fifty questions of philosophy - sorites paradox

Supposed that you have a full head of hair. That means that you probably have around 100,000 individual hair. Now pull one of them out. Does that make you bald? Of course not. As single hair doesn't make any difference. 99,999 hairs still make a full head of hair.

滿頭黑髮的你，會因為不小心拔了一根起來就成了禿頭嗎？當然不會。

Indeed, we would surely all agree that, if you are not bald, removing just one hair could never make you bald. And yet, if you pull out another hair, and another, and another... Eventually, if you carry on long enough, you will have none left and you will indubitably be bald. So you apparently move from a state of unquestionable non-baldness to a state of unquestionable baldness by taking a series of steps that can never on their own have that effect. So when did the change come about?

如果你很確定你不是禿頭，那麼拔掉一根頭髮當然不會使你變成禿頭。而如果，你拔了一根頭髮，又一根頭髮，又一根頭髮。。。直到你的頭髮都拔完為止，那麼你當然就成了一個禿頭。這個禿頭的推論，是典型的堆垛悖論形式。而我們能從這個推論形式得到些甚麼？

This is a version of a famous puzzle, usually attributed to the ancient Greek logician Eubulides of Miletus, know as the sorties paradox. 'Sorties' comes from the Greek word soros, meaning a 'heap', as the original formulation of the puzzle features a heap of sand. Expressed in terms of addition rather than subtraction, the argument looks like this:

1 grain of sand does not make a heap.

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

If 2 grains do not make a heap, then 3 grains do not.

[and so on until . . ]

If 99,999 grains do not make a heap, then 100,000 grains do not.

So 100,000 grains of sand do not make a heap.

But everybody would surely baulk at this conclusion. So what can have gone wrong?

從一顆穀粒不能構成一堆，我們會推論出二顆穀粒也不構成一堆，。。。進而推論得到一萬顆穀粒也不構成一堆。但是，很顯然我們會同意一萬顆穀粒是一堆，那麼是我們推論時出了甚麼錯嗎？

Problem of vagueness

Faced with an unpalatable conclusion of this kind, it is necessary to track back over the argument by which it has been reached. There must be something wrong with the premises on which the argument is based or some error in the reasoning. In fact, in spite of its great antiquity, there is still no clear consensus on how best to tackle this paradox, and various approaches have been taken.

我們回去檢查這個悖論的推論形式與前提，似乎也對於我們釐清這個悖論沒有多大用處，反而是企圖削弱這個悖論的理論不停地被研究著。

One way out of the paradox is to insist, as some have done, that there is a point at which adding a grain of sand makes a difference; there is a precise number of grains of sand that marks the boundary between a heap and a non-heap. If there is such a boundary, clearly we do not know where it is, and any proposed dividing line sounds hopelessly arbitrary: do 1001 grains, say, make a heap, but not 999? This really is a bog slap in the face for common sense and our shared intuitions.

倒有一件事和研究這個悖論相關的，那就是：加上一顆穀粒和減少一顆穀粒，比如10000和9999之間的差別，造成它是一堆或者不是一堆。而我們是不是真的能夠畫出這般清楚的界線呢? 假如這個界線是很清楚的，那我們現在仍不知道這界線在哪裡，或者我們任意地畫出來罷了。

More promising is to take a closer look at a major assumption underlying the argument: the idea that the process of construction by which a non-heap becomes a heap can be fully and reductively analyzed into a series of discrete grain additions. Clearly there are a number of such discrete steps, but equally clearly it seems that these steps are not fully constitutive of the overall process of heap-building.

最重要的部分則是在於這個論證的前提：減少不是一堆的那堆的穀粒數量是一個離散的(不連續)增加數值方式。很顯然的，這個模糊理論的方式是，藉由一步一步的方式建構出"一堆"的意思。

This faulty analysis fails to recognize that the transition from non-heap to heap is a continuum, and hence that there is no precise point at which the change can be said to occur. This in turn tells us something about the whole class of terms to which the sorites paradox can be applied: not only heap and bald, but also tall, big, rich, fat and countless. All of these terms are essentially vague, with no clear dividing line separating them from their opposites - short, small, poor, thin, ect.

因為建構一堆與不是一堆是一個連續的過程，因此我們無法劃出一個很明顯的點來區別一堆與不是一堆的改變。

One important consequence of this is that there are always borderline cases where the terms do not clearly apply. So, for instance, while there may be some people who are clearly bald and others who are clearly not, there are many in between who might, according to context and circumstances, be designated as one or the other. This inherent vagueness means that it is not always appropriate to say of a sentence such as 'X is bald' that is (unequivocally) true or false; rather, there are degrees of truth. This at once creates a tension between these vague terms that occur in natural language and classical logic, which is bivalence (meaning that every proposition must be either true or false).

我們總是無法從這種邊界事例當中找出清楚劃清界限的方式。不過，我們還是可以在日常生活或是實際的例子找出哪些人是禿頭，而且很明確；哪些人不是禿頭。而模糊性的意思就是，我們在使用「X是禿頭」這類句子的時候，「是禿頭」是程度上的真，而非全部的真。一旦我們僅止用二值原則來處理這類語詞時，模糊性的問題就會出現。

The concept of vagueness suggests that classical logic must be overhauled if it is to fully capture the nuances of natural language. For this reason there has been move towards the development of fuzzy and other multivalued logics.

研究這論證的細微末節，使我們發現與研究除了二值以外的其他邏輯。