philosophy discussion 3

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Epistemic View

Proposal: There are sharp boundaries between, say, “bald” and “not bald”, but we can never know where they are. Losing one hair can make someone go from not bald to bald, but we can’t ever know when this happens.

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1) A man with 1 hair on his head is bald.

2) If a man with 1 hair on his head is bald, then a man with 2 hairs on his head is bald.

3) If a man with 2 hairs on his head is bald, then a man with 3 hairs on his head is bald.

….

C) Therefore, a man with 100,000 hairs on his head is bald. Everyone is bald.

• On the epistemic view, one of the premises other than the first is

false. Suppose it is number 125:

125) If a man with 124 hairs on his head is bald, then a man with 125

hairs on his head is bald.

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Problems with the epistemic view:

• According to the epistemic view, no one knows which premise in

the sorites paradox is false. So facts about whether our word “bald”

applies to someone with, say 130 hairs is forever unknowable. But

is it plausible to think that there are unknowable facts of this sort

about the application of our own words?

• Words like “bald” have the meanings they do because of the way

that we use them. But how could we use our words in ways which

determined standards if we could not know the cut-off points?

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Truth-Value Gaps

• Proposal: it is impossible to know the sharp cut-off point for

words like ”bald“ because there is no sharp cut-off point.

• Some people are clearly bald and others are clearly not bald.

But there are also some people in the middle. If you say that

one of them in the middle is bald, you haven't said anything

true; but you haven't said anything false either. The rules for

applying the word “bald” just don't deliver a verdict for these

people -- it is “undefined' when it comes to them.

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• A proponent of truth-value gaps claims that some number of the

premises in a sorites argument will fail to be true.

• Consider again premise (125): If a man with 124 hairs on his

head is bald, then a man with 125 hairs on his head is bald.

• Suppose that it is neither true nor false to say that someone with

124/5 hairs is bald. In this case, premise (125) is an example of

an “if-then” sentence both of whose constituent sentences are

neither true nor false, but rather “undefined.” It follows that

premise (125) is not true.

• This may be enough to explain why the conclusion of the sorites

argument is false.

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Problems with the truth-value gap view:

• Suppose Bob has 125 hairs. Consider these sentences:

– If Bob is bald, then with one less hair he would still be bald.

– If Bob is bald, then with one more hair he would still be bald.

• Intuitively, the first sentence is definitely true. The second looks not as

clearly true as the first. The truth-value gap view cannot explain the

asymmetry between the two sentences.

• This sentence looks like a logical truth:

• Either it is raining or it is not raining.

• But if ”is raining“ is a vague predicate, then

this sentence may turn out to be neither true

nor false. This is strange.

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Supervaluationism

• Proposal: there are lots of middle cases of thinly haired men.

The rules for “bald” don't dictate that it is true to say of them

that they are bald, but also don't dictate that it would be false

to say of them that they are bald. So, in a certain sense, it is

“up to us” to say what we want about such cases.

• The act of “drawing the line” between the bald and non-bald

can be called a sharpening of “bald.” There are many

possible sharpenings of “bald” which are consistent with the

rules governing the word.

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• A sharpening associates each predicate with a partial function that

maps the set of individuals onto a positive extension, a negative

extension, or neither (for the borderline cases).

– A sentence is true if and only if it is true with respect to every

sharpening.

– A sentence is false is and only if it is false with respect to every

sharpening.

– A sentence is undefined if and only if it is true with respect to

some sharpenings, and false with respect to others.

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Mark Sainsbury, Vagueness, p. 35

• Q: How does supervaluationism escape the sorites paradox?

• A: Many premises in the typical instance of the sorites

paradox will be true on some sharpenings, but false on

others. So, some of these premises will be undefined.

• This makes room for the view that the reasoning is valid and

the conclusion false.

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The supervaluationist proposal is similar to the truth-value

gap view but it has certain advantages:

• The supervaluationist can capture the asymmetry

between these sentences:

– If Bob is bald, then with one less hair he would still be bald.

– If Bob is bald, then with one more hair he would still be bald.

• Given supervaluationism, this sentence is true in every

circumstance:

• Either it is raining or it is not raining.

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Problem of higher-order vagueness:

• Just as there are borderline cases between “bald” and “not

bald,” there are also borderline cases between cases

where “bald” applies and cases in which it is undefined. But

the truth-value gap view and supervaluationism assume

that there is a sharp dividing line between the cases where

“bald” applies and the cases in which it is undefined.

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• Instead of positing a sharp dividing line between the cases

to which “bald” applies and the cases for which it is

undefined, why not simply posit such a sharp dividing line

between “bald” and “not bald”? It gives us the same result

and is simpler. (But, of course, then we would be stuck with

the idea that there is some number of hairs such that, if you

have that number you are not bald, but if you lost just one,

you would be bald.)

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Degrees of Truth

• Suppose a man with around 120 hairs on his head starts becoming

bald. Consider once again:

– 125) If a man with 124 hairs on his head is bald, then a man with 125 hairs

on his head is bald.

• Proposal: The if-part (antecendent) of premise (125) is nearly but not

quite true. It‘s degree of truth is, say, 0.96. The then-part (consequent)

of premise (125) is also nearly true but not quite so nearly true as the if-

part. Say it‘s degree of truth is 0.95.

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• When the antecedent of a conditional

has a higher degree of truth than its

consequent, then the conditional as a

whole has a degree of truth less than 1.

(For a conditional is clearly false if its

antecendent is true and consequent

false.)

• So the conditional premise (125), though very nearly true,

is not quite true. This is enough to stop the sorites

argument being sound, for not all of its premises will be

strictly true.

• The tiny falsities of many of the premises in the sorites

argument propagate through the long chain premises to

yield a wholly false conclusion.

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Problems with degrees of truth:

• The assigment of degrees of truth is often artificial. For example,

what value should we assign to ”France is octogonal“?

• Like the truth-value gap view and supervaluationism, the degree

of truth approach assumes there is a sharp dividing line between

cases to which “bald“ applies and cases for which it is undefined.

But why should we not simply instead posit a sharp dividing line

between “bald“ and “not-bald“? This is once again the problem of

higher-order vagueness.

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