The process is continued. The logic remains the same. At each stage a heap remains and removing one more grain makes no appreciable difference and so a heap always remains.

Finally, when all the grains have been removed, a heap of sand still remains.

This is of course a paradox. It is a paradox of the Sorites or little by little type.

A Simple Approach to Sorities

A new approach I have reflecting on and am writing on February 8, 2018.

An essential point is that we are not employing a clear definition of a ‘heap’. Let us therefore consider a range of definitions.

1. A heap is 5,000 grains of sand. In this case the Sorites argument obviously fails.

Question. Why 5,000?

Answer. The point was to have a precise definition of a heap. 5,000 was just for example. We could have chosen any number that made the grains look like a heap.

Q. Then we could also have chosen, say, 10,000. And in that case there would be no definite point at which the heap ceased to be a heap.

A. That is not true. You can choose 5,000 or 10,000 or any other reasonable number. But you must choose one and only one number if you want a clear definition.

Q. But surely we don’t want a definite number?

A. As far as this approach goes we do want a definite number. We can go to an alternate kind of definition but we may not conflate different definitions. That is, if you want a definition that is more satisfactory, we can do that, but we cannot then say ‘but this new definition conflicts with the old and so resurrects the Sorites problem”.

Q. So what is a more satisfactory definition? Surely we want one that corresponds to our notion that a heap is not a definite number of grains.

2. Well another possibility is that a heap is 10,000 grains and more (for example), 100 grains or less are definitely not a heap, and the mid range is a semi-heap. Again, the Sorites paradox does not arise.

Q. But the cut-offs are still ‘too precise’ to correspond to the notion of a heap.

A. Fine, let us try something else.

3. Define a fractional heap. Heap-hood = number of grains divided by 10,000. Again Sorites does not arise.

Q. That is better but not good enough. We really want to account for the fact that a heap is not well defined.

A. Well let’s rethink. But keep in mind that we want one definition, not multiple. We don’t mind indefiniteness in the object but we must exclude indefiniteness in the concept of a heap.

4. Let’s use intuition. For the first few grains taken off the ‘heap’ it is still looking like a heap. For the last few it looks like stray grains. But there’s a mid range in which I cannot say for sure that is a heap or for sure that is not a heap. Again there is no Sorites paradox because we can say that for a few grain removals it’s still a heap, but we cannot say that this can be continued for any number of grains.

Q. But my intuition might be different.

A. True but if you introduce your intuition together with mine, you are confusing two different definitions (if you want Sorites to go through), rather than relying on one definition. In fact if I use my intuition on different occasions, I might be effectively using different definitions. Therefore, when I said let’s use my intuition I should have said let’s use my intuition right now.

5. Conclusion—the Sorites paradox arises when we use a vague or graded concept but demand a clear or sharp concept.

I will leave the rest of the document standing, untouched for now.

History of Sorites

The Sorites paradox is well known—in Greek thought: Eubulides - Wikipedia, the free encyclopedia; and there is interest in it in modern times: Sorites Paradox - Stanford Encyclopedia of Philosophy.

Formalizing the Sorities argument

Let P (n) be the statement ‘Removal of the one grain of sand at stage n from an amount of sand will not change it from a heap into a non-heap’.

The Sorites argument is as follows.

1. P (1) is true (first premise).

2. P (n) implies P (n + 1) for all integers ‘n’ (second premise).

To complete the proof of the Sorites argument we rely on the principle of mathematical induction according to which if (1) and (2) hold then P(r) is true for r = 1, 2, 3… without end—i.e. P(r) is true for all positive integers r. We thus get:

3. Therefore P(r) is true for all positive integers r.

That is, no matter how many grains are removed a heap remains.

Do we really need the principle of mathematical induction? Well, not since the number of grains is finite. However, if we started with a no heap and added grains – one at a time, we could use the same argument to say a heap would never accumulate. Then we would need mathematical induction for examination of any number of finite cases would not be enough to generalize.

The value of mathematical induction is further that it shows why Sorites is fallacious. Although P(1) implies P(2) and so on, it does not follow that P(n) implies P(n+1) for all n (and we will see why below).

But in the case of removal, the naïve argument shows that a heap would remain after an infinite number of grains of sand had been removed—when the heap had become ‘negative’. Mathematical induction avoids this ‘paradox’.

Disproving the Sorites argument

That is, the conclusion (3) is clearly false.

Therefore at least one of the premises is false. Since (1) is true, (2) must be false. That is P(n) does not imply P(n+1) for all n.

That is, there must come a stage at which removal of a single grain of sand does change the heap into a non-heap.

We have of course disproved the argument but we have not quite unpacked it because though disproved the argument still seems to make sense.

Informal unpacking of the Sorites argument

There are two aspects of a situation that make for a Sorites paradox.

First, the distinction between being a heap and not being a heap is vague. If a heap is defined as a definite number of grains or a precise height then there will be a definite point at which the heap becomes a non-heap.

Second, it is implicit in the argument that the removal of a single grain of sand does not make an appreciable difference. Consider, on the other hand, removing noticeable amounts of sand at each stage. Then even if what remains seems like a heap after the first few removals, we will not be inclined to argue that a heap will invariably remain. The Sorites argument depends on the removal not making an appreciable difference to the heap.

What is wrong seems to be that we are projecting the initial judgment that there is no appreciable change in heap-hood due to removal of single grains of sand indefinitely. This projection is clearly lacking in validity. It is precisely the same as projecting a series based on any finite number of terms or observations.

What we do know is that there is no seeming change early in the process—for the first few removals. But the Sorites argument generalized this to ‘no change for all removals’. Is this true?

If we were to do an experiment and ask a subject whether a heap remained after removal of each grain of sand there would come a point when the subject would begin to wonder—to say something like ‘Hmm, I’m not sure’—and another point when the response would be ‘No, that’s not a heap’. The precise point would be different for the same subject in different trials and for different subjects but there would be definite ranges in which most of the changes in judgments would occur.

The error(s) of the Sorites argument are, apparently, (1) extrapolation or induction and (2) substituting the extrapolated judgment for experience (experiment).

Formal resolution

The notion of a heap is vague. But it is more than just vague. The Sorites argument is not clear on what kind of concept the concept of a heap is. Is it a subjective concept or is it objective?

Begin by considering a heap to be ‘defined’ subjectively. We find the Sorites argument to apply to an initial number of grain removals. However, we cannot say that the argument generalizes to any number of grain removals because we know that at some point most experimental subjects will no longer find ‘that a heap remains’. If an objection is made to the term ‘most subjects’ the response is that it is in the nature of the subjective case that you may not expect to do better (and still have a definition).

On the other hand consider an ‘objective’ definition. What would it be? In the end, it would have to be some definite number—a definite number of grains, or a definite height and so on and further, we would have to specify our choice and not leave it at ‘a definite number of some quantity’. In this case the Sorites argument does not even get off the ground.

Relevance

The Sorites arguments have been called ‘little by little’ arguments. Although the underlying assumption has been proven false it retains an appearance of being intuitively true.

The relevance of Sorites is practical and analytical.

Practical Relevance

A practical relevance is that Sorites like arguments are present, if implicitly, in discussions and debates of social and other importance. An example occurs in the discussion of abortion. A live newborn is obviously a living human being and feels pain. The fetus is therefore alive just before birth, one day before birth; one more day before birth… the false conclusion now follows: the freshly fertilized egg is also a living human being and feels pain.

‘Little by little’ situations abound: one power plant or one industry will not result in environmental damage… one wood stove does not result in smog…

Analytical Relevance

The analytical relevance is the need for care in argument generally; and specifically in the case of vague notions. Heap-hood is an example of a vague notion. The extreme cases—heap-hood and definite non-heap-hood are easy to identify. A problem arises in the border case.

The problem of border cases arises for many concepts. What is life? We have clear cases of life and non-life. There are in-between cases, for example we can ask whether a virus living and whether society can be considered a living organism.

There are two concerns (1) Do the in-between or border cases invalidate the core concept? (2) What is the status of the in-between cases?

Let us address these questions. We begin with some examples.

Why is a virus difficult to classify? It possesses some characteristics that we associate with life but not others. Therefore, strictly a virus should be judged as not living. However, intuition suggests that perhaps viruses are living. Thus we are faced with apparent conflict. However, the root of the conflict is that we want to have a formal definition as well as an intuitive one. The resolution is simple. Choose some formal definition. If a virus satisfies that definition it is living. Alternatively, insist on retaining intuition. You will then have to live with ambiguity.

However, the ambiguity is perhaps a good thing. It is important to ask why we want to be so sure whether a virus is living or not. It is living according to one set of criteria and not according to another set—so what? It remains true that there is a host of organisms are living and many entities that are not and some border cases of vagueness. Why is this undesirable? Why is it desirable to know that a virus is or is not living?

There are two answers to the last question. When we have a biological conception of life it empowers understanding; then if we can say of some possibly living entity ‘yes it is life’ or ‘no it is not’ then the understanding does or does not transfer to the entity (so the real point is not whether a virus is living but what are its characteristics and interactions). The second answer is that the question whether a virus is living is a challenge to sharpen our conception of life. It remains, however, that with a given definite conception of life a virus has a definite status.

Richard Dawkins has argued that automobiles are living because they have many of the attributes of living organisms. He further argues ‘words or concepts are our servants, not our masters’ implying of course that we have play in the formulation of concepts. However, I would argue that words are neither servants nor masters but are instruments in negotiating reality. There is, in this negotiation, neither absolute fixity (words as masters) nor absolute play (words as servants). That is the general case.

I say that you gain nothing by saying that ‘an automobile is living’ unless you have a conception of life. I may feel pleased with myself to think ‘cars live’ but someone else may feel offended that they are equated to a mechanical object—but we are both wrong. Given a concept of life, cars do or do not live—and either way we gain understanding. Now we can also allow vagueness because it makes for creativity and later and more powerful precision. However, it remains true that current precise concepts are currently the most useful and we gain nothing by confusing precise concepts with suggestive ones.

Let us consider another example. Is a society or culture living? Why should we care? One reason concern with what happens to my person after I die. If culture is living I may be pleased—or displeased—to know that I continue on in the form, say, my cultural contribution. Now there is nothing about culture to think that it is a conscious organism. Therefore the vague thought ‘culture is living’ might deceive me into thinking that my consciousness somehow continues on. It is important that I am not arguing that it does or that it does not but that ‘culture is living’ implies nothing regarding my personal continuity.

Thus for concepts that purport to define classes, the border cases do not necessarily invalidate a core concept but show that the concept may be in transition; and they show that what is important about a concept is not so much whether a case falls under it but what falling or not falling under it tells us about the case.

These comments should not be regarded as suggesting that we should not play with concepts and that there is nothing to be gained by so doing. There may be much to be gained. (It is good to be precise and specific; and it is good to be intuitive and allow indefiniteness; but it is not good to confuse one with the other.) What is to be gained is at least twofold (a) in the generalization and therefore empowering of concepts and (b) in the enhanced understanding of objects, new and old, that fall under the concept. How will we know when our extension is fruitful in these ways? The history of science suggests that a fruitful extension occurs when there is simultaneous extension of both individual concepts and relations among concepts, i.e. of concepts and laws or theories of science.

A Greater Relevance

Sorites is rather uninteresting in itself—except of course that the paradox is perhaps unexpected. However, it might seem to have little relevance to our lives. We say however, that there are two kinds of relevance. There are simple practical applications such as the question of when a fertilized egg is human—when it feels pain. And then there is an analytical relevance—analyzing Sorites teaches us about the need for care in thought and consequently about core and border cases of concepts and the dual uses and needs of precision as well as intuition in concepts (and the need to not confuse the two).

The greater relevance is that thought on problems apparently removed from immediate concerns can have immense consequences. This should not be surprising for care in thought arises in practical situations and therefore it is natural that we should enjoy care in thought (evolution is not Calvinistic). We then play with ideas and the play has practical consequences.

The history of philosophy has many such examples. Some are great to the point of defining our civilization. Thales of Miletus (c. 600BC), perhaps the first western metaphysician, speculated that the world was water. This does not seem to be an interesting thought. However, it was remarkable (a) in breaking with religious cosmology and (b) in looking to explain the world in terms of something simple in the world. Thales' metaphysics begins a process of rational and imaginative inquiry one of whose other endpoints is modern science.