. Nothing about our language use intuitively suggests that this would be the case, in fact, it seems to be the opposite, more in line with the linguistic theorist. Some vagueness in sense seems actually to be a feature of language use. For example, our use of something like “go roughly there” coupled with a gesture surely does not mean we are referring to a strictly bounded zone but actually that we are indifferent to where you go given its roughly in that area.
Schiffer thinks that a ‘happy-face solution’ is required to solve any paradox. For the sorites paradox a happy-face solution is one which not only tells us which of the sorites propositions is false but also tells us why the paradox looked to be true.[6] The epistemicist would argue that it seems true at first because our inevitable ignorance of borderline cases leads us to take the principle of tolerance as plausible when it is not true. There are many things we could not know in principle but have good reason to believe exist regardless. Schiffer argues, and I agree with him, that this explanation is unsatisfying. There are scientific claims for example, that also seem in principle unverifiable, but we have compelling reasons to think they are true. What is incredible about the epistemic account is that it offers no compelling reason to believe that use alone would determine ‘the facts’ about vague concepts. Especially considering our use seems to indicate there is no fact of the matter (as per the last point).
A more common-sense account of vagueness can be found in linguistic theories. They are more in line with our intuitions about how malleable language use seem to be. But they have their own problems in answering why vagueness seems like a problem.
Speaking in Tongues
This section will argue why purely linguistic (as opposed to epistemic or metaphysical) solutions to sorites paradox are unsatisfactory. Without embracing either an epistemic account or a metaphysical account, it fails to explain what is interesting or why vagueness arises. That we seem to be deprived of knowledge or information about the world from vagueness.
The most popular form of the linguistic approach to solving sorites paradox is supervaluationism. Supervaluationism posits that vague terms are incomplete in their meaning. Therefore, in borderline cases, they are incapable of determining whether or not they apply to a particular state of affairs. It is then said that there is a plurality of appropriate ways, ‘sharpenings’, in which the expression could be made precise (each a sort of disambiguation). Truth of when to apply the vague word becomes the instances in which it is true on all sharpenings (supertrue), while the false is those false on all sharpenings (superfalse). Sentences that apply vague predicates to borderline cases in which only some sharpenings are true, are deemed to be neither true nor false.
Clever as this solution is, in denying there is epistemic or metaphysical vagueness, it seems to do nothing other than offer an analysis about the way we use language.[7] Take the earlier example, in the form:
(1) Rowan is drunk.
The supervaluationist must begin by saying there is no determinate truth value in borderline cases of drunkenness, it is neither true nor false. Next, they would outline all acceptable sharpenings highlighting the sharp boundaries separating the borderline numbers of intoxicating sips from the supertrue and superfalse numbers. The question they must answer now is: what makes the borderline cases vague? Holding onto their commitment that there is no determinate truth value means it cannot be epistemic, that is, borne out of ignorance of the facts. Not even a God could shed light on the matter.
What this means is that there is no fact of the matter whether the predicate “drunk” is appropriately applied to Rowan at a borderline instance of drunkenness. This seems intuitively plausible – language is not perfectly fine grained, drunk is not a well-defined word, it would be foolish for someone to fix a sharpening as the one definition. But moving from the realm of language for a moment, what about Rowan? Does this not mean that there is no fact of the matter about whether Rowan is drunk, whether Rowan exemplifies drunkenness? Is Rowan vaguely drunk? This would be metaphysical vagueness, which supervaluationists also reject. The question then becomes why would a supervaluationist be motivated to use any predicate in the first place? It either has to be because there is truth of the matter about its applicability (semantic decision, or community activity for the epistemicist, resulting in knowledge) or because there is a some existential relationship between the predicate “drunk” and “Rowan” as they have to know how to apply it in paradigm cases (where surely they are saying something about the world). In other words, what are the standards for an effective utterance if they deny both? If they do deny both, I argue the next point follows.
In denying both epistemic and metaphysical accounts, supervaluationist and other linguistic theories seem to both be positing a world made up of sharply bounded objects but referring to it with predicates necessarily empty of any referent. They seem to be saying, for example, there is a supertrue concept of “Mount Taranaki” but no fixed actual mountain it can be applied to (metaphysical) nor any standard to which it could applied against (epistemic). If this is right, holding a purely linguistic theory is no different in form from a nihilism about vagueness as it essentially just says there are no heaps in the world just that we use the word to communicate (somehow).[8] What then is the point of supervaluationism – what does it mean to be supertrue of something if it is not about the thing, nor something that could be known? This does not seem right.
My view is that there is truth in saying that there is vagueness in language, but it does not require a theory of supervaluationism, just that we can communicate is enough.[9] (I think a Wittgensteinian shrugging of the shoulders and metaphysical agnosticism is all one needs.) It does on my view though, require some either some substantial metaphysical or epistemic explanation for why vagueness arises, and why, if it is just semantic indecision, it appeared to be so enticing.
Vagueness as Process
So far, I have outlined why the epistemic account, while tidy, is not convincing to me because it does not seem compatible with how language is used. Before that I gave an argument that linguistic accounts may either be implicitly relying on metaphysical vagueness or in denying that, it gives a solution that is unsatisfactory – it does not end up with a proper explanation of why vagueness arises. As per the simplified version of the approaches I mention, this leaves us with a metaphysical approach. A potential solution, and the one I tentatively offer, is that not only is language vague, but the world is.
I conjecture that: (1) the world as a whole is vague and (2) our language therefore fails to capture the world exactly, which is (3) why we find the sorites paradox compelling. These ideas are heavily drawn from philosopher Henri Bergson’s text An Introduction to Metaphysics.
If linguistic vagueness is also metaphysical one might posit that we live in a world of objects. Mountains begin and end at a certain millimetre. This seems unlikely for the same reason the epistemicist account is unsatisfying; any sorites series can reduce these definitions to absurdity. It seems much more likely to me that any concept such as mountain is a human projection of form upon matter. The concept mountain really is semantic in the sense that we have constructed its meaning in use; it is a purely pragmatic labelling of the world that allows us to inhabit and communicate it to each other. Linguistic vagueness usually stops here and that is why it fails to answer why we find sorites paradox interesting. I argue this segmentation into discrete objects is useful and perhaps a necessary condition of human action, but ultimately an illusion. Any metaphysical partitions drawn in nature seem ultimately to be arbitrary for the same reason the predicate ‘drunk’ is. Behind it all the actual world is vague, but I mean this in a radical sense. I do not even think there are objects, let alone 'vague objects', at all.[10] The world is wholly vague. It is, at bottom, a movement - change is absolutely ubiquitous. Even what seems like the most elementary constituents of the world, atoms, and subatomic particles, are not the final divisions. Think about it: there is no empty space that true individuation would require, everything is relational. They are highly predictive and useful models about the structure and relations of things in the world as abstracted from something more basic. There is always the outstanding question: “what is an atom, a quark, a photon?” We can only say what they do as we have constructed them.
This picture of the world I think explains why vagueness in language is compelling. We are first situated temporally in this vague and constantly moving world, we have access to it and wish to communicate our experience with others. Language is then an abstraction from it. Drawing on Bergson, it is a form of analysis which he defines as:
a translation, a development into symbols, a representation taken from successive points of view from which we note as many resemblances as possible between the new object which we are studying and others which we believe we know already[11]
In abstracting from the world in-itself as we experience it, language divides the world into simple idealised discrete points that could never in principle actually capture it nor anything real. If you imagine the Louvre in Paris, say someone took photos of the building from all mathematically possible angles, it will never compare to simply walking through it. In other words, the world is not reducible to any set of representations. If you accept these points it seems to me an entirely satisfactory account of the existent of vagueness. Language (and even scientific models) as a tool of representation will necessarily fail to be exact, always an approximation. It will always only ever be a picture of the Louvre, never the Louvre itself.
In terms of the original sorites argument, this account embraces the linguistic theorists’ view, vagueness is just semantic indecision. The things that we label ‘drunk’, ‘red’, ‘heap’, ‘mountain’, are purely pragmatic human ways of carving up the world. The benefit of this account is that it gives a positive metaphysical answer by embracing what I partially accused linguistic theories of – nihilism about vagueness in the form of a vague world with no real objects (in my qualified sense). It also answers why that kind of vagueness could ever occur in the first place if it is not a matter of knowledge – because language is an imperfect abstraction that attempts and fails to represent that world in discrete packages. The paradox seems falsely tempting at first because we are habituated into our cherished world of objects when in reality our experience of the world is pure temporal continuity. The repeated application of the principle of tolerance premise is justified only insofar as it is pragmatic. Metaphysically, there is no basis for carrying it out – there are no heaps.
That which is commonly called a fact is not reality as it appears to immediate intuition, but an adaptation of the real to the interests of practice and to the exigencies of social life. Pure intuition, external or internal, is that of an undivided continuity. We break up this continuity into elements laid side by side, which correspond in the one case to distinct words, in the other to independent objects...For the living unity [the universe], which was one with internal continuity [consciousness], we substitute the factitious unity of an empty diagram as lifeless as the parts which hold it together.[12]
That there are, in a sense, multiple objects, that one man is distinct from another man, tree from tree, stone from stone, is an indisputable fact; for each of these beings, each of these things, has characteristic properties and obeys a determined law of evolution. But the separation between a thing and its environment cannot be absolutely definite and clear cut; there is a passage by insensible gradations from the one to the other: the close solidarity which binds all the objects of the material universe, the perpetuity of their reciprocal actions and reactions, is sufficient to prove that they have not the precise limits which we attribute to them. Our perception outlines, so to speak, the form of their nucleus; it terminates them at the point where our possible action upon them ceases, where, consequently, they cease to interest our needs. Such is the primary and the most apparent operation of the perceiving mind: it marks out divisions in the continuity of the extended, simply following the suggestions of our requirement and the needs of practical life.[13]
Notes
[2] Another thing to note is that the conclusions of these arguments do
not necessarily have to be at fixed bounds. It could be concluded from the
first argument that one could never get drunk drinking beer. From the second
you could conclude that you could get drunk from drinking zero beers, though of
course this is actually possible.
[3] This does
not mean there cannot still be borderline cases where an ordinary speaker will
still be unsure of whether to assent to the vague term or not.
[4] What I mean here is
that we, in our experience, do not think small changes to a state of affairs is
important to whether we apply a vague concept.
[6] Schiffer, Stephen. "The Epistemic Theory of Vagueness." Philosophical
Perspectives 13 (1999): 481-503.
[7] Merricks (2001) has inspired the following argument.
[8] There are of course
complications here, some (most?) supervaluationists would hold that there is
some precise language and therefore only a partial nihilism; about vague terms
but perhaps not about mathematics or identity statements for example. Another
explanation that could be given is that the world is made up of precise objects
but language will always fail to capture them precisely, here again they do not
give up precision but rightly point out the limits of language. This is all to
say that it would not be a complete nihilism but certainly one about vague concepts.
[9] What I mean here
anticipates my next point. Language is vague because it is an inexact
representation of the world and ultimately only a pragmatic project. I cannot
see how supervaluationism would contribute to communication, in fact, by trying
to make terms precise we lose a feature of language – that it can expand to
mean different things (maybe Lewis would agree here “no one is fool enough”
because it would frustrate language use). It is superfluous if you accept that language
is pragmatic alone.
[10] I mean this
metaphysically, not epistemologically or pragmatically. A chair exists as an
object in the sense that it helps us achieve our goals and we can successfully
convey meaning while using the word in communication with others truthfully but
it is not an Aristotelian substance or something, it is just a vague hunk of
matter with human meaning imposed on it.
Bibliography
Bergson, Henri. An Introduction to Metaphysics. Translated by T. E Hulme. New York: G.P Putnam’s Sons, 1907.
Bergson, Henri. Matter and Memory. Translated by Nancy Margaret Paul and W. Scott Palmer. New York: Dover Publications, 1912.
Hyde, Dominic, and Diana Raffman. “Sorites Paradox.” The Stanford Encyclopedia of Philosophy. Accessed March 17, 2020. https://plato.stanford.edu/entries/sorites-paradox/.
Merricks, Trenton. "Varieties of Vagueness." Philosophy and Phenomenological Research 62, no. 1 (2001): 145-157.
R.M. Sainsbury. Paradoxes. 3rd Edition. Cambridge: Cambridge University Press, 2009.
Schiffer, Stephen. "The Epistemic Theory of Vagueness." Philosophical Perspectives 13 (1999): 481-503.
Williamson, Timothy. Vagueness. Routledge, 1994.
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