3. The Heap 英文版 悖论 逻辑学教材

The Paradoxes of Eubulides of Miletus (4th century B.C.)

3. The Heap

(a.k.a. The Bald Man, a.k.a. The Sorites Paradox)

I.The Paradox

Consider this obvious truth: addition of 1 kernel of wheat is too insignificant to turn what is not a heap of wheat into a heap of wheat and, similarly, subtraction of 1 kernel of wheat is too insignificant to turn what is a heap of wheat into what is not a heap of wheat.

The obvious truth has the paradoxical consequences shown by the following Soritical argument:

1. 1 grain of wheat is not a heap.

2. If 1 grain of wheat is not a heap, then 2 grains of wheat are not a heap.

[by the obvious truth] 3. If 2 grains of wheat are not a heap, then 3 grains of wheat are not a heap.

[by the obvious truth] .

.

.

10,000. If 9,999 grains of wheat are not a heap, then 10,000 grains are not a heap.

[by the obvious truth] Therefore, 10,000 grains of wheat are not a heap.

The argument is valid. (In effect, it just uses 10,000 applications of the truth-preserving inference rule modus ponens.) And, its premises all seem to be true: (1) is clearly true; (2) is true by the obvious truth that addition of 1 little kernel can’t turn a non-heap into a heap; and (3)-(10,000) are true for the same reason. But, the conclusion is obviously false. So, the paradox here is, again, a seemingly sound argument with a false conclusion.

II.The Heart of the Matter

The heart of the Heap paradox is captured in three assertions each of which seems true:

a. small changes don’t make for a difference in the application of soritical predicates like

‘…is a heap’ or ‘…is bald.’

b. large changes do make for a difference in the application of such predicates,

c. a large change is nothing more than a large number of small changes.

But as the Soritical argument seems to show, the triad (a)-(c) is inconsistent, i.e., all of its members cannot be true together.

III.Formalization

Let’s symbolize the Soritical argument:

1. Fa1

2. Fa1? Fa2

3. Fa2? Fa3

.

.

.

n. Fa n-1? Fa n

n

‘F’ is the predicate letter for the soritical predicate—in this case ‘…is not a heap’—and ‘a1’, ‘a2’, etc. are individual constants denoting the objects to which the pr edicate applies, e.g., 1 kernel of wheat, a collection of 2 kernels of wheat, etc.

The general conditions for a Sorites argument are:

i. the series a1-a n is ordered consecutively (e.g. by increasing number of kernels),

ii. ‘F’ satisfies the foll owing conditions:

a. it is true of a1

b. it is false of a n

c. each adjacent pair in the series, a j and a j+1, are similar enough so as not to differ in

respect of ‘F.’

[Thus, we can construct a more emotionally compelling Sorites ar gument with ‘F’ interpreted as ‘is not a person’ and the series a1-a n denoting the stages of embryonic development from

zygote to infant.]

III.Vagueness

The source of the Sorites paradox is the semantic phenomenon of vagueness: a word’s or concept’s having a gray area of application, i.e., borderline or indeterminate cases, circumstances in which the word or concept neither clearly applies nor clearly fails to apply.

If every word or concept were as precise as, say, rational number, which has no borderline cases, then there would be no Sorites paradox. That is, if there were a sharp boundary for ‘…is not a heap’—a number k such that k grains clearly were not a heap but k+1 grains clearly were a heap—then one of the premises in the Sorites argument would be false. Specifically, the k+1st premise—Fa k? Fa k+1—would be false, for ‘Fa k’ (‘k grains of wheat is not a heap’) would be true but ‘Fa k+1’ (‘k+1 grains of wheat is not a heap’) would be false.1 So, we’d no longer have a sound argument with a fa lse conclusion.

1 Remember, a conditional with a true antecedent and a false consequent is false.

But, ‘…is not a heap,’ ‘…is bald,’ etc. certainly don’t seem to have such sharp boundaries. So, we seem stuck with the paradox.

IV . Reactions

A . Natural Language isn’t Logical

One proposal is to deny that formal logic characterizes natural language —at least until its vague predicates are replaced with precise ones —for the symbolic languages of formal logic are deliberately precise while natural languages are lousy with vagueness.2

I don’t care for this approach. For, arguments w hose premises and conclusions are expressed in natural language can be valid or not. Indeed, natural language arguments constructed with vague predicates can be valid or not. And validity is certainly within the scope of formal logic.

B . Ignorance of Sharp Meanings

Another proposal is to claim that, despite appearances, one of the premises in

Soritical arguments is really false. It follows that there really is a sharp boundary for every soritical predicate like ‘…is a heap’—some specific number which sharply distinguishes heaps from non-heaps.

To defend this idea, let me recast the form of the Soritical argument from a series of conditional premises to a single universal premise: 2 This was Frege’s attitude as well as Russell’s.

1. Fa 1

2. (?n )(Fa n ? Fa n + 1) 10,000

1. The man with no hair on his head is bald.

2. For any number n , if the man with n hairs is bald, then the man with n+1 hairs is bald. Therefore, (3) the man with 10,000 hairs on his head is bald. The argument is valid. Since (1) is undeniably true and (3) is undeniably false, avoiding the paradox requires that premise (2) be false. Therefore:

~(?n )(Fa n ? Fa n + 1).

Hence:

(?n )~(Fa n ? Fa n + 1).

[by QN] Thus:

(?n )(Fa n & ~Fa n + 1). [by material implication and De Morgan’s law]

That is, there does exist a particular number, call it k , such that the man with k hairs on his head is bald but the man with k +1 hairs is not bald. It’s just that we don’t (or can’t) know

what k is. In other words, there really are no vague words or concepts, and what seems like vagueness is not a semantic phenomenon but rather the result of our ignorance about the precise meanings of our own words and concepts.

I don’t like this reply either. Language is our creation, and there isn’t more to it than we put in it. So, it is very implausible that the meanings of our terms should be so far beyond our knowledge.

C. Fuzzy Set Theory and Fuzzy Logic

There is an intuitive connection between an object’s membership in a set and the truth of a sentence that predicates a property of the object, viz. if a belongs to the set of F-things—a∈ {x∣x is F}—then the sentence ‘F a’ is true. Intuitively, then, it seems that any object is either in or out of a given set.

According to fuzzy set theory, on the other hand, there are an (uncountably) infinite number of degrees of set membership, as many degrees as there are real numbers in [0, 1]. So, a can be completely non-F, …, a little F, …, a little more F, …, pretty F, …, quite F, …, very F, …, or fully F. An d, there are correspondingly many degrees of truth: if a∈ to degree n {x∣x is F}, then ‘F a’ is true to degree n. For example, as a man loses more and more hair, he becomes more and more bald and the assertion ‘S is bald’ becomes more and more true.

In theory, the concept of degrees of truth implies degrees of validity, and, with further development, the fuzzy theorist hopes to show that the conclusion of a Soritical argument does not follow from its premises with a sufficient degree of validity to be paradoxical.

I won’t pursue the development of this solution because I already find it enormously implausible that our concept of truth should be so unbelievably refined as to include a non-denumerably infinite number of distinct truth-values.

D. Truth-Value Gaps.

In the classical logic that we’ve studied, there are two, and only two, truth-values: the true and the false. Fuzzy logic is an example of non-classical (or deviant) logic of the sort called many-valued.

A logic of truth-value gaps is a different sort of non-classical logical theory. For example, if ‘71 kernels is not a heap’ is neither clearly true nor clearly false, then it has no truth-value—the sentence is a truth-value gap. This promises to resolve the paradox, for if there are premises in a Soritical argument that get no truth-value, then the argument is no longer sound.

However, it is not obvious that truth-value gaps will resolve the paradox. Imagine that we adopt a gappy logical theory where some sentences with vague predicates—like

premise (1) of the Soritical heap-argument—are true, some—like that argument’s conclusion—are false, and some—the borderline cases—are truth-value gaps.

This proposal leaves us with a new and very similar problem: precisely which sentences get no truth-value? That is, precisely which cases are the borderline cases? The situation here is not that fewer than, say, 500 grains clearly is not a heap, more than 1000 grains clearly is a heap, and between 501 and 999 grains clearly is borderline. On the contrary, it is very unclear whether 900 grains is a clear heap or a clear borderline case.

In other words, the concept borderline case is itself vague—it also has borderline cases! So, even if we treat the problem of vagueness by acknowledging three truth statuses—true, false, and neither true nor false—we still face the vagueness problem of determining which sentences have which truth statuses.