Supervaluationism and the sorites paradoxes

7/29/2019 Supervaluationism and the sorites paradoxes

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Does supervaluationism provide a satisfying solution to the sorites paradoxes?

The sorites paradox is the name given to a class of paradoxical arguments, an example ofwhich is the heap paradox. Consider a heap of sand: intuition tells us that removing a

single grain cannot turn the heap into something that is not a heap. If we had two

quantities of sand grains that differed in number by one, then both or neither are heaps.But this seemingly obvious and uncontroversial supposition appears to lead to the

paradoxical conclusion that a single grain of sand is a heap. Consider a diminishing

quantity of sand. A collection of 100,000 grains of sand is a heap. By our supposition, somust 99,999 grains. However if this is a heap, so must a collection a grain less in number,

and so on indefinitely, until we are forced to say that even a single grain of sand is a

heap.

There are many situations that when this logic is applied to them arrive at an apparently

paradoxical conclusion (an adult of 66 is short, a man with a full head of hair is bald

etc.). The feature all these arguments have in common is a vague predicate: heap, short,

bald etc. Vague concepts such as heap dont appear to have any sharp boundaries, andthe consequent indeterminacy surrounding the extension of the predicate is a heap, is

what results in there being no sharp distinction between is a heap and is not a heap. Itis the phenomena of vagueness which is the source of the paradoxical reasoning (1). In

this essay I will discuss if supervaluationism adequately defuses the sorites paradoxes.

There are three responses for any paradox: accept the conclusion, reject the reasoning as

invalid or reject one or more of the premises. In this case the first two responses are

unappealing: the false conclusions such as those given above seem unacceptable and the

reasoning behind them is simple, using only the fundamental logical principle modusponens. Supervaluationism uses semantics in order to reject a condition premise. The

supervaluationists theory treats the indeterminacy of vague predicates as real, through aprocess called sharpening. By fixing a sharp boundary between a predicates positiveand negative extensions, vagueness is replaced by precision (2). Therefore it can be

decided which way to classify each of the borderline cases.

Supervaluationists do not intend that vagueness should apply to an object as a property

does, but instead believe that vagueness resides in language, and can be avoided through

the use of semantics (3). By the supervaluationists account, a statement is true, is a

heap, if it is true for all sharpenings, false if it is false for all sharpenings. Statements inthe penumbra, on the other hand, are neither true nor false. As it is subjective as to where

sharpenings are made, the truth or falsity of a statement falling within the penumbra will

only be so relative to a sharpening.

Take the case of sharpening the vague predicate heap. For those objects for which the

predication of heap is intuitively true is definitely a heap fall under its positiveextension, and those that are definitely not a heap fall under its negative. Those that do

not fall under either extension are fall under the prenumbra (4). This account claims to

resolve the paradox by showing that not all of the premises of the arguments are true,


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particularly the principle of tolerance. Also, this account doesnt break or alter any laws

of formal logic, such as the instances of the principle of bivalence.

An argument against supervaluationism comes from considering a sentence central to the

sorites paradoxes:

For some number n, a collection of n grains is a heap but a collection of (n-1) grains is

not.

Intuition tells us that this sentence (and its analogues) is false. The definition ofvagueness is that it has no sharp boundary, so it can be argued that by the definition of a

vague predicate the above sentence is false. But in supervaluationism this sentence is true

due to the process of sharpening (5).

It also may not be such a positive factor that supervaluationism preserves classical logic.

Vagueness often throws doubt on the law of the excluded middle:

P v P

An example of this law is that either he is an adult or he isnt But there are cases whereborderline cases should be considered, when an arguments can exploit the law. For

example should a young offender be imprisoned as long as an adult would for the same

crime, or not at all if as a child he was too young to understand the consequences of hisactions. There is a definite case to argue that neither response is appropriate, and

borderline cases need to be considered.

Another problem for the supervaluational theory is hat it uses an inadequate conceptionof vagueness (6). The first of these inadequacies is an insufficient characterisation of

vagueness. The assumption in supervaluationism is that a vague term can be classified in

one of three ways: its positive extension, its prenumbra and its negative extension. Thereare, however, predicates which do not seem vague and yet remain classifiable by these

three sets. We could define a adult by the following clauses;

People who have not reached their 18th birthday are not adults

People who have reached their 21st birthday are adults.

In this case adult can be classified in three ways: its negative extension bounded byclause 1, its positive extension bounded by clause 2, and the rest (18-20 year olds) fall

under its prenumbra (7). But in the way it was defined above adult cannot said to be

vague. The clauses that define adult are precise, but the definition isnt an exhaustiveone which leads to the case of three classifications. As shown, the supervaluationists

theory cannot tell between a vague term and one whose definition is incomplete, since

any predicate with three possible classifications will be treated the same. So it meanssupervaluationism shouldnt be used unthinkingly to apply or deny a predicate, the

borderline cases need to be considered.


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A second inadequacy in a supervaluationism is its failure to account for higher order

vagueness (8). In its creation of the three sets, supervaluationism makes the

presupposition that there is a sharp defining boundary between a vague predicatespositive extension and its prenumbra, and again between its prenumbra and negative

extension. In other words, in the sharpening of a vague predicate there is as much

vagueness affecting the boundaries between prenumbra and either extension as there wasoriginally between the positive and negative extensions. So the sharpening doesnt really

clear things up with regards to vagueness. For example, a sixteen year old might be

considered a borderline case for a child, and belonging under the prenumbra, but a fifteenyear old might not be a clear cut choice for borderline children. They may be children, or

they may be borderline children; the doubt and consequent vagueness remains.

There would in theory be no difficulty in dividing the extension of a predicate into threeprecise classifications if higher order vagueness wasnt an issue. But if a predicate has a

higher order vagueness, then the process of sharpening would itself be vague - some

cases would be on the borderline for the prenumbra. In order for the supervaluationists

theory to truly defuse the paradoxes, it would have to be altered to account for higherorder vagueness. For the theory to work, there would have to be a clear case of

sharpening that makes one of the conditional premises untrue. However, this changewould create its own difficulties and wouldnt solve the other problem for

supervaluationism.

Word Count: 1265

References

1. Waismann, Friedrich (1968) "Verifiability", inLogic and Language, Anthony

Flex (ed.), Oxford: Basil Blackwell.

2. Williamson, Timothy (1994) Vagueness, London: Routledge.

3. Boolos, George (1991) "Zooming Down the Slippery Slope",Nous, 25: 695-706.

4. Endicott, Timothy (2000) Vagueness in the Law, (Oxford University Press).

5. Evans, Gareth (1978) "Can there be Vague Objects?",Analysis, 38: 208.

6. Fine, Kit (1975) "Vagueness, truth and logic", Synthese, 54: 235-59.

7. Vagueness: A Reader , Rosanna Keefe and Peter Smith (eds.), Cambridge: MIT

Press, 1996, 119-150.

8. Graff, Delia (2000) "Shifting sands: an interest-relative theory of vagueness",

Philosophical Topics, 28: 4581.

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Supervaluationism and the sorites paradoxes