Sorites paradox

The paradox of the heap, also Sorites paradox (from the Greek Soros: heap ) is a phenomenon that occurs in vague terms. The paradox is found when attempting to determine something as heap: it can not find any, do not arbitrarily decided number of elements specify from which a pile should at least exist, because the term " heap " implies that something that is a heap is also a heap of remains when removing some of its elements. One hand, this idea around so it is difficult to say at what point a collection of elements in a heap may apply. The term " heap ", understood as the accumulation of similar parts, can not seem to clearly define. With other similar stored vague predicates is spoken by Sorites cases, for example when the bald -headed paradox.

The formulation as a heap paradox is probably due to Eubulides, or on Zeno of Elea, as well as a number of other famous paradoxes.

Problem

There are several variants of the heap paradox, but they all point to the same problem. By way of example they should be presented here is a version with sand grains:

The paradox can also be the other way around wind; by namely assume that a grain of sand is not a heap and there is no sand pile can not be by adding a grain of sand turn into a pile of sand. Then never created a heap, even if we add as many grains of sand. This is also counter-intuitive, because there are heaps.

The same paradox also arises when we try terms such as " large" and " small", for example in relation to define on body sizes, or if colors are to be defined: Sorites paradoxes are a typical feature of all vague predicates.

The method used in the problem definition is similar to the complete induction: The paradox does not consist in drawing a single conclusion, but in the series of many uniform conclusions. The result is a chain lock, 'If n grains are a heap, then ( n - 1 ) grains of a heap; n grains are a bunch, so ... - If ( n - (n - 1)) grains are a bunch, then 1 grain is a heap; ( n - (n - 1)) grains are a bunch, so one grain a heap of ' chain conclusions are therefore also referred to in the tradition as Sorites conclusions. . However, the property of being a pile, not distributed over the individual grains.

In principle, the conclusion would also accept simple: We could already define a grain of sand in a heap, as it is a semantic rather than a mathematical definition. From the philosophy of language point of view, however, appears less attractive: this is just a matter of using terms such that they capture intuitions.

Rather more common solutions are those that deny the second premise. This would mean that we sometimes resolve by the removal of a grain of sand from a heap the heap as such. This position has a serious problem: Where exactly is the line between a heap and an array of sand grains that can no longer be called a heap?

Resolutions

There are different ways to deal with the problem. On the one hand may be argued that there is a clear criterion that an accumulation of sand grains can be described as clusters. Second could be argued that there is a transition region in which a collection can be referred to either as clusters or as a non- heap. Third, can the problem criticize the approach and interpreted as criticism of the ambiguity of our natural language.

Uniqueness of the concept of scale

Here it is claimed that a certain number of grains of sand a heap is formed. A certain number is here but almost never mentioned; a statement like " 40 grains of sand are a heap, however, are not a bunch of sand 39 " would be too difficult to justify. However, Gottlob Frege had the hope that such a number but somehow could find:

Frege criticized in this context violently John Stuart Mill, who did not consider the notion of the heap for clearly defined. He is of the opinion that research can also aid in the determination of terms that build on each other, move forward and could unravel these concepts gradually.

Unlike Frege is Timothy Williamson does not think that a concrete border could ever find; yet there they are. Also colors are to be distinguished only by people outside of a "margin for error"; that is, two very similar shades are perceived by us as equal, even if they have physical differences - at larger differences between the hues also people notice differences. Similarly, it was also in vague terms: If there is a big difference there, we are capable of distinguishing for example between " heap " and "non- heap ". For small differences such as 39-40 grains of sand is our ability to distinguish not fine enough to arrive at a conclusion. This position is referred to as Epistemizismus.

Gray areas

If an exactly certain threshold point is rejected, can also be argued that certain collections can neither be described as a bunch, still can be said of them that they were not a bunch. For example, it can be said that the statement " 40 grains of sand are a heap " is neither true nor false, but her zukomme another truth value. Such a solution can be represented by means of a multivalued logic.

A first attempt is to introduce a penumbra. This is an area that lies between the positive and the negative extension of the term " heap ". This area can not be said that the accumulation of sand grains is a heap, still that she was not a bunch. These statements would then be in terms of a trivalent logic associated with an indeterminate truth value. Variants of this solution can be represented with various intermediate stages, so for example with a pentavalent logic, or more truth values.

However, when starting from a limited number of truth values ​​leads to yet another problem: Where is the boundary between a true example as a pile to be designated accumulation of sand grains and a cluster, from which one can say this is neither true nor false as? To justify this limit is hardly easier than in the classical observation with two truth values. Moreover, the problem can not be solved, but only analyze in more and more areas of gray by adding a limited number of other truth values ​​.

Even more of a solution is the use of fuzzy logic, in which there are infinitely many truth-values ​​between "true" and "false". Then the question of the exact limit is no more. The heap paradox is often cited as an argument for fuzzy logic, but this logic is because of their other consequences quite controversial.

Criticism

It can be argued that the above statements are made with their conclusions in a formal system, and have nothing per se to do with the real world, the thoughtfulness of the philosopher remains nonetheless. The separation of formal, primarily intended for the exact description and conclusion of the system of real-world meaning it does allow, to defuse the heap paradox resolved it shall not be. There is, for example, provided that only the number of grains of sand decides what is a pile of sand. But even 100 grains of sand when they are in a row next to each other, newly forming no pile, that is, a particular arrangement in space and the presence of the gravitational force are necessary to produce something close to what colloquially might the term clusters.

If one stays within a formal system, accepts a term word, only a symbolic reference. Only man has the ability to assign a formal structure a more real-world reference. The symbol heap allocates man intuitively a real-world significance, which can make it within a formal system only accept if this reflects the real world comprehensively and in detail. The formal system can not afford this when the detail has been omitted. Herein, the paradox is rooted. Can be Exchanged by the symbol bunch of mountain, which may take the same meaning in a formal sense, the same paradox would result, since the transition from "mountain" to " hills " via the removal of small amounts of material take place.

If terms of our colloquial language - in this case heap - subjecting exact methods with stringent inferential, apparent problems may arise and possibly incorrect results are produced. After the critical philosophy spoke of the early Ludwig Wittgenstein believes one can only resolve such problems by analyzing the misuse of our language. Colloquial terms have a vague scope and need for their use in formal systems, such as are characteristic of mathematics and logic, be clearly defined in their meaning, are thus possibly also redefined.

An ideal definition colloquial terms can not create or enforce scientific standards. Their meaning is always followed by the expediency in the respective area of ​​use. For the late Ludwig Wittgenstein, the meaning of the word is its use of language, in this case it is convenient characterization of " pile " and not about the name of "non- heap " because this word has no use as a term in everyday language. In everyday language, the term pile is mainly used when counting its elements is impractical or seems impossible for the speaker, just think of piles of sand or flock of chickens. A call to count the collection, so that the " non- heap " can be determined, but it violates the initial condition, which excludes counting. Herein lies the subtlety of the heap paradox.