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Abstract for:

Aesthetic Sorites: A Peircean Resolution

            Peter Unger takes the sorites (from the Greek for heap, soros) paradox to entail a radical skepticism and nihilism.  Arthur Danto, in The Transfiguration of the Commonplace, presents an example of a statue of a chained cat that seems be a kind of aesthetic sorites and it seems to support Unger’s conclusions about skepticism and nihilism vis a vis the art versus reality question.  I use Peirce’s Logic of Relations as a way of addressing sorites paradoxes, in general, and Danto’s chained cat in particular.



Aesthetic Sorites: A Peircean Resolution


            Eubilides of Megara is taken to be the inventor of the sorites paradox (from the Greek for heap, soros).  The paradox is very simple, yet confounding.  It can work in two directions.  If there is a heap of sand and one grain of sand is removed from it, it is still a heap of sand.  The paradox is that there would seem to be no condition under which the removal of one grain of sand would ever convert a heap of sand into a non-heap, yet, at some point, there is no longer a heap of sand.  The paradox also works in the other direction.  One grain of sand is not a heap.  If one adds one grain of sand to the first, it is still not a heap.  There is no condition under which the addition of just one grain of sand will transform what is not a heap into a heap, and yet, there is a heap.  This paradox seems to challenge the validity of basic principles of logic like the validity of the function of modus ponens or even of the validity of proof, in general.[1]  This paradox is generated by the vagueness of the concept of a heap.  It just so happens that most of our predicates are susceptible to this kind of vagueness.  A swizzle stick is.[2]  A frog is.[3]   A color is.[4]  A number is.[5]  Aristotle’s doctrine of the mean is.[6]  A person is.[7]  The Mona Lisa can generate a sorites paradox.[8]  The significance of this paradox can, as it were, explode in a person’s mind so that it leads a philosopher like Peter Unger to conclude that Eubilides is one of the greatest (and most underappreciated) philosophers of all time and that the inevitable conclusion of considering this paradox is a radical skepticism and nihilism.[9]  I agree with Unger that the paradox does have some real epistemological and ontological implications, although I come to somewhat different conclusions than Unger. 

First, however, I would like to examine sorites-prone phenomena in the aesthetic realm.  By sorites-prone phenomena I mean those phenomena that are susceptible to engaging sorites-like logical conundrums.  I will argue that any phenomenon that seems to call forth a judgment based on a basic dyadic opposition will be a sorites-prone phenomenon.  Aesthetics is rife with such judgments.  Some basic dyadic oppositions that are central to aesthetics are, for example, the following: art vs. reality, art vs. craft, artwork vs. material counterpart, form vs. content, beautiful vs. not beautiful, beautiful vs. sublime, genius vs. mediocrity, original vs. derivative. 

            A beautiful example of the sorites-like dangers that these dyadic oppositions can get us into is presented by an example, given by Arthur Danto in his book The Transfiguration of the Commonplace.  It is a statue of a chained cat.  It is a somewhat constructed example because the example is based on an artwork that is in all probability not the artwork of the example.  That is, Danto takes a statue of a cat that happens to be chained to a railing, probably for security reasons, it is located on Manhattan, for a statue of “Chained Cat,” i.e., the chain for Danto’s purposes, is an intentional part of the artwork, put there by the artist, not a accidental property of reality put there by a security person.  In any event, here is Danto’s description of how this artwork works: 

it is…a sculpture of a chained cat, one end of which is wittily attached to a piece of reality (a chain from art to reality is what we have been looking for).  Of course what we take to be a bit of reality can in fact be part of the work, which is now a sculpture of a cat-chained-to-an-iron-railing, though the moment we allow it to be a part of the work, where does or can the work end?  It becomes a kind of metaphysical sandpit, swallowing the universe down into itself.[10]

A “metaphysical sandpit,” indeed.  What has called forth this metaphysical sandpit?  It is the logic of the sorites that calls it forth.  That is, just as a heap of sand remains a heap of sand after the removal of one grain of sand, and, by logic, even after the removal of all the grains of sand, the artwork remains an artwork even with the addition of a railing, or the building to which the railing is attached, etc.  Of course, the chain is really an unnecessary addition since every artwork is connected to reality by its base or a nail and wire, or some screws in a wall.  It is as though a non-artwork piece of reality can be transformed into an artwork, part of the “artworld,” by a simple change of one’s mind.  This is part of Danto’s game with the not “Chained Cat” “Chained Cat.”  That is, he is demonstrating not just how a piece of reality can be transformed into a piece of art by a kind of change of mind on the part of the viewer, but that an artwork itself can be changed from one artwork into another by a similar change of mind.  These are dizzying conclusions.  Are they right?

Bertrand Russell introduces an interesting term for the vague region that, he argues, all predicates possess, he refers to it as the ‘penumbral’ area.[11]  The penumbra is the area of maximum vagueness for any term.  A heap of sand is not apparently a problematic phenomenon, but at some point it will become problematic, people may disagree about whether a particular aggregate of sand grains constitutes a heap or not.  Between the tadpole and the frog, there is a penumbral state that is more than a tadpole but not quite a frog.  Between orange and red there is the penumbral spectrum of reddish-orange or orange-ish-red.  Between the cat and the rail falls the shadow.  What is needed is a logic for guiding us through these penumbral areas. 

            Peirce’s metaphysics and semiotics provides just such a logic of the penumbra.  First of all, nouns, i.e., many predicates, work, but can be misleading.  For Peirce, every noun contains an implicit verb component.  That is, there are real individual things, and they are revealed through their reactions to other real individual things, but this very description of individuation depends on an activity, so that a table is an individual table only insofar as it is reacting to the floor, the surrounding air, the chair, the book that is sitting on it.  To capture this verb component, one might say, somewhat unnaturally, the table tables.  To capture this on-going activity one might switch to the gerund form, the table is always, as long as it is a table, tabling.  In the gerund form of any noun, the sorites paradox dissolves.  Two grains of sand may be already ‘heaping,’ even if they are not yet quite a heap, and so is the heap of sand heaping, still.  It is also the case, however, that two grains of sand may not be heaping, but only maintaining their two-ness, just as a fifteen year old boy may not be going bald at all, while a twenty six year old man, with exactly the same number of hairs as the fifteen year old, may, in fact, be balding.  So, in other words, this verb-al component of nouns suggests that the meaning of these nouns has its esse in futuro, as Peirce says.[12]  That is, the appropriate use of a noun may need to include the way the noun anticipates a future condition. 

This does not mean that we must convert all of our nouns into gerunds in our everyday language, although there are, apparently, languages that function in that way in their ordinary forms.[13]  To speak of a table as simply a table, as in, “put the book on the table,” is fine.  But, just as we mostly use Newtonian physics, but know that Einstein’s relativity and quantum mechanics provide a much more detailed and precise description, perhaps it would be useful to keep in mind, as a kind of habit, that nouns always contain a verb-al component.  This idea of the verb-al component of nouns will be quite helpful when we find ourselves in the penumbral area of a sorites-like situation.

            So, what is the logic that is operative here?  First of all, it is, as Peirce says, a Logic of Relations.  That is, it is a logic of things in relation to other things (with which they react).  These things involve singulars (identified by their reactions with other singulars) and generals (which determine the nature of those reactions).  For Peirce, we directly perceive generals in the relations of particulars and we directly perceive relations between things.[14]  What we are looking for when we are looking for what something means is the dynamic system of relations that obtains among any group of particulars.  Logic, for Peirce, is a normative science; it is the normative science of drawing conclusions from premises, or, in other words, of making arguments.[15]  Hence there are good and bad arguments.  So the question becomes, what is the sound argument contained in any particular situation.  The penumbral area is the place of maximum vagueness and of maximum import in our thinking.  That is, our construction of the argument for this situation, our assessment of what the true nature of this dynamic system of relations is, will be proven good or bad by how well I negotiate the penumbral region.

            So the appearances, viz a viz sorites-like situations, are misleading.  It seems like the question should be, what general should apply to a given particular?  That is, is this aggregate of sand a ‘heap,’ or, in Danto’s example, is the railing ‘art’?  From this Peircean perspective, however, the question now becomes, what is the system of relations in which this particular can be situated, and in what direction is that system developing?  This way of posing the question undoes the contradiction producing, reductively binary, approach to the problem.  It also preserves the deeper structure of our language, its ability to match the evolving character of the world and our experience in it.  This way of putting the question sets up a perspective on the problem that admits of a real solution to the problem, that is, a solution that will give us something new to work with to help us find our way to act, a way to find some traction in the apparently tractionless penumbra.  This, for Peirce, is always the goal of pragmatism.

            The nature of the penumbra is similar to the nature of a sign or the nature of the world.  For Peirce, the world is continually evolving.[16]  The concept of evolution has two implications.  One is simply that things, systems of relations, change.  The second is that this change is governed by or obeys general principles that allow for some degree of predictability.  What things mean, what the meaning of any particular system of relations is, is determined by the nature of its evolution, by how it is evolving.  Another way to say this is that meanings emerge.

            The idea that meanings emerge implies two corollaries: first, that the meaning of a situation is not immediately given, is not immediately there; and second, that it will become apparent with time, that it will emerge.  One must be attentive and figure out what the relevant relations (relevant in terms of the situation’s development, and relevant in terms of the consequences that will be most important for the observer), but if one is, the meanings will emerge.  Furthermore, this process of emergence is continuous and, ultimately, infinite.  In Peircean terms, every representamen, that is, every sign, has an interpretant; and every interpretant becomes a representamen for a further interpretant.  Our work as thinkers and doers in this world is to continually move from representamen, the signs, to their interpretents, so that we know what is going on and what to do next.

             Understanding will consist of being able to anticipate the emerging  meaning, the emerging pattern.  “Getting it” will mean seeing the, as it were, reasoning of the emergence, seeing it as intelligible, which just means seeing the pattern of relevant relations.  One will achieve this understanding by, at first, making guesses about what may emerge next (what Peirce calls abduction).  That is, you will develop a theory (a kind of intuited guess) for the directionality of the emergence which will predict the direction of the emergence’s development, and then your theory is tested, in time, by whether the emergence develops as your theory predicts or not.  The theory is revised with new abductions until one at arrives at an explanation that seems to consistently match the emerging relations of what you are examining.  Emotionally, this process will be accompanied by surprise, confusion, puzzlement, and then delight and satisfaction.

            How, however, does one even begin to read a sign in the dark, that is, in the penumbra?  It is a cognitive darkness that I refer to here, and the answer is, somewhat ironically, by not thinking.   That is a bit misleading, but the point is that, as in Kant’s second moment of the experience of the beautiful, we find that we have no concept to fit our experience.  What we are looking for in the penumbra is an emergent pattern, an emergent intelligibility, an emergent unity, and unity, for Peirce, is something we feel before we think it.[17]  An abduction will begin with an sense of unity that will call for an “emotional interpretant.”[18]  This is as true, for Peirce, in mathematics as it is in aesthetics, so that Peirce claims that the validity of a new theorem is felt before it is proven,[19] which is a way of saying that mathematics is also aesthetics.

            This feeling of a unity, it seems to me, emerges via they dynamic of an indexical sign.  A weathervane is an example of an indexical sign.  The sign is physically determined by what it is a sign of.  We feel the emergent unity or intelligibility or pattern when something in the system of relations gets registered in the system of the neurons in our brain.  One literally ‘sees’ the emergent pattern all of a sudden as one may suddenly ‘see’ the rabbit in the duck/rabbit.[20]

            This process can occur with, for example, a completed painting.  Everything in the painting is still active, is still verb-al in the sense that each part of the painting reacts with each other part and the sum of these reactions is a system of relations that has a logic, that is, something intelligible about it.  We respond to this constant activity with our own, the saccades we make with our eyes and with our mind as we try to trace out the lines of activity in the painting, as we try to figure out via developing theories, what relations may obtain and to what they may be leading.  The meaning of the painting will not be immediately given, but will emerge given our attentive and thoughtful examination of it according to the logic of relations.  Our activity must include a kind of receptivity, just as one must expose one’s face and turn toward the wind to feel the breeze.[21]  When on does this, relations will begin to emerge, abductions will pop in to one’s head.  Abductions are followed by deductions, tested inductively, and deductions function, most frequently, as erasure.[22]  That is, as we develop an abductive theory we test it deductively by seeing if what we would be led to expect of the painting because of our theory about it in fact holds up.  Those aspects of our theory that do not hold up must be ‘erased,’ and our theory revised.  As our theory is refined, surprise gets transformed into satisfaction.  The same process occurs in listening to music or reading a poem or looking at a statue of a cat.  Understanding an artwork means seeing what makes it intelligible, it means being able to, more or less, anticipate the ways it will develop in the future.

            What does this Peircean analysis of a logic for the penumbra do for aesthetic sorites, in general, and Danto’s chained cat, in particular?  Aesthetic sorites are metaphysical sandtraps only if one approaches them with the assumption that they represent a problem that is essentially static and that must be resolved dyadically: either art or not art, original or derivative, sculpture or painting.  What it suggests is a particular attitudinal stance, an attitude of receptivity as well as thoughtfulness, an expectation that the meaning of an artwork will not be given immediately, but will emerge, and that each meaning that emerges will become material for further meanings.  Danto’s interpretation of the chained cat is brilliant, and the artist may very well have meant to indicate an art-versus-mere-real-thing sorites with her or his artwork.  That, however, is not a metaphysical sandpit, but an interpretant that becomes a representamen that will call for a further interpretant.  In other words, the cat is, as it were doing art, and we are doing art in responding to it.  To think that the work ends in a metaphysical sandpit is to think too much and fails to trust that further meanings will emerge, which is, after all, to trust our feelings, which can be a difficult proposition for a philosopher.  According to Peirce’s Logic of Relations, meanings will emerge, our trust is warranted.  What we should trust, however, is not that a decisive certainty will be achieved, but that a way toward more and more intelligibility will be found.  This may be a far cry from Cartesian knowledge, but neither is it Unger’s radical skepticism and nihilism.  It is a form of ontological and aesthetic realism, which is to say, it posits that there really are meanings given.  In the end, the sand is either heaping or it is not, and it is a statue of a chained cat or it is not, whatever we might think about it.


[1] Michael Dummett, for example, says in his essay “Wang’s Paradox”:  “The only alternative left us…therefore appears to be to deny that , in the presence of vague predicates, as argument each step of which is valid is necessarily itself valid.  This measure seems, however, in turn, to undermine the notion of proof (= chain of valid arguments), and, indeed, to violate the concept of valid argument itself….”   Michael Dummett, Truth and Other Enigmas (Cambridge: Harvard University Press, 1978), 252.

[2] Peter Unger, “Skepticism and Nihilism,” Nous, Vol. 14, No. 4, Special Issue on Epistemology.  (Nov., 1980), 519-21.

[3] James Cargile, “The Sorites Paradox,” The British Journal for the Philosophy of Science, Vol. 20, No. 2.  (Oct., 1969), 193-4.

[4] Delia Graff, “Phenomenal Continua and the Sorites,” Mind, Vol. 110, No. 440.  (Oct., 2001), 908-10.

[5] There are at least two ways in which numbers can engage sorites-like paradoxes.  One is Wang’s paradox, discussed by Michael Dummitt in Truth and Other Enigmas (248-268).  Wang’s paradox is simply o is small; If n is small, n + 1 is small: Therefore, every number is small (Dummett, p. 250).  There is also a sorites like paradox with respect to numbers, as I read it, in Plato’s Phaedo when Socrates is describing the mysteries of 1 + 1 = 2 (Phaedo, 96e-97b).  That is, a piece of chalk on the table is one piece of chalk (but not the only piece of chalk in the room).  When some other piece of chalk in the room begins to approach the original single piece of chalk, the original piece of chalk is still only one piece of chalk.  The distance n - 1 inch, does not make the independently single pieces of chalk two.  Neither does n – 2 inches, and the sorites paradox says they will never be two, and yet, at some point, we want to say, there are two pieces of chalk there.

[6]Jon Moline, “Aristotle, Eubilides and the Sorites,” Mind, New Series, Vol. 78, No. 311.  (Jul., 1969), 394-97.

[7] Unger, 523ff.

[8] Laurence Goldstein, “The Sorites as a Lesson in Semantics,” Mind, New Series, Vol. 97, No. 387.  (Jul., 1988), 449ff.

[9] Unger, 518.

[10] Arthur C. Danto, The Transfiguration of the Commonplace: A Philosophy  of Art (Cambridge: Harvard University Press, 1981), 102.

[11] Bertrand Russell,

[12] Charles S. Peirce, The Essential Peirce: Selected Philosophical Writings, Volume 2 (1893-1913), edited by the Peirce Project (Indianapolis: Indiana University Press, 1998).  Peirce says of the word “man,” for example, “The word itself has no existence although it has real being, consisting in the fact that existents will conform to it”  (“Sundry Logical Conceptions” (1903), 274).  He speaks of laws of nature as having “a sort of esse in futuro” ( “On Phenomenology” (1903), 153)

[13] Peirce says, “Whenever I speak of a term I always mean a rhema.  The difference is that a term is the equivalent of a common noun and cannot form the predicate of a proposition unless a verb is inserted, while a rhema contains the verb within itself….In primitive languages there is strictly no such thing as a common noun.  Old Egyptian and Arabic are instances.”  Essential Peirce, “The Nature of Meaning,” (1903), 220.

[14] Essential Peirce, “The Nature of Meaning,” (1903), 223-4.

[15] Essential Peirce, “The Nature of Meaning,” (1903), 218.

[16] For a very helpful account of Peirce’s concept of evolution see Doug R. Anderson, Creativity and the Philosophy of C. S. Peirce (Boston: Martinus Nijhoff Publishers, 1987), 85-121.

[17] Essential Peirce, “The Seven Systems of Metaphysics” (1903), 190-94.

[18] See Felicia Kruse, “Emotion in Musical Meaning: A Peircean Solution to Langer’s Dualism,” Transactions of the Charles S. Peirce Society, Fall 2005, Vol. 41, No. 4, 269-70.

[19] See Shannon Dea, “’Merely a Veil Over the Living Thought’: Mathematics and Logic in Peirce’s Forgotten Spinoza Review,” Transactions of the Charles S. Peirce Society, Fall 2006, Vol. 42, No. 4, 511.

[20] For Peirce, we literally see both generals and relations.  I use scare quotes only because when looking for the rabbit in the duck/rabbit, for example, there is a sense in which one sees everything that is there, but one cannot put it together in a way that brings forth the pattern of the rabbit.  The same problem can happen with a joke one does not get or a painting or a piece of music that one does not get.  The rhythm in a Jackson Pollock drip painting takes some time to see, for example, even though there is no part of the painting that is actually hidden.

[21] Peirce uses the metaphor of a breeze in his description of the feel of the way our thoughts are directed in musement, “’Enter your skiff of Musement, push off into the lake of thought, and leave the breath of heaven to swell your sail.’” Essential Peirce, “The Neglected Argument for the Reality of God” (1908), 437.

[22] Essential Peirce, “Sundry Logical Connections,” (1903), 288.