Paradox of the heap

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The paradox of the heap , also Sorites paradox (from the Greek sorós : heap), is a phenomenon that occurs with vague terms. The paradox becomes apparent when an attempt is made to define something as a heap: It is not possible to specify a specific, non-arbitrarily determined number of elements of which a heap should at least consist, because the concept of a heap implies that something is a heap is, even a heap remains if some of its elements are removed. If you reverse this idea, it becomes difficult to say when a collection of elements can count as a heap. The term “heap”, understood as an accumulation of similar parts, cannot apparently be clearly defined. Other similar vague predicates are also referred to as Sorites cases , e.g. B. in the paradox of the bald .

The formulation as a cluster paradox probably goes back to Eubulides or Zeno of Elea , as well as a number of other famous paradoxes .

Problem

There are several variants of the cluster paradox, but they all point to the same problem . A version with grains of sand is presented here as an example:

We assume that 100 grains of sand are a pile (if this is disputed, you can start with a higher number). The second premise (which is also to be understood as an axiom ) is decisive : If we remove a grain of sand from a pile of sand, then the remaining grains of sand continue to form a pile. Thus it can be concluded: because 100 grains of sand are a heap, 99 grains of sand are also a heap; but because 99 grains of sand are a heap, 98 grains of sand are also a heap, etc. Ultimately, we arrive at the statement that one grain of sand is a heap. That is a statement that we intuitively do not want to accept.

The paradox can also be raised the other way around; namely, by assuming that a grain of sand is not a heap and that adding a grain of sand does not turn a heap of sand into a heap of sand . Then there is never a heap, even if we add any number of grains of sand. This is also counterintuitive since there are heaps.

The same paradox arises when we try terms like "large" and "small" e.g. B. to be defined in relation to body size, or if colors are to be defined: Sorites paradoxes are a typical property of all vague predicates.

The method used to pose the problem is similar to complete induction : the paradox does not consist in drawing a single conclusion, but in stringing together a large number of uniform conclusions. There is a chain link : 'If grains are a pile, then grains are a pile; Grains are a pile, so ... - If grains are a pile, then 1 grain is a pile; Grains are a heap, so 1 grain is a heap. ' Chain links were therefore also referred to in tradition as Sorites links. However, the property of being a bunch, not the individual grains distributed .

In principle, the conclusion could also be easily accepted: We could already define a grain of sand as a heap, since it is a semantic and not a mathematical definition. From the point of view of the philosophy of language, however, this does not seem very attractive: The point here is to use terms in such a way that they capture intuitions.

Solutions that dispute the second premise are more common. That is, by removing a grain of sand from a heap, we sometimes break up the heap as such. This position has a serious problem: where exactly is the boundary between a heap and an arrangement of grains of sand that can no longer be called a heap?

Resolutions

There are several ways to deal with the problem. On the one hand, it can be said that there is a clear criterion according to which a collection of grains of sand can be described as a heap. Second, it can be argued that there is a transition area in which a cluster can not be described as either a pile or a non-pile . Third, the problem can be criticized in its approach and understood as a criticism of the ambiguity of our natural language.

Uniqueness of the scope

Here it is claimed that if there are a certain number of grains of sand a pile is created. A certain number is hardly ever mentioned; a statement like “40 grains of sand is a heap, whereas 39 grains of sand are not a heap” would also be difficult to justify. However, thank God Frege had the hope that such a number could somehow be found:

“Through intellectual work [...] it is often only possible to recognize a concept in its purity, to peel it out of the foreign coverings that hid it from the spiritual eye. [...] Instead of finding a particular purity of terms where one believes to be close to their source, one sees everything blurred and unseparated as if through a fog. "

In this context, Frege strongly criticizes John Stuart Mill , who did not consider the concept of the heap to be clearly definable. He is of the opinion that research can also make progress in defining terms, building on one another, and unraveling these terms step by step.

Unlike Frege, Timothy Williamson does not believe that a concrete limit can ever be found; nevertheless they exist. Even color tones can only be distinguished from people outside of a “margin for error”; d. In other words, we perceive two very similar color tones to be the same, even if they have physical differences - people only notice differences when there are slightly larger differences between the color tones. It is similar with vague terms: As far as there is a big difference, we are able to differentiate between “pile” and “not pile”. In the case of small differences such as between 39 and 40 grains of sand, our ability to distinguish is not fine enough to arrive at a result. This position is also known as epistemicism .

Gray areas

If a precisely defined transition point is rejected, it can also be asserted that certain collections can neither be designated as heap, nor can they be said to be not a heap. For example, it can be said that the statement “40 grains of sand are a heap” is neither true nor false, but that it has a different truth value. Such a solution can be represented with the help of a multi-valued logic .

A first attempt is to introduce a gray area or penumbra . This is an area that lies between the positive and negative extensions of the term "pile". In this area it cannot be said that the accumulation of grains of sand is a heap, nor that it is not a heap. These statements would then be assigned an indefinite truth value in the sense of a three-valued logic . Variants of this solution can also be represented with different intermediate stages, for example with a five-valued logic or even more truth values.

If, however, a limited number of truth values ​​are assumed, another problem arises: where is the line between a collection of grains of sand that can truly be called a heap and a collection of which this cannot be said, either true or false? Justifying this limit is hardly easier than in the classic approach with two truth values. In addition, the problem cannot be solved by adding a limited number of other truth values, but only broken down into more and more gray areas.

The use of fuzzy logic is more of a solution , in which there are an infinite number of truth values ​​between "true" and "false". Then the question of an exact limit no longer arises. The cluster paradox is often used as an argument for fuzzy logic, but this logic is controversial because of its other consequences.

criticism

It can be stated that the above statements with their conclusions are made in a formal system and have nothing to do with the real world per se, the thoughtfulness of the philosopher nevertheless remains. The separation of the formal system, which is primarily intended for the exact description and conclusion, from the real-world meaning makes it possible to defuse the cluster paradox, but it does not resolve it. It is z. B. provided that the number of grains of sand alone decides what a pile of sand is. But even 100 grains of sand, if they are in a row next to each other, do not form a heap at all, that is, a certain arrangement in space and the presence of gravitational force are necessary to produce something that is colloquial and similar. U. corresponds to the term pile .

If you stay within a formal system, a concept word only takes on a symbolic reference. Only humans have the ability to assign a further real-world reference to a formal structure . The human being intuitively assigns a real-world meaning to the symbol heap , which can only be assumed within a formal system if it depicts the real world comprehensively and in detail. The formal system cannot do this if the specification is missing. This is where the paradox is rooted. Replacing the symbol heap for mountain , which can have the same meaning in the formal sense, would result in the same paradox, since the transition from "mountain" to "hill" can also take place by removing the smallest amounts of material.

If one subjects our colloquial terms - in this case heaps - to exact methods with stringent conclusions, pseudo-problems can arise and possibly also wrong results are produced.

According to the philosophy of language criticism of the early Ludwig Wittgenstein , such problems can only be resolved by analyzing them as misuse of our language. Colloquial terms have a vague scope and, for their use in formal systems, such as are characteristic of mathematics and logic, their meaning must be clearly defined, i.e. also redefined if necessary.

An ideal definition of colloquial terms can neither be justified nor enforced with scientific guidelines. Their meaning always follows the expediency in the respective area of ​​use.

For the late Ludwig Wittgenstein ( Philosophical Investigations ), the meaning of the word is its use of language, in this case it is an appropriate characterization of “pile”, and not the designation of “not pile”, since this word is not used as a term in the Has colloquial language. In colloquial language, the term heap is mainly used when counting its elements is inexpedient or seems impossible for the speaker, think of piles of sand or piles of chickens. A request to count the accumulation so that the “non-heap” can be determined violates the initial condition, which excludes counting. Herein lies the subtlety of the heap paradox.

Individual evidence

  1. ↑ According to Timothy Williamson, Vagueness, London 1998, p. 8 and Dominic Hyde:  Sorites Paradox. In: Edward N. Zalta (Ed.): Stanford Encyclopedia of Philosophy .
  2. ↑ According to Friedrich Kirchner, Carl Michaelis u. Johannes Hoffmeister, Dictionary of Philosophical Terms, Leipzig 1944, p. 647 (with reference to Aristotle, Physik, 250b )
  3. ↑ See Ulrich Pardey, Unscharfe Grenzen. On the Heap Paradox, Darwinism and Recursive Grammar , Journal for General Philosophy of Science 12-2002, Volume 33, Issue 2, Springer, Berlin 2002, pp. 323-348.
  4. ↑ Gottlob Frege, The Basics of Arithmetic, Reclam p. 21.
  5. ↑ See Gottlob Frege, The Basics of Arithmetic, Reclam p. 21 and John Stuart Mill, System of Deductive and Inductive Logic, Volume 2, Braunschweig 1877, pp. 249–252.
  6. ↑ See Timothy Williamson, Vagueness, London 1998, pp. 216-247, especially pp. 230-234.
  7. ↑ See RM Sainsbury, Paradoxien, Reclam pp 49-53.
  8. ↑ Timothy Williamson, Vagueness, London 1998, pp. 111-113.
  9. ↑ See Timothy Williamson, Vagueness, London 1998, pp. 113f., Pp. 120-122 and pp. 127-131.
  10. ^ Ludwig Wittgenstein: Tractatus logico-philosophicus, Logisch-philosophische Abhandlung. Suhrkamp, ​​Frankfurt am Main 2003, section 4.0031
  11. ↑ Ludwig Wittgenstein: Philosophical Investigations. Critical Genetic Edition. Ed .: Joachim Schulte. Scientific book society. Frankfurt 2001. § 43

literature

  • Ulrich Pardey: Blurred boundaries. On the heap paradox, Darwinism and recursive grammar , Journal for General Philosophy of Science 12-2002, Volume 33, Issue 2, Springer, Berlin 2002, pp. 323-348.
  • Piotr Łukowski: Paradoxes , Studia Logica Library, Trends in Logic Vol. 31, Springer, Dordrecht a. a. 2011, pp. 131-170.
  • Richard M. Sainsbury : Paradoxien , transl. By Vincent C. Müller, Reclam, Stuttgart 1993, 2nd ed. 2001, pp. 39-72. (Translated by: Paradoxes , Cambridge University Press, Cambridge, New York et al. 3rd ed. 2009, pp. 40-48).
  • Timothy Williamson : Vagueness , Routledge, London 1998, pp. 8–35 and the rest of the book on various approaches to solving the problem.

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