Cognitive Science
and Natural Language

I. Objects

1) One is able to detect various objects and separate one from the other although it is not easy to describe how such detection works in all the detail except that ultimately the object is perceived and as any percept it is stored in memory as its remembrance idea.

Sorts of objects according to senses involved

If the visual faculty is used to detect an object it is called a space object which is an object in the usual sense of the word.

If the hearing faculty is used then the detected object is in general a sound and may be a word of natural language in particular in which case it may be reproduced by pronunciation.

Further possibilities may occur if the faculty of taste, smell or touch is involved but we are not interested in them here.

Partial definition of a spoken natural language of an individual human being (HB):

When a space-object is detected, thus also perceived and later a word is perceived by the same human being (HB) then these two percepts of object and of word may be associated psychologically in HB's mind and memory permanently. The word is called a name of the object concerned and one says that the word refers to the object concerned and one also says that the object concerned is the meaning of the word. The correspondence established between words and objects may be called the basic semantics of words which always underline a natural language.

Ambiguity

Basic semantics and the corresponding natural language itself is called ambiguous if there exists a word of the language concerned which is a name of two different objects, thus a homonym (like the word 'page' which refers on the one side to a sheet of paper in a book , but on the other one it refers to a young servant at King's court). Thus, English is ambiguous and so are many other natural languages although it is quite unclear why any ambiguity is admitted: it is certainly unnecessary and evidently undesirable, at least, from the communication point of view. In addition, various types of ambiguity may be differentiated according to which category or categories the two meanings belong. Then the example above shows a categorical ambiguity because both meanings are of the same category, namely they are

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objects. English is ambiguous also not categorically because e.g. the word 'comb' means on the one side a particular objects but on the other one also an attribute, namely the verb denoting the corresponding activity performed by the comb.

Forms of natural language

As said in the partial definition of natural languages of a human being it is assumed that the spoken form is the first one as it is accepted by linguists everywhere. The written or printed form of natural language is derived from the spoken one by replacing the heard words by visual perceptions of written or printed words. These two sorts of form of natural languages are very frequent but there is even a third sort of form of natural language, namely the sign-language of deaf people where the words are replaced by the corresponding signs and gestures. In this case there are some differences between American and Japanese sign languages, but this is behind the scope of this paper.

On the one side it should be reminded that the first fact 1) is very well known and the correspondence between words and objects is reflected even in the title of a book "Word & Object" by logician and philosopher Willard Van Orman Quine (The M.I.T. Press 1960).

On the second hand side it seems obvious that not all words of a natural language refer to objects because many words refer to attributes (predicates) which concerns the second fact (sub 2)) although it is not discussed in the Quine's book at all.

II. Attributes

2) One is able to differentiate various attributes of objects. With an attribute usually a cognitive action is associated which leads to a perception of that attribute in a similar way as perception of objects. Let us consider as an example the attribute red (or to be red) the cognitive action of which is very simple and consists just of having a look at the object concerned under the normal daylight sunshine condition when perceiving the redness (or red colour). In fact the same cognitive action is used when any other colour as green, or black or white is concerned when the perception of greenness, or blackness or whiteness is assumed, respectively. It should be reminded that the perception of a colour is as objective as the perception of an object because the objectivity consists of an independence of human wishes and not of the cognizing subject himself; obviously all perceiving depends of the cognizing subject concerned.

Although one cannot justify saying that two different persons perceive the same redness (as there is no way how to compare their perceptions themselves). Nevertheless all normal people correctly differentiate various colours (obviously an abnormality of a daltonist or a colour-blind person must be excluded). The traffic control by traffic-lights on a crossroad is a clear manifestation of the objectivity of colours.

Finally, let us consider an another example of attribute "longer than" consisting in fact of two consecutive words, the cognitive action of which is also very simple and consists of putting one object

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next to the other and then observing whether the perception of exceeding occurs (obviously that objects which exceeds is longer than the other one). E.g. in my room after putting the chair next to the desk the observation says that the desk exceeds and the chair does not exceed. Thus the desk is longer than the chair.

Now basic semantics can and should be extended to all words which name various attributes and again a correspondence among words of attributes and attributes themselves is established. Usually all words naming objects are called nouns in contradistinction to the names of attributes where some of them are adjectives and others are verbs.

Sorts of attributes and their characteristics.

An attribute which concerns (or requires) one single object is called a property; if it concerns (or requires) more than one object it is called a relation. If two or three objects are required the relation is called binary or ternary, respectively.

The most characteristic feature of an attribute At requiring n>1 objects consists of the following fundamental facts considering an n-tuple of objects (Obj₁, Obj₂, ..., Obj_n). There are two possibilities:

either the attribute At may be attributed to the n-tuple concerned, in the words At holds for that n-triple and At is true for it,

or the attribute At may not be attributed to the n-tuple concerned, in other words At does not hold, and At is false for that n-tuple. It should be probably reminded that the concept of an n-tuple is used here in the usual mathematical sense, that is it is a sequence of some elements such that they may be provided with an integer i, 1 Ł i Ł n, telling us that the i-th element is concerned and obviously as it is a sequence, its elements need not to be mutually different. In mathematics the integers i are usually provided by the indexing directly, but in natural languages no indexing is allowed; instead various syntactical or grammatical means with some conventions are exploited to replace the indexing.

Examples of syntactical or grammatical means

(1) a fixed word order within language phrases (in English)

(2) the declination of nouns and pronouns (in Latin and Czech)

(3) the use of prepositions (in English and Czech)

Another characteristic feature of an attribute At which requires n-objects consists of a possibility to negate it, that is to determine another attribute NEGAt which requires the same number of objects as At does but is complementary to it according to the following two requirements:

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(4) NEGAt (Obj₁, Obj₂, ..., Obj_n) holds if and only if At (Obj₁, Obj₂, ..., Obj_n) does not hold

(5) NEGNEGAt = At, that is the double negation of At is again the original attribute At

III. Knowledge

An elementary piece of knowledge concerning an attribute At which requires n ł 1 objects is a language phrase, that is a sequence of words which contains the attribute name At concerned and besides that exactly n noun-phrases NP₁, NP₂, ..., NP_n such that At(NP₁,NP₂,..., NP_n) is true. Such phrase is usually called a positive proposition.

The second sort of elementary piece of knowledge concerning the attribute At, or more specifically its negation NEGAt is a language phrase which contains the attribute At concerned and besides that exactly n nounphrase NP₁, NP₂, ..., NP_n such that At (NP₁, NP₂, ..., NP_n) is false. Such a phrase is called a negative proposition.

In each proposition either positive or negative there is exactly one attribute which is concerned. It is called the main attribute of the proposition. All other attributes which may occur in a proposition are called secondary and they occur in particular noun-phrases.

Examples of propositions concerning various sorts of verbs.

(6) A transitive verb (like to see) is an attribute which is a binary relation, therefore it requires two objects and the corresponding proposition contains two noun-phrases and the verb stands between them. The first noun-phrase (usually called grammatical subject) stands at the beginning of the proposition and would be in the nominative if declination is available. The second noun-phrase would be in the accusative and stands at the end of the proposition (it is usually called grammatical object); in English an example of a positive proposition could be "My mother sees a green tree." where sees is its main attribute while my and green are secondary attributes.

(7) A verb (like to walk or to go) is intransitive which means it is not a relation but only a property requiring just one single object. Therefore the proposition contains only one single noun-phrase standing at the beginning of the proposition and followed by the verb. A negative proposition concerning the main attribute walk will be "My wife does not walk.".

(8) English is peculiar in that sense that the verb walk (but not go) may be used as transitive. Then according to (6) two noun-phrases occur in propositions like "My wife walks her dog.".

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(9) The most complex propositions of natural language (so far we know) concern double transitive verbs (like to borrow, or to lend, or to write) because they are ternary relations which require three objects. Thus in corresponding propositions there must be three noun-phrases. The first one is at the beginning of the proposition, usually called the grammatical subject, followed by the verb 'borrow' or its negation, which is followed by the second noun-phrase called the first grammatical object. The third noun-phrase is at the end of the proposition and is preceded by the preposition "from". This is an orthodox English word order as in the following example "I borrowed a new car from my son.". The power of using preposition (3) is demonstrated by the following phrase "I borrowed from my son a new car." which is in an unorthodox wordorder but which is quite understandable as equivalent to the first one.

IV. Learning natural language

Any human being HB₀ which masters a natural language NL is able to teach any other human being HB₁ the language NL through the following learning process.

(10) The teacher HB₀ detects, that is perceives an object or attribute and waits until the pupil HB₁ perceives the same object or attribute, respectively (thus Russell's ostensive definition [Russell] is applied and only after that the teacher HB₀ says the corresponding name which is a word from NL and the pupil HB₁ repeats its pronunciation (eventually several times until HB₀ considers it as correct and the word pronounced by HB₁ begins to be associated with the object or attribute concerned in HB₁'s mind. Obviously the same process must be repeated for all objects and attributes concerned in NL.

Very often the teacher HB₀ is a mother and the pupil is her child. Then the language NL learned by the child is called its mother-tongue. The learning process (10) also supports the idea of the biblical common great grandmother Eva although it seems to be unclear how Eva could master her natural language NL as she could not learn it according to (10) because she is understood as the first speaker of NL. An answer may be provided by the following

(11) determination of a new natural language

Any human being is able to determine a new natural language NL^* by the following process: the HB detects an object or an attribute and then HB chooses (arbitrarily!) a sound which HB can reproduce and will consider as a word of NL^*. The choice of sounds or words is in general quite arbitrary but usually the following two requirements are satisfied.

(12) easy pronunciation of chosen sounds should be demanded and therefore e.g. the consonants at the end of words should be voiceless because if they would be voice they would be

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pronounced as voiceless by any careless or sloppy pronunciation (therefore English good is often pronounced as goot by foreigners).

(13) in order to exclude any ambiguity each new word should be chosen only once to name an object or attribute.

In a similar way as is the determination of new natural language the extension of a given natural language by adding new words for new objects or attributes is defined.

These extensions of a natural language correspond to its various phases of historical development. In the first phase it was, probably, some "family language" when only members of family, domestic animals and pets, family house and its furniture were objects concerned but later various artisan objects were added etc.

V. Translation

A natural language NL is called translatable into another natural language NL^* if both language NL and NL^* concerns the same objects and attributes and if they contain the same sorts of language phrases as they may be determined by various transformations according to Chomsky grammars. Here only propositions are discussed but in general there are also orders or questions and obviously the sort of a phrase should be preserved by translation. Thus propositions should be translated as propositions more specifically, NL is translatable into NL^*

if the following two requirements are satisfied:

a) to each object which has its name Obj in NL there exists its name Obj^* in NL^*; and similarly

b) to each attribute which has its name At in NL there exists its name At^* in NL^*.

Now a correspondence, that is a function TR, called the translation of NL into NL^*, can be defined by the following three requirements:

(14) TR (Obj) = Obj^* for each object name Obj from NL

(15) TR (At) = At^* for each attribute name At from NL

(16) TR (NEGAt) = NEGAt^* ; the translation of negated attribute is the negation of its translation

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then the following assertions are valid:

(17) If At (NP₁, NP₂, ..., NP_n) is a positive proposition in NL and if we denote by NP^*_i=TR(NP_i) for each i = 1, 2, ..., n and At^* = TR (At) then

At^* (NP^*₁, NP^*₂,..., NP^*_n) is a positive proposition in NL^*

(18) If NEGAt (NP₁, NP₂, ..., NP_n) is a negative proposition in NL and if we denote by NP^*_i = TR (NP_i) for each i = 1, 2, ..., n and NEGAt^* = TR (NEGAt) then

NEGAt^* (NP^*₁, NP^*₂,..., NP^*_n) is a negative proposition in NL^*

finally if NL is a natural language of Eskimos from Alaska and NL^* a natural language of a Central Africa's tribe NL is not translatable into NL^* because there are almost no existent common objects of NL and NL^*. Further, a seal is a very important component of Eskimo's food and its importance is reflected by several different words for it according to special circumstances. E.g. let us assume that SEALON is a word for a seal on a floating iceberg with a good likelihood to be successfully hunted. It is unclear how a positive proposition "I see a SEALON" could be translated into the Africa's tribe language.¹

VI. Numbers

Although facts 1) and 2) show the empirical source of natural language there exist words without such direct reference to experience which are neither names of objects nor names of attributes. In English words of this sort are e.g. "one" or "two" or "ten" or "hundred", thus numbers, or more specifically word-numbers to differentiate them from ciphers or figures. Now two circumstances are of great importance:

On the one hand it seems very likely that full arithmetics with addition has been developed only during some later phase of historical development, in particular when the written form of the language concerned was available and also ciphers or figures have been invented. It means that in the early phase of historical development of natural languages only the word-numbers were available.

On the other one the counting of elements of a set was discovered and found useful probably in very early phase of the language development. One can imagine in a family language that the mother has counted the numbers of pieces of bread her children have eaten during the dinner in order to decide who should get more. It means that counting is using word-numbers only.

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Word-number definition:

Let LN be a natural language in its very early phase of historical development, thus without numbers and without ciphers or figures. Then word-numbers of LN will be quite arbitrary words which are new (thus do not belong to LN) and will be added to NL to determine its extension NLW with word-numbers. The only circumstance concerning new words to be considered as word-numbers is the following convention:

it has to be remembered for ever the order of choosing the particular word-numbers that it is to be noticed which is the first, which is the second, or just the next one, etc. As choosing the new words is a human activity, it would stop sooner or later but certainly after a finite number of choices. It means that there always will be only a finite number of word-numbers or other words the ordering of word-numbers, called their natural order will have its last element GR. According to this ordering a word-number W_a is called smaller than another one W_b (if W_a proceeds W_b thus each word-number different from GR is smaller than GR or in other words GR is the greatest word-number available in the extension NLW. Obviously considering another natural language NL^* and its extension with word-numbers it may happen that its greatest word-number GR^* may be smaller or greater than GR in an obvious sense, namely when we silently identify the i-th word number of NL^*W with the i-th word-number of NLW. In fact this difference between the greatest word-numbers is of no importance because if necessary for some purpose each NLW may be further extended beyond its GR just by choosing a new word and putting it at the end of natural order of NLW.

(19) Cognitive algorithm of counting a set S:

Each element from S must be considered (in arbitrary order but only once) and a word-number is assigned to it as follows, at the beginning of considering always the first word-number is assigned and then to each element which has been considered the next word-number is assigned. The process of considering and assigning must be continued until all elements from S are exhausted. Two cases are possible: either

1) the process could be always continued and the algorithm terminates when the last elements from S to be considered is concerned and the next word-numbers W_r is assigned to it. Then the algorithm is successful and W_r is its result which is called the number of elements of S;

or

2) the process cannot be continued because there is no more the next word-number in the natural order of word-numbers, which means that the greatest word-number GR has been already assigned thus before exhausting all elements from S; then one says that the algorithm

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failed and there will be no exact result, only an estimate, namely that the number of elements of S is greater than GR.

Considering a translation of NLW into NL^*W it seems obvious that word-numbers of NLW must be translated as word-numbers of NL^*W, the only question remains which word-number should be taken. Obviously very different requirements must be used than those when the translation of NL into NL^* was defined because there are no direct references either to objects or to attributes. Instead the translation of word-numbers should preserve their natural order.

Intuitively it means that the i-th word-number of NL^*W is translated as the i-th word-number of NLW although it is not easy to prescribe for which value of it is valid. It is certainly valid for very small values of i as i=1 or i=2 but GR may be not translatable into NL^*W because it could be greater than GR^*, etc. More accurately the preservation of natural order may be expressed by the following requirements exploiting the function NEXT of word-numbers:

(20a) the first word-number of NLW is translated as the first word-number of NL^*W, and

(20b) if W_a and W_b are word-numbers of NLW such that NEXT (W_a) = W_b and their translations are TR(W_a) = W^*_a and TR(W_b) = W^*_b of NL^*W then NEXT (W^*_a) = W^*_b.

It is clear why the counting algorithm of a set S may succeed within one language with word-number but fail within another one. It is a negative result associated with the restrictive number of word-numbers, but there are also some positive results as the observation (21) showing the uniqueness of the result if the counting algorithm is successful.

(21) If a set S is successfully counted with the result W_a within NLW and with the result W^*_a within another language NL^*W then either TR(W_a) = W^*_a or TR(W^*_a) = W_a.

Note

1. This topic was discussed by prof. J. L. Fischer in his regular course on theory of knowledge at the Masaryk University in Brno (Czech republic), probably in 1946 or 1947.

Reference

[Russell] B. Russell: Human Knowledge. Its Scope and Limits. London 1960⁴ (first ed. 1948), part II, ch. II, p.78ff.

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