Quantum Heap—The Sorites Paradox

In this paper, I will address an ever-present dilemma inherent to the issue of vagueness: when does a heap of sand become not a heap of sand, does Unger exist, where did Theseus’ Ship go? We seem to think that we know when a heap is not a heap and when it is, or a ship not a ship, but the transition between being and not being is still unclear and under much debate. Peter Unger goes as far as to doubt his entire existence, as well as the existence of composite objects in general.[1] I will explain the idea of vagueness as well as posit an argument as to why we cannot seem to find the elusive point of transition within the haze of vagueness, as well as answer some pertinent objections to my view.

There is a certain appeal to being general and vague. It allows us to make a commitment without being specific. Take, for instance, a request for a kind of desert. Being general allows the specificity a type of desert (ice cream, cookies, cake), without committing to a specific one (mint-chocolate chip, oatmeal and raisin, champagne cake). We take risks when being vague, however. In the desert scenario, we might get a type ice cream we do not like, or might even be allergic to. But there is a kind of satisfaction when we receive something we like unexpectedly, more than receiving something that was specified and known beforehand. It is what makes mysteries so appealing, popping the question so nerve-wracking, and other mom’s meatloaf so portentous.


Roy Sorensen defines vagueness as, “… the possession of borderline cases”—a case that is unclear whether it is true or false (i.e. when a heap is not a heap). Included within borderline cases are more borderline cases: At what point does a case become borderline? This higher order vagueness[2]—‘vague’ is vague—seems to reach into infinity. Where exactly the ‘borderline case’ (Ba) begins is also borderline (Bb). This second ‘borderline case’ is also unclear as to when it becomes borderline (Bc), and so on. It might seem that either: 1) Everything is vague and nothing is definite. 2) Nothing is vague and everything is definite. However, neither choice seems to entirely coincide with how we experience the universe. We perceive definite things (that which can be responded with ‘yes’ or ‘no) and we perceive vague things (that which can be responded with ‘yes’ and ‘no’). This conflict is most prominent when attempting to ascertain a generality and a specificity at the same time (i.e. pinpoint when the ‘borderline case’ becomes borderline), and also arises in linguistic ambiguity—right (direction) and right (correct).

There are at least four schools of thought regarding vagueness: Many-valued Logic (MVL), Supervaluationism, Subvaluationism, and Contextualism (Sorensen). MVL posits that, “borderline statements are assigned truth-values that lie between full truth and full falsehood” (Sorenson). This view is widely used in psychology, in which truth-values are set at 0 (absolute falsehood) and 1 (absolute truth). Higher order vagueness is inherent in MVL, since there are an infinite amount of numbers between any two numbers of unequal value.[3] Where MVL falls short of truly grasping ‘vagueness’ is by attempting to establish definite truth-values for everything, including when a heap is not a heap. Supervaluationism eliminates the problem encountered by attempting to stipulate a definite value for a generality, by stating that there are no truth-values—“a demonstration that a statement is not true does not guarantee that the statement is false” (Sorensen). Subvaluationism views statements as being true (tautology), false (contradiction), or both true and false (borderline cases). Both sub and supervaluationism appeal to certain aspects of ‘vagueness,[4]’ yet fail to entail it completely. Contextualism suggests, ‘truth’ and ‘falsehood’ rely on the context of a statement. Thus, as the context of a statement changes, so too does the truth or the falsehood of the statement. It may seem that contextualism eliminates generality, which would pose a problem to this view; however, this can be answered by making the context for the situation general. For this paper, I will focus on Contextualism, since it seems to better agree with the theory of relativity, which I will attempt to explicate shortly, which would fundamentally make it a stronger claim.

Uncertainty Principle

The seemingly impossible hurdle is in solving the sorites paradox[5]. In order to achieve this, I feel inclined to include a small amount of quantum mechanics; namely the uncertainty principle[6]. This principle can also be applied to help understand why an arbitrary line between certainty and vagueness cannot be found. Just as we can know the position or velocity of a particle, we can know when a heap is a heap or when it is not, but not both. The closer we get to absolute certainty (it is or is not a heap), the further we get from absolute vagueness (when a heap is not a heap). This principle can be applied to most, if not all, instances of a sorites paradox where the transition from one state to another is unclear.

  1. Absolute Certainty—Absolute Specificity
  2. Absolute Uncertainty—Absolute Vagueness
  3. Absolute Certainty—Absolute Generality

By ‘absolute specificity’ I mean, that which can apply to only a single thing (a tree); ‘absolute generality,’ that which applies to something as a whole (a forest); ‘absolute vagueness,’ that which cannot be determined true or false with certainty. The ‘absolute specificity’ of the universe, according to modern science, is that it is entirely made up of quarks—the smallest particle that constitutes neutrons and protons and, therefore, everything within the universe (Encyclopedia Brittanica). The ‘absolute generality’ of the universe is the universe regarded at once. Imagine ‘echelons of reality,’ each consisting of things that share the same reality[7]—much like biological classification.[8] More specific echelons can be established (one-armed humans), as well as more general echelons (mammals). The checks and balances of specificity and generality[9] apply here, as well. The most specific echelon would necessarily consist of largest amount of composites, yet have a single variable (only quarks reside in their echelon of reality), conversely but the most general echelon would have the largest amount of variables (galaxies, solar systems, planets, animals, etc.) but only one composite—itself. Vague echelons are those that share their echelon of reality with others but not all (humanity shares its reality with certain other mammals, but not all). These principles can apply to metaphysics as well. Consider whether lying is good or not. Many parents tell their children to never lie, that lying is bad and always wrong. However, we can see how that premise becomes unclear in the following scenario:

World War II, Nazi Germany—one hundred heavily armed Schutzstaffel (SS) are at the front door of a house, asking if there are any Jews inside. If there are, they (SS) will kill the Jews and everyone in the house for harboring them. If there are not, the SS will leave. There are Jews in the house (friends and/or family members). According to the view that lying is always wrong, and assuming the course of action the SS took hinged on the words of the person at the door, he/she must tell the SS that there are Jews in the house, in which case everyone would die. If the person at the door lies, however, and tells the SS there are no Jews, everyone survives.

In that scenario, it is difficult to imagine ‘lying’ as a bad thing, or even wrong. However, it does not follow that, because ‘lying’ was arguably justified in that scenario, it is justified in every scenario. In general, we can say that lying is wrong; however, when specific scenarios are examined, the generality of lying and being wrong no longer applies, and each scenario must be considered individually. Here we can see how ‘right and wrong’ are not as ‘black and white’ as we would like. However, when we think about what is generally right and generally wrong, an arbitrary line seems to exist that clearly separates the two.

Sorensen states, “The more specific a claim, the less likely it is to be true.” This claim is only correct with regard to quantifying the amount of truth—the specificity of a claim will apply to a fewer quantity of variables (humans vs. mammals). It seems that more specific claims hold a stronger qualitative truth (the cup holds liquid seems weaker than the cup holds twelve ounces of liquid).

Sorites Paradox

Suppose the following sorites paradox: There is a pile of one thousand grains of sand a, and next to it is a pile of sand that is one grain of sand less, and so on, down to a single grain of sand c. We are certain that a is a heap of sand, and we are also certain that c is not a heap of sand. At some point the ‘heap’ becomes ‘not a heap’ b, but the exact point of transition is uncertain. Peter Unger believes, according to modern science, there is little difference between a heap and almost anything else (p. 101).

The ‘absolute specificity’ of the universe, according to modern science, is that it is entirely made up of quarks—the smallest particle that constitutes neutrons and protons and, consequently, everything within the universe. The ‘absolute generality’ of the universe is the universe regarded as a whole. Likewise, with metaphysics, the more general a premise, the more it will apply universally; conversely, the more specific a premise, the less it will apply universally.

Peter Unger uses the sorites paradox to conclude that he does not exist. At first, this might seem nonsensical—after all, how could someone that does not exist conclude anything, much less develop an argument with premises and a conclusion? His argument is as follows:

  • I exist
  • If I exist, then I consist of many cells, but only a finite number
  • I survive the removal of a single cell (so long as I have any cells left to remove from), n-1.[10]
  • So, I would exist even if I had no cells
  • I could not exist if I had no cells
  • So, I do not exist

His sorites paradox mirrors the heap by simultaneously attempting to find certainty in a specificity (‘Unger’ is nothing but a collection of cells) and generality (Unger exists). Trying to determine where borderline cases start, follows the same the same uncertainty principle laid out above. It is not a matter of solving the impossible, rather, it is a matter of understanding and accepting that it is impossible—akin to finding the end of infinity, showing nothingness, or walking to the end of the horizon. We can demonstrate these impossibilities in closed systems. By ‘closed systems,’ I mean a system that has been given quantifiable limits.[11] In the case of infinity, we can demonstrate a closed system between any two given numbers that are not equal. Between 1 and 2, for example, are an infinite amount of numbers, but what is more interesting is that we have a beginning to those infinite numbers (1), and an end (2). In this closed system, infinity is given quantifiable limits. Without those limits, however, the end of infinity is impossible to reach. Nothingness can be demonstrated within an empty container; the container acts as the closed system in this case.

Unger’s scenario provides a simple way to establish a truth-value for each iteration of ‘Unger’ as he loses his cells. Since there are a finite amount of cells x each iteration would be a single unit less in truth-value x-1. Suppose Unger had a million cells—under the MVL model, Unger with all his cells would be given a truth-value of 1; Unger with n-1 cells would have a truth-value of .999999, and so on until 0.000001, Unger with one cell. According to MVL, the arbitrary point at which Unger passes from being to not being is when the 500,000th cell is removed. There seems an inherent wrongness when saying that Unger exists at 500,000 cells, but not 499,999 cells. The Supervaluationist has his own dilemma when trying to explain how Unger exists with all his cells, but not the removal of the first cell, while the subvaluationist needs to explain how Unger can both be and not be at the same time. A contextualist, however, can posit a solution to the problem. The context under which ‘Unger’ does not exist, is when he is regarded at the cellular or atomic level. At that echelon of reality, ‘Unger’ is nothing but a mass of independent cells, or a collection of atoms, that are oblivious to anything but their reality (if it can be said that they are aware of anything). This is analogous to an army; at an individual echelon, there are individual soldiers that act alone. When we expand the echelon to the unit, we can see that the collection of all the soldiers makes up an army, thus at one level the army does not exist (only individual soldiers) while at another level the army exists (composed of many individual soldiers). When the context from showing non-existence to existence, the idea that Peter Unger (professor at New York University that people see and hear) does not exist might seem ridiculous—particularly to his students. The absolute specificity of being human is, as Unger points out, that we are nothing but cells[12]; however the absolute generality of humanity is the being that all those cells make up as a whole. Proof of Unger’s existence is relative to the echelon of reality that is being referring to.

Another sorites paradox to consider is the horizon paradox.[13] As with all sorites paradoxes, the horizon attempts to achieve the impossible through possible means. Imagine the Earth was a smooth sphere, like a billiards ball. The horizon would be equidistant in all directions from the observer[14]. Now, consider walking toward the horizon. For each step taken toward it, the horizon retreats the same distance. It would not matter how fast or slow the person ran or walked, or in which direction he or she went. It would not matter how strongly the person claimed that they would reach the horizon, and simply stating that it should be possible would not make it so. The horizon would always remain five miles away. There is a sense, however, in which a person might be able to reach a horizon, much like at a certain relative perspective, Unger does not exist. If a person claimed they would walk to a certain point, which, at their current location, was the horizon, they could then walk to that point and say that they are at the spot of the horizon stated previously, but not at the current horizon. This, however, is another example of a closed system. The horizon, like the end of infinity, is only achievable when it has been given limits. We can find a more modern, cinematographic example of a horizon in a closed system—The Truman Show, when the boat Jim Carey is navigating punches a hole through the ‘horizon.’

A subvaluationist might reply, it is both true and false that the horizon in the movie was, in fact, a horizon. There are instances when it is true—at around five miles distance it becomes the horizon; and instances when it is false—it is a painting and not the actual sky, Jim Carey’s boat punctured a hole through it, therefore it is not the horizon. The problem of vagueness is a fault of limited linguistics. Our lexicon is insufficient to portray all of the possibilities within vagueness.

The fault, however, may not lie with an inadequacy to properly explicate a concept. Certainly much fault lies in “trying to reach the horizon” instead of accepting the nature of its existence—that which makes a horizon also makes it unattainable. It is not simply a lack of vocabulary or fluke in physics that we cannot properly say or calculate when a heap is not a heap. It is also no fault of either that we can only effectively ascertain the position or velocity of a particle. Calculating both demands a certain amount of indeterminacy in the resulting answer.

Indeterminacy seems to be inscribed into the universe and permeates every aspect within it, even if but a little—“[it] is found in quantum mechanics, analyses of the open future, fictional incompleteness, and the continuum hypothesis” (Sorensen). Indeterminacy is requisite in computer science and the production of semiconductors, and satisfies Leibniz’s Law[15]. Comprehension of the sorites paradox depends on understanding indeterminacy and its role in the physical and metaphysical realms.

Metaphysical principles that agree with physical principles have stronger foundations and veracity. It might be difficult to tell someone that you do not exist, especially since, by telling him or her anything, you inadvertently interact with that person—quite an accomplishment for someone who does not exist. For a contextualist, however, an explanation is but a matter of relativity, and quite feasible: At the echelon of reality shared by cells, no one exists; however, at our echelon of reality, we necessarily exist. It might be that these echelons of reality have an inherent sorites paradox, where one echelon shares a bit of its reality with another. Ultimately, all the echelons must flow seamlessly together to form an all-encompassing echelon, formed by everything yet relative to only one thing—the universe regarded at once.

The union of the uncertainty principle to metaphysics might help to explain certain paradoxes, or at least why they are unsolvable. It is a necessary generality imbedded in the system that gives life a spark. We get surprised at the outcome of a situation that was unknown. It is intriguing and mysterious, impossible to achieve by its very nature, yet we still expend countless hours trying to determine the indeterminable. In this paper, I have attempted to explain the sorites paradox and show how a principle of quantum mechanics might be applied metaphysically to help better understand the paradox. I posited an order of ‘echelons of reality’ to help clarify why it seems that certain aspects of the universe seem so alien to the reality that we experience. Stronger philosophical foundations are created when considering an approach that is more harmonized with scientific discoveries and theories.

[1] I Do Not Exist (Unger)

[2] Higher order vagueness is the observation that when a case transitions from definite to borderline is, in itself, unclear.

[3] i.e. between .1 and .2

[4] High order vagueness makes truth-values improbable; borderline cases can seem both true and false, yet there is a sense that there is a definite line between truth and falsehood.

[5] “A class of paradoxical arguments… which arise as a result of the indeterminacy surrounding limits of application of the predicates involved.” (Hyde)

[6] “The uncertainty principle (for position and momentum) states that one cannot assign exact simultaneous values to the position and momentum of a physical system.” (Hilgevoord)

[7] For example: All quarks share the lowest echelon of reality, while the universe encompasses the highest echelon of reality. Likewise, all humans share a ‘human echelon.’

[8] i.e. life, domain, kingdom, phylum, etc.

[9] As a premise becomes more specific, the less it can be applied to general cases; the more general a premise becomes, the less it can be applied to specific cases.

[10] It is important, I feel, to allow Unger a type of “frozen in time” perspective, in which cells do not reproduce. Otherwise, a new cell would simply replace the removal of a single cell. In such a frozen reality, it is easier to follow Unger’s argument and understand why he questions his existence.

[11] An ‘open system’ is a system that has no set boundaries: infinity, perfection, noting, etc. It is impossible to demonstrate open systems without assigning them limits, or ‘closing the system.’ Closed systems are such that they allow us to demonstrate open systems: perfection can be demonstrated by scoring 100% on a test; however, this does not make the test perfect, it makes a subset of perfection attainable.

[12] It could be argued that we are not even cells, but rather atoms, or quarks. However, the same concept of vagueness applies either way.

[13] For the purpose of this paper, ‘horizon’ means: The line at which the earth’s surface and the sky appear to meet

[14] The approximate distance of the horizon, due to the curvature of the earth, is approximately five miles.

[15] The Identity of Indiscernibles states, no two distinct things exactly resemble each other. (Turner)


Encyclopedia Brittanica. N.p.. Web. 7 Dec 2012. <http://www.britannica.com/EBchecked/topic/486323/quark>.

Hilgevoord, Jan and Uffink, Jos, “The Uncertainty Principle”, The Stanford Encyclopedia of Philosophy (Summer 2012 Edition), Edward N. Zalta (ed.), URL = <http://plato.stanford.edu/archives/sum2012/entries/qt-uncertainty/>.

Hyde, Dominic, “Sorites Paradox”, The Stanford Encyclopedia of Philosophy (Winter 2011 Edition), Edward N. Zalta (ed.), URL = <http://plato.stanford.edu/archives/win2011/entries/sorites-paradox/>.

Michael, Rea. Arguing about Metaphysics. New York: Routledge, 2009. Print.

Sorensen, Roy, “Vagueness”, The Stanford Encyclopedia of Philosophy (Summer 2012 Edition), Edward N. Zalta (ed.), URL = <http://plato.stanford.edu/archives/sum2012/entries/vagueness/>.

Turner, Raymond and Eden, Amnon, “The Philosophy of Computer Science”, The Stanford Encyclopedia of Philosophy (Winter 2011 Edition), Edward N. Zalta (ed.), URL = <http://plato.stanford.edu/archives/win2011/entries/computer-science/>.