Confronting the Sorites Paradox

8/6/2019 Confronting the Sorites Paradox

1/3

Confronting the Sorites Paradox

Shahab Raza

Of the many paradoxes that have baffled human languages, one of the strongest in its potency as a

problem is the Sorites Paradox. In this paper I will outline what exactly the Sorites paradox is, and

then go on to describe and critically analyze the effectiveness in absolving the problems that weseem to have. I will offer one common account that try to counter the problem, explain why it runs

into problems, and then propose my own account trying to resolve the paradox.

There have been many versions of the paradox being offered by various philosophers in history, and

the one I will present was certainly not the first one, but I feel it is the best for sake of clarity. Here is

a standard argument:

500,000 grains of sand are a heap. (B)

Ifn grains of sand are a heap, then so are n 1 grains. (I)

...

therefore, one grain of sand is a heap. (C)

This is clearly a deductively valid argument; the only rule of inference being used here (admittedly

repeatedly) is modus ponens. If P then Q, and P, therefore Q. And the argument is also sound; the

first premise certainly seems true (and if you have problems with it, then you can replace 500,000

with an arbitrarily large n that will satisfy you). It also seems like the second premise is true. Imagine

a heap of sand if you take one grain out of it, surely it is still a heap.

Where, then, is the paradox? Weve constructed a sound, valid argument above so by basic rules of

logic, the conclusion should be true. But we can safely say that a grain of sand does notconstitute a

heap.

Before we try to come up with ways to resolve this problem, lets take a deeper look at the

phenomenon that is at work here. Its clear that the issue is with the word heap, whose meaning itseems is not so clearly defined. In technical terms, the predicate heap is an example of a vague

predicate predicate for which there are borderline cases, or cases in which it is not clear whether

the predicate can be said to apply. Clearly there are collections of grains of sand that may or may not

be (to avoid begging the question) considered a heap.

One of the most common ways of answering the question raised by the Sorites paradox is by

creating an account of supervaluation. Supervaluation relies on there being lots of ways to make

precise a vague predicate. For instance, child can be made precise by setting it to mean under 18

years of age. It can also mean under 15 years of age under a certain valuation1. Supervalution

theory maintains that a vague predicate is true if it is true under all valuations, and false if it is false

under all valuations. Of course there will be certain statements using vague predicates that areneither true nor false, such as Justin Bieber is a child (P). (Bieber is now 17 years old).

Whether or not this account protects the law of the excluded middle is not entirely clear. P or not P,

and all statements of that form will be true if the interpretation of child is the same. But P is true if

Justin Bieber is a child under every valuation, and that isnt true. And P, by the same reasoning, is

1Valuation here is defined as a function from the domain of objects to predicates. Since we are only interested

in one particular predicate, a valuation can also be viewed as a set of objects satisfying the predicate.


2/3

also not false. So there seems to be an issue here; sentences of the form Qor notQ need not be

true under supervaluation theory; if that is the case, then that is a problem for the theory.

Its not just the law of the excluded middle that is at odds with supervaluation theory. Though the

definition of validity can be defined using this tweaked notion of truth, and protected through

modus ponens, other forms of reasoning such as proof by cases and proof by contradiction cease to

be validity-preserving. More pertinently, the supervaluationist is committed to there being a sharp

distinction between objects that satisfy a predicate (regardless of valuation), ones that fall in the

gray area, and ones that are false. A two year old is a child is true, but P isnt (it isnt false either), so

where along that spectrum is the distinction? Not only does the very same problem arise again, it

seems the supervaluationist now has to offer explain away two distinctions rather than one!

I will now present my accunt, for which there are three basic tenets.. First, vague predicates are

observational. They are also subjective. And when we talk about the truth of claims that use vague

predicates, truth is merely how closely the valuation coincides with the social valuation.

Lets take one of these at a time. Any vague predicate is observational, meaning that the truth

function that the predicate defines is completely dependent on our observation, and nothing else.

So while it may seem like the parameters of what constitute a child or red are determined by age

and frequency of light waves respectively but in actual effect it is entirely governed by observational

evidence. For instance, with two boys of the same age, it may be very apparent that one is a child

just because he is shorter, has less facial hair etc.

Its this facet that manages to free us of the paradox. In the original argument, the inductive step (I)

was based on there being no distinguishable difference between, in this case, a heap of sand and the

same heap after the removal of one grain. It seems very likely that if one was to make each iteration

distinguishable from the first, there wouldbe a distinct point at which the subject could claim that

the pile of sand wasnt a heap anymore. In psychophysics this term is known as just noticeable

difference (jnd), and this difference may not be constant. (When there is less sand, it would take the

absence of less grains for the change to be noticeable).

Since there is no observational difference between a heap ofn grains, and a heap ofn 1 grains, (I)

must be true. But there is a problem here with the way the Sorites argument proceeds. Assume for

simplicitys sake that in the case of a heap jnd is 10 grains of sand. If 10 grains are taken away one-

by-one, what we observe is different to what we would observe had they been taken away all at

once, and is probably a lot more similar to the original heap. Therefore, if iterated, we can say

(truthfully) that the pile of sand is a heap, because it looks like a heap. Whats going on here is

analogous to the floor or ceiling function, which rounds any rational either up or down to the

nearest integer. Our observation works in the same way, it merely perceives the integer closest to

the rational. So while the rational maybe incrementally decreased, because its not distinguishable

we just consider it to be the integer.

It is very convenient to have the observational account of the meaning of vague predicates, but it

would seem to lead to disturbing conclusions. First, since the truth of a claim like (P) hinges on my

observation, how could I ever be wrong? And would that mean that all our discussion about the

truth of such claims is meaningless, since neither of us have any access to any others valuation. In

the next part of the essay, I embellish my account to deal with these problems. Certainly I am


3/3

committed to subjectivity, but I will also provide an objective way of looking at the valuation of

vague predicates, and the truth of claims involving them.Subjectivity implies that the range of objects that satisfy a particular predicate differs from person to

person. This seems fairly obvious; whether or not there is a sharp distinction in objects that satisfy a

vague predicate, older people for instance may plausibly call someone a child who I think isnt one. Since everyones ranges are different, it gives rise to the idea of a consensus-based conglomeration

of everyones individual range, which forms a social valuation. Any comments regarding the truth (or

lack thereof) of claims involving a vague predicate are actually just a measurement to how close the

person-in-questions valuation is to the social valuation.

There seems an issue here. For surely if I make a claim (lets say, P) then you can negate the claim,

and yet both of us will still be making true claims. It also seems like there will then be no objective

answer to the question of the truth ofP.

This is a fair response, but I can avoid it with a minor tweak but still maintaining the core of the

account. Lets reconsider what happens when a claim involving a vague predicate is made. When Iutter P, what I really am saying is Justin Bieber is a child, according to social valuation. This still

allows for the subjectivity (since my knowledge of the social valuation may be erroneous) and also

avoids the previous problems. So when my friend rejects my utterance ofP, he is also claiming that

Bieber is not a child according to the social valuation (from his perspective), and consequently only

one of us can be correct. It may be impossible to know exactly all statements that are true according

to social valuation, but by definition it does exist. And for any claim (containing a vague predicate) C,

Cis true if and only if it is true according to the social valuation.

In this essay, I explored the nature and form of the sorites paradox, and offered possible

explanations for it. I also discussed the problems faced by these explanations, and then posited my

own account of reconciling the paradox. I explained how it avoids the paradox, and then consideredone worry, which I responded to. My account may seem a little arbitrary, but it explains away the

paradox without calling into question classical logic, or the way we use language.

Documents

Confronting the Sorites Paradox