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# Two or three grains of sand ''do not'' make a heap. # A million grains ''do'' make a heap. # If n grains of sand ''do not'' make a heap, neither do (n+1) grains. # If n grains of sand make a heap, so do (n−1) grains. The paradox is that, contrary to (3), we might add grain after grain to our collection of sand until it truly does become a heap. And according to (4), if we began with a heap, and took single grain after single grain away from the collection, it would never stop being a heap; even if there were no grains of sand left at all. So common language suggests that heaps of sand have the properties described in (3) and (4), since the difference of one grain of sand is thought to be negligible. But it seems that these properties are actually mutually inconsistent. That is the paradox. The paradox is also known as the Undetermined Boundary Paradox since it relates to situations in which a continuous dataspace can be divided into sections, but the boundary between the sections cannot be identified. POSSIBLE SOLUTIONS Many philosophers and logicians have confronted this puzzling argument and registered their analysis. Some, like Bertrand Russell , simply deny that logic works with vague concepts. Others go so far as destruction of all arguments of this form, including mathematical induction (which may or may not be a Sorites argument). Setting a fixed boundary One technique for resolving the paradox is to set a fixed boundary, say 10,000 grains. If there are less than 10,000, then it's not a heap; if there are 10,000 or more, then it is a heap. Such solutions are philosophically unsatisfactory, as there seems little significance to the difference between 9,999 grains and 10,001 grains—the boundary, wherever it may be set, remains as arbitrary as its precision is misleading. Nevertheless, just such arbitrarily precise distinctions are often drawn in the real world—for example, in setting the boundaries between Exam grades. Trivial solutions Another method is to call any set of grains that has two or more grains in it a heap. While this solves the paradox, it does not really give any insight into the dilemma. This can be considered a special case of the fixed boundary. Another trivial solution is to deny that any number of grains will make a heap -- in other words, that the word "heap" is meaningless, since the precise conditions under which it can be verified cannot be produced. Induction Applying Mathematical Induction shows that the first property combined with the third implies that a million grains of sand ''do not'' make a heap, in contradiction with the second property. Similarly, a combination of the second and fourth properties shows that two or three grains ''do'' make a heap, in contradiction with the first property. (If one was willing to write out each of the million intermediate premises of the form, "If 999999 grains do not make a heap, then 999998 do not," one could formulate the argument without relying on induction.) This contradiction arises from the interaction of the above properties. The second two fairly clearly express the idea that there is no clear line between "is a heap" and "isn't a heap". Note, however, that the four taken together also imply that any pile of sand can non-problematically be classified as "heap" or "non-heap". (This again follows from Mathematical Induction .) What the paradox seems to show is that these two ideas are contradictory. That is, one cannot simultaneously claim, when classifying X's: # That there is no clear line separating the X's that are Y from the X's that are not Y # That every X is either a Y or not a Y Another, similar paradox is: # 1 is a small number # Add 0.1 to that. 1.1 is still small. # 1,000,000 is a large number # Subtract 0.1 from that. 999,999.9 is still large. The following can be derived by mathematical induction: If X is a small number, then X + 0.1 is also a small number. If Y is a large number, then Y - 0.1 is also a large number. Applying rule 2 to the number 1 will eventually get to one million. By induction, one million must be small, which is against rule 3. Similarly, the process works in reverse. Applying rule 4 to one million eventually gets to 1. By induction, 1 must be large, which is against rule 1. Flaw with induction For a mathematical induction argument to hold, the induction step must hold ''for all'' values. It can be argued that in the Sorites the inductive steps (3 and 4) do not actually hold ''for all'' values. Some methods to resolve the paradox center around changing the induction step so that it no longer holds ''for all'' values. Multi-valued logic Another approach is to use a multi-valued logic. Instead of two logical states: ''heap'' and ''not-heap''. A three value system can be used, for example ''heap'', ''unsure'', ''not-heap''. Three valued systems do not resolve the paradox as there is still a dividing line between ''heap'' and ''unsure'' and also between ''unsure'' and ''not-heap''. Probability The definition is based upon how many people think if it is a heap or not. When the number is 1, 100% probability says it is not a heap When the number is 10, 50% probability says it is not a heap When the number is 1000000, 0% probability says it is not a heap Consensus and vagueness One attempt to clarify matters goes as follows: Many of the examples of this argument use words which refer to members of a vaguely defined set with an underlying quantitative scale which can be used to make precise analogs. For example, one could define a p-heap which has at least p grains of sand. One would then have a precise analog for which the Sorites argument would clearly fail because statement 2 above could not be applied to all p-heaps. There would be a "least p-heap" to which the item could be applied. Consider the "height" form of the argument. # A man whose height is seven feet is tall. # Reducing the height of a tall man by one inch leaves him still tall. # A man whose height is four feet is tall. Now consider this argument: # A man whose height is seven feet is considered tall by everyone. # Reducing the height of a man considered tall by consensus may change the consensus or not. If the reduction is small, then the consensus may only change slightly. # A man whose height is four feet is considered tall by very few human people. The usefulness of language is the ''consensus'' we share on the definitions of terms. Precise terms have a mechanism by which one can persuade others that a specific application of the term is valid. Vague terms have no such mechanism. If a person insists on calling a seven foot man short, one might suspect that their reference set includes many professional basketball players who play the center position, but we would hardly accuse them of a logic error. Vague terms are useful to the extent that we have consensus, but when used out of context, vague terms generally confuse. The Sorites paradox merely illustrates logical analysis of how one uses vague language. It indicates that it is a fallacy to assume that everybody agrees on the definition of a vague term. Some people may agree in its application to but not all members of the universe of discourse will as a matter of course. A consensus method essentially changes the definition of a heap from being a Subjective definition to an Objective one because the vague use of the term in the first example leaves it open to be subjectively defined by each individual person involved in the situation, the second example can actually be measured and determined EXAMPLES Real world examples of Sorites type effects can be found whenever there is a need to translate from a Continuous or many-valued domain (such as the large number of grains of sand) into a system with only two states. The film The Englishman Who Went Up A Hill But Came Down A Mountain deals with a Sorites type situation in the classification of mountains as being over 1,000 feet. The hill in question was just under 1,000 feet and the local community took earth up the hill so that it would be over 1,000 feet and classified as a mountain. Some Optical Illusions , in particular Multistable Perception and the Necker Cube , illustrate some extreme Sorities type problems. The two dimensional drawing of the Necker cube is perceived as being in one of two states, either pointing inwards or pointing outwards, akin to a borderline state of heapness/not-heapness. There is a poor man who has no money at all to his name. A rich man decides, in an extreme act of generosity, to give the poor man one million dollars. He gives it to him a dollar at a time, giving him one dollar every second. By the end of the process, the poor man has become rich; however, at what time did this first happen? Addictions are an example of the paradox - an individual can know that sustained use of a substance or performance of a behaviour can be harmful, yet continues using the substance or performing the behaviour because "one more won't make any difference" (and then it still will not do so next time, or the time after that, etc.). SEE ALSO EXTERNAL LINKS
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