Paradox sorite

The first of the paradoxes sorites is the paradox of the heap (sorites is an adjective derived from sõros which in old Greek means “ tas ”). He was formulated at fourth century BC by the Greek philosophers of the megaric École. He tends to show impossibility that there is to constitute a heap by the accumulation of grains.

This type of Paradoxe is based on the reasoning by recurrence and the semantic blur inherent in the definitions of the words of the usual language.

Statements

In its original form, the paradox of the heap states

*un isolated grain does not constitute a heap.

*l' addition of a grain does not make a not-heap, a heap.

One from of deduced that

* one cannot constitute a heap by the accumulation of grains.

For of convraincre, it is enough to reason by the absurdity, one obtains a contradiction by recurrence then. In addition, to denounce the second premise implicitly amounts stating

*il exists a number N such as: N grains does not form a heap, n+1 grains form a heap.

If one postulates

now * a heap remains a heap if a grain is removed to him.

Then, considering a heap, one can deduce from it by recurrence that

* a single grain or even the absence of grains always constitutes a heap.

the paradoxes sorites are often more simply formulated like thorny simple matters:

* How much grains is it necessary to make a heap?

Alternatives

the Paradox of egg and the hen
the paradox of bearded the :

What constitutes a beard is not judged except for a hair.

Or:

One can remove a hair of barb to bearded, it will remain bearded; however, after a certain number of removed hairs it will not be it any more. Starting from how much hairs it will change statute?

the paradox of the batrachian (formulated by James Cargile):

If one films the evolution of a tétard in Grenouille, it is necessarily on the film an image of tétard followed by a frog image. However practically such a couple of images cannot be.

the paradox of the rich man :

a man without the penny hopes to grow rich by the begging. But a foreigner ruins his hopes while saying to him: “ a man does not pass from poverty to the wealth of receiving a sou. ”. The beggar will thus remain fatally poor.

Starting from which age is one old?
Combien is it necessary some so that there is much of it?

The alternatives can be multiplied ad infinitum by considering any word of the French language taken in various contexts.

Answers

For the ancient philosophers, the paradox sorite existed only in one vision essentialist of the world, where it was supposed that the categories of the reason preexisted to the exercise of this one (vision of Plato opposed to that of Aristote).

For the analytical philosophical such as Bertrand Russell, this type of paradox highlights the semantic laxism of the definitions in the usual language. One could deduce from it that one cannot reason healthily out of the framework of the formal languages. However, this framework being too restrictive for the everyday usage, one will recommend the use of the fuzzy Logique.

The answer of the logicians consists in saying that any usual definition of the “ tas ” is an axiom such as the formula “ a heap remains a heap if a grain is removed to him; ” or not modélisable is either contradictory.

One cannot apply quantitative considerations to a qualitative concept. However the concept of heap cannot be regarded as quantitative (from a mathematical point of view) because the question “ N objects do it form a tas ? ” does not have clear answer independent of the choice of N (except in Algorithmique). The definition of a heap as a quantity is thus invalid. In addition, “ a heap remains a heap if a grain is removed to him; ” does not have direction if heap is a qualitative concept. The Analogie with the concept of Voisinage in Topologie is particularly lighting.

In current language, the definition of a word is not given beforehand; it builds by the use of the mot. the use of “ tas ” in a sentence this word defines, relatively within the meaning of the other words of the sentence, and in so far as the sentence is relevant. Conversely, the relevance of the sentence results from the direction of the word “ tas ”. The language is then autoreferential.

Let us consider for example that if a grain nourishes a bird, then a heap of grains nourishes a cloud of birds. The assertion “ a heap of grains nourishes a cloud of oiseaux ” is all the more acceptable (as a plain truth) that terms “ tas ” and “ nuée ” are fuzzy. It is included/understood whereas the “ tas ” is with the “ grain ” what the “ nuée ” is with the “ oiseau ”. A more precise definition of the “ tas ” would invalidate the assertion in the absence of adequate precision on the “ nuée ”. This example illustrates the possibility of reasoning in a not-fuzzy way on concept fuzzy. It also shows the relativistic character that the sentence gives direction of the word “ tas ”. It shows finally that the concept of Paradigme is more adequate than that of Définition.

However, in current language, the “definition” of a word is not single; it is contextual. Concept of “ tas ” is not the same one according to whether one considers a “ heap of grains ” (entity énumérable), a “ heap of boue ” (entity not énumérable, but quantifiable), a “ heap of ennuis ” (nonquantifiable entity) or quantifying concept of “ heap of éléments ” in Algorithmic.

Thus the sentence “ Does one need more troubles than of grains to make a tas ? ” is a nonsense created by a contextual amalgam. It is the same for “ an isolated grain does not constitute a heap. the addition of a grain does not make a not-heap, a tas. ”

This last interpretation brings closer the paradoxes sorites to the Paradoxe of the Gruyere.

Comment by Hegel

According to Hegel ( Encyclopedia of philosophical sciences , Logic, Doctrines to be it, Measurement, § 107-111, in particular in the Addendum with the § 108), the argument sorite, far from being simply “a oiseux chattering of school”, is of great importance. After opposehaving indeed opposed the concepts of “quality” and “quantity”, Hegel sees in “measurement” the synthesis in act of these two concepts: the argument of the sorite has this of invaluable which it makes it possible to show how quality can transform the quantity, and reciprocally.

By adding one by one grains of sand, one indeed carries out a simply quantitative increase, which does not involve qualitative change. The change being insensitive, the reciprocal externality of the concepts of quantity and quality seems checked: the quantitative modification does not involve qualitative modification.

And yet, a material change leads well to a qualitative change, when it is not simply any more the number of units which differs, but truly the order of magnitude . Hegel gives a political example of it: if a constitution is adapted to a small country, a minor modification of extended from its territory will not call a modification of the constitution. On the other hand, if the extension is important, the constitution will not be necessarily any more adequate. One would be well in sorrow to find the limit exact of surface which made us pass from an adequate constitution to an inadequate constitution.

The paradox sorite watch that the radical separation of the concepts of quality and quantity does not make it possible to positively think the passage of the one in the other, that one however observes in this type of experiments. This is why Hegel substitutes for the opposition of these two concepts the concept of measurement, unit of the quantity and quality, likely of instancier in quality, quantity, or to allow the passage of quality the quantity, of the quantity to quality.

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