suddenly changes: they start to suppose that actually all existing languages are related since they are supposed to be derivates of same proto-language that existed in a very distant epoch in past and due to this fact we can’t prove unrelatedness but can just state that a language doesn’t belong to a stock.
2.2.2. Concepts of relatedness and unrelatedness from the point of view of other sciences
In order to clear the meaning of the concept of relatedness it’s useful to pay some attention to other sciences where this concept also is used. If we take a look at, for instance: biology, physics or technical sciences we can see that many items are distributed by classes/classified despite they obviously have common origin; and considering them it is completely normal to speak about relatedness and unrelatedness. All being have common origin and so they all are relatives in a very deep level but this fact doesn’t mean they cannot be classified into kingdoms, phylums, classes, orders, suborders, families, subfamilies; the fact that ant, bear, pine tree, whale, sparrow have common ancestor doesn’t mean it is impossible to distinguish bear from whale and whale from pine tree.
However, as far as languages aren’t self replicating systems like biological systems and are closer to artifacts so any parallels between biological systems and language always should be made with certain degree of awareness since they are more allegories than analogies while correlations between languages and some artificial items are more precise, for instance: all existing cars are derivates of steam engine that existed in the middle of 19th century, but it doesn’t mean we can’t classify cars/engines and speak of relatedness and unrelatedness of certain types.
These examples evidently show us the following:
1) When they say about an item that is related with another it means “they both belong to the same class”.
2) It is possible to speak about relatedness and unrelatedness of certain items even though all classes of them have common origin.
2.2.3. Concepts of relatedness and unrelatedness from point of view of set theory and abstract algebra
Concept of relatedness is actually equivalence relation since it meets necessary and sufficient requirements for a binary relation to be considered as equivalence relation:
1) Reflexivety: a ~ a: a is related with a;
2) Symmetry: if a ~ b then b ~ a: if a is related with b then b is related with a;
3) Transitivity: if a ~ b and b ~ c then a ~ c: if a is related with b and b is related with c then a is related with c.
If an equivalence relation is defined on a set then it necessarily supposes grouping of elements of the set into equivalence classes and these classes aren’t intersected (Hrbacek, Jech: 1999).
2.2.4. Particular conclusions on the concepts of relatedness and unrelatedness for linguistics
When it is said that certain languages are genetically related (or simply related) it means that these languages belong to the same stock or even to the same group.
Taking into the consideration what has been said in 2.2.2 we should keep in mind that in the case of languages there are actually no positive evidences that all languages existing nowadays originated from the same ancestor, i.e.: monogenesis is still an unproved hypothesis, though anyway even if all languages can be reduced to the same proto-language that existed in a very distant past it doesn’t mean yet we can’t speak of their relatedness/unrelatedness.
Then, taking into consideration what has been said in 2.2.3 we can say the following:
The set of languages existing nowadays on the planet is rather well described: we know that there are about 7102 languages and about 151 stocks and 83 isolated languages (Ethnologue: 2015), so we can speak about 234 stocks; and we hardly can expect discovering of some new unknown languages. Thus, we can say that we have rather complete image of set of languages and that there are about 234 classes of equivalence/relatedness.
If we take an X stock, we obviously can show many languages which don’t belong to the stock, i.e.: languages which are not related with language x (a random language of X stock), for example: in the case of Indo-European stock there are many languages which are not related with English: Arabic, Basque, Finnish, Georgian, Turkish, Chinese, Japanese, Hawaiian, Eskimo, Quechua and so on. In the case of Sino-Tibetan stock there are many languages which aren’t related with Chinese: Arabic, English, Eskimo, Finnish, Japanese, Turkish, Vietnamese and so on.
Thus, we can conclude the following:
1) Relatedness means “language belongs to a stock” unrelatedness means “language doesn’t belong to a stock”.
2) If set of 234 classes/stocks has been set up then it obviously supposes that there should be a possibility of classification, i.e.: we can say whether a language belongs to a stock; moreover, we always can show some languages which don’t belong to the stock. If possibility to prove unrelatedness is denied then we actually can’t establish scopes of stocks and can’t distinguish one stock from another; then even a single stock hardly could have been assembled.
3) Any two randomly chosen languages can be related or not related, i.e.: there can be no “third variant” since relatedness/unrelatedness supposes the existence of classes which don’t intersect. If a language of X stock is related to a language of Y stock it means that these stocks are related.
4) Possible objection can be the following: one can probably say that it is impossible to make precise conclusions in linguistics. Actually, I don’t think someone can seriously say this, however, if someone would speak out something like this I can only point on the fact that very long ago people thought that precise conclusions are impossible in physics. Possibility of precise estimations and precise conclusions depends on scholars’ will and on scholars’ intellectual courage only, but not on material itself; any material can be represented as item that can’t be formalized, and many items have already been successfully formalized.
2.2.5. An important consequence: transitivity of relatendness/unrelatedness
If we have proven unrelatedness of an x language belonging to X stock with y language that belongs to Y stock then, due to transitivity of unrelatedness, it means that x is not related with the whole Y stock.
2.3. Applying PAI method to some unsettled hypotheses
2.3.1. PAI against Nostratic hypothesis
Basing on comparison of lexis adepts of Nostratic hypothesis state that Indo-European stock, Kartvelian stock, Uralic stock and Turkic stock are relatives. Let’s see and test whether, for instance, Indo-European and Turkic stocks could be relatives.
PAI of Indo-European is about 0.5 (calculated after data represented in 2.1.4.2);
PAI of Turkic stock is about 0.012 (calculated after Yazyki mira. Tyurkskiye yazyki 1996).
Values of PAI differ more than tenfolds so these stocks evidently can’t be genetically related.
2.3.2.
