On Vagueness - when is a heap of sand not a heap of sand?
Imagine a heap of sand. You carefully remove one grain. Is there still a heap? The obvious answer is: yes. Removing one grain doesn’t turn a heap into no heap. That principle can be applied again as you remove another grain, and then another…
After each removal, there’s still a heap, according to the principle. But there were only finitely many grains to start with, so eventually you get down to a heap with just three grains, then a heap with just two grains, a heap with just one grain, and finally a heap with no grains at all. But that’s ridiculous. There must be something wrong with the principle. Sometimes, removing one grain does turn a heap into no heap. But that seems ridiculous too. How can one grain make so much difference? That ancient puzzle is called the sorites paradox, from the Greek word for ‘heap’.
There would be no problem if we had a nice, precise definition of ‘heap’ that told us exactly how many grains you need for a heap. The trouble is that we don’t have such a definition. The word ‘heap’ is vague. There isn’t a clear boundary between heap and no heap. Mostly, that doesn’t matter. We get along well enough applying the word ‘heap’ on the basis of casual impressions. But if the local council charged you with having dumped a heap of sand in a public place, and you denied that it amounted to a heap, whether you had to pay a large fine might depend on the meaning of the word ‘heap’.
More important legal and moral issues also involve vagueness. For instance, in the process of human development from conception to birth to maturity, when is there first a person? In a process of brain death, when is there no longer a person? Such questions matter for the permissibility of medical interventions such as abortion and switching off life-support. To discuss them properly, we must be able to reason correctly with vague words such as ‘person’.
You can find aspects of vagueness in most words of English or any other language. Out loud or in our heads, we reason mostly in vague terms. Such reasoning can easily generate sorites-like paradoxes. Can you become poor by losing one cent? Can you become tall by growing one millimetre? At first, the paradoxes seem to be trivial verbal tricks. But the more rigorously philosophers have studied them, the deeper and harder they have turned out to be. They raise doubts about the most basic logical principles.
Traditionally, logic is based on the assumption that every statement is either true or false (and not both). That’s called bivalence, because it says that there are just two truth-values, truth and falsity. Fuzzy logic is an influential alternative approach to the logic of vagueness that rejects bivalence in favour of a continuum of degrees of truth and falsity, ranging from perfect truth at one end to perfect falsity at the other. In the middle, a statement can be simultaneously half-true and half-false. On this view, as you remove one grain after another, the statement ‘There is a heap’ becomes less and less true by tiny steps. No one step takes you from perfect truth to perfect falsity. Fuzzy logic rejects some key principles of classical logic, on which standard mathematics relies. For example, the traditional logician says, at every stage: ‘Either there is a heap or there isn’t’: that’s an instance of a general principle called excluded middle. The fuzzy logician replies that when ‘There is a heap’ is only half-true, then ‘Either there is a heap or there isn’t’ is only half-true too.
At first sight, fuzzy logic might look like a natural, elegant solution to the problem of vagueness. But when you work through its consequences, it’s less convincing. To see why, imagine two heaps of sand, exact duplicates of each other, one on the right, one on the left. Whenever you remove one grain from one side, you remove the exactly corresponding grain from the other side too. At each stage, the sand on the right and the sand on the left are exact grain-by-grain duplicates of each other. This much is clear: if there’s a heap on the right, then there’s a heap on the left too, and vice versa.
Now, according to the fuzzy logician, as we remove grains one by one, sooner or later we reach a point where the statement ‘There’s a heap on the right’ is half-true and half-false. Since what’s on the left duplicates what’s on the right, ‘There’s a heap on the left’ is half-true and half-false too. The rules of fuzzy logic then imply that the complex statement ‘There’s a heap on the right and no heap on the left’ is also half-true and half-false, which means that we should be equally balanced between accepting and rejecting it. But that’s absurd. We should just totally reject the statement, since ‘There’s a heap on the right and no heap on the left’ entails that there is a difference between what’s on the right and what’s on the left – but there is no such difference; they are grain-by-grain duplicates. Thus fuzzy logic gives the wrong result. It misses the subtleties of vagueness.
There are many other complicated proposals for revising logic to accommodate vagueness. My own view is that they are all trying to fix something that isn’t broken. Standard logic, with bivalence and excluded middle, is well-tested, simple and powerful. Vagueness isn’t a problem about logic; it’s a problem about knowledge. A statement can be true without your knowing that it is true. There really is a stage when you have a heap, you remove one grain, and you no longer have a heap. The trouble is that you have no way of recognising that stage when it arrives, so you don’t know at which point this happens.
A vague word such as ‘heap’ is used so loosely that any attempt to locate its exact boundaries has nothing solid and reliable to go on. Although language is a human construct, that does not make it transparent to us. Like the children we make, the meanings we make can have secrets from us. Fortunately, not everything is secret from us. Often, we know there’s a heap; often, we know there isn’t one. Sometimes, we don’t know whether there is one or not. Nobody ever gave us the right to know everything!
https://aeon.co/ideas/on-vagueness-when-is-a-heap-of-sand-not-a-heap-of-sand
The sorites paradox originated in an ancient puzzle that appears to be generated by vague terms, viz., terms with unclear (“blurred” or “fuzzy”) boundaries of application.
‘Bald’, ‘heap’, ‘tall’, ‘old’, and ‘blue’ are prime examples of vague terms: no clear line divides people who are bald from people who are not, or blue objects from green (hence not blue), or old people from middle-aged (hence not old).
Because the predicate ‘heap’ has unclear boundaries, it seems that no single grain of wheat can make the difference between a number of grains that does, and a number that does not, make a heap. Therefore, since one grain of wheat does not make a heap, it follows that two grains do not; and if two do not, then three do not; and so on. This reasoning leads to the absurd conclusion that no number of grains of wheat make a heap.
The puzzle can be expressed as an argument most simply using modus ponens:
1 grain of wheat does not make a heap.
If 1 grain doesn’t make a heap, then 2 grains don’t.
If 2 grains don’t make a heap, then 3 grains don’t.
…
If 999,999 grains don’t make a heap, then 1 million grains don’t.Therefore,
1 million grains don’t make a heap.
The argument is a paradox because apparently impeccable reasoning from apparently impeccable premises yields a falsehood. The argument can be run equally in the opposite direction, from the premise that one million grains make a heap: if one million grains make a heap, then one million less one grain make a heap; and if one million less one grain make a heap, then one million less two grains make a heap; etc. It follows, absurdly, that even a single grain makes a heap. Thus soritical reasoning appears to show both that no number of grains make a heap and that any number of grains make a heap.
https://plato.stanford.edu/entries/sorites-paradox/
The Fallacy of the Heap (left/right media favorite)
The fact that there is no dividing line determining either or sides of a continuum as not being the right kind of evidence for ruling in or out 'verifiable trends' taking place along that continuum - see 'balding' below;
Alias:
The Argument of the Beard
The Fallacy of the Beard
The Fallacy of the Continuum
Slippery Slope
Forms:
A differs from Z by a continuum of insignificant changes, and there is no non-arbitrary place at which a sharp line between the two can be drawn.
Therefore, there is really no difference between A and Z.
A differs from Z by a continuum of insignificant changes with no non-arbitrary line between the two.
Therefore, A doesn't exist.
Example:
A single grain of sand does not make a heap of sand. Also, a single grain of sand won't turn a non-heap into a heap.
Therefore, there are no heaps of sand.
Exposition:
The Fallacy of the Heap plays upon the vagueness of the distinction between two terms that lie on a continuum. For instance, "bald" is a vague word, and a man who is a borderline case of baldness is a familiar sight: it isn't clear whether he is bald or not, so we say that he is "balding".
"Bald" and "non-bald" lie at opposite ends of a spectrum of hairiness, and there is no precise point where baldness turns into non-baldness. We could, of course, decide to count, say, 10,000 hairs or less as the definition of "bald", but this would be arbitrary. Why not 10,001 or 9,999? I know of no answer other than the fact that we prefer round numbers.
So, there is no non-arbitrary line between baldness and hairiness, but it does not follow from this fact that there really is no difference between the two.
A difference in degree is still a difference, and a big enough difference in degree can amount to a difference in kind. For instance, according to the theory of evolution, the difference between species is a difference in degree.
Similarly, the lack of a bright line between contrary concepts does not mean that one of the concepts is a myth―that is, there is nothing to which it refers.
For example, some people have argued that there is no such thing as life, since the line between animate and inanimate thing is fuzzy. However, we can all easily identify many living things and nonliving things, and the fact that there are some things which fall into a gray area―viruses, for instance―does not mean that the concept of life is without reference.
Exposure:
The mathematically-inclined reader may notice that the heap and beard arguments are reminiscent of proofs by mathematical induction. As a matter of fact, a modern version of these arguments is known as "Wang's paradox":
One is a small number.
If you add one to a small number, the resulting number is small.
Therefore, all numbers are small.
The conclusion follows from the premisses by mathematical induction. However, you should realize by now that the problem is that "small" is vague, and the moral of the paradox is that we must be careful when reasoning mathematically with imprecise concepts.
The Fallacy of the Heap is primarily a philosopher's fallacy, though it occasionally arises in legal or moral contexts. A great deal of ink has been spilled in fruitless philosophical debates over exactly where to draw the line between concepts that lie on continua.
This might be called the "legalistic" side of philosophy, for it is primarily in the law that we are forced to decide hard cases that lie in gray areas. For instance,
if the legislature were to decide that baldness is a disability deserving of certain benefits, then the courts might be forced to decide the issue of whether a certain person is bald.
In everyday life, we are seldom faced with decisions of this kind, and
we continue to use the concept of baldness without worrying about borderline cases.
When someone falls into the fuzzy area between bald and hairy, we just say that he is "balding", thus avoiding the issue of whether he is now bald.
One reason that so many philosophical debates are seemingly endless and undecidable is because they involve a search for a mythical entity, namely, a non-arbitrary distinction between concepts that lie upon continua in conceptual space. The logical attitude towards such problems is to avoid them if at all possible; but
if a decision cannot be avoided, then draw an arbitrary line in the gray zone and stick with it. Don't be drawn into defending the decision against the charge that it is arbitrary; of course it's arbitary, for any such decision will be arbitrary.
For this reason, it is not a criticism of such decisions to point out their arbitrariness. (arbitrariness is not sufficient to rule in or out a verifiable trend)
Philosophers, naturally, are uneasy about arbitrariness, but when we are dealing with conceptual continua, it is an unavoidable fact of life. Where there is only gray, there are no black-and-white distinctions to be made.
https://www.fallacyfiles.org/fallheap.html
The sorites paradox - (sometimes known as the paradox of the heap) is a paradox that results from vague predicates. A typical formulation involves a heap of sand, from which grains are removed individually. With the assumption that removing a single grain does not cause a heap to become a non-heap, the paradox is to consider what happens when the process is repeated enough times that only one grain remains: is it still a heap? If not, when did it change from a heap to a non-heap?
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