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SORITES ANIL MITRA © AUGUST 27, 2010. REVISED April 21, 2013 OUTLINE
SORITES—NATURE AND RELEVANCE An ExampleA heap of sand rests on a table. If one grain of sand is removed, there is no appreciable difference and so a heap remains. If a second grain is removed there is, again, no appreciable difference and so a heap still remains. 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 type. History of SoritesThe 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. A First SolutionThere 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. If we were to remove a single atom every five seconds the experiment would barely get going in a lifetime. On other hand if I remove noticeable amounts of sand the Sorites argument that a heap will not turn into a non-heap is not secure for even if what remains is judged to be a heap remains at the first few removals, each removal diminishes the ‘heap-hood’ of the heap. Now clearly, even when the removal of a single grain of sand is not noticeable, as grains are removed there will come a point at which most people will begin to question the heap-hood of the ‘heap’ and another point at which most people will say ‘its not a heap anymore.’ Therefore it is obvious that the Sorites argument is wrong. The problem is to elucidate what is wrong about it. 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. It is the same as the projection or extrapolation of a law when every observation so far fits the law. If we knew that removing a grain of sand does not change heap-hood then there would indeed be a paradox. However we do not know this. What we do know is that there is no seeming change for some heaps. If we were to do an experiment and ask an experimental subject whether a heap remained after removal of each grain of sand there would come a point when the subject would begin to scratch his or her head and say something like ‘I don’t know’ or ‘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). InductionInductive reasoning is generalization. We know that there are fallacies of induction: generalization from a finite data set—even to a slightly more inclusive finite sample space—is never necessarily valid. Sorites is induction and induction is never necessary. However, the question precisely where the error is has not been resolved. It may seem to have been resolved by the foregoing argument but we may be left with a nagging thought that the Sorites argument has not been dealt a death blow. Mathematical InductionLet P (n) = 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. P (1). P (n) implies P (n + 1). Now, the principle of mathematical induction now implies P(r) for all positive integers. Which is clearly false. The error in the argument from mathematical induction is that P (n) implies P (n + 1) only for some initial range of n—i.e., it is not true that P (n) implies P (n + 1) for all n. The error in the Sorites argument that introduced this piece is its tacit but erroneous of an argument that, for rigor, would need to be tantamount to a mathematical induction argument—which we have just seen to be invalid. RelevanceThe 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 (and of course, in the end, the analytical is practical and the practical informs the analytical). Practical RelevanceA 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 RelevanceThe 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 is a virus living… can a society 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 first address whether difficult to classify border cases invalidate the core concept. 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. What is the source of the conflict? To attempt resolution we should address the question ‘What is life?’ Let us further ask what the question presupposes. In asking it we may presuppose that there is something definite, i.e. life, and our problem is one of ignorance which may be resolved by elucidation. However, given that there is no universally accepted notion of life it may be the case that there is no definite thing, ‘life’, such that all entities are definitely living or definitely non-living. Clearly, functional and evolutionary biology can get along without this definiteness and it remains true that the concept of life definitely applies to most cases. It will of course now be said ‘But surely we would like to be able to say whether a virus is living or not.’ This is the question of the in-between cases. What of a culture? Or an automobile? 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. Let us come back to culture. Is it living? Why should we care? One reason to care about whether a culture is living is the 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 of my contribution—small or large—to culture. If I know that culture is living I may then derive satisfaction in knowing that I may leave a legacy that is in fact my continuity after my death. In fact, however, I know no such thing from the thought that culture is living. If I define life I may define it in terms of characteristics. One characteristic, obvious in human life, is continuity of consciousness. However, the extension of the notion of life to cover culture just because there are similarities or from the notion that ‘concepts are our servants’ or the idea that ‘there is no reason to restrict the notion of life to its known biological or material forms’ does not imply that every characteristic from the definition, especially the desirable ones, continue over. So we can, if we choose, consider a culture to be living. However, we cannot thereby conclude that a culture will have all the characteristics that we otherwise associate with living organisms. In particular we cannot assume that because a culture is a human culture and humans have continuity of consciousness that cultures will have continuity of consciousness. Let us also consider the example of the automobile. We may regard automobiles as living. One reason to regard automobiles as living is that the forms of automobiles evolve. However, we cannot then conclude that all our understanding of evolution from what we have traditionally considered living will translate automatically to the case of automobiles. Thinking of automobiles as living may make us happy—perhaps because we seem to have a certain facility with concepts that we may have earlier thought as rigid. However, if we cannot deduce anything from this facile extension of the concept of life to cover automobiles then the extension of the concept has not helped in any significant way (except of course that it makes us happy). 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. 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 RelevanceThe relevance of sorites reveals a greater relevance—a kind of relevance: the relevance of reflection, imaginative and critical, on ideas that have no immediate relevance. The history of philosophy and mathematics has many examples of such thought of which some are simply examples of specific relevance and interest but some are of great relevance to the point of defining our civilization, defining the nature of science and its great ideas. As an example, the roots of modern science are many and include apparently unrelated trends that begin with Greek thought—for example, the break with supernaturalism in the thought of the 600BC philosopher Thales of Miletus who argued that the substance of the world is water: even though the thought is simplistic it is an essential break from earlier thought in which the explanation of the world lies outside the world. Two conclusions may be made. First, there is an obvious conclusion regarding the value of thought that has no obviously direct application. The long term adaptability that such thought confers may explain its institutionalization and perhaps even its origin in our species. A second conclusion is not so obvious. It is that there is often a connection, sometimes a gradual transition and often not obvious, from thought that is relatively abstract to thought that is very connected to the world and very practical. ‘Theoretical thought’ is practical precisely for this reason—while some such thought will not bear fruit, some will and without allowing the ‘failures’ will be excluded from the successes. A more concrete general relevance of the Sorites paradoxes concerns the nature of concepts and their application. Refer back to the discussion of life. There are conclusions for the precision of concepts. Regardless of the definition we use and whether we use a formal definition or just intuition we are bound to find borderline cases that cannot be removed by definition. One approach of course is to ask why we want to define a concept. Is there a compelling reason to have a precise definition of life? It is not clear that there is. We have seen that the lack of perfect precision does not imply that our ideas are unsatisfactory. The discussion may be taken one level higher. Some kinds of thing may have precise boundaries and therefore have no vague border area; other notions may have imprecise boundaries and therefore have essentially vague border areas. Is there a precise distinction between vague and non vague things? Perhaps a partial answer to the question lies in the distinction between concrete and abstract objects. This thought is continued in essays on my website http://www.horizons-2000.org where it is seen that it is not precisely the abstract-concrete distinction that is at issue. Instead, some objects transcend context and are known precisely while others are contextual and indefinite with regard to range of application precisely because contexts are open rather than closed. |