ABSTRACT: Subjective science, as a pattern of an objective science, is based on a dialectical triad: language-logic-philosophy, which is the language of dialectics. The first level of language is its grammar, whose basis constitutes lexemes (name-words) and lexas (relation-words). Dialectical logical factor sets (logical parts of speech) are the basis of dialectical logic and logical morphology. They form the second level of the language of dialectics. The logical factor sets are represented by dialectical forms of thinking, reflecting the contradictory nature of reality. In the first approximation, any face of a state or a phenomenon of nature has at least two sides of comparison. All variety of these sides are joined together by the common name oppositi. Dialectical philosophy is the third generalized level of language. The main qualitative postulates of dialectical philosophy constitute two postulates: a postulate of existence and a postulate of evolution. A simple description of an object of thought usually contains a statement and a judgment. In this paper, we consider the simplest meanings of judgments, namely, some dialectical combinations-judgments of Yes and No. Judgments are postulated on the basis of a material-ideal dialectical field.

Dialectics is a moving soul of every scientific evolution of thought. It represents a principle, which introduces in contents of science an immanent bond and necessity... - Hegel

I. Introduction

Natural science is, in point of fact, a dialectical science. It is a subjective image of a definite class of objects and a class of relations and processes related to the objects. The class of processes and relations should be treated as an ideal class with respect to which the class of objects is a material class. Both these classes form a material-ideal class of objects, which represents an original of the definite science. We term the original an objective science or an objective pattern of natural (subjective) science. The subjective science, as the pattern of the objective science, is built first of all on the basis of a dialectical triad: language-logic-philosophy, which in the broad sense of the word are the language of dialectics.

II. The Dialectical Triad

2.1. Grammar of the Language

The first level of the language of dialectics and logical thinking is its grammar. A basis of the grammar constitutes words-lexemes (names-words) and words-lexas (relations-words) which are not separated by a clear-cut boundary.

All variety of lexas in languages of the high level is disintegrated into a series of parts of speech or grammatical factor sets. Words equal on the definite functional criterion are elements of these factor sets. It is impossible precisely to demarcate such sets because they are always rather fuzzy. The grammatical factor sets reflect definite sides of nature, which should consider as objective factor sets of nature itself. Thus, they are images of factor sets of originals. Characteristic sets reflecting definite relations of the real World are a part of the grammatical factor sets as well.

Relying on Latin language, we will consider the sets of originals and images. Nature, as a complex of objective basis-superstructure (base-adbase or base-superstructura), induces in human consciousness

a triad of basis: form (forma), contents (continens), process (processus); and

a triad of superstructure: a common name of an object of thought (commune nomen),

abstract quantity (abstractum quantitas),

quality of a relatively common character (generale qualitas) (Fig. 1).

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Fig. 1. A graph of basis-superstructure.

At a morphological level, objective basis and superstructure form a morphological subjective structure of the basis-superstructure (Fig. 2).

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Fig. 2. A morphological graph of basis-superstructure of the grammatical level.

At the level of superstructure, common names form a factor set of pronomen (pronoun), the abstract quantity forms a numerale (numeral) factor set, the quality forms an adverbium (adverb) factor set.

At the level of basis, material form is expressed by a factor set of nomen adjectivum (adjective) and ideal form is presented by a participium (participle) characteristic set.

Contents symbolizes a factor set of nomen substantivum (noun).

Action process is expressed by a verbum (verb) factor set and decomposed into two subsets of action (actio) and state (statio) verbs.

A verb of a activum verbum subset (a subset of action) is transitium verbum (transition verb), whereas a verb of a stativum verbum subset (a subset of state) is intransitivum verbum (intransitive verb).

State process is described by characteristic sets: gerundio (verbal adverb, gerundio), gerundium (verbal noun, gerund), and gerundivum (verbal adjective, gerundive).

Verbs of motion and state express definite self-relations of a subject, object, environment (ambitus) and relations between them. The main peculiarity of these relations and self-relations is a definite directedness. We call tendencia (tendency) a set of verbal forms of the directedness of a process. In more or less extent, the tendencia set is presented in contemporary languages by different forms of voice. Verbs of zeroth directedness should be referred to the tendencia set, because the set will not be complete without them.

Basis and superstructure, as the morphological level of a language, have the proper superstructure — the syntactical level (Fig. 1.3).

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Fig. 3. A syntactical graph of the grammatical level.

The syntactical level, as the complex superstructure, has own basis: a set of subjective contents, subjectum (subject); a set of objective contents, objectum (object); a set of form, attributum (attribute); a set of motion and state, praedicatum (predicate). Circumstantia (adverbial modifier) represents the superstructure of the syntactical level, which decomposes into a series of subsets with own functions similar to adverb.

2.2 Logical Grammar

Dialectical logic and logical grammar (the last includes a logical morphology and syntax) are related to the second level of the language of dialectics whose basis constitute dialectical logical factor sets or logical "parts of speech". The logical factor sets are represented by dialectical forms of thinking reflecting contradictory nature of reality. Let us consider the main of them.

Cognition of the World proceeds first of all on the basis of comparison. In the first approximation, any face of a state or a phenomenon of nature has at least two sides of comparison. If sides of such faces coincide in some features, it is fixed by lexemes-tautologies such as direct-direct, curved-curved, etc. If there are some differences, lexemes-synonyms are used. Opposite sides are expressed by lexemes-antonyms: direct-curved, simple-complex, etc. Polar-opposite notions are expressed by lexemes-antilogies such as material-ideal, rest-motion, etc. We join tautologies, synonyms, antonyms, and antilogies by the common name oppositi (sg. oppositus) (from the Latin, oppositus=opposite). Oppositi are lagical parts of speech.

Apart from these groups of words, there are many words-homonyms, which being identical in form express different notions. Words contained in an oppositus, we term elements of the oppositus or semioppositi (in particular, semitautologies, semiantilogies, etc.). Definite relations take place between semiopposoti and oppositi. We denote the common name of the relations by the symbol " ", which is a symbol-pronoun of any relations.

Oppositi of similarity-difference, analysis-synthesis, deduction-induction, general-particular, contents-form, quality-quantity, truth-lie, etc.,

logical pronominal oppositi-tautologies Yes Yes and No   No,

logical pronominal oppositi-antonyms Yes   No and No   Yes,

logical pronominal oppositi-antilogies Yes   No and No   Yes

and their combinations are the most important elementary oppositi-judgements of dialectical logic.

A set of sentences, describing an object of thought without sufficient clearing up of its essence, constitutes a description of the object of thought. The description is the first logical cyntactical factor set. The description is followed by determination of its essence based on the statements.

A statement is meant as a set of sentences that characterizes, to a certain extent, the essence of the object of thought on the basis of ideas and concepts. Of course, there is no distinctive boundary between a description and a statement. The statement is the second logical cyntactical factor set. Generalized statements, or pronoun statements form complex judgements constituting the third factor set.

A dialectical complex judgement is a form of thinking, in which the object of thought of the objective-subjective reality is described through the whole spectrum of contradictions and noncontradictions by affirmation (Yes), negation (No), affirmation-affirmation (Yes-Yes), negation-negation (No-No), affirmation-negation (Yes-No), negation-affirmation (No-Yes), and more complicated combinations of Yes and No.

Statements and judgements are usually combined into an unified statement-judgement set that is called an reasoning. Descriptions, statements, judgements and reasonings could be united by one common name as reflections.

Reflections are concerned with the level of knowledge, below of which there is a level of ignorance. On the interface between these levels, the intuitive processes of consciousness and thinking occur. They are induced by experience, experience and Cosmos, and only by Cosmos. Logic induced by experience, experience and Cosmos are human-controllable to some extent. Intuition induced by Cosmos alone occurs on the subconscious level. Intuition is not human-controlled and deep understanding of the World is impossible without it. Intuition generates its own ideas, concepts, statements, judgements and reasonings.

A set of logical means of thinking realizing transition from one level of knowledge-premise to another new level of knowledge-conclusion is a deduction. The deduction forms a logical functional bond between a premise and conclusion.

In dialectics, an estimation of an extent of truth and falsity of oppositi and deductions, describing objects of thought, is realized by oppositi-judgements whose components of affirmation and negation express measures of truth and falsity, respectively. We term these logical judgements about judgements as oppositi-opinions or simpler, opinions (from the Latin, opinio=view). Evidently, opinions themselves can be analyzed from point of view their truth-falsity. Therefore, it makes sense to consider also logical judgements about the judgements-opinions. Such judgements we call oppositi-gnomes (from the Greek, gnomh=opinion) or simpler gnomes. Opinions and gnomes are peculiar the first and second logical derivatives of oppositi-judgements, describing objects of thought. In real life, the judgements about fairness of any affirmations, opinions, and theories are not restricted solely by the logical derivatives of the second orders.

2.3. Dialectical Philosophy

The third generalized level of the language of dialectics is dialectical philosophy. The main qualitative postulates of the dialectical philosophy constitute two postulates:

1. The Postulate of Existence. It exists () the material-ideal, objective-subjective, finite-infinite, continuous-discontinuous, quantitative-qualitative the dialectical Universe () whose sides () are linked by the system-set of dialectical relations and with the relations of material-ideal exchange Image21.gif (75 bytes):

,

where kN, sk and fk are contradictory sides of facets-oppositi of the Universe,

is the symbol of the infinite universal bond,

is the union-conjunction "and".

2. The Postulate of Evolution. Any object or process A in the Universe is equal and not equal to itself:

III. Quantities and Measures of Dialectical Judgements

We will consider the simplest meanings of judgements-semioppositi. If Z is an elementary dialectical affirmative (or negative) judgement, its possible meanings are:

Si (No), a neutral affirmation (neutral negation) or briefly, an affirmation (negation);

+Si (+No), a positive affirmation (positive negation); and

Si (- No), a negative affirmation (negative negation).

Affirmations +Si, - Si (negations +No, - No) are related to affirmation Si (negation No) as particular and general. Below are some examples of the elementary judgements:

Si = " a body moves with the velocity Si between the points A and B";

+Si = "a body moves with the velocity Si from the point A to the point B";

- Si = "a body moves with the velocity Si from the point B to the point A".

No = "a body does not move with a velocity No between the points A and B";

+No = "a body does not move with a velocity No from the point A to the point B";

- No = "a body does not move with a velocity No from the point B to the point A".

Natural objects and phenomena of nature induce in dialectics additive and multiplicative oppositi-judgements of the following kinds: Si+Yes and SiYes, Si+No and SiNo, No+Si and NoSi, No+Not and NoNot.

We denote measures of judgements Si and No by sumbols M(Si) and M(No) or briefly by the symbol M. Oppositi-judgements Si and Yes (No and Not) are synonyms. A measure of a judgement represents a composite complex of basis and superstructure: , where q is basis, a core of the measure, quantitative value; the sign "" is superstructure, an envelope of the measure, which is expressed by a set of various signs that refer to the basis.

Elementary judgements of affirmation with an unit measure of affirmation will be termed unit affirmations and denoted by 1. The number a of the units will be assumed to be a measure of the affirmative judgement: M(Si) = a•1 or briefly M(Si) = a, aR. The affirmation is positively if a > 0, otherwise it is negative.

Elementary judgements of a polar negation with a unit measure of the polar negation will be named unit polar negations and denoted by i. The number b of the units will be assumed to be a measure of the polar negation judgement: M(No) = bi or briefly M(No) = ib, bR. The negation is positively if b > 0, otherwise it is negative.

Judgements with measures a and ib are polar-opposite judgements-antilogies, whereas judgements with opposite by sign measures a and b (correspondingly ia and ib) are simply opposite judgements, as judgements-antonyms.

According to the above designations, additive and multiplicative contradictory judgements with polar-opposite semioppositi have the following form, correspondingly: Z= a+ib and Z= aib, at that the multiplicative judgement is a particular case of the additive judgements.

Let us agree an a-semioppositus to term the material component of an oppositus, meaning the word "material" as a general name of a-semioppositi. The adjective "material" can take the definite meaning in every specific case. Correspondingly, a polar-opposite ib-semioppositus will be termed the ideal component of the oppositus, assuming the word "ideal" as a general name of ib-semioppositi. Here, the word "ideal" can take the definite sense also in every concrete case. For example, if "material" = quantitative then "ideal" = qualitative.

Oppositi Z= a+ib form a field of material-ideal measures-numbers, syntax of which is described by the following algebras (1, 2) of

affirmation (Yes): (Si1) • (Si2) = +Si3 , and negation (No): (No1) • (No2) = - Si

(Si1) • (Si2) = - Si3 ; (No1) • (No2) = - Si

The algebras of affirmation and negation are models of objective algebras of physical fields of exchange. If an electric field is a field Yes then a magnetic field, as a negation field of the electrical field, is a field No. Therefore, interactions of electric charges are described by the algebra of affirmation, but electric currents - by the algebra of negation: two charges of the same sign repel each other, i.e. (Si1) • (Si2) = +Si3 , and currents of the same sign mutually attract, i.e. (No1) • (No2) = - Si , etc.

The algebra of dialectical judgements shows one’s worth in social processes as well. It formally coincides with the algebra of complex numbers but in contents the dialectical algebra of judgements does not equal to the algebra of complex numbers.

A simplest material-ideal number-oppositus has the following form

schp5.jpg (5897 bytes)

where iy is an argument of the number, schp6.jpg (3778 bytes) is an index of expansion of the basis b in the base of the natural logarithm e. (2) The semioppositus a expresses generally a nonrecurring facet of the process at the quantitative side and the semioppositus biy describes a periodical aspect of the process with the ideal period

, where ,

at its qualitative side. If b = 10, we arrive at Yb = Image22.gif (62 bytes) p. A graph of such dialectical judgement of affirmation-negation of the wave character is shown in Fig. 4.

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Fig. 4. schp7.jpg (23861 bytes)

The ideal period Image22.gif (62 bytes) p and its rational parts lie at the base of ancient measures, at the mass spectrum of elementary particles and at the series of fundamental constants of physics. (2) For example, an ancient Roman ounce is equal to 2.7288 decagram. The bond of the material-ideal dialectical field of measures-judgements with the material-ideal World is exhibited here.

IV. Conclusion

The three-level representation of the language of dialectics and its mathematical principles, whose basis constitute oppositi-judgements of the three-dimensional structure, more complete reflect objective oppositi-judgements.

The numerical field of dialectics and its logical apparatus significantly simpler and naturally describe and solve problems of a scientific-technical character, many of which are not solving within the classical analysis.

It is impossible without dialectical analysis an essential theoretical and practical advance deep into an atom and elementary particles, which are adequately described only on the basis of variable wave oppositi-judgements (2).

References

(1) L.G. Kreidik, G.P. Shpenkov, The Material-Ideal Numerical Field, Proceedings of the Conference "CONTACT’95", Sofia, 1995, pp. 34-39.

(2) L.G. Kreidik, G.P. Shpenkov, Alternative Picture of the World, Bydgoszcz, 1996.

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