The Sorites In History

The name "sorites" derives from the Greek word "soros" (meaning "heap")

and originally referred, not to a paradox, but rather to a puzzle known

as The Heap: Would you describe a single grain of wheat as a heap? No.

Would you describe two grains of wheat as a heap? No. ... You must

admit the presence of a heap sooner or later, so where do you draw the

line?

It was one of a series of puzzles attributed to the Megarian logician

Eubulides of Miletus. Also included were: The Liar: A man says that he

is lying. Is what he says true or false? The Hooded Man: You say that

you know your brother. Yet that man who just came in with his head

covered is your brother and you did not know him. The Bald Man: Would

you describe a man with one hair on his head as bald? Yes. Would you

describe a man with two hairs on his head as bald? Yes. ... You must

refrain from describing a man with ten thousand hairs on his head as

bald, so where do you draw the line?

This last puzzle, presented as a series of questions about the

application of the predicate ‘is bald’, was originally known as the

falakros puzzle. It was seen to have the same form as the Heap and all

such puzzles became collectively known as sorites puzzles.

It is not known whether Eubulides actually invented the sorites

puzzles. Some scholars have attempted to trace its origins back to Zeno

of Elea however the evidence seems to point to Eubulides as the first

to employ the sorites. Nor is it known just what motives Eubulides may

have had for presenting this puzzle. It was, however, employed by later

Greek philosophers to attack various positions. Most notably by the

Sceptics against the Stoics’ claims to knowledge.

These puzzles of antiquity are now more usually described as paradoxes.

Though the conundrum can be presented informally as a series of

questions whose puzzling nature gives it dialectical force it can be,

and was, presented as a formal argument having logical structure. The

following argument form of the sorites was common:

1 grain of wheat does not make a heap.

If 1 grain of wheat does not make a heap then 2 grains of wheat do not.

If 2 grains of wheat do not make a heap then 3 grains do not.

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

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10,000 grains of wheat do not make a heap.

The argument certainly seems to be valid, employing only modus ponens

and cut (enabling the chaining together of each sub-argument which

results from a single application of modus ponens). These rules of

inference are endorsed by both Stoic logic and modern classical logic.

Moreover its premises appear true. Some Stoic presentations of the

argument recast it in a form which replaced all the conditionals, ‘If A

then B’, with ‘Not(A and not-B)’ to stress that the conditional should

not be thought of as being a strong one, but rather the weak Philonian

conditional (the modern material conditional) according to which ‘If A

then B’ was equivalent to ‘Not(A and not-B)’. Such emphasis was deemed

necessary since there was a great deal of debate in Stoic logic

regarding the correct analysis for the conditional. In thus judging

that a connective as weak as the Philonian conditional underpinned this

form of the paradox they were forestalling resolutions of the paradox

that denied the truth of the conditionals based on a strong reading of

them. This interpretation then presents the argument in its strongest

form since the validity of modus ponens seems assured whilst the

premises are construed so weakly as to be difficult to deny. The

difference of one grain would seem to be too small to make any

difference in the application of the predicate; it is a difference so

negligible as to make no apparent difference in the truth values of the

respective antecedents and consequents.

Yet the conclusion seems false. Thus paradox confronted the Stoics just

as it does the modern classical logician. Nor are such paradoxes

isolated conundrums. Innumerable sorites paradoxes can be expressed in

this way. For example, one can present the puzzle of the Bald Man in

this manner. Since a man with one hair on his head is bald and if a man

with one is then a man with two is, so a man with two hairs on his head

is bald. Again, if a man with two is then a man with three is, so a man

with three hairs on his head is bald, and so on. So a man with ten

thousand hairs on his head is bald yet we rightly feel that such men

are hirsute, i.e. not bald. Indeed, it seems that almost any vague

predicate admits of such a sorites paradox and vague predicates are

ubiquitous.

As presented, the paradox of the Heap and the Bald Man proceed by

addition (of grains of wheat and hairs on the head respectively).

Alternatively though, one might proceed in reverse, by subtraction. If

one is prepared to admit that ten thousand grains of sand do make a

heap then would can argue that one grain of sand does since the removal

of any one grain of sand cannot make the difference. Similarly, if one

is prepared to admit a man with ten thousand hairs on his head is not

bald, then one could prove that even with one hair on his head he is

not bald since the removal of any one hair from the originally hirsute

scalp cannot make the relevant difference. It was thus recognised even

in antiquity that sorites arguments come in pairs: non-heap and heap;

bald and hirsute; poor and rich; few and many; small and large; and so

on. For every argument which proceeds by addition there is another

reverse argument which proceeds by subtraction.

The paradox attracted little subsequent interest until the late

nineteenth century when formal logic once again assumed a central role

in philosophy. With the demise of ideal language doctrines in the

latter half of the twentieth century interest in the vagaries of

natural language and the sorites paradox in particular has greatly

increased.