Sorites paradox

The paradox of the heap (or the Sorites paradox, σωρος (sōros) being Greek for "heap" and σωριτης (sōritēs) the adjective) is a paradox that arises from reasoning with predicates which seem not to have a sharp cutoff, such as 'bald', 'tall', and the like.

For instance, it seems plausible that no heap of sand will stop being a heap just because one grain of sand is removed. This, however, leads to seemingly odd results.

One million grains of sand make a heap.
If some collection of grains of sand make a heap, then that collection minus one grain will still make a heap.
So, 999,999 grains of sand make a heap.

Repeated applications of premise 2 (each time starting with one less number of grains), will eventually allow us to arrive at the conclusion that 1 grain of sand makes a heap. On the face of it, there are three ways to avoid that conclusion. Object to the first premise (deny that one million grains makes a heap, or more generally, deny that there are heaps), object to the second premise (it is not true for all collections of grains that removing one grain cannot make the difference between it being a heap or note), or accept the conclusion (1 grain of sand can make a heap). Few, if any, reply by accepting the conclusion. In addition to advocating a response, philosophers who work on this paradox also try to explain why it is that the premise one would have to deny seems so plausible, despite being false.

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.

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.

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 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.

The predicate "rich" can also be the subject of a Sorites paradox. Suppose 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, which takes about eleven and a half days, 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.).

External links

References

"Margins of Precision" by Max Black
Boguslowski
Kit Fine
Peter Unger