Logic: Sorites Paradox


Our classic Stoics often spent a good deal of time on Logical problems.  As I wrote yesterday about the question regarding ‘right reason’ (Gr: ὀρθὸς λόγος), the foundation of Stoic epistemology requires that true understanding is possible, via the idea of katalepsis (Gr: κατάληψις).  The idea of the Sage necessitates it, and without the Sage there isn’t a measure for our own knowledge and progress.

One such issue is the “Sorites Paradox,” so named for the Greek word for ‘heap’ which is σωρίτης.  The basic paradox has two forms.

If I place down a single grain of sand, is it a heap?  “No,” you will say.  I will continue placing down grains and asking the question, until at some point you admit, “yes, that’s a heap.”  Then I remove one.  ‘Is this still a heap?’  Thus the paradox, that one grain of sand cannot determine heapness.

I start with a heap of 10,000 grains of sand, since the absence of one grain cannot unmake a heap (see above), we will recursively remove grains until there is 1, and then 0.  Both of these would necessarily qualify as heaps per the above.  Paradox.  We can even logically go further to negative numbered grains still being heaps if Heap-Number minus 1 always yields a new Heap-Number.


We can see this same problem with baldness, plucking the hairs from the head one-by-one, at what point would we call him bald? The English form of the word “balding” might provide us with a logical escape here, in that he is in the process of becoming bald, but that’s neither here nor there.


We can also see it in the issues of collective nouns for groups of animals, as shown by this joke image, on the right.  A murder being the collective noun for a group of corvidae, this image presents the question and pun of ‘attempted murder.’

This problem is not localized to quantities of sand, hairs, and crows, as we will see shortly.

Chrysippus’ answer to the heap paradox is recorded in Cicero’s Lucullus/Academic Prior, and amounts to suspending judgment:

“You value the art [of logic], but remember that it gave rise to fallacies like the sorites, which you say is faulty. If it is so, refute it. The plan of Chrysippus to refrain from answering, will avail you nothing. If you refrain because you cannot answer, your knowledge fails you, if you can answer and yet refrain, you are unfair.”

—Cicero, Lucullus/Academic Prior §§ 91—98.

Chrysippus suggests that before the vagueness of the question causes doubt, one should withhold judgment until it’s sure.  This prevents the incongruency between 17 not being a heap, but 18 being one.

However, this is not really a solution, merely a way of avoiding the dialectal trap, as History of Philosophy notes.  In the podcast, the example is given that before one is forced into the logical corner of arguing that 24 is not a heap, and 25 is; we should begin to withhold judgment sometime around 20, before the doubt is clear.  I suspect any argument partner would infer, however, the logical paradox in silence; but Chrysippus was more concerned with protecting the epistemology of the Stoics than he was at winning 6th Grade debate points.

The issue at hand is one of vagueness, and the imprecision that is manifest in human language.  Language is made up of arbitrary symbols, for instance, nothing about the sounds of the English word ‘tree’ ( /t͡ʃɹi:/) contains anything which carries a universal understanding of the conception of ‘tree.’  It’s a symbol, agreed upon by all English speakers, but it is arbitrary.

Some of the solutions to the paradox rely on this trait of human language.  Some, by means of technical resolution, affirm a boundary which is fixed (like 10,000 units makes a heap), and others posit that there are boundaries for heaps, but they are unknowable.  Still more rely on specific types of many-value logics, and similar types of reasoning.

The colloquial phrase, “I know it when I see it” is often disparaged as simplistic understanding or ‘folksy cleverness’, but in fact it relates a truth about vagueness, subjectivity, and the symbols available to us through human language.

It is possible to make a case for the subjectivity of a heap:
Say we have boulders the size of a mini-van.  Five of these would make quite a formidable pile… one we could reasonably describe as a heap.  50 sesame seeds, however, might not be a heap.  What about 500 motes of dust?

That is not my position, however.  Rather, I want to look past the sign of the word ‘heap,’ and try to get at the thing which it symbolizes.

In grammar and linguistics we can discuss ‘mass nouns,’ which are also called no-count nouns.  Liquids tend to fall in this category.  Many languages have a partitive case (sometimes a function of the genitive) which deals with these.  See: English “some tea,” or Russian “чаю.

The core premise of the paradox is that a heap is a certain number of objects grouped together, but this premise is not explicitly stated, and its suppression causes the logical issues seen here.  So, I will bring that out, and state that such a definition is not accurate, and show how a more accurate definition alleviates the paradox.

‘Heap,” I argue, is a similar no-count word as above.  A heap describes the manner of ordering and/or generally parabolic shape of the bodies of the items in question, and in which the specific number of items is not the operative determiner of the disposition.  Example, 10 shirts in the corner of my bedroom are deemed by my girlfriend to be a heap, as in “Can you please clean up that heap of clothes.”  The very same number of shirts, (even the exact same shirts themselves) folded and stored in a stack in the closet, are no longer a heap, it seems.  The operative determiner, then, is the relatively unordered manner of stacking, and the parabolic shape which results.

Remembering that bodies according to the stoics can even be “matter disposed in a certain way,” as in the difference between ‘a hand’ and ‘a fist’, ‘heap’ seems to be one such disposition.  Thus, heaps exist, and do have an objective definition.

The issue which then needs to be explicitly pointed out is the count-requirement of the paradox.  Applying a count-criteria to a no-count problem necessarily creates a paradox, and it’s not that this particular paradox in questions needs a count-resolution, it’s simply an inappropriate question.

Inappropriate questions are easily formulated, such as “How many waters does that bottle hold?”  or “What is the number five’s favorite color?”  These are certainly sayable, and even intelligible utterances.  Yet, they lack any relevance to the universe as we know it.  They have no clear answer, because the type of answer requested doesn’t fit the proposition.

Whether one agrees that heap is a count or no-count word, the paradox provides an interesting avenue of exploration.  The chance to apply Stoic ontology, that of bodies and disposition, to the subject was a fun thought experiment.  I don’t recall ever seeing this position stated before, possibly because relying on definitions is a weak point in propositional logic.  As this is my first attempt to wrestle with a classical paradox, I’ll accept that it’s a baby step.  So far as surety can go with the Sorites Paradox, the thing I’m most sure of is that I ought to fold my shirts before they become a heap in the corner.  (:


One thought on “Logic: Sorites Paradox

  1. The paradox is faulty. Once we have decided on an objective definition of heap, then the paradox will be answered. The same applies for when a person becomes a person during gestation. A baby 1 second from being born is viewed as a person, but what about the second before that? Is is still a person? Then there’s a cellular perspective… if a baby has like 80,000,000 cells the second she/he comes out of the vagina it will be considered “life” and has rights. What about if it had 80,000,000 cells while inside the vagina? What if a baby in the womb had 80,000,000 cells and a baby fresh out of the womb has 79,999,999 cells? So basically, how many cells can you take away before a person is not considered a person? TO even think a single cell is what determines between life and not life is like saying “this nth grain makes a heap a heap”. That’s why we need objective scientific definitions, like fertilization, or nth amount of grains being officially a heap.

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